Download New circuit principles for integrated circuits

FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ
VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ
New circuit principles for integrated
circuits
Part 1: Theory of nonlinear circuits
Preliminary text
Author:
Kamil Vrba, František Kouřil
Brno
2.11. 2006
Obsah
Introduction to problems of non-linear and parametric circuits ..................................................................... 5
1.1
CLASSIFICATION OF CIRCUIT ELEMENTS AND CIRCUITS ....................................................... 5
1.2
IDEALIZATION OF PROPERTIES OF ELECTRONIC ELEMENTS AND CIRCUITS .................... 9
1.3
PROCEDURES IN THE ANALYSIS AND SYNTHESIS OF NON-LINEAR AND PARAMETRIC
CIRCUITS ............................................................................................................................................ 10
1.4
THE VALIDITY OF SOME LAWS FOR NON-LINEAR AND PARAMETRIC CIRCUITS ........... 14
1.5
BASIC SIGNAL CONVERSIONS IN NON-LINEARAND PARAMETRIC CIRCUITS ................. 16
Circuit elements and their models ..................................................................................................................... 18
2.1
IDEALIZED CIRCUIT ELEMENTS ................................................................................................... 18
2.1.1
Linear, non-linear, non-controlled, and controlled elements ................................................................ 18
2.1.2
Characteristics and parameters of two-terminal elements..................................................................... 20
2.1.3
Resistors................................................................................................................................................ 25
2.1.4
Capacitors ............................................................................................................................................. 28
2.1.5
Inductors ............................................................................................................................................... 31
2.1.6
Multiterminal elements ......................................................................................................................... 34
2.1.7
Ideal sources of electrical energy .......................................................................................................... 39
2.2
APPROXIMATION OF THE CHARACTERISTICS OF NON-LINEAR
2.2.1
Problems of analytically expressing non-linear characteristics ............................................................ 41
2.2.2
A survey of the most frequently used approximation functions ........................................................... 44
2.2.3
Determining the coefficients of an approximation function ................................................................. 46
2.2.4
Dimensionless form of approximation functions .................................................................................. 50
ELEMENTS ................. 41
Modelling of circuit elements ............................................................................................................................. 52
3.1
THE PHILOSOPHY OF MODELLING .............................................................................................. 52
3.2
CLASSIFICATION OF CIRCUIT ELEMENT MODELS .................................................................. 53
3.3
MATHEMATICAL MODELS OF CIRCUIT ELEMENTS ................................................................ 55
3.4
CIRCUIT MODELS OF ELEMENTS ................................................................................................. 56
3.5
EXAMPLES OF STATIC MODELS OF NON-LINEAR RESISTORS.............................................. 60
3.6
DYNAMIC MODELS OF CIRCUIT ELEMENTS ............................................................................. 64
3.7
DESIGNING SYNTHETIC CIRCUIT ELEMENTS WITH THE AID OF AFFINE
TRANSFORMATIONS ....................................................................................................................... 66
Spectral transformations in non-linear and parametric circuits .................................................................... 79
4.1
THE FOURIER SERIES AND ITS COEFFICIENTS ......................................................................... 79
4.1.1
Expressing a periodic function by the Fourier series ............................................................................ 79
4.1.2
Expressing a quasi-periodic function by the Fourier series .................................................................. 80
4.2
SPECTRAL ANALYSIS OF SIGNALS IN NON-LINEAR CIRCUITS WITH A HARMONIC
INPUT SIGNAL ................................................................................................................................... 81
4.2.1 The graphical method ............................................................................................................................ 81
4.2.3 Analysis of signals when approximating the characteristic by a piecewise linear function ................... 83
4.2.4 Analysis of signals when approximating the characteristic by a power polynomial.............................. 87
4.2.5 Analysis of signals when approximating the characteristic by an exponential function and an
exponential polynomial.......................................................................................................................... 88
4.6.1
Basic non-linear or parametric circuits ................................................................................................ 90
4.6.2
Polyphase systems ............................................................................................................................... 93
4.6.3
Structural synthesis of symmetrized systems ....................................................................................... 95
4.6.4
Some examples of symmetrized systems of non-linear circuits .......................................................... 97
Methods for analysing non-linear and parametric circuits........................................................................... 101
5.1
GRAPHICAL METHODS ................................................................................................................. 101
5.1.1
Resultant characteristics of one-ports consisting of several elements ................................................ 102
5.1.2
Graphical methods for solving first- and second-order non-linear circuits ......................................... 106
5.2
ANALYTICAL METHODS .............................................................................................................. 113
5.2.1 The method of equivalent linearization .............................................................................................. 113
5.2.2 The piecewise solution method ........................................................................................................... 116
Signal amplification .......................................................................................................................................... 119
6.1
THE PRINCIPLE OF AMPLIFIERS WITH CONTROLLED RESISTORS .................................... 119
6.2
TRANSISTOR AMPLIFIERS ............................................................................................................ 120
6.2.1 Transistor amplifier with a resistor load ............................................................................................. 120
6.2.2
Emitter and cathode follower .............................................................................................................. 121
6.2.3
Differential amplifier .......................................................................................................................... 124
6.2.4 Types of transistor amplifier ............................................................................................................... 125
6.2.5
Amplifier operation classes................................................................................................................. 127
6.3
AMPLIFIERS WITH NEGATIVE RESISTANCE ............................................................................ 132
6.3.1 The principle of amplification ............................................................................................................ 132
6.3.2
Selective amplifier with a tunnel diode............................................................................................... 134
Signal rectification and shaping ...................................................................................................................... 135
7.1
RECTIFIERS ...................................................................................................................................... 135
7.1.1
Non-linear rectifiers ............................................................................................................................ 135
7.1.2
Parametric rectifiers ............................................................................................................................ 145
7.2
WAVE-SHAPING CIRCUITS ........................................................................................................... 148
7.2.1
Clippers ............................................................................................................................................... 148
7.2.2
Functional converters .......................................................................................................................... 154
7.2.3
Pulse shaping by means of a transformer............................................................................................ 156
7.3
FREQUENCY MULTIPLIERS .......................................................................................................... 157
7.3.1
Frequency multipliers with non-linear two-terminal elements ........................................................... 157
7.3.2
Frequency multipliers with non-linear three-terminal resistors .......................................................... 159
7.3.3
Parametric frequency multipliers ........................................................................................................ 159
7.3.4
Frequency multiplication by means of wave-shaping circuits ............................................................ 160
7.4
FREQUENCY DIVIDERS ................................................................................................................. 162
7.4.1
Concept of frequency dividers ............................................................................................................ 162
7.4.2
Frequency dividers with a time filter .................................................................................................. 162
7.4.3
Feedback frequency divider ................................................................................................................ 164
Signal mixing, modulation, and demodulation ............................................................................................... 166
8.1
FREQUENCY MIXERS AND CONVERTERS ................................................................................ 166
8.1.1
Non-linear (additive) mixers ............................................................................................................... 167
8.1.2
Parametric (multiplicative) mixers ..................................................................................................... 172
8.1.3
Self-oscillating mixers ........................................................................................................................ 174
8.2
MODULATORS ................................................................................................................................. 175
8.2.1
Modulated signals ............................................................................................................................... 175
8.2.2
Modulators for amplitude modulation ................................................................................................ 176
8.2.3
Modulators for frequency modulation ................................................................................................ 183
8.2.4
Modulators for phase modulation ....................................................................................................... 186
8.2.5
Modulators for pulse modulation ........................................................................................................ 188
8.3
DEMODULATORS ........................................................................................................................... 189
8.3.1
Demodulation of modulated signals ................................................................................................... 189
8.3.2
Demodulators of amplitude-modulated signals .................................................................................. 189
8.3.3
Demodulators of frequency-modulated signals .................................................................................. 191
8.3.4
Demodulators of phase-modulated signals ......................................................................................... 192
8.3.5
Demodulators of pulse-modulated signals .......................................................................................... 192
Generation of oscillations ................................................................................................................................. 194
9.1
GENERAL CHARACTERISTICS OF GENERATORS OF ELECTRICAL OSCILLATIONS ...... 194
9.2
GENERATORS OF HARMONIC OSCILLATIONS - OSCILLATORS .......................................... 194
9.2.1
LC oscillators with negative resistance ............................................................................................... 194
9.2.2
Feedback LC oscillators...................................................................................................................... 198
9.2.3
Basic types of LC oscillators .............................................................................................................. 207
9.2.4
RC oscillators...................................................................................................................................... 214
9.3
GENERATORS OF NON-SINUSOIDAL OSCILLATIONS ............................................................ 218
9.3.1
Explanation of the operation of the relaxation generator .................................................................... 219
9.3.2
Examples of relaxation generators ...................................................................................................... 224
Non-linear feedback and resonance phenomena............................................................................................ 229
10.1
NON-LINEAR FEEDBACK SYSTEMS ........................................................................................... 229
10.1.1 Amplifiers with a non-linear feedback network.................................................................................. 229
10.1.2 Feedback networks with a non-linear amplifier .................................................................................. 237
Introduction to problems of non-linear and
parametric circuits
1.1 CLASSIFICATION OF CIRCUIT ELEMENTS AND CIRCUITS
In the theory of electric and electronic circuits, it is usual to distinguish between lumped
and distributed circuits or elements [23]. In most cases, we do not need to consider the
existence of delay phenomena in circuits. The idea of lumped circuits is therefore sufficient
to serve the objectives of this book. Thus we shall restrict ourselves to lumped circuits and
elements only.
The basic building components of linear electric circuits are resistors, capacitors,
inductors, and transformers. Electrical properties of these elements are characterized by their
parameters. Thus for a resistor R it is its resistance R, for a capacitor C its capacitance C, for
an inductor L its inductance L, and for a transformer Tr its inductances L1, L2 and mutual
inductance M (see Fig. 1).
Fig.1. Symbols of basic linear electrical elements
The parameters R, C, L of circuit elements, however, are not always constant. In
electrical engineering practice, we work with a number of circuit elements, especially the
electronic ones, whose parameters vary markedly. In these elements, the cause responsible
for their parameter changes must therefore be taken into consideration. The parameter can
vary owing either to changes in the current flowing through the element, or to changes in the
voltage across it, or to an external quantity, which can be not only electrical but mechanical,
luminous, thermal, etc. In the first case, the parameter is a function of the current or voltage,
in the second case a function of time, since an external control quantity is always a certain
function of time.
In these cases, however, it is of no advantage to characterize the variability of electrical
properties of circuit elements by the variability of their parameters. It is advantageous to start
directly from the interdependence of the circuit quantities, i.e. the voltage v and the current i
(or their integrals - the flux linkage -r and the charge q). Such dependences then constitute
the basic characteristics of circuit elements (e.g. i = f(v), v = f(q), = f(i), etc.).
Classification of elements. On this basis, all the elements of electronic circuits are
divided into linear, non-linear, linear parametric, and non-linear parametric elements. The
parameters of linear elements are constant; they depend neither on the current flowing
through them, nor on the voltage across them, nor on any external quantity. The characteristics of linear elements are thus linear. In non-linear elements, their parameters depend
on the current through them or on the voltage across them, and a change in this current or
voltage (the cause) entails a change in the parameter (the effect). These parameters vary
according to a certain law which is characteristic for the given element. The characteristics of
non-linear elements are thus not straight lines but curves. The parameters of controlled
(parametric) linear elements depend on an external control quantity. Their characteristics are
therefore straight lines with each discrete value of the control quantity having its
corresponding straight line. In parametric representation, the controlled element is thus
characterized by a family of straight lines. In non-linear elements which are at the same time
controlled, their parameters depend on both the current flowing through them or the voltage
across them, and on an external control quantity. Such elements are therefore characterized in
a plane with rectangular co-ordinates by a family of curves. Basic types of characteristics of
non-linear elements. Depending on the waveform of the circuit quantities, we distinguish
static and dynamic characteristics. The static characteristics are measured at time-invariant
or very slowly changing quantities of the circuit. Such a static characteristic
(1.1)
is illustrated in Fig. 2a; here, Y and X are do quantities.
Fig. 2. Static and dynamic characteristics of a non-linear element
The dynamic characteristics express the interdependence of instantaneous values,
amplitudes, or effective values of circuit quantities. Unlike the static characteristics, they are
determined at quantities changing rapidly with time (usually harmonically). A dynamic
characteristic expressing the dependence of instantaneous values of the circuit elements
(1.2)
is, for ideal non-inertial elements, identical to the static characteristic, but it can differ from it
considerably (Fig.2b), e.g. owing to inertial phenomena which show up more markedly only
at higher frequencies. Amplitude characteristics express the dependence between the amplitudes of circuit quantities of the same frequency, most frequently between the amplitudes of
the first harmonic components
(1.3)
C1assification of circuits. In keeping with the above classification of circuit elements into
four groups, electrical circuits are also classified into four groups. They are the linear, nonlinear, linear parametric, and non-linear parametric circuits. A circuit is non-linear or
parametric if, in addition to linear elements, it contains at least one non-linear or parametric
element. Non-linear parametric circuits contain at least one parametric (controlled) non-linear
element. However, circuits containing simultaneously non-linear and parametric elements (at
least one of each) are also non-linear parametric circuits [13], [49].
The task of linear circuits is to transfer a signal and, if need be, to select a certain
frequency band from its spectrum. In linear circuits, the signal spectrum can be made only
poorer, not richer. The task of non-linear and parametric circuits consists in converting the
frequency spectrum and changing the waveform (the shape) of the signal. The conversion of
the frequency spectrum is accompanied by a shifting of electrical energy in the spectrum.
Fig. 3. Generalized block diagram of a circuit
In circuit theory, a circuit can be represented by a generalized block diagram as shown in
Fig. 3. A signal x(t) (the cause) is applied to the circuit input, and a response y(t) (the effect)
appears at its output. The circuit response to the input signal depends on both the type of
input signal and the circuit properties, i.e. on the characteristics or parameters of its elements.
The dependence between the input and the output signals of the circuit is expressed by the
non-homogeneous differential equation
(1.4)
In the general form, this dependence between the cause and the effect can be written in the
form
(1.5)
where the coefficients a0, al , ... , aN depend on the parameters of the circuit elements.
Linear circuits. The parameters of all elements of a circuit, and thus also the coefficients
in Eq. (1.4), are constant (time-invariant). The phenomena in these circuits are described by
linear equations with constant coefficients. Unlike all other circuits, in these circuits no
enrichment of the frequency spectrum of signals is possible. The principle of superposition
holds for these circuits.
Linear parametric circuits. The parameters (at least one) of circuit elements, and thus
also the coefficients of Eq. (1.4), do not depend on the voltage and current in the circuit; they
are functions of time t. The phenomena in these circuits are therefore described by linear
differential equations with varying coefficients, and the functional dependence (1.5) assumes
the form
(1.6)
Though the principle of superposition holds for parametric linear circuits, the frequency
spectrum of signals is enriched in them.
Non-linear circuits. The parameters of elements (at least one) and the coefficients of Eq.
(1.4) depend on the voltage or current in the circuit. This results in a non-linear functional
dependence
(1.7)
The principle of superposition does not hold for non-linear circuits, and the frequency
spectrum of signals is converted in them.
Non-linear parametric circuits. The parameters of elements and thus also the coefficients
a0, al , ... , aN depend both on the voltages and currents in the circuit and on time.
Consequently, the functional dependence of a non-linear parametric circuit has the form
(1.8)
The principle of superposition does not hold for these circuits either, and in these circuits,
too, there is a transformation of the frequency spectrum of signals.
In reality, all devices are non-linear and parametric, since all causal relations in nature
and technology are non-linear and, at the same time, parametric. Under certain conditions,
however, e.g. when the circuit operates over a limited range of voltages and currents or when
the effect of external quantities on element parameters is negligible (e.g. that -of
temperature), we can neglect the majority of effects and regard the circuit as either linear or
parametric linear or non-linear, depending on which causal dependence is more pronounced.
Depending on the number of separate storage elements in the circuit, the order of
complexity of the circuit is determined. If a circuit (or a system) contains only non-inertial
resistors (either linear or also non-linear), it is called the zero-order circuit, or the noninertial circuit. The phenomena in zero-order circuits are described by algebraic or
transcendental equations (not by differential ones). The circuits containing one independent
storage element and an arbitraty number of resistors are called first-order circuits. The
phenomena in these circuits are described by first-order differential equations. Circuits of Nth order contain N storage elements (linear, controlled, and nor-linear elements) and an
arbitrary number of resistors (linear, controlled or also non-linear resistors). At the same time,
it is assumed that the topological structure of the circuit contains neither a series nor a parallel
combination of storage elements of the same nature. The phenomena in circuits of the N-th
order are described by N-th order differential equations.
The properties of the above circuits and the methods for solving the above types of
equations are markedly different, and it is this fact that underlies the traditional classification
of circuits into linear and non-linear.
The fundamental principle of the methods applied in investigating the properties of linear
circuits is the principle of superposition. It holds for circuits with both constant parameters
and time-varying parameters. But while linear differential equations with constant
coefficients can be solved exactly, the situation is much more complicated when solving
linear differential equations with time-varying coefficients. This circumstance and also the
fact that in linear circuits with varying parameters the spectrum is transformed and enriched,
have usually led us to include these circuits among non-linear circuits.
Comparatively good results have been obtained by means of the theory of first-order
differential equations with varying coefficients. There are also several second-order equations
with varying coefficients that have been investigated in detail, enabling us to extend our
knowledge of their solution. The Mathieu equation [4], [29], [62]
(1.9)
is of particular importance in the theory of parametric circuits.
It is a feature of non-linear equations that their coefficients depend on the function y (see
Eq. (1.7)). An example of non-linear equations of the second order is the so-called van der
Pol differential equation for the oscillator [19], [45], [49]
(1.10)
Solving equations of this type is very difficult. There are no general solutions to non-linear
differential equations; only a few special cases of these equations can be solved exactly. We
must therefore apply approximate methods. A number of different methods have been
developed to find approximate solutions to non-linear equations. Each of them has its own
specific area of application in which its advantages prevail and its disadvantages can be
neglected. One of the basic tasks in solving a nonlinear circuit is the selection of the most
suitable method for the analysis of the given particular circuit.
When employing approximate methods for solving non-linear equations, not just one but
rather two or three approximate methods should be used, and the results compared. This will
eliminate any major errors, evaluate errors caused by the approximate solution, and possibly
reveal something new that will throw light on the origin of errors that are due to the
application of an unsuitable method. In addition to analytical and graphical methods,
numerical methods are very frequently applied in the analysis of non-linear circuits.
1.2 IDEALIZATION OF PROPERTIES OF ELECTRONIC ELEMENTS AND
CIRCUITS
In an analysis of electronic circuits, the dynamic properties of a circuit under
examination can never be evaluated with absolute accuracy. If all the influences were taken
into consideration, the analysis of the phenomena in the circuit would be very complicated
and practically impossible. Therefore effective simplifications of the circuit and its model are
adopted at the beginning of the analysis. We must introduce limiting assumptions.
The assumptions permitting an effective simplification of the analysis concern, above all,
the nature of the signal that excites the circuit or which is produced in the circuit. There are
two basic limitations to be imposed: the limitation of the rate of time changes (the frequency)
of the signal, and the limitation of the magnitude of the signal.
Fig. 4. Model of an actual one-port RC
The justification for introducing the above limitations and the resulting possibility of
idealizing the element and circuits will be demonstrated on a simple example of the one-port
RC (Fig. 4). Assume that this one-port represents an equivalent circuit, or a model of an
actual resistor having a resistance R and an undesired (parasitic) capacitance C. Let the
voltage v supplied by the source be harmonic and let its frequency be , i.e. v = Vcos (t).
Then the amplitude IR of the current iR flowing through the resistor R is equal to IR = V/R,
and the amplitude IC of the current iC flowing through the capacitor C is IC = CV.
It can be seen that the amplitude IR does not depend on frequency, or to put it more
generally on the rate of time changes of the signal, while the amplitude IC does depend on
frequency, and is smaller at lower frequencies, i.e. the slower the time changes of the signal.
For co 0 there will be IC0, and the influence of the capacitor will not appear. In this
case, the actual resistor behaves as an ideal element with resistance R; its model will be
represented only by the ideal resistor R.
At high frequencies, however, the capacitance of the capacitor is manifested. In such
cases we therefore prescribe a certain maximum error that can be admitted in the solution,
e.g. that the current IC be less than 1% of the value of the current IR, i.e. IC < 10-2IR. Hence
follows the condition  < 10-2/(RC) = m. Thus if the angular frequency of the signal satisfies
this condition, the capacitor C can, with a chosen allowable error of solution, be left out and
the circuit can be solved as if it contained a linear resistor R only. This holds for all
frequencies  ℮ <0, m> . Any signal whose frequency satisfies this condition can be
regarded as a relatively slow signal for the circuit under consideration.
With more complicated circuits, the theoretical procedure in examining the criterion of
the signal rate is often more difficult than the actual solution of the circuit; therefore we
usually rely on experimental findings as to whether the given signal is relatively slow, and
thus whether some circuit parameters can be neglected.
A similar procedure is used when examining the problem of signal magnitude, when we have
to decide whether the signal is relatively small or large. In so doing, we start from a definition
by which a signal in a given circuit can be considered relatively small if under the given
requirements on solution accuracy the current components produced by the circuit nonlinearity are negligible.
Note that if we did not limit in advance the rate of changes in the circuit quantities, we
should often have to apply non-linear models with distributed parameters. Their analysis,
however, is in practice very difficult and, in most cases, impossible. We will therefore
assume that the circuits under examination behave as non-linear or parametric circuits with
lumped parameters.
1.3 PROCEDURES IN THE ANALYSIS AND SYNTHESIS OF NON-LINEAR AND
PARAMETRIC CIRCUITS
In principle, there is no difference between the approach to the analysis or the synthesis
of linear and non-linear or parametric circuits. In circuit analysis, the task can be formulated
as follows: there is a known circuit, i.e. we know the parameters and the characteristics of its
elements and their circuit configuration. For non-autonomous circuits we also know the input
signal x(t). It is necessary to find what the output signal y(t) will be, or what the dependence
of the output signal on the input signal, the so-called transfer characteristic y(x), will be.
Conversely, in some cases the output signal y(t) will be given and the input signal x(t) must
be found.
In circuit synthesis, both the input signal x(t) and the output signal y(t) are usually given.
The task consists in examining what elements (including their parameters and characteristics)
the circuit must contain, and how these elements must be connected (i.e. finding the
configuration of the circuit).
Fig. 5. A non-linear signal-shaping converter
The synthesis of circuits is much more complicated and difficult than their analysis, and
there is not always a solution in the general form. Moreover, the solutions are not unique. At
present, we can solve by synthesis only some, mostly simple, problems, e.g. synthesizing the
characteristics of frequency multipliers such that only the, useful product is obtained at the
multiplier output, or synthesizing a converter shaping a triangular signal into a harmonic one
(Fig. 5). Since a synthesis of circuits is in almost all cases a complicated task (with the
exception of the simplest cases), and the scope of our book is limited, we shall mostly
concern ourselves with circuit analysis.
The state of a circuit can be defined as a set of instantaneous values of circuit quantities
in the circuit under examination; the set of the above quantities characterizes the
instantaneous state of the circuit. The term "process" will be used to denote a certain time
sequence of instantaneous states of circuit quantities in the circuit examined. In electronic
circuits, we distinguish two kinds of process:
a) Processes in which the individual instantaneous states do not recur; they are called
transient processes.
b) Processes characterized by the recurrence of instantaneous states in the circuit: they
are collectively called steady-state processes.
Steady-state processes can further be subdivided into two basic types: a do steady state,
also called the quiescent state, which occurs in the circuit when the circuit quantities do not
change with time; and the periodic steady-state process, in which all the circuit quantities
have a periodic waveform with a finite and non-zero period.
A periodic steady-state process can occur in the circuit either owing to a periodic signal
applied to its input terminals or, in the case of unstable circuits, spontaneously. If there is no
time-varying signal acting on the input terminals of a circuit with time-invariant parameters,
its differential equation does not contain any explicitly expressed time. Such circuits are
referred to as autonomous. Under certain conditions (if the circuit is unstable, which will be
discussed in Chapter 9), free oscillations appear in it. If an external time-varying signal is
acting on the circuit or if the circuit parameters vary with time, then an explicitly expressed
time value appears in the respective equation, and such circuits are called non-autonomous.
The processes in non-autonomous circuits are thus expressed by non-homogeneous equations,
the processes in autonomous circuits by homogeneous equations.
The above classification holds for both linear and non-linear and parametric circuits in
which element parameters change periodically (from t = -  to t = + ). If, however, the law
changes according to which the parameters of a parametric element vary, a transient process
arises which is excited by these changes. Hence in parametric circuits, we must also take into
consideration transient processes which are due to a change in the parameter of the element.
When solving circuits with periodic steady-state processes, it is usually possible to
analyse their waveforms by means of the Fourier series expansion. This permits us to solve
and investigate phenomena in non-linear as well as parametric circuits by spectral methods in
the frequency domain. The solution in the frequency domain is applied in those cases where
we are primarily interested in problems of the conversion (transformation) of the signal
spectrum. If, on the other hand, we are primarily interested in the change of shape (i.e. of the
waveform) of the signal, we use the methods of solution in the time domain. Between the
shape (the waveform) and the frequency spectrum of the signal there is always a unique
dependence expressed by the Fourier transform. Any change in the shape entails a change in
the spectrum, and vice versa.
In the general case of a non-linear converter, the task can be formulated as follows: a
signal x(t) is applied to the input of a non-linear converter with the transfer characteristic
y(x), Fig. 5. The converter shall convert this signal so that a signal y(t) is obtained at its
output, which differs from the signal x(t) by its shape and, consequently, also by its spectrum.
When solving a problem in the time domain, the converter must operate in such a manner that
its output signal is
(1.11)
Thus, for example, a signal of triangular waveform, as in Fig. 5, is converted to a signal of
harmonic waveform at the output.
When solving a problem in the frequency domain, the problem is similarly formulated.
We know the frequency spectrum of an input signal x(t) given in the form of its Fourier series
or its spectral function. The converter must change it into an output signal y(t) that has a
spectrum of the desired form.
The method for solving Eq. (1.11), which describes the dependence between the
functions x(t) and y(t), depends on what is given and what remains to be determined: if a
synthesis of the converter is required, or if the performance of the given converter is to be
analysed. In the former case, for a known input signal x(t) and an output signal y(t) their
interdependence y(x) is to be found; in other words, a converter with a transfer characteristic
y(x) is to be constructed. In the latter case, a converter and its transfer characteristic are
given, and for a known input signal x(t) we seek the response y(t), i.e. we analyse the transfer
properties of the circuit.
The above tasks of changing the shape (the spectrum) can be solved either by graphical
or by analytical methods. The graphical methods are usually employed for a qualitative
examination of the studied phenomena. They are clear, they are suitable for simple signals,
and they do not require an approximation of non-linear characteristics in analytical form. The
solution results, however, are of no general validity: they hold only for a given particular
case. With graphical methods, the solution accuracy is usually not very great. The analytical
methods, on the other hand, yield solutions in a general form (not for particular values of
circuit parameters). They are applicable even to complex signals, and the solution accuracy
can be high.
In the graphical solution of simple problems, the method of three planes with rectangular
co-ordinates is usually applied. Thus, for example, Fig. 6a shows the non-linear transfer
characteristic y(x) of a converter, Fig. 6b the waveform of an input signal x(t), and Fig. 6c the
response (the output signal) of the circuit y(t). In the analysis, we construct, for the given
signal x(t) and the non-linear characteristic y(x), the dependence y(t), so that in the third
plane we derive, with the help of the curves x(t) and y(x), the respective points of the curve
y(t) as indicated in Fig. 6.
Fig. 6. Illustrating the method of three planes
If the signals x(t) and y(t) are given (in the case of synthesis), then with the help of these
curves the respective points of the characteristic y(x) of the converter can be derived.
The transfer properties of a parametric linear converter (Fig. 7) can be expressed by the
functional dependence
(1.12)
where P(t) is the time-varying parameter of the converter. In Fig. 7, p(t) denotes the
waveform of the control signal acting on the circuit and affecting its parameter P(t) = P(p(t)).
If in the converter analysis, the input signal x(t) and the law of the time change in the
parameter P(t) are known, the response y(t) is easy to find. Designing a converter with the
time-varying parameter P(t), by means of the given signals x(t) and y(t), is more difficult. In
this case, a convenient circuit configuration must first be found (this task has no unique
solution), and only then can the dependence P(t) be sought.
Fig. 7. A parametric linear signal converter
It is usually necessary to provide a filter that will suppress all the undesired components
of the output of the converter, leaving only the desired signal. The unwanted components of
the output signal are characterized as non-linear distortion; the better the non-linear or
parametric converter, the fewer undesired components appear at the output, i.e. the non-linear
distortion is minimal. (Recall in this connection that by one of the definitions used, the nonlinear distortion of a harmonic signal is given by the ratio of the effective value of all the
higher harmonic components to the effective value of the fundamental harmonic component,
i.e.
, where V1 is the amplitude of the fundamental harmonic
component of the signal, V2 , V3 , ... the amplitudes of the second, third, etc. harmonic
components.)
1.4 THE VALIDITY OF SOME LAWS FOR NON-LINEAR AND PARAMETRIC
CIRCUITS
Kirchhoff's 1aws. Both Kirchhoff's laws are valid for linear and nonlinear as well as
parametric circuits, i.e.
(1.13)
Equations for a non-linear circuit are thus constructed in the same way as for a linear
circuit, i.e. on the basis of Kirchhoff's laws. When constructing equations for a circuit
containing non-linear resistors, the given ampere-volt characteristics of the resistor must be
taken into consideration. If the resistor is voltage-dependent, with the ampere-volt
characteristic i(v), the equation must be constructed with respect to the voltage v. If the voltampere characteristic v(i) is known, the equation is constructed with respect to the current i.
The same recommendations also hold for circuits containing non-linear capacitors with the
characteristics q(v) or v(q) or non-linear inductors with the characteristics (i) or i().
If the circuit contains both a non-linear resistor and a non-linear capacitor (or a nonlinear inductor), the differential equation must be written with respect to the variable which
forms the argument in the characteristic of the non-linear capacitor (or the non-linear
inductor); this is the so-called state variable. The other variables can be determined when the
above variable has been found.
The princip1e of superposition. The principle of superposition holds for all linear circuits
and thus also for parametric linear circuits. We shall prove this statement.
In parametric linear circuits, the dependence between an output signal y(t) and an input
signal x(t) is expressed by the relation
(l.14)
If the input signal consists of N components,
(1.15)
the output signal (the response)
(1.16)
will be equal to the sum of responses to each component of the input signal
(1.17)
It can be seen that, as in the case of linear circuits with constant parameters, the response
of a parametric linear circuit to the action of a sum of signals is equal to the sum of
responses to the action of each signal taken separately. It is in this that the principle of
superposition consists.
For non-linear circuits, the principle of superposition does not hold. This is easy to
demonstrate by the example of a simple non-linear dependence
(1.18)
If the input signal consists of two components, x = xl + x2, then the response
(1.19)
differs from the sum of responses to each component taken separately (i.e.
),
namely by a new component 2ax1x2. The response to the sum of two components of an input
signal thus does not equal the sum of responses to each component taken separately.
The fact that the principle of superposition does not hold for non-linear circuits is one of
the most important features of non-linear circuits.
Fig. 8. A non-linear resistor replaced by a voltage source
The principle of compensation . On the basis of the theorem of compensation, a
non-linear current-dependent resistor in a non-linear circuit can be replaced by a currentcontrolled voltage source without entailing any change in the state of the circuit. The
magnitude of the voltage of the source is equal to the voltage drop across the non-linear
resistor, and its direction is identical with that of the current flowing through the non-linear
resistor (Fig. 8).
To prove this theorem, we select from the circuit D one branch with a non-linear resistor
(Fig. 8a). Its resistance is current-dependent. As shown in Fig. 8c, we can insert into this
branch two current-controlled voltage sources vC(i), whose voltages are of the same
magnitude but opposite polarity; this does not alter the state in the circuit. If
(1.20)
then the potential difference between the points 1-1' is zero, so that these points can be shortcircuited (Fig. 8d) and only the current-controlled voltage source vC(i) will remain in the
branch (Fig. 8b).
Similarly, a voltage-dependent resistor can be replaced by a voltage controlled current
source. By the principle of compensation, controlled voltage or current sources can be used to
correspondingly replace nonlinear inductors or capacitors also.
1.5 BASIC SIGNAL CONVERSIONS IN NON-LINEARAND PARAMETRIC
CIRCUITS
The use of non-linear and parametric circuits allows the realization of a number of
conversions which are of great practical value. A brief survey and a general characteristic of
these circuits will now be given.
1. Conversion of ac to do voltage is accomplished by rectifiers.
2. Conversion of do to ac voltage is accomplished by oscillators or generators,
chopping circuits, etc.
3. Multiplication of the frequency of a harmonic signal, i.e. the conversion of a signal
of frequency co to a signal of frequency n, where n = 2, 3, . . . , is performed by
frequency multipliers.
4. Division of the frequency of a harmonic signal is realized in frequency dividers. It
is the conversion of a signal of frequency c- to a signal of frequency /n, where
n = 2, 3, 4, ....
5. Transformation of the frequency of a harmonic signal (m/n)-times, where m = 2, 3,
4, . .. and n = 2, 3, 4, . .. , with m  n, e.g. m/n = 3/2, is performed by frequency
converters.
6. Transposition of the spectrum of a signal, or mixing, is performed in mixers. In this
operation, two signals of frequencies 1 and 2 are applied to the mixer input (one
of them can be modulated), and after filtration the output signal is of the
combination frequency n1  m2.
7. Regulation (or stabilization) of voltage or current (either do or ac) is realized by
means of voltage or current regulators. At the do regulator output. we obtain a
nearly constant voltage or current even if the voltage across its input or the load
parameters vary within a certain range.
8. Changing the shape of a signal, e.g. of a harmonic signal to a rectangular or
triangular signal, is performed by shaping circuits, or shapers (signal-clipping
circuits, function converters).
9. Limiting the amplitude of a signal is carried out with the aid of amplitude limiters.
A signal of a certain constant amplitude is required at the limner output even if the
input signal amplitude varies within a certain range.
10. Modulation of amplitude, frequency, phase, or pulses is performed by modulators.
11. Signal demodulation, i.e. restoring the modulating signal by demodulating the
modulated signal, is performed by demodulators.
12. Amplification of voltage, current, or power of a signal is performed by amplifiers.
13. Logarithmic amplification and increasing the power of a signal as
a function of time, where the voltage or current at the output is proportional to the
logarithm or a given power of the input voltage or current, is performed by
logarithmic amplifiers.
14. Multiplication of two (or more) signals as functions of time is performed by signal
multipliers. At the multiplier output, a signal is obtained whose instantaneous
value is proportional to the product of instantaneous values of the two input
signals.
15. Division of two signals, i.e. obtaining the quotient of two time functions, is
performed by signal dividers.
16. Synthesis of circuits with negative resistance, capacitance, or inductance, or of
circuits which behave as non-linear (possibly controlled) capacitors or inductors.
Such circuits are called synthetic resistors, synthetic capacitors, or synthetic
inductors.
17. Realization of logic functions in logic circuits (logic circuits realizing logic
addition, multiplication, inversion and also more complex logic functions).
The list of special converters and functions that can be realized by non-linear and
parametric circuits could be continued. Only the most important ones have been given here,
but even this suffices to demonstrate the versatility of the functions these circuits can
perform. As well as these useful and widely exploited phenomena, there are a number of
undesired (parasitic) phenomena in non-linear and parametric circuits which owe their origin
to frequency dependences or to circuit instability and which can interfere with normal
operation.
Circuit elements and their models
2.1 IDEALIZED CIRCUIT ELEMENTS
2.1.1 Linear, non-linear, non-controlled, and controlled elements
The basic electrical properties of a two-terminal or, in general, multi-terminal circuit
element are given by the relations between their terminal quantities, i.e. the voltage v, the
current i, the charge q, and the flux linkage fir. These quantities are found by measuring, with
the help of meters connected to the terminals of a two-terminal or multi-terminal element.
Terminal voltages and currents can be measured directly, while charges and flux linkages can
be measured indirectly by integrating the current i(t) or voltage v(t), since
(2.1)
For simplicity, we shall first deal only with two-terminal elements (one-ports). By
measurement, we can establish relations between an arbitrary pair of quantities (as long as
such relations exist), excepting the pair i and q and .the pair v and , which are governed by
relations (2.1). The remaining combinations of the basic quantities thus form four basic
relations, namely a relation between v and i, between v and q, between i and , and between q
and . These relations correspond to four basic types of two-terminal elements. The last
given relation has as yet been of little practical importance, so that we shall limit ourselves to
the first three cases [13].
A two-terminal element characterized by the relation between v and i, that is to say either
v(i) or i(v), is called the ideal resistor. A two-terminal element characterized by the relation
between v and q, namely v(q) or q(v), is called the ideal capacitor. The relation between the
quantities  and i, i.e. (i) or i(), characterizes the ideal inductor. (Note that the attribute
"ideal" is often omitted.)
The above basic relations, represented graphically as the functional dependences
(2.2)
where y and x represent some of the basic quantities v, i, q, , are called the characteristics of
two-terminal elements. In general, these dependences are non-linear, and such elements are
called non-linear elements. If these dependences are linear, then such elements are called
linear. To denote non-linear two-terminal elements, we shall use the symbols as given in
Fig. 9.
Fig. 9. Diagram symbols of a) non-linear resistors, b) nonlinear capacitors, c) non-linear inductors
In practice, there are a number of two-terminal elements whose properties depend
markedly on a certain external physical quantity p. Such elements are called controlled
elements, and the independent variable physical quantity p is called the control quantity. This
quantity .can be electrical (current, voltage) or non-electrical (temperature, illumination,
pressure, force, velocity, etc.).
The basic characteristics of controlled two-terminal elements can thus be expressed in
general by the functional dependence
(2.3)
which describes a curved surface in three-dimensional representation. To facilitate the
measuring and graphical representation of these dependences, we usually express them by a
family of curves with constant values of the control quantity p, or by a family of curves with
constant values of the independent variable x.
A controlled two-terminal element can then be defined as an element whose basic
characteristic at any instant depends on the value of the control quantity p. In this way, three
types of controlled elements are obtained:
Fig. 10. Diagram symbols of a) non-linear controlled resistors, b) non-linear controlled capacitors, c) non-linear
controlled inductors
Fig. 11. Diagram symbols of linear controlled a) resistors, b) capacitors, c) inductors
1. Controlled resistor characterized by the dependence i(v, p) or v(i, p);
2. controlled capacitor characterized by the dependence q(v, p) or v(q, p) and
3. controlled inductor characterized by the dependence i(, p) or (i- p)
For these elements, the diagram symbols will be used as given in Fig. 10.
For some two-terminal elements, characteristic (2.3) can be expressed in the form
(2.4)
where the quantity y (for a constant value of the quantity p) is directly proportional to the
magnitude of the quantity x. such elements will be called linear controlled elements and
denoted in diagrams by the symbols given in Fig. 11.
Note that in the general case, the properties of an element can depend on several external
physical quantities (p1, p2, ..., pN). The basic characteristics will be expressed by functional
dependences of the type y = f(x; p1 - p2- ..., pN).
2.1.2 Characteristics and parameters of two-terminal elements
Linear e1ements. Linear non-controlled two-terminal elements are characterized by a
straight line going through the origin of rectangular co-ordinates in the y - x plane (see Fig.
12). In the general case, the characteristics are thus expressed by the analytical relation
(2.5)
where P = y/x is a constant quantity called the parameter of the element. This parameter can
be either positive or negative. If it is positive, we speak of an element with positive
parameter; if it is negative, we speak of an element with negative parameter (Fig. 12b). Note
that linear two-terminal elements are characterized completely by their parameter P alone.
Fig. 12. The characteristics of linear elements: a) with a positive parameter (P > 0), b) with a negative parameter
(P < 0)
Control1ed 1inear e1ements. Regarded as linear controlled two-terminal elements are those
elements whose parameters depend distinctly on a certain physical quantity p but not on
circuit quantities. In general, the characteristics of linear controlled elements can be
expressed mathematically by the functional dependence
(2.6)
where the element parameter P(p) is a linear or non-linear function of the control quantity p.
In practice, `the control quantity is usually considered a function of time, i.e. p ≡ p(t), so
that we can also write
(2.7)
Controlled linear elements are therefore also referred to as elements with time-varying
parameters.
Linear controlled two-terminal elements are characterized in the y - x plane by a family
of straight lines, with each straight line corresponding to a certain value of control quantity p
(Fig. 13) or to a certain time t. Fig. 14 shows an example of a family of characteristics of a
linear controlled element whose parameter changes harmonically in time
(2.8)
Fig. 13. The characteristics of a linear controlled element. The arrow of the control quantity p gives the direction
of the increase in this quantity
Fig. 14. An example of the characteristic of a linear controlled two-terminal element and the waveform of the
parameter P(t)
Non-linear e1ements. The parameters of non-linear two-terminal elements depend on the
voltage acting on the element or on the current flowing through it. Thus for non-linear
elements, the dependent variable y - y(t) is a non-linear function of the independent variable
x - x(t). Expressed mathematically, we have
(2.9)
In the y - x plane, such a two-terminal element is characterized by a curve and not by a
straight line going through the origin of co-ordinates (Fig. 15).
The basic properties of non-linear two-terminal elements can be characterized in several
ways. The static parameter P is defined as the ratio of the dependent variable y to the
independent variable x (Fig. 15a), i.e.
(2.10)
Fig. 15. Illustrating the definitions of the parameters of a non-linear two-terminal element
The differential parameter Pd of a two-terminal element is defined as the ratio of the
differentials of the quantities x and y at the given point A or as the derivative of the
characteristic at the point considered (Fig. 15b):
(2.11)
Introducing in place of infinitesimal increments the finite increments x and y, we
obtain the difference parameter P of the two-terminal element (Fig. 15c)
(2.12)
It must be stressed that all these parameters are a function of the independent variable x.
Note also that for linear elements we have Pd(x) = P(x), while for non-linear elements there is
(2.13)
The more this ratio differs from one, the more pronounced is the element non-linearity. The
dimensionless quantity k can thus be employed to evaluate the non-linearity of the element
characteristic.
Two-terminal elements with a negative differential parameter. If the static characteristic
y = f(x) contains a falling portion (see Fig. 16), the differential parameter Pd is negative in
this part of the characteristic, i.e.
(2.14)
Fig. 16. The characteristics of a two-terminal element with a negative differential parameter; a) the N-type
characteristic, b) the S-type characteristic
The shape of the characteristics with a falling portion resembles the letter N or S.
Consequently, such characteristics are usually termed N-type or S-type characteristics. At
points of inflection of the characteristic (i.e. at points A, B), the differential parameter equals
zero (in the case of N-type characteristic) or is infinitely large (in the case of S-type
characteristic).
Controlled non-linear elements. Non-linear controlled two-terminal elements are
characterized by the functional dependence
(2.15)
The parameters characterizing the properties of such elements can be derived from
functional dependence (2.15). The static parameter P is defined as the ratio of the dependent
variable y to the independent variable x with the control quantity p constant, i.e.
(2.16)
From the total differential of Eq. (2.15)
(2.17)
we obtain the differential parameter Pd and the so-called transfer differential parameter Kd,
with
(2.18)
Rewriting Eq. (2.17) in the form
(2.19)
we obtain another differential parameter
(2.20)
All these parameters of non-linear controlled two-terminal elements are functions of the
quantities x and p.
As in the case of non-linear elements, we often work in engineering practice with the
difference parameters of these elements:
(2.21)
The required difference and static parameters of a non-linear controlled element are
comparatively easy to determine at the given operating point
Fig. 17. Determination of the parameters of a non-linear controlled two-terminalelement from the characteristics
y(x, p) by a graphical method
with the aid of a family of static characteristics by a graphical method. The substance of this
method is clear from Fig. 17. At the given operating point M(x0, p0) the parameters P, P, K
and A are determined.
In the following, we shall deal in brief with the basic characteristics and parameters of
two-terminal resistors, capacitors, and inductors and give some typical examples of these
elements [13], [23], [58].
2.1.3 Resistors
Linear resistors. A two-terminal element characterized by a straight line going through the
origin of co-ordinates in the i - v or v - i plane is called the linear resistor (Fig. 18). Its
characteristic can be expressed mathematically by the linear relations
(2.22)
Fig. 18. The ampere-volt characteristics of passive and active resistors
The parameter R = v/i is called the resistance [Ω] and the parameter G = i/v the
conductance of the resistor [S]. When R > 0 or G > 0, we speak of a passive resistor. When
R < 0 or G < 0, we speak of an active resistor or of a resistor with negative resistance
(conductance).
Passive-linear resistors are commercially available components. Linear resistors with
negative resistance do not exist, but we are able to produce them synthetically.
Linear control1ed resistors are characterized by a family of straight lines in the i - v
or v - i plane, with each straight line corresponding to a certain, value of the control quantity
p or time t. A linear controlled resistor can hence be characterized by the relations
(2.23)
Fig. 19. An example of the characteristics of a photoresistor
Fig. 20. An example of the characteristics of a magnetoresistor.
and similarly, a time-varying linear resistor by the relations
(2.24)
Here, R(t) = u/i is a time-varying resistance [Ω] and the parameter G(t) = i/u is a timevarying conductance [S].
Linear controlled resistors are e.g. photoresistors, magnetoresistors, carbon microphones,
field-effect transistors operating at very low voltages (v < 1 V), etc.
A photoresistor is a semiconductor element whose resistance depends on illuminance E
[lx]; some of these elements are linear (Fig. 19).
A magnetoresistor is a semiconductor element whose resistance can be controlled by
magnetic inductance B [T]. The dependence of the current i flowing through the element on
the applied voltage v is linear. The dependence R(B) is linear only within a certain range of B (Fig. 20).
Non-1inearresistors are characterized by non-linear dependences
(2.25)
Fig. 21. Diagram symbols and typical shapes of the ampere-volt characteristics of a) voltage regulator (Zener)
diodes, b) tunnel diodes, c) diode thyristors (four-layer diodes)
In the first case, the resistor parameters are the voltage-dependent conductance G(v) and
the differential conductance Gd(a), with
(2.26)
while in the second case it is the current-dependent resistance R(i) with the differential
resistance Rd(i):
(2.27)
There exist a number of non-linear resistors such as semiconductor diodes, voltage
regulator (Zener) diodes, varistors, tunnel diodes, four-layer diodes, glow discharge tubes,
etc. A feature of the last three elements is that they have regions of negative differential
resistance or conductance in their characteristics. The diagram symbols and typical shapes of
the characteristics of several such non-linear resistors are given in Fig.21. Note that the tunnel
diode has a type N ampere-volt characteristic, while the four-layer semiconductor diode has a
type S characteristic.
Fig.22. Diagram symbols and typical ampere-volt characteristics of a) semiconductor photodiodes, b) fieldeffect transistors, c) thyristors
Non-linear controlled resistors are characterized by a family of curves in the i - v
or v - i plane, with each curve corresponding to a certain value of the control quantity p. This
family of curves is expressed mathematically by the functional dependence
(2.28)
In the first case, the parameters of a non-linear controlled element are the voltagedependent conductance G(v, p) and the differential conductance Gd(v, p):
(2.29)
in the second case, the current-dependent resistance R(i, p) and the differential resistance
Rd(i, p), with
(2.30)
Frequently, for electrically controlled non-linear resistors we also employ the transfer
differential parameters. In a resistor with the characteristics i = i(v, v1), where v1 is the
control voltage (e.g. the voltage of the control grid of a valve), we consider the transfer
differential conductance
(2.31)
and the amplification factor
(2.32)
(the negative sign owes its origin to the derivation - see Eq. (2.20)).
Non-linear controlled resistors are e.g transistors, field-effect transistors (MOS
transistors), semiconductor photodiodes, thyristors, etc. The diagram symbols and typical
shapes of the characteristics of a few non-linear controlled resistors are given in Fig. 22.
2.1.4 Capacitors
Linear capacitors. A two-terminal element characterized by a straight line going through the
origin of co-ordinates in the q - v plane is called a linear capacitor. This characteristic can be
expressed analytically by the relation
(2.33)
the constant parameter C = q/v is called the capacitance [F]. The relation between the current
and the voltage of a linear capacitor is obtained by differentiating Eq. (2.33):
(2.34)
Thus a linear capacitor is characterized completely by a single quantity, its capacitance.
Linear contro11ed capacitors are characterized by a family of straight lines going
through the origin of co-ordinates in the q - v plane, with each straight line corresponding to a
certain value of the control quantity p. Expressed analytically,
v
or, for the time-varying linear, capacitor,
(2.35)
(2.36)
where C(p) = q/v is a capacitance depending on the control quantity p and C(t) is a timevarying capacitance.
The current i(t) flowing through a linear capacitor with time-varying capacitance C(t)
equals
(2.37)
A controlled linear capacitor is e.g. a motor-driven variable capacitor or a capacitive
microphone. In both cases, the control quantity is non-electrical (mechanical and acoustic)
[13].
Non-1inear capacitors are characterized in the q - v plane by the curve
(2.38)
The static parameter of a capacitor with the characteristic q(v) is the voltage-dependent
capacitance C(v), its differential parameter the voltage dependent differential capacitance
Cd(v), with
(2.39)
For a capacitor characterized by the relation v(q), the static parameter is the chargedependent elastance D(q), the differential parameter is the charge-dependent differential
elastance Dd(q).
The waveform of the current i(t) for a voltage-dependent capacitor can be expressed
either by means of differential capacitance in the form of
(2.40)
or by means of static capacitance in the form of
(2.41)
As examples of non-linear capacitors most frequently employed in practice, we can
mention capacitive diodes, MOS capacitors, or capacitors with ferroelectrics (Fig. 23) [13],
[25].
Note that in non-linear capacitors, the easiest to measure is the differential capacitance
(e.g. by the resonance method). For these elements, the characteristics Cd(v) are therefore
usually given (see Fig. 24). The coulomb-volt characteristic q(v) is then obtained by
integrating Eq. (2.39):
(2.42)
Fig. 23. Typical shapes of the coulomb-volt characteristics of a) capacitive diodes, b) MOS capacitors, c)
capacitors with ferroelectrics
Fig. 24. Examples of the farad-ampere characteristics of a) capacitive diodes, b) M OS structure capacitors
Non-1inear contro11ed capacitors. These two-terminal elements are characterized by a
family of curves in the q - v plane, with each curve corresponding to a certain value of the
control quantity. These characteristics can be expressed mathematically by the functional dependence
(2.43)
The parameters of a voltage-dependent non-linear controlled capacitor are the capacitance
(2.44)
the differential capacitance
(2.45)
and the transfer differential parameter
(2.46)
On the basis of these parameters, the current i(t) which is flowing through a voltagedependent capacitor can be calculated. If static capacitance is employed, then
(2.47)
With the help of differential parameters, we obtain
(2.48)
As examples of controlled non-linear capacitors, we can quote photovaricaps of the
diode type or with a MOS structure controlled by illuminance E [13], and capacitors with
ferroelectric dielectric controlled by pressure or-temperature [13], [25]. Fig.25 shows typical
shapes of the characteristics of a semiconductor capacitive diode controlled by illuminance E.
Fig. 25. Diagram symbol of a photovaricap, the coulomb-volt and the farad-volt characteristics of a
photovaricap
2.1.5 Inductors
Linear inductors are characterized in the  - i plan a by a straight line going through the
origin of co-ordinates. Their weber-ampere characteristic can thus be expressed analytically
by the relation
(2.49)
where the parameter L =  /i represents the inductance [H]. The relation between the voltage
and the current is in the case of a linear inductor given by the relation
(2.50)
Linear contro11ed inductors are characterized in the  - i plane by a family of straight
lines going through the origin of co-ordinates; each straight line corresponds to a certain
discrete value of the control quantity p.
These characteristics can be described analytically by the equation
(2.51)
in the case of the controlled element, or by the equation
(2.52)
in the case of the linear inductor with time-varying inductance. Here, L(p) = / i is an
inductance depending on the control quantity p, and L(t) is the time-varying inductance. The
waveform of the voltage v(t) developed across the inductor by the current i(t) will be
calculated from the equation
(2.53)
Fig. 26. A linear controlled inductor and its characteristics
As an, example of the linear controlled inductor, we can give the coil with movable
ferrite core (Fig.26). Here, the control quantity is the depth, designated 1, to which the core is
set into the coil. Note that in this case the dependence L(l) is non-linear.
Non-1inearinductors are characterized by the non-linear characteristic
(2.54)
In the first case, the inductor parameters are the current-dependent inductance L(i) and
the differential inductance Ld(i), with
(2.55)
Similarly in the second case, the inverse inductance Γ() and the inverse differential
inductance Γd() can be defined.
The voltage v(t) developed across the current-dependent inductor by the current i(t) will
be determined by applying the inductance law, either with the help of the differential
inductance
(2.56)
or with the help of the static inductance
(2.57)
Fig. 27. Basic characteristics of a non-linear inductor with a ferromagnetic core
An example of the non-linear inductor is the coil wound on a core of ferromagnetic
material. Neglecting the hysteresis phenomenon, the characteristic  (i) has the typical shape
as given in Fig. 27b; which also shows the dependences L(i) and Ld(i) derived from it.
Non-linear controlled inductors are characterized in the  - i plane by a family of curves;
each curve corresponds to a certain discrete value of the control quantity p:
(2.58)
The basic parameters of a current-dependent non-linear controlled inductor are the static
inductance
(2.59)
the differential inductance
(2.60)
Fig. 28. A non-linear inductor controlled by current lo, and its ampere-weber characteristics
and the transfer differential parameters
(2.61)
The voltage v(t) across a non-linear controlled inductor through which a current i(t) is
flowing can be calculated either from the static inductance
(2.62)
or from the differential inductance and the transfer differential parameter Kd(i-p)
(2.63)
An example of the non-linear controlled inductor is the coil L with a toroidal
ferromagnetic core and a control winding CW (Fig.28a). Varying the direct current to in the
winding of the control coil, the weber-ampere characteristics (i, I0) are shifted vertically, as
shown in Fig. 28b.
2.1.6 Multiterminal elements
In technical practice, there exist a number of non-linear elements which are controlled
electrically (by voltage or current). Such elements have three or more terminals. If an element
has three terminals, we speak of a three-terminal element; if, in general, it has M terminals,
we speak of an M-terminal element.
When analysing circuits with such elements and when investigating their properties, we
are interested not only in the output circuit of these elements but also in their input circuit, in
which the control signal is operative, because a non-linear controlled element also affects the
source of the control signal. For these reasons, we regard electrically controlled non-linear
elements as multi-terminal elements.
Fig. 29. Representation of a three-terminal element
Consider first a three-terminal element as given in Fig. 29a. Of the six variables given
here, only two currents and two voltages are independent. Thus we can regard one terminal as
common and define only two currents (il , i2) and two voltages (vl , v2) according to Fig. 29b.
In theoretical considerations and analyses, any of the given terminals can be regarded as
common. In practice, however, we prefer the terminal which facilitates the measuring of the
characteristics of the element and which is most frequently used as the common terminal.
When measuring the characteristics of a three-terminal element, the procedure is similar
to that applied in two-terminal elements. We connect the element to a voltage or current
source and measure the response. The response, however, need not be only a current or a
voltage.
Measurement can yield the charge qj on the terminal j as a time integral of the current ij, since
(2.64)
or the flux linkage j as the integral of the voltage vj:
(2.65)
Any independent combination of the obtained variables vj, ij, qj and j, where j = 1, 2,
forms a valid family of characteristics characterizing the basic properties of the element. As
with two-terminal elements, there are three that are of great practical importance and are the
most frequently applied combinations of circuit variables, namely
1. v1, v2, i1, i2,
2. v1, v2, q1, q2
3. il, i2, 1, 2
The first combination characterizes a three-terminal resistor, the second a three-terminal
capacitor, and the third a three-terminal inductor. The three-terminal resistor, capacitor, or
inductor is characterized by two sets of characteristics measured with respect to an arbitrary
common terminal, since of the four variables, only two are independent. In the literature, the
three-terminal element is sometimes regarded as a degenerate two-port in which one input
and one output terminal are connected to the common terminal of the three-terminal element.
The above considerations and definitions can be easily extended also to multi-terminal
elements (M > 3).
Three-termina1 resistors. Two families of characteristics of a three-terminal element will be
obtained in the following fashion. Connecting an independent source of the variable x 2 (i.e.
the source of voltage v2 or the source of current i2) to the terminals 2-3 (Fig. 30a) and
measuring on the input terminals 1- 3 the characteristics i1(v1) or v1(i1) for a number of values
of the quantity x2, we obtain a family of the so-called input characteristics. An example of
such a family of input characteristics is illustrated in Fig. 30b.
The other system of characteristics is obtained if we connect an independent source of
the variable xl (i.e. the voltage vl or the current il) to the input terminals 1-3 as shown in Fig.
30c and measure the family of characteristics i2(v2), or v2(i2) on the terminals 2-3 for a
number of values of the quantity xl. This family of characteristics is called the output
characteristics of a three-terminal resistor. Fig.30d shows an
Fig. 30. Measuring the input and the output characteristics of a three- terminal resistor
example of such a family of characteristics. Note that both the input and the output
characteristics of a three-terminal resistor can be regarded as the ampere-volt characteristics
of a controlled resistor (with the control variable x2 or xl). In this sense, a three-terminal
resistor is equivalent to two controlled two-terminal resistors.
Since each of the variables xl, y1, x2, y2 can be either a voltage or a current, there exist
four basic combinations of the characteristics of a three-terminal resistor, corresponding to
four methods for measuring their characteristics:
1. The conductance characteristics
(2.66)
2. The resistance characteristics
(2.67)
3. The conductance-resistance hybrid characteristics
(2.68)
Fig. 31. The method of measuring the conductance characteristics of a three-terminal resistor
4. The resistance-conductance hybrid characteristics
(2.69)
Note that knowing one family of input and output characteristics, the other three sets can
be derived from it by the graphical method [13].
Examples of three-terminal resistors. For the sake of generalization, we have up
to now considered an arbitrary three-terminal resistor of an internal structure that has not
been defined. There are, in practice, a number of such elements: for example, transistor, fieldeffect transistor, thyristor, etc. Typical shapes of the input and output characteristics of some
of these elements are given in Fig. 31.
The three-terminal capacitor and inductor. It follows from the definition of the
three-terminal
Fig. 32. A three-terminal inductor characterized by two sets of characteristics
capacitor that this element is characterized by two sets of characteristics or by two equations
with the variables v1 , v2, q1, q2. They are usually given in the form
(2.70)
A three-terminal inductor is characterized by two families of characteristics or by two
equations with the variables i1, i2, 1, 2. The properties of these elements are usually
characterized by the relations
(2.71)
As an example of the three-terminal inductor, we can take the coil with a toroidal core and
three taps as shown in Fig. 32, together with the typical weber-ampere characteristics.
Fig. 33. An (M + 1)-terminal circuit element
(M + 1)-terminal circuit elements. Now consider an (M + 1)-terminal element, with
(M + 1) > 3. The element has M independent terminals designated in Fig. 33 by the symbols
1, 2, ... , M, the terminal designated M + 1 is common. As in the preceding cases, the
electrical properties of such an element are characterized by a family of M characteristics
which give the dependence of M dependent variables yl , y2, ... , yM on M independent
variables xl, x2, ..., xM, namely
(2.72)
As before, the quantities xm and ym stand for vm or im or also for qm or m.
An (M + 1)-terminal resistor can also be regarded as a three-terminal non-controlled
resistor if between the remaining M - 2 terminals and the common terminal we connect
sources of constant voltage (or current). An (M + 1)-terminal element then behaves as the
ordinary three-terminal resistor described above.
2.1.7 Ideal sources of electrical energy
Independent (non-controlled) sources. Idealizing the properties of actual sources of electrical
energy, we obtain two ideal two-terminal elements, namely the ideal source of voltage and
the ideal source of current [13], [25]. These sources can supply electrical energy permanently
to a connected load, so that they are also called active elements. The voltage of the ideal
voltage source does not depend on the current supplied by the source to the connected load,
and the current of the ideal current
Fig. 34. Diagram symbols and ampere-volt characteristics of ideal independent voltage and current sources
source does not depend on its terminal voltage. In general, the voltage or the current of these
sources can be a certain function of time, v = v(t), i = i(t). In special cases, these quantities
can be constant, v = . V0 or i = I0 ; in that case they are sources of do voltage or do current.
Fig. 34a gives diagram symbols of the sources of voltage v(t) and current i(t). In the circle
marking the source, we sometimes draw the symbols of the voltage or current waveform
( =, ~ ) or, in the case of sources supplying harmonic voltage or current, the symbol of its
frequency cc-. Figs 34b, c show the diagram symbols of the sources of do voltage V- and do
current to and their corresponding characteristics in the i - v -plane. The characteristics of
sources of ac voltage or current can also be illustrated in a similar fashion. In this case,
however, we shall plot the amplitude, the effective value, or some other characteristic
quantity of alternating voltage or current.
Contro11ed sources. As with the parameters of passive elements, the voltage or current
of ideal sources can also be controlled by some control quantity. This control quantity can
again be either non-electrical or electrical. If it is non-electrical, we speak of controlled
sources in the general sense of the word; if it is electrical, then such sources are called
electrically controlled sources [13]. In some sources controlled by a nonelectrical quantity,
e.g. by illuminance, temperature, pressure, etc., the non-electrical energy of the control
quantity is converted into electrical energy. Such sources represent a special group of
controlled sources known as energy converters.
A control quantity p can control both a voltage source and a current source; thus we
obtain two ideal controlled sources:
A voltage source controlled by the quantity p is characterized by a family of vertical
ampere-volt characteristics (Fig. 35a). The terminal voltage of this source does not depend on
the terminal current but only on the control quantity, i.e. v = v(p). A current source controlled
by the quantity p is characterized by a family of horizontal characteristics (Fig. 35b). The
current of this source does not depend on the terminal voltage, but only on the control
quantity, i.e. i = i(p). In a special case, the dependence v(p) or i.(p) may be linear.
Electrically controlled ideal sources. An important group of controlled sources are ideal
sources of voltage and current which are controlled linearly by a voltage vl or by a current il .
There are four types of linear electrically controlled sources:
1. The voltage-controlled voltage source, v2 = Av1
2. The current-controlled voltage source, v2 = Wi1
Table 1.
Ideal electrically controlled sources
Fig. 35. The ampere-volt characteristics of ideal controlled sources of a) voltage, b) current
3. The current-controlled current source, i2 = Bi1
4. The voltage-controlled current source, i2 = Sv1
The diagram symbols and the corresponding characteristics in the i - v plane for these four
types of electrically controlled sources are given in Table 1. Since in all cases there is either
i1 = 0 or v1 = 0, the input power delivered to these controlled sources is also zero, i.e. the
control of the sources takes place without any energy being consumed. All these ideal fourterminal elements are linear and non-reciprocal. The voltage-controlled voltage source
represents the ideal voltage amplifier, the current controlled current source represents the
ideal current amplifier.
2.2 APPROXIMATION OF THE CHARACTERISTICS OF NON-LINEAR
ELEMENTS
2.2.1 Problems of analytically expressing non-linear characteristics
During the operation of a non-linear circuit, the instantaneous operating point moves within a
certain region of the characteristics. The region employed in this manner is referred to as the
operating region of the given element; its selection is .greatly affected by the intended
application of the element (e.g. the region of substantial non-linearities in circuits designed
for the transformation of spectra or, conversely, the region of small non-linearities in
amplifiers, the saturation and cut-off regions of electric current in switching elements, etc.).
Note that the operating region must be within the area defined by allowable operation
parameters, that is by allowable voltages, currents, powers, temperatures, or other limiting
factors (as specified by the manufacturers).
Generally speaking, the characteristics of an (M + 1)-terminal nonlinear element are
expressed by the vector function of the vector variable
(2.73)
giving the dependence of M dependent quantities (such as currents) yl , y2 , . . . , ym on
M independent quantities (such as voltages) x1, x2 , . . . , xM, that is to say
(2.74)
From the geometrical point of view, this vector function corresponds to a system of M curved
(analytically unspecified) surfaces (hyper-surfaces) in the (M + 1)-dimensional Euclidean
space.
If we want to investigate the properties of non-linear circuits analytically, we must
replace the "natural" non-linear characteristics of circuit elements in the assumed operating
region by a mathematical model expressing these characteristics by analytical expressions
with reasonable accuracy. In some cases, we can express the- non-linear characteristics of
elements analytically on the basis of a description of the principles governing the physical
phenomena occurring in these elements. These phenomena: are usually not simple, and their
corresponding mathematical model is therefore mostly complicated and unsuitable for further
mathematical operations with signals. Non-linear characteristics of elements are therefore
usually found experimentally by measuring the required dependences. The results are then
tabulated or plotted in graphs (e.g. the familiar characteristics of valves, semiconductor
diodes, transistors, etc.). The manufacturer gives statistically average characteristics of
commercially available non-linear elements, usually in the form of graphs.
For simplicity, let us first consider the simple example of expressing analytically the
characteristic of a two-terminal non-controlled non-linear element. Its actual characteristic
y = f(x) is usually known at only N + 1 discrete points, for which we have found by
measurement (or from a graph) the values y0 , y1, . . . , yN of the dependent variable
corresponding to the values x0, xl, ..., xN of the independent variable, respectively. Now we
are faced with the task of finding for the known characteristic f(x) a suitable analytical
function g(x)  f(x), or approximating the characteristic f(x) by a function g(x), which we call
the approximation function.
The purpose of the approximation, however, can sometimes be different. The characteristic
f(x) can be given analytically as the mathematical model of some physical phenomenon, and
in this case the sense of the approximation lies in replacing the complicated, and for further
calculations unsuitable, analytical expression f(x) by a simpler approximation function g(x).
Similar problems are encountered when approximating the characteristics of multiterminal non-linear and parametric elements, where an arbitrary actual characteristic ym =
fm(x1, x2 , . . . , xM) in an (M + 1) dimensional space is replaced by a simpler approximation
function ym = gm(x1, x2 , . . . , xM). The characteristic of the given element is usually known
only as a set of discrete values obtained by measurement, or in the form of a graphical
representation of the characteristics.
The problem of approximating a non-linear characteristic is solved in two steps:
1. we choose a suitable approximation function, and
2. we find the coefficients of this approximation function.
There are a number of approximation functions suitable for concrete practical
applications, so that the first step given above has no unique solution. The approximation
function is always chosen such as to express the given characteristic with sufficient accuracy
and be at the same time simple and suitable for further mathematical operations. The more
accurately the approximation function expresses the actual non-linear characteristic of the
element, the greater is our expectation of accurate solution of the properties of the
investigated non-linear circuit. It is, of course, no use trying to achieve a greater accuracy
than that which corresponds to the accuracy with which the characteristics have been
measured. Naturally, when choosing the approximation function, we also take into
consideration the purpose of the approximation, that is to say, what further mathematical
operations with signals it is designed for, whether only a qualitative idea as to the properties
of the given circuit is to be obtained or whether a quantitative evaluation of the signals in the
circuit is also being sought. Thus the choice of the approximation function is, to a certain
degree, a matter of compromise, and we often prefer a definite function on the basis of
experience and intuitive estimation, and with a view to the possibilities provided by modern
computer techniques.
After defining the operating region on the characteristic and after the choice of a suitable
approximation function, we start determining the coefficients of this approximation function.
First, however, we must specify the conditions of approximation, i.e. specify quantitatively to
what extent and in what fashion the approximation function g(x) is to approximate the given
characteristic f(x). The measure of approximation can be formulated in several ways.
Fig. 36. The criterion of uniform approximation
Fig. 37. Approximation by an interpolation curve
In the case of uniform approximation, the approximation function g(x) must not deviate
from the approximated characteristic f(x) by more than a certain value  (see Fig. 36). Thus
in the operating region of the characteristic, the condition
(2.75)
must be fulfilled.
In the case of mean squares approximation, the function g(x) must not deviate from the
given function f(x) by a mean square error greater than δ, i.e. it must satisfy the condition
(2.76)
where (X1, X2) is the approximation interval corresponding to the given operating region.
When approximating, we are often content with simple interpolation (fitting); in which
the function g(x) coincides with the function f(x) only at the given discrete (e.g. tabular)
points (points of coincidence, or nodes M1, M2 , . . . in Fig. 37). No requirements are laid
down for the course of the approximation function between these points (it is only given by
the type of the approximation function chosen). Note that the conditions for the coincidence
of the functions g(x) and f(x) need not concern only the values of these functions at selected
points but also e.g. their derivatives
(x) and
(x) at these points.
When approximating non-linear characteristics, the so-called "piecewise fitted"
approximation function is often used. In this case, the characteristic is expressed segment
wise by several functions which link up with one another at points of coincidence (nodes M1,
M2, ...) with a desired smoothness (at the node of the two mutually linking segments, the
coincidence in derivatives is required to include the derivative of the order of n). When these
"piecewise fitted" approximation functions are given by a number of smoothly fitted power
polynomials, they are called the splines [2].
2.2.2 A survey of the most frequently used approximation functions
1. The power polynomial
(2.77)
has constant coefficients a0 , a1, . . . , aN which depend on the shape of the characteristic. This
group of approximation functions also includes approximations by a straight line (by linear
function), by several straight line segments (by piecewise linear line), and approximation by
incomplete polynomials. The degree N of this polynomial is given, mainly, by the shape of
the characteristic and by which details of the phenomenon under examination we are
interested in.
2. The exponential polynomial
(2.78)
where an, bn are again constant coefficients of the approximation polynomial. The
approximation by exponential polynomial has considerable advantages since for a
satisfactory approximate expression of actual characteristics it is usually sufficient to apply
only two or three terms of the exponential polynomial.
3. The trigonometric polynomial (the Fourier series)
(2.79)
where an and bn are constant coefficients of the Fourier series, and X is the width of the
interval in which the characteristic is approximated (X is of the same dimension as x).
Experience has shown that rendering exactly a non-linear characteristic by the Fourier series
usually requires a comparatively large number of terms of the series. It is therefore seldom
that this approximation is applied unless a computer is used for the calculation.
4. The fractional rational function
(2.80)
This function is used for the approximation of non-linear characteristics of elements, usually
in the simplest form, i.e. when M = 1. Quite frequently, this function is applied when
approximating the characteristics of parametric circuits.
5. The power function
(2.81)
where m and n are natural numbers. This function is usually employed in special cases, e.g.
for vacuum valves.
6. The transcendenta1 function. These functions include various direct and inverse circular
and hyperbolic functions such as y = a sinh bx, y = a tanh bx, etc.
The approximation functions given in the above survey under items 1 to 4 can obviously
be used for approximating characteristics of any shape (provided we take a sufficient number
of polynomial terms). In contrast to this, the last two groups of functions are usually applied
only to the characteristics of a certain shape which the approximation function satisfies.
In some cases, we are able to derive analytical expressions for non-linear characteristics of
circuit elements straight from the description of the physical phenomena taking place in these
elements. Then we either work directly with these theoretically derived functions or we
approximate them by another, more suitable function, e.g. by Taylor's series.
Sometimes, we approximate a characteristic in the given operating region not by a single
approximation function but in such a way that we divide the operating region into two or
more partial areas, in which we approximate the characteristic by a suitable simple method
corresponding to the shape of the characteristic in the given partial area. The approximation
functions must fit each other on the borders of the partial areas. A typical example of this
procedure can be seen in the replacement of an actual characteristic y = f(x) by two or more
linear (i.e. straight-line) segments; we also encounter, however, more complicated
combinations, for instance a linear segment in combination with a parabolic segment, etc. We
can also have multidimensional cases; for example y = f(xl , x2), which in the geometrical
sense represents a surface in three-dimensional space, can be replaced by a system of
mutually fitting planar triangles in this space, whose vertices lie at the given points in the
plane y = f(x1 x2).
Based on similar reasoning is the so-called local approximation. In this type of
approximation, we start from the known discrete values of the characteristic in the close
vicinity of the instantaneous operating point, and in this not very extensive area we
approximate the characteristic by a simple function (usually by a power polynomial of not
more than second degree). The local approximation is often used when solving non-linear
tasks with the aid of computers.
2.2.3 Determining the coefficients of an approximation function
For the evaluation of the coefficients of approximation functions, several methods are used.
The most convenient to use are:
a) the interpolation method,
b) the method of least squares,
c) the rectification method.
The choice of the method for calculating the coefficients depends especially on the
approximation function applied and on the selected conditions of approximation given above.
The substance of these methods will now be explained.
The interpo1ation method. The basic idea of this most frequently applied method
consists in choosing several suitably selected, e.g. measured (tabular) points of the
characteristic through which the approximation function must pass. The number of selected
points must be equal to the number of coefficients in the analytical expression. The coordinates of these points are substituted into the approximation function, which gives us a
system of equations. The number of these equations equals the number of coefficients.
Solving the obtained system of equations yields the required coefficients [4].
The spacing of the known points is chosen such as to cover the whole operating region of
the characteristic; the values x0 , x1, . . . , xN of the independent variable often form an
arithmetic series. If only a small number of points are necessary for the calculation, we
choose them such as to lie in the significant areas of the curve. When choosing the tabular
points, it is necessary to proceed with caution since with an unsuitable spacing of the
abscissas of these points we obtain an approximation (interpolation) function which is
severely and sometimes unacceptably rippled.
The application of the method of selected points to the calculation of the coefficients of
an approximation function will be demonstrated by two examples.
Let the characteristic y = f(x) be expressed by N + 1 points, i.e. the values .x0 , x1, x2 , . . .
, xN of the independent variable have their corresponding values of the dependent variable y0,
y1, ..., yN (obtained e.g. by measurement). We shall approximate this characteristic by the
power polynomial
(2.82)
Substituting the co-ordinates (x0, y0), (x1, yl) through (xN, yN) into the preceding equation, we
obtain a system of N + 1 equations
(2.83)
Solving this system of equations, we obtain the desired coefficients a0, al, a2, ..., aN of the
approximation function. Note that these coefficients have different dimensions.
Determining the coefficients is usually more difficult when approximating by
transcendental functions. For example we are to approximate the characteristic y = f(x) by the
function
(2.84)
(see Fig. 38). For the calculation of two unknown coefficients a, b we choose on the curve
y = f(x) two characteristic points with the co-ordinates (xl, yl) and (x2, y2), through which the
analytically expressed curve must pass. Substituting the co-ordinates of these points in the
above approximation function, we obtain a system of two equations
(2.85)
Solving this system of equations, we obtain the desired coefficients a, b. To determine the
coefficient b, we write the transcendental equation
(2.86)
Fig. 38. Approximating the characteristic by hyperbolic sine
which is best solved graphically: for a few suitably chosen values b we calculate the value of
the right-hand side of the equation and, on the basis of the points obtained, we construct an
auxiliary curve sinh (bx2)/sinh (bxl) = f(b). On this auxiliary curve, the desired value b is
given by the point whose ordinate has the value y2/yl. The coefficient a can then be calculated
from the relation a = y1 /sinh (bx1). In this case, the coefficient a has the dimension of the
quantity y, while b has the reciprocal dimension of the quantity x (since bx must be a
dimensionless number).
The method of 1east squares. Again, let us take the characteristic y = f(x), which we
know at the tabular points (x0, y0), (x1, y1), ..., (xN, yN). Instead of seeking an approximation
function defined as in the preceding case by N + 1 coefficients and going exactly through the
given N + 1 points, let us now seek a simpler curve (defined by M + 1 < N + 1 coefficients),
which goes as close to these N + 1 points as possible. This will be the case if the sum of the
squares of the deviations of the approximation function from the nominal values yn is
minimum.
If g(xn; a0, a1, ..., aM) is the value of the approximation function for the argument xn, and
f(xn) the actual (measured) .value pertaining to the same argument xn, then the values of the
coefficients a0, al, ..., aM must be such that the sum S of the squares of deviations is
minimum, i.e.
(2.87)
This sum has a minimum when the values of the coefficients a0, a1, . . . , aM are such that the
first derivatives of this sum with respect to am equal zero, i.e.
(2.88)
For the calculation of M + 1 coefficients, we thus obtain M + 1 equations
(2.89)
This method is more involved than the method of selected points, but it usually yields better
results. The values of the arguments xl , x2 , ... , xN are suitably chosen over the whole
operating region; they need not form an arithmetic series.
When applying this method, special attention must be paid to the choice of the degree M
of the approximation polynomial. This degree should be as low as possible, but such as to
express with sufficient accuracy the global shape of the characteristic approximated. The
polynomial degree can with advantage be determined with the help of differences [4].
We shall take as an example the calculation of the coefficients by the method of least
squares while approximating the characteristic y = f(x) by a power polynomial of the second
degree
(2.90)
and using for the determination of the coefficients four points of the curve, corresponding to
the arguments xl through x4.
The respective system of three equations for the determination of the three desired
coefficients a0, a1, a2 will in this case be in the form
(2.91)
The rectification method. The substance of this method consists in choosing co-ordinates
in which the non-linear characteristic f(x) is represented in such a way that the chosen
approximation function g(x) plots as a straight line. Thus it is a case of transforming the coordinates. If after the transformation the given points of the characteristic (x0, y0), (x1, y1), - - , (xN , yN) are on this straight line or if they approximate it sufficiently in keeping with the
required criterion, then the applied approximation function is satisfactory. The desired
coefficients will be found from the plot.
Let us approximate a non-linear characteristic of a function with two unknown
coefficients a, b:
(2.92)
To determine these unknown coefficients, we transform Eq. (2.92) into a linear relation
between the quantities X and Y
(2.93)
with the quantities
(2.94)
being functions of x and y but not of the unknown coefficients a, b. From Eq. (2.92), the
coefficients A, B can be determined which are in a certain relation to a, b:
(2.95)
From Eq. (2.94), the desired coefficients a, b can now be calculated. We shall give two
examples.
1. Consider a function y = axb. This function is easy to rectify by taking its logarithm:
(2.96)
Denoting Y = ln y, X = ln x, A = b, B = ln a, we obtain the linear relation
(2.97)
The graph of the equation y = axb plotted in the rectangular system of co-ordinates with
logarithmic scales on the axes obviously represents a straight line with a slope proportional to
the coefficient b. Thus it is easy to determine from the graph the two desired coefficients a, b.
2. The rectification method is also of advantage when calculating the coefficients of the
exponential function y = a ebx. This dependence, too, is easy to rectify by taking its logarithm
(Fig. 39):
(2.98)
The coefficient a will be determined from the condition that for x = 0
Fig. 39. Determining the coefficients of an exponential function by the rectification method
there is y = y0 and hence a = y0 . The coefficient b will be determined by constructing in semilogarithmic representation the straight line
(2.99)
and then b ~ tan α.
2.2.4 Dimensionless form of approximation functions
The individual coefficients of an approximation function expressing a non-linear
dependence between physical- quantities are of different dimensions. This circumstance
usually makes operations with such functions difficult. It is therefore of advantage to first
transform the approximation function into a dimensionless form and, when the solution is V
Transforming an approximation function into a dimensionless form is called normalizing the
characteristic. This normalization usually yields not only a simplification of the calculation
but also a generalization of the results obtained [49].
Consider, for example, the characteristic i(v) expressed by the approximation function
(3.00)
where the coefficients have these dimensions: a0[A], a1[A/V], a2[A/V2], a3 [A/V3]. This
approximation function will be transformed into dimensionless form as follows. First, we
divide both sides of the equation by the coefficient a0 ; we obtain
(3.01)
The coefficient b1 = a1/a0 has the dimension [1/V]so that the new variable
(3.02)
is dimensionless. With the help of this dimensionless quantity, Eq. (3.01) can be rewritten as
(3.03)
Denoting now
, we obtain Eq. (3.01) in the form
(3.04)
In this equation y, x, b2, b3 are dimensionless quantities; y represents the normalized current, x
the normalized voltage. It can be seen that by normalizing we have reduced the number of
parameters of the non-linear equation; instead of four coefficients, the equation contains only
two. Moreover, this form of equation makes it possible to evaluate the effect of dimensionless
coefficients and the importance of the individual terms of the polynomial. For example, with
b2
1 and b3
1 we can, for small values of x, neglect the third and fourth terms with
respect to one. Conversely, with b2
1 and x of the order of one, the first two terms can be
neglected.
When transforming from normalized quantities into physical quantities, we proceed in
reverse order. Thus, for example, in the above example, the desired physical quantities
i(t) and v(t) will be obtained from Eqs (3.01) and (3.02):
(3.05)
A substantial advantage of normalizing thus consists of constructing a dimensionless nonlinear function. This will be achieved when all physical quantities such as currents, voltages,
charges, time, etc. are related to a certain constant quantity. It is convenient to choose as the
reference quantity specially significant values such as the initial values (for t = 0) or the finite
values (for t = ), possibly also the amplitudes or the maximum admissible values of the
respective quantities, etc.
Modelling of circuit elements
3.1 THE PHILOSOPHY OF MODELLING
Physicists and technicians rarely analyse the properties of a physical system under
examination in its entire complexity. What they usually do is to create for the analysis a
model of the system considered, in which all the factors are preserved which affect decisively
the operation of the system but where, on the contrary, a number of factors are eliminated
which greatly affect the complexity of the system but have only a small, and hence
negligible, effect on its properties for the assumed type of operation [13], [23], [37].
When analysing the properties of electronic circuits, we usually proceed not by solving
an actual circuit including all the finesse of the dynamic properties of the elements it is made
up of, but rather by solving a simplified circuit, in which the individual circuit elements have
been replaced by their models. In many cases, we can make do with the simplest models of
circuit elements - the ideal circuit elements, which were dealt with in the preceding chapter.
In many applications, however, we cannot avoid the necessity of including here also other
phenomena caused by various (and often undesired) effects of electrical and magnetic fields
associated with the given element, by the transport delay of charge carriers in the medium in
which the motion of these carriers takes place, etc. These phenomena are taken into
consideration in models of higher complexity. As a simple example, let us take the
semiconductor diode. At low frequencies (e.g. 50 Hz), it can be replaced by a simple model
as shown in Fig. 40a, with the ampere-volt characteristic as defined by Eq. (3.2). At a
comparatively high frequency, the dynamic ampere-volt characteristic of the diode differs
considerably from the static characteristic (Fig. 40b), and a new model must be constructed
for the diode, such as that in Fig. 40c. This
Fig.40. a) Diagram symbol of the semiconductor diode, b) the dynamic ampere-volt characteristics of a power
model takes into consideration the resistance RS of the semiconductor diode measured at frequencies of 50, 500,
2500, and 5000 Hz (curves (1), (2), (3), (4), respectively), c) model of a semiconductor diode
semiconductor material and of the diode contacts as well as the voltage-dependent
capacitance Cd(v) of the PN junction.
The models of circuit elements should
a) permit an adequate qualitative description of electrical properties of an element in
assumed operating conditions;
b) have suitable parameters that are comparatively easy to identify by measuring or by
calculating from known physical properties of the element (e.g. its geometry, distribution
of doping impurities in the semiconductor, etc.);
c) satisfy the expected techniques of analysing the circuit (e.g. the numerical analysis).
The truer (and thus also more complex) the model, the greater is the accuracy of the
resulting solution that can be expected. This will, of course, entail an increase in the demands
on the extent of analysis. The model of an actual device is thus always a certain compromise
between reality and simplicity. Its complexity is affected on the one hand by requirements on
the accuracy of the results of analysis, on the other hand by a reasonable limitation of the
extent of necessary calculations.
3.2 CLASSIFICATION OF CIRCUIT ELEMENT MODELS
Mathematical and circuit models. Mathematical models are represented by mathematical
equations expressing the relations between the respective quantities. They lend themselves
especially to analytical solution and to numerical analysis. A very good mathematical model
of the semiconductor junction diode is the shifted exponential function; shifted exponential
functions are also made use of in the Ebers-M oll mathematical model of the bipolar
transistor.
A circuit model is basically an equivalent circuit of an actual device consisting of ideal
elements, which has roughly the same properties as the actual device. Circuit models are
suitable for both qualitative and quantitative analysis of a circuit or a system, and also for
understanding the physical phenomena occurring in the actual device. Thus, for example, the
circuit given in Fig.40c is the circuit model of a semiconductor junction diode.
Fig. 41. Examples of the ampere-volt characteristics of a hypothetical non-linear controlled resistor with its
operating region and quiescent operating point P as marked
Global and local circuit models. Non-linear circuit models can further be
subdivided into global and local models. A global model must approximate the characteristic
of an element in a wide range. A local model approximates the properties of an element in
only the required operating region of the characteristic (Fig. 41). The local model is a certain
part of the global model and, consequently, it is always simpler. This operating region is
often so small that the curves within this region can be replaced by parallel straight lines. In
this case the local model is represented by a linear circuit. These models are called local
linear models, or simply linear models. Linear models thus render the properties of an actual
device only in the close neighbourhood of the quiescent operating point.
Static and dynamic circuit models. Resistor circuit models derived from static
characteristics will be called static models. These models do not contain inertial (i.e. storage)
elements (e.g. Fig. 40a). If, on the other hand, the model renders the beli-aviour of an element
in dynamic conditions, we speak of a dynamic model. Dynamic models contain inertial
elements (see Fig. 40c).
Components of non -linear models. Linear models are made up of ideal linear
resistors, capacitors, and inductors (with both positive and negative parameters) and ideal
voltage and current sources (both non-controlled and controlled). When constructing nonlinear models, further circuit elements are made use of, namely ideal non-linear resistors,
capacitors, and inductors, possibly also non-linear controlled ideal voltage and current
sources, or also charge and magnetic flux sources (charge models [5]).
When constructing non-linear models of resistive elements, we most frequently employ
as the ideal non-linear resistor the ideal diode and the idealized semiconductor diode.
The ideal diode can be defined as an automatically operating ideal switch sensitive to voltage
polarity (Fig. 42). In the forward direction, the ideal diode behaves as a short. circuit, and in
the reverse direction as an open circuit. Electrical properties of an ideal diode are described
analytically by the equations
(3.1)
Fig.42. The diagram symbol of an ideal diode
and its ampere-volt characteristic
Fig. 43. The diagram symbol of diode and its ampere-volt
characteristic
The idealized semiconductor diode (Fig. 43) is a diode whose ampere-volt characteristic
is described by the equation
(3.2)
where VT = kT/qe is the thermal voltage, k is the Boltzmann constant, T is the absolute
temperature, qe is the charge of electron. This relation has been derived theoretically from the
physical phenomena occurring in the semiconductor diode. Eq. (3.2) gives a comparatively
good approximation to the characteristic i(v) of an actual junction semiconductor diode over a
comparatively wide range of current (i = 10-8 to 10-2 A).
3.3 MATHEMATICAL MODELS OF CIRCUIT ELEMENTS
Mathematical models of two -terminal elements. For illustration, mathematical
models of a few of the most widely used elements will be given. As will be seen, the
respective equations basically represent approximation functions of the fundamental
characteristics of an element.
a) The mathematical model of a junction semiconductor diode is Eq. (3.2).
b) The mathematical model of a varistor is the simple equation
(3.3)
where A is a constant depending on the dimensions and properties of the material, and a is a
material constant depending on the material.
c) The mathematical model of a semiconductor photodiode and of a solar battery is
the equation
(3.4)
where i- is a current proportional to the luminous flux .
d) The mathematical model of a lossless varactor diode is the relation
(3.5)
where C0 is the differential capacitance at v = 0, V is the contact potential, and n is a
characteristic exponent depending on the type of PN junction.
Mathematical models of three -terminal elements. As an example we can take
the mathematical model of a transistor. When studying the physical phenomena inside a
transistor, Ebers and Moll
Fig. 44. Comparison of the measured input and output characteristics of a typical NPN transistor in commonbase connection (the solid curves) with the characteristics corresponding to the Ebers-Moll equations (the
dashed curves)
derived two equations [22]
(3.6)
where iE and iC are the emitter and collector currents, respectively; vEB and iCB are the emitterto-base and collector-to-b-.se voltages; IES, ICS, αR, and αF are parameters which depend on
the construction of individual transistors and must be measured for every transistor
separately. The Ebers-Moll equations represent a very good mathematical model of the
transistor. An example of the good agreement between the ampere-volt characteristic of the
mathematical model and the ampere-volt characteristics of a typical NPN transistor is given
in Fig. 44. The solid curves represent the ampere-volt characteristics of an actual transistor;
the dashed lines give the ampere-volt characteristics corresponding to the Ebers-Moll
equations (with the parameters IES, ICS, αR and αF measured on the same transistor).
3.4 CIRCUIT MODELS OF ELEMENTS
Principles of the synthesis of static models of resistors. Static models of resistors
are made up of ideal resistors, non-controlled and controlled sources, and non-linear resistors
(especially ideal diodes and
Fig. 45. Shifting the ampere-volt characteristics of a resistor in the i-v plane a) horizontally with the aid of a
non-controlled voltage source, b) vertically with the aid of a non-controlled current source
idealized semiconductor diodes). The purpose of the synthesis is to make such a model that
its characteristics are roughly the same as the characteristics of an actual resistor obtained by
measurement. In the synthesis of á model we thus start from the known characteristics of an
actual device [13], [30].
There is no general method for the synthesis of element models. We usually proceed by
successively making up the model of an actual resistor from the above ideal elements. While
doing so, we shift the ampere-volt characteristics in the i - v plane horizontally and vertically
with the help of ideal non-controlled voltage and current sources (Fig. 45), or we may
construct sets of characteristics with the aid of ideal controlled voltage or current sources
(Fig. 46). If a set of characteristics has arisen
Fig. 46. Sets of ampere-volt characteristics and their corresponding models
by an equidistant shifting of a certain line characteristic along an axis of the i - v plane, the
model consists of a non-linear resistor and a linear controlled source. If the characteristics are
distributed non-uniformly, then the applied source must be controlled non-linearly.
Fig. 47 shows a hypothetical set of characteristics of a model of a controlled resistor MR.
If this set of characteristics is .to be limited to one of the quadrants of the i - v plane, we use
the respective limiting conditions in the calculation (e.g. for the first quadrant i ≥ 0, v ≥ 0). In
circuit models, these limiting conditions
Fig. 47. The set of ampere-volt characteristics of a controlled resistor model with the control quantity x
Fig. 48. The technique of limiting the ampere-volt characteristics to one quadrant
are satisfied by inserting ideal diodes at suitable points of the circuit as shown for example in
Fig. 48 (again for the first quadrant).
Synthesis of global models for non -linear three-terminal resistors. A threeterminal resistor is characterized by two families of ampere-volt characteristics. An arbitrary
three-terminal resistor can hence be modelled by a convenient combination of two controlled
resistors. If the characteristics in the two families are distributed in a more or less parallel
fashion, the given three-terminal resistor can be modelled by a pair of coupled models
corresponding to controlled non-linear resistors. Two simple examples will be given.
Consider first a three-terminal resistor with the characteristics as given in Figs 49a, b. It
can be seen that the family of both input and output characteristics has been formed by
shifting the basic line characteristic (shown by heavy solid line) along the axis of the voltage
vl or v2. Both the input and the output sides of the three-terminal element are therefore
modelled by a series combination of a non-linear resistor and a controlled ideal voltage
source. The resistors R1 and R2 have their characteristics shown in heavy lines in Figs 49a
and 49b, respectively.
Since both the input and the output characteristics are spaced uniformly along the voltage
axis, the voltages of the two sources in the model are also directly proportional to the currents
controlling them. The proportionality factors
(3.7)
have been derived from the quantities indicated in Figs 49a, b. Obviously, they have the
dimensions of a resistance.
Let us further have a three-terminal element with the characteristics arranged as given in
Fig. 50. In this case, both the input and the output characteristics are spaced uniformly along
the current axis. The model will therefore consist of two non-linear resistors (R1 and R2, in
Fig. 50c) with the ampere-volt characteristics marked by heavy lines in Figs 50a and 50b, and
of two current sources controlled linearly by .voltage. The factors
(3.8)
have the dimensions of a conductance.
Fig.49. The input and output characteristics of a three-terminal resistor and its model
Fig. 50. Sets of input and output ampere-volt characteristics of a three-terminal resistor and its replacement by a
model
3.5 EXAMPLES OF STATIC MODELS OF NON-LINEAR RESISTORS
Models of non -linear two -terminal resistors are most frequently made with the aid of
ideal diodes or idealized semiconductor diodes. The static characteristics of some resistors
have such a shape that they can be approximated by a characteristic consisting of several
straight line segments (by a piecewise linear line). In such cases, ideal diodes can be applied
in the resistor model.
Consider a two-terminal resistor which has the characteristic as given in Fig. 51a. First
we approximate the characteristic of the actual resistor by several straight line segments.
Then we embark upon the synthesis of a model that has an identical characteristic. The
substance of the synthesis consists in realizing successively the individual segments of the
characteristic. Proceeding from left to right, we add to the model the respective building
block at each such step. For example the characteristic i(v) given
Fig. 51. The procedure in synthesizing the model of a non-linear two-terminal resistor
in Fig. 51a by three segments (1), (2), (3), with the slope corresponding to the resistances
R(1), R(2), R(3), respectively, will be divided into three partial characteristics a, b, c, shown in
Fig. 51c, with the slope corresponding- to the resistances
(3.9)
respectively. This is because for segment (1) it holds R(1) = Ra, for segment (2) it holds
R(2) = 1/(1/Ra + 1/Rb), and for segment (3) it holds R(3) = R(2) + Rc. For each such partial
characteristic, a corresponding building block will be made as shown in Fig.51d. combining
these building blocks will yield an overall model (Fig. 51b) with the characteristic as given in
Fig. 51a. The whole procedure for the synthesis of the model is clear from Fig. 51. Note that
in the first two segments of the ampere-volt characteristic their slope gradually increases,
with the resistance in the respective part of the model decreasing gradually (the individual
branches are connected in parallel), while in the third segment the slope of the characteristic
is smaller again, with a corresponding resistance increase in the model. This will be obtained
by adding building block (3) in series with the first two building blocks.
Three examples will now be given:
The voltage reference diode (the Zener diode). A typical shape of the static ampere-volt
characteristic of such a diode is shown in Fig. 52a (in dashed line). As can be seen from the
same figure, it can be approximated quite closely by three straight-line segments. In the
synthesis of the model, building blocks with ideal diodes will be used again. Fig. 52b shows
the resultant model, Fig. 52d the respective building blocks in which, as in the preceding
case, Ra = R(1) and Rb = R(3).
The tunnel diode. A typical shape of the static ampere-volt characteristic of a tunnel
diode (GaAs) is given in Fig. 53a. In this case, too, the characteristic is sometimes
approximated by three straight-line segments
Fig. 52. The static ampere-volt characteristic of the voltage reference (Zener) diode and its static non-linear
model
Fig. 53. The static ampere-volt characteristic of tunnel diode, its approximation by three straight-line segments,
and the respective model
marked in Fig. 53c. The respective model is illustrated in Fig. 53b. Since there is a segment
with negative slope on the characteristic, a linear resistor with negative resistance (Rb < 0)
must be used in the model.
The semiconductor photovoltaic cell. A typical shape of the characteristic of such a
photodiode with PN junction is given in Fig. 54a. The total current i flowing through the cell
consists of the current ip(v) and the current i(). The current iD(v) is the current flowing
through the non-illuminated diode (with  = 0). The current i() appears when the diode is
illuminated; within a wide range of luminous flux it is almost directly proportional to the
luminous flux . In the theory of semiconductor elements [54] an equation has been derived
for the total current
(3.10)
This equation represents the mathematical model of the photodiode. It is obvious that the
shape of the characteristics corresponds closely to the shape of the characteristic of the
idealized semiconductor diode. It will therefore be used as a non-linear resistor in the circuit
model. The required shifting of the characteristics will be achieved by a current supplied
from an ideal current source connected in parallel to the diode and controlled by the luminous
flux  (Fig. 54b).
Fig. 54. The static ampere-volt characteristics and the non-linear circuit model of a semiconductor photovoltaic
cell
Models of non-linear three-terminal resistors. The above principles can also be
used for the synthesis of static models of threeterminal elements, and we shall illustrate this
by a few examples.
Models of unipolar transistors. As the next example we shall give the non-linear model
of a junction FET. Fig. 57 gives the diagram symbol of the unipolar transistor with N-type
channel and its typical input and output characteristics approximated by straight-line
segments. Since the input characteristic iG(vGS) does not depend on the output voltage, the
input circuit of the model will consist of three ideal elements, viz. the ideal diode D1, the
ideal resistor RG (Fig. 57d), and the source of voltage VGS0 , which can be left out since the
quantity VGS0 is comparatively small. When .modelling the output characteristic iD(vDS, vGS),
we shall restrict ourselves to the first quadrant only. This restriction is achieved by the ideal
diodes D2 and D3. The resistance RD = R´D + Rm determines the slope of the characteristics,
the resistance Rm affects the slope of the boundary line defined by the equation iD = vDS/Rm,
while the controlled current source_ forms a family of characteristics. The coefficient S is
given by the spacing of the characteristics
(3.11)
Since unipolar transistors are controlled by electrostatic field, they exhibit properties similar
to those of a vacuum pentode in three-terminal connection. Hence the model given in
Fig. 57d can also be applied to pentodes.
Models of bipolar transistors. As an example, consider an NPN transistor in commonbase connection, whose diagram symbol is given in Fig. 58a. The input and the output
characteristics of such a transistor can be approximated by straight-line segments as shown in
Figs 58b, c. Applying the procedure as given in Section 3.4, we can construct a global model
as indicated in Fig. 58d. The coefficients
(3.12)
will be determined from the input characteristics in the third quadrant and from the output
Fig. 58. Two models of NPN transistors in common-base connection
characteristics in the first quadrant. The characteristic of the non-linear resistor R in the
model is shown in Fig. 58c by the heavy solid line. In practical applications, the quantities
kvCB, RE, and Rm, can usually be neglected, which makes the model even simpler; for large
signals we thus obtain the model given in Fig. 58e. The same procedure is adopted in the
model synthesis of the NPN or the PNP transistor in common-emitter connection (see [13]).
3.6 DYNAMIC MODELS OF CIRCUIT ELEMENTS
In modelling non-linear and controlled resistors we have so far started from the assumption
that the electrical quantities are do quantities or that they change comparatively slowly. If the
electrical quantities change
Fig. 58. Two models of NPN transistors in common-base connection
comparatively quickly, there appear in the elements inertial phenomena which are modelled
by ideal storage elements (capacitors and inductors) and which we add to the basic static
model of the element [38].
Fig.59. The dynamic model of a semiconductor diode
The dynamic model of a semiconductor diode is given in Fig. 59. It consists of
an idealized semiconductor diode, a resistor Rs, and two non-linear capacitors. one of which
is voltage- and the other current dependent. The resistance RS represents the resistance of the
material of the semiconductor and of the diode leads. The capacitance CD is diffusion
capacitance; CT is the barrier capacitance. The characteristic iD(vD) of the idealized diode is
described by the equation
(3.13)
The dynamic properties of the PN junction in the reverse direction are modelled by the
barrier non-linear capacitance
(3.14)
where Co is the capacitance of the junction at vD = 0, V is the contact potential, vD is the
reverse voltage of the PN junction, and n is an exponent depending on the type of junction
(n = 1/2 for an -abrupt junction, n = 1/3 for a graded junction). The dynamic properties of the
PN junction in the forward direction are modelled by the diffusion .nom-linear capacitance
(3.15)
where tD is the lifetime of minority carriers, VT = kT/qe .
The dynamic model of a bipolar transistor . When solving electrical circuits with
bipolar transistors, especially when applying digital computers, dynamic models of the
bipolar transistor are often resorted to, which are derived from the Ebers-Moll static model of
the transistor.
So-called injection and transport models of the transistor are referred to in this connection.
The injection model is shown schematically in Fig. 60. It consists of two controlled
current sources, two idealized semiconductor diodes, and two pairs of non-linear capacitors.
The ampere-volt characteristics of the diodes DF and DR are expressed by the equations
(3.16)
(3.17)
where IEF, ICR are saturation currents, vBE, vBC are voltages across the emitter and the collector
junctions.
Fig. 60. The injection model of a transistor
The equations for the injection model are in the form
(3.18)
(3.19)
where the diffusion capacitances CDE, CDC model the charge accumulation in quasi-neutral
regions of the base, and the barrier capacitances CTE, CTC model the charge in the reverse
regions of PN junctions. The coefficients αR, αF are the current amplification factors of the
transistor in reverse and forward directions, respectively. The above capacitances can be
expressed by the relations
(3.20)
(3.21)
(3.22)
(3.23)
Substituting from (3.16) and (3.17) into (3.22) and (3.23), we obtain for the diffusion
capacitances the expressions
(3.24)
(3.25)
The quantities IEF, ICR, αF, αR, CE, CC, VE, VC, nE, nC, TEF and TCR are the parameters of the
injection model.
The transport model of the transistor consist-s of the same ideal elements as the injection
model. The difference consists in the choice of reference currents. Both variants of the model
of the bipolar transistor are equivalent if they are derived under identical initial assumptions
[38].
3.7 DESIGNING SYNTHETIC CIRCUIT ELEMENTS WITH THE AID OF AFFINE
TRANSFORMATIONS
One of the main problems of circuit theory has in recent years been the synthesis of nonlinear circuits. Conventional non-linear circuit elements will in many cases serve this
purpose, but some requirements can be satisfied only if circuit elements with unusual,
"unnatural" characteristics are used. Elements possessing such characteristics are made
synthetically. Usually, the same methods are applied here as when simulating actual devices,
and the model obtained is then realized by using commercially available devices and
components. In this connection, further considerable possibilities can be found in the
application of the linear acne transformation, which we know from linear algebra.
Affine transformation in a plane. For the affine transformation of an original in a
plane with co-ordinates x1, x2 into an image with co-ordinates x1, x2 the following relations
hold [41]:
(3.26)
or, in matrix notation,
(3.27)
where
(3.28)
is a regular matrix 'of the centroaffine transformation satisfying the condition det A  0, and
(3.29)
is the matrix of the translation vector co-ordinates.
If the determinant is det A > 0, we are concerned with the proper affine transformation
during which the orientation of the transformed geometrical configurations does not change.
In the case of det A < 0 the transformation is improper, and the orientations of the original
and the image of the transformed planar configuration are opposite.
A survey of some special cases of affine. transformations in a plane can be found in
Table 2.
Application of affine transformation in circuit theory. When applying the
rules of affine transformation to electrical circuits, we must always take into account that the
quantities we are concerned with have their physical dimensions. Consequently, the elements
of the transformation matrix A will also have specific physical dimensions.
We must further bear in mind that when using the transformation we frequently
differentiate quantities with respect to time or, as the case may be, we integrate them with
respect to time. We have come across time derivatives or integrals of circuit quantities v, i, q,
and  already in the elementary description of capacitor and inductor properties.
Table 2 - continued
The following well-known relations hold for these circuit quantities:
(3.30)
(3.31)
where the operator s has the usual meaning of differentiating with respect to time, and s-1 the
meaning of integrating with respect to time. As for conventional circuit elements, the
exponent values of the operator s are restricted to the set (-1, 0, 1). Observe in this connection
that modern circuit theory also applies higher derivatives or multiple integrals of circuit
quantities; this brings us to the field of unconventional circuit elements, which are usually
designed synthetically.
Linear transformation networks that realize the required geometrical affine mapping of
circuit element characteristics and which, as the case may be, also permit simultaneous
changes in their physical nature, form an independent class of linear circuits; we shall refer to
them under the general term of affinors. In simple cases, they are general transformation twoports with voltage and current orientations as shown in Fig.61. Their transformation
properties are expressed by cascade equations which. in matrix notation, are in the form
(3.32)
In this case, the elements of the matrix A have a physical dimension; namely all a11[-],
a12[Ω], a21[S], a22[-].
Fig.61. Voltage and current orientations in a transformation two-port (in keeping with the recommendations of
the IEC meeting in Prague in 1967)
In this two-port, we require an affine transformation from the plane (0´; sav1, -sbi2) into the
plane (0'; scv2, -sdi2) where the exponents a, b, c, d of the operator s are integers. We shall
assume this transformation to be bilateral
(3.33)
The matrix equation corresponding to this transformation relation is
(3.34)
and from this we can derive Eq. (3.32) in the form
(3.35)
By comparison we find that the individual coefficients of α have the dimensions α11[sc-a],
α12 [Ωsd-a], α21 [Ssc-b], α22 [sd-b]. From the discussion of Eq. (3.35) it follows that the absolute
magnitude of the coefficients α11, α12, α21 α22 determines the geometrical motion (such as the
change in scale, rotation, reflection) of characteristics in the transformed plane, while the
exponent differences c - a, c - b, d - a, d - b determine the qualitative change in the physical
nature of the circuit element connected across the affinor output and transformed into its
input.
Though on examining the possibilities of realizing affinors we find that there are no
objections in principle to a realization that would implement directly the transformation as
indicated in Eq. (3.34), it is reasonable to prefer successive transformation by way of a
cascade of specialized simpler affinors. Cascaded in this case are, on the one hand, motion
affinors, which perform geometrical affine motions (scaling, rotation, reflection) in the same
plane (e.g. in the ampere-volt plane, less frequently in the coulomb-volt or ampere-weber
plane); on the other hand, so-called mutators, which realize the transformation of the physical
nature of the connected circuit elements without changing the geometrical shape of their
characteristics.
In the category of motion affinors, it is again of advantage to consider separately twoports for scaling transformation, called scalors, two-ports for rotation transformation, called
rotators, and two-ports for reflection transformation, called reflectors.
Sca1ors. Transformation two-ports implementing the affine mapping as given under
items 1, 5, and 6 of Table 2 are called scalors. They are linear active non-reciprocal two-ports
which perform the elongation (dilatation) or compression (contraction) of an originally nonlinear characteristic of a resistive, capacitive, or inductive circuit element in the direction of
one or the other co-ordinate axis or, as the case may be, in the direction of both axes of the
given plane. Accordingly, we distinguish voltage, current and power scalors [15].
For the voltage scalor, whose block-diagram symbol is given in Fig. 62a, the following
set of equations holds
(3.36)
in which the coefficient kv. > 0 determines the voltage scaling value during. the
transformation. If a non-linear resistor is connected across the output terminals of a voltage
scalor, the scalor will from the viewpoint of input terminals act as a resistive element whose
characteristic i1(v1) as compared with the original characteristic i(v) = -i2(v2) of the connected
element is elongated or compressed in the direction of the voltage axis depending on whether
kv > 1 or kv < 1 (see Fig. 62b).
The possibility of realizing a voltage scalor while using one "floating" or two "grounded"
controlled sources is illustrated in Figs 62c and 62d, respectively.
Fig. 62. Voltage scalor: a) diagram symbol, b) scaling in the direction of the voltage axis during transformation,
c) and d) optional realizations by means of either one or two controlled sources, respectively
Fig. 63. Current scalor: a) diagram symbol, b) scaling in the direction of the current axis during transformation,
c) and d) realization models with either one or two controlled sources, respectively
The block-diagram symbol of the current scalor is given in Fig. 63a. The set of
transformation equations of this two-port has the form
(3.37)
On connecting a non-linear resistive element to a current scalor, the shape of its characteristic
is changed in the direction of the current axis, i.e. elongated if ki > 1 (Fig. 63b) or compressed
if ki < 1. The current scalor can be realized with the aid of either one or two controlled
sources, as shown in Figs 63c, d.
Connecting a voltage scalor and a current scalor in cascade, we obtain a power scalor
characterized by the coefficients kv and ki .
Rotators. Linear active reciprocal two-ports implementing the affine transformation
called rotation (item 3 in Table 2) are called rotators. Characteristically, they rotate the
characteristic of the connected non-linear circuit element with respect to the origin of the
given system of co-ordinates through a given constant angle. The shape of the given
characteristic does not change so that we have to do here with an orthogonal transformation
in a plane. Unlike scalors, rotators require a different connection depending on whether the
properties of resistors, capacitors, or inductors are being transformed. Accordingly, we
distinguish resistive, capacitive, and inductive rotators [16].
For the resistive rotator the following system of equations holds:
(3.38)
Its block-diagram symbol is given in Fig. 64a. This rotator rotates the ampere-volt
characteristic i(v) = - i2(v2) of the connected non-linear resistor in the ampere-volt plane
through an angle , as shown in Fig. 64b. The parameter R in Eq. (3.38) has the dimension of
a resistance, and its magnitude will be found from the ratio of the current scale mi [mm/A] to
the voltage scale mv [mm/V]
(3.39)
Fig.64. Resistive rotator: a) diagram symbol, b) rotation of the characteristic in the volt-ampere plane through an
angle  during transformation, c) and d) two examples of realization models
It is the resistance of a resistor whose ampere-volt characteristic plotted in the given system
of co-ordinates with the scales mi and mv is represented by a straight line going through the
origin and making an angle of 45° with the voltage axis. A resistive. rotator can be realized
by means of resistive two-ports containing two mutually coupled controlled sources or a T- or
-network in which at least one resistor has a negative resistance. Two examples are shown
in Figs 64c, d.
The capacitive rotator rotates the characteristic q(v) of the connected non-linear
capacitor in the coulomb-volt plane counterclockwise through an angle . Its realization
models are capacitive two-ports [16].
Fig. 65. Resistive reflector: a) diagram symbol, b) reflection of the characteristic about a straight line making an
angle /2 with the voltage axis, c) and d) two examples of realization models
The inductive rotator performs rotation in the ampere-weber plane through an angle . It
is implemented by means of inductive two-ports [16].
Ref1ectors are linear non-reciprocal active two-ports realizing the improper affine
transformation called reflection (see item 4 in Table 2). An image of the characteristic of the
connected circuit element is in this transformation obtained by its reflection about the line
going through the origin of co-ordinates and making an angle  with the axis of independent
variables. In a similar way to rotators, we distinguish resistive, capacitive, and inductive
reflectors [15].
The transformation properties of a resistive ref lector, whose block diagram symbol is
given in Fig. 65, are expressed by a system of equations
(3.40)
in which R has the same meaning as the corresponding parameter of the resistive rotator
(Eq. (3.39)). Fig.65 shows what characteristic i1(v1) is obtained on the input terminals of a
resistive reflector if a resistor with the characteristic i(v) = - i2(v2) is connected across its
output. Figs 65c and 65d give two examples of realization models of a resistive reflector.
In the special case when  = 45°, the resistive reflector represents a two-port
characterized by conductance equations i1= Gv2 and i2 = - Gvl , and is called the gyrator. Its
gyration conductance is G = 1 /R.
In the case of  = 90°, Eqs (3.40) are simplified to v1 = -v2 and i1 = - i2 . A resistive
reflector characterized by these two equalities is called the voltage negative impedance
converter (VNIC). It is characterized by reflecting the characteristic i(v) = - i2(v2)
symmetrically about the current axis (i.e. changing the voltage sign).
Another special case of a resistive reflector is obtained when choosing  = 0°. Eqs (3.40)
rewritten as v1 = v2 and il = i2 characterize the current negative impedance converter (CNIC).
It reflects the characteristic i(v) = -i2(v2) symmetrically about the voltage axis and thus
changes the current sign.
For the transformation of the characteristics of capacitors or inductors we shall use
capacitive or inductive reflectors, respectively. Similar to rotators, realization models are in
this case capacitive or inductive two-ports [15].
Mutators. Two-ports realizing the change in the physical nature of the connected circuit
elements without changing the geometrical configuration of their characteristics (apart from
scaling or possibly interchanging the independent and dependent variables) are called
mutators [15]. Since in this type of transformation no complicated geometrical motions of the
characteristics are required, the matrix A in Eq. (3.35) will be in a simpler form, as given
under items 1 and 2 in Table 2.
In the first of these two cases, the transformation equations will be in the form
(3.41)
Fig. 66. Mutator with proper transformation in accordance with Eq. (3.38): a) diagram symbol, b)
transformation of the characteristics from the plane (0'; scv2, -sdi2) into the plane (0; sav1, , sbil), c) and d) optional
realizations by means of two controlled sources
The transformation will be proper if the product αllαl2 > 0. The transformation properties of
this mutator are expressed completely by its cascade matrix
(3.42)
where n = c - a, m = d - b. The change in the physical nature of the connected circuit element
is determined by the exponents n and m of the operator s. The arrangement of the matrix A
and the meaning of the exponents n and m have led to the block-diagram symbol for this
mutator as given in Fig. 66a. An idea of the transformation of a characteristic from the plane
(0'; scv2, -sdi2) into the plane (0; savl, sbil) can be gained from Fig. 66. The basic circuit models
given in Figs 66c, d contain sources controlled by derivatives or integrals of the voltage or
the current.
In the second case considered, the independent and dependent variables are interchanged.
This can be seen from the transformation equations
(3.43)
Fig. 67. Mutator with improper transformation (interchanging the independent and the dependent variables) in
accordance with Eq. (3.40): a) diagram symbol, b) transformation of the characteristic from the plane (0'; scv2, sdi2) into the plane (0; scv2, sbil), c) and d) realization models with two controlled sources
With α12α21 > 0 this transformation is improper. The transformation properties of this type of
mutator are expressed by the cascade matrix
(3.44)
where the exponents n = d - a and m = c - b are defined differently from the identically
denoted exponents in matrix (3.42). The block-diagram symbol of a mutator with
transformation equations (3.43) is shown in Fig. 67a; Fig. 67b illustrates the transformation
of a characteristic from the plane (0'; scv2, -sdi2) into the plane (0; sav1 , -sbil) involving an
interchange of the independent and the dependent variables. In this case, too, sources
controlled by the derivatives or integrals of the voltage or current are applied in the basic
circuit models (Figs 67c, d).
The plane of exponents x and y, shown schematically in Fig. 68, gives an idea as to what
transformation properties and abilities the mutators have. In this plane, each point Px,y with
integer co-ordinates x, y characterizes a specific plane (0; sxv, syi) with typical physical
features. The double circles denote the points
Fig. 68. The distribution of identification points PX,Y in the plane of exponents x and y
corresponding to the three conventional planes. The point P0,0 characterizes the plane (0; v, i)
where we place the characteristics i(v) of resistive elements. The point P-l,0 corresponds to the
plane (0; , i) where the characteristics i() of inductive elements are to be found. The
characteristics q(v) of capacitive elements pertain to the plane (0; v, q); its corresponding
point in Fig. 68 is the point P0, -1.
In the sense of the preceding considerations, the mutator with a cascade transformation
matrix (3.42), characterized by the exponents n, m of the operator s, represents a
transformation from an arbitrary identification point Px,y to another identification point
Px + n , y + m. If we wish to transform a resistive nonlinear element into an inductive non-linear
element, we shall use for this purpose a mutator with n = -1, m = 0, and this will perform the
transformation from the point P0,0 into the point P-1,0 . Such a mutator, loaded at the output by
a resistor with the non-linear characteristic i(v), shows at the input terminals as an inductor
with the non-linear characteristic i() geometrically similar to the characteristic i(v). The
transformation works in both directions: an inductive non-linear element with the
characteristic i() connected to the input terminals transforms at the output terminals into a
non-linear resistor with the characteristic i(v) geometrically similar to the characteristic i().
The same mutator can be used for a number of other transformations into non-conventional
regions. Thus, for example, a capacitor with the non-linear characteristic q(v) will transform
at the input as an unconventional circuit element with the characteristic q() geometrically
similar to the original characteristic q(v) (here we pass from the identification point P0,-1 to
the point P-1,-1). The properties of an element with the characteristic q() will be discussed
later. If an inductor with the characteristic i() is connected not to the input but to the output
terminals, it will transform at the mutator input as an unconventional element with the
characteristic i(s-2v), where the variable s-2v itself is unconventional (a double time integral of
the voltage v).
Obviously, the transition is in this case from the point P-1,0 to the point P-2,0 . Also the
two remaining transformations corresponding to the transitions from P0,0 to P-1,0 (resistor at
mutator input) and from P0,-l to P1,-1 (capacitor at mutator input) take us into unconventional
regions.
A mutator with n = 0, m = -1 permits in the conventional region the transformation of a
non-linear resistor into a non-linear capacitor (transition from the point P0,0 to the point P0,-1)
and vice versa. In addition, however, a number of transformations into unconventional
regions are possible with this mutator (from P0,0 to P0,l , from P-1,0 to P-1,1 and to P-1,-1, from
P0,-1 to P0,-2). Designed for the transformation of a nonlinear capacitor into a non-linear
inductor and vice versa is the mutator with n = -1, m = 1 (the transition from P0,-1 to P-1,0). It
can also be used to realize transformations into unconventional regions (from P-1,0 to P-2,1,
from P0,0 to P-1,1 and to P1,-1, from P0,-1 to P1,-2).
The memristor. To illustrate the unusual dynamic properties of unconventional circuit
elements, a simple example will be given. On two earlier occasions we considered the
transition to the point P-1,-1 in Fig. 68. This point characterizes the plane (0; s-lv, s-li) or, in
conventional notation, the plane (0; , q), in which the conventional variables  and q are
used but which does not rank with conventional planes. Chua [14] has shown that an element
with the coulomb-weber characteristic q() is of a resistive nature but, in addition, it has
certain inertial or memory properties. Thus he has denoted this element as the memristor (a
coinage from memory resistor). Introductory information as to the basic properties of the
memristor can be obtained from studying the elementary case with
the quadratic characteristic
(3.45)
where the proportionality factor k has obviously the dimension of CWb-2 or AV-2 s-1. The
dependence between the current and the voltage is in this memristor given by the relation
(3.46)
It can be seen that a memristor with the characteristic expressed by Eq. (3.45) behaves as a
resistor with time-varying conductance G(t) = 2k(t). The magnitude of this conductance is
directly proportional to the flux linkage (t) or, in other words, to the time integral of the
voltage v(),  -, t, which until time t was acting on the given memristor.
For a memristor whose characteristic q() is expressed in the plane (0; , q) by a power
polynomial of degree P
(3.47)
the dependence between the current and the voltage can in general be derived in the form
(3.48)
A discussion of this relation reveals that also in this case the memristor behaves as a resistor
with time-varying conductance G(t). The memory properties are again expressed by the flux
linkage (i.e. by the time integral of the voltage), namely in the term of the p-th degree by its
(p - 1)-th power. It is interesting to note that in the linear memristor this memory property
disappears; this is bečause for P = 1, Eq. (3.48) will be in the form
(3.49)
A memristor with the linear characteristic q = k0 + kl obviously behaves as a linear resistor
with constant (time-invariant) conductance G = k1, and they cannot be distinguished from
each other.
Further interesting properties could also be found in the other unconventional elements
obtained by transformation with the aid of mutators. Typical features here are the time
dependences of the parameters; these dependences can be derived from both integrals (as in
the case of the memristor) and derivatives of the variables.
Spectral transformations in non-linear and
parametric circuits
In the solution of electrical properties of non-linear and parametric circuits operating in
steady state, we start from an analysis of the signal spectrum in the circuit. For this purpose
we employ the familiar possibility of expressing a periodic (quasi-periodic) function by the
Fourier series in the trigonometric or in the exponential (complex) form [4].
4.1 THE FOURIER SERIES AND ITS COEFFICIENTS
4.1.1 Expressing a periodic function by the Fourier series
Periodic signals satisfying the condition
(4.1)
with period T can by Fourier be expressed by a series of trigonometric functions
(4.2)
Here, 1 = 2 /T. In Eq. (4.2) the following relations hold:
The Fourier coefficients A0, Ak, Bk are given by the relations
(4.3)
and they express the magnitude of the do component of the signal x(t), the amplitude of the
cosine component of harmonic oscillations of frequency kccy , and the amplitude of the sine
component of harmonic oscillations of frequency k1, respectively.
If the function x(t) is even, i.e. x(t) = x( - t), then all the sine components of series (4.2)
must be zero (since they are odd functions). The Fourier series of the even function contains
therefore in addition to the do component only the cosine components
(4.4)
By contrast, the Fourier series of the odd function satisfying the condition x(t) = - x( - t) will
contain only the sine components (without the do component)
(4.5)
At times, it is of advantage to recast the Fourier series of a real function x(t) in the complex
form [26]
(4.6)
where the Fourier coefficients
(4.7)
It follows that between Eq. (4.6) and Eq. (4.2) the following relations hold:
(4.8)
For many purposes, it is advantageous to consider not the periodicity of the function in time
but its periodicity in angle. If for the instantaneous argument (instantaneous phase) of the
fundamental harmonic component
we introduce the designation αl = lt, then e.g. Eqs (4.6) and (4.7) will change to the forms
(4.9)
(4.10)
respectively. Eqs (4.2) and (4.3) can also be rewritten in a similar fashion. If there is a singlevalued real function y = f(x) of the periodic variable x = x(αl), then this function
y = y(αl) = f(x(αl)) will also be periodic and it can be expressed by the Fourier series.
4.1.2 Expressing a quasi-periodic function by the Fourier series
It often happens in engineering practice that acting on a non-linear (parametric) element in a
non-linear (parametric) circuit are several independent harmonic signals of different mutually
incommensurable frequencies (e.g. in mixing, amplitude modulation, demodulation). The
output (and sometimes also the input) signal is then given not by the periodic but by the
quasi-periodic time function.
Quasi-periodic functions are non-periodic functions with a discrete spectrum whose
spectral components have frequencies from the set of combination frequencies
(4.11)
where the frequencies l , 2 , ... , N are linearly independent and where the coefficients kl ,
k2 , ... , kN are integers. From the mathematical point of view, we are concerned here with a
class of almost-periodic functions whose base l, 2, ..., N is integral and finite (see [36],
pp. 67, 118, or [40], p. 145).
A direct method of notation and spectral analysis of quasi-periodic functions (as given
e.g. in [4]) is not of much advantage in practice. Of greater advantage in expressing quasiperiodic signals is the application of a multidimensional Fourier series. In that case the
procedure is such that for the whole calculation the instantaneous phases cant are regarded as
linearly independent time functions
(4.12)
and only when the calculation is finished do we return to the original expressions. The Ndimensional Fourier series has the form
(4.13)
The complex Fourier coefficients of this series have the magnitude 1 2n 2n
(4.14)
If the quantity x is real, then -for the Fourier coefficients the following relations hold:
(4.15)
If there is a single-valued real function y = f(x) of the quasi-periodic variable x = x(αl , α2 , . ..
, αN), this function will also be quasi-periodic, and it can be expressed by the N-dimensional
function y = y(αl, α2, ..., αN) = f(x(αl, α2, ..., αN)) which is periodic with period 2 in all the
variables an.
4.2 SPECTRAL ANALYSIS OF SIGNALS IN NON-LINEAR CIRCUITS WITH A
HARMONIC INPUT SIGNAL
4.2.1 The graphical method
A periodic output signal y(t) or y(α) (where α = t) is very often derived graphically from the
known transfer characteristic y = f(x) of a non-linear element (circuit) and from the known
input signal x(t) or x(α). The procedure of such derivation is illustrated in Fig. 69. The curve
y(α) is derived point by point. Thus, for example, point Ql with the co-ordinates (αl , yl) in
Fig. 69c corresponds to point Pl with the co-ordinates (αl , xl) in Fig. 69b. The co-ordinate xl
is shared, of course, by further points P2, P3, P4, ... of the curve x(α) in Fig. 69b whose
corresponding points Q2, Q3, Q4, ... with the co-ordinate yl lie on the curve y(α) in Fig. 69c.
Integrals (4.3), which express the Fourier coefficients of a periodic function y(α), are in
this case usually calculated by an approximate method, the essence of which will be given
here in brief.
Let there be a periodic function y(α), which we know for instance from the graphical
derivation in accordance with the above explanation; this function has in the argument an
angle a and is periodic with period 2 . The period of the function y(α) will be divided
uniformly into N parts. Corresponding to the n-th part (n = 1, 2, . . . , N) is the abscissa
αn = 2n/N and the ordinate yn = y(αn). Integrals (4.3) expressing the Fourier coefficients of
the given function (in a notation which respects the new variable α instead of t) will be
substituted by the approximate sums
(4.16)
(4.17)
(4.18)
The higher the harmonic component we want to know, i.e. the larger the k we require, the
greater the number of parts N must be. It can be derived that N ≥ 2k (an analogue to the
Shannon-Kotelnikov theorem for expressing a continuous function with a bounded spectrum
by a series of discrete values). The above method is known as the discrete Fourier transform,
and it is widely applied in computer-aided processing of signals. A very effective calculation
apparatus to determine the Fourier coefficients of a periodic function y(α) is provided by the
efficient algorithms of the so-called fast Fourier transform (FFT) [10], [46].
4.2.3 Analysis of signals when approximating the characteristic by a piecewise linear
function
Replacing the non-linear characteristic y = f(x) of a non-linear element or circuit as indicated
in Fig. 72 by two straight-line segments, we can express this characteristic analytically in the
form
(4.26)
where S is the slope of the straight-line characteristic in the region x > Xp . The frequency
spectrum of an output signal y(α) will be sought in the case the input signal x(α) consists of
the do and one harmonic component
(4.27)
The waveform of the output quantity will be found by substituting for x in the approximation
function (4.26); we obtain
(4,28)
The boundary between these two cases is represented by a state which obviously arises at the
special value α =  and for which the condition holds
from which we can determine the magnitude of the angle , called half the conduction angle,
namely from the relation
(4.29)
Now we can determine the individual components of the output signal y(α) or y(t). These
components are calculated by means of modified Eqs (4.3). The magnitude of the do
component y0 will be
and substituting for Xo - Xp from Eq. (4.29), we obtain
(4.30)
In this formula for the calculation of the do component of the output signal, the quantity X1
can be substituted by the maximum value of the output quantity y, which in Fig. 72 is denoted
as Ymax . This maximum value obviously occurs for cosα = 1 so that by Eq. (4.28) it holds for
this value
(4.31)
Substituting now from here into expression (4.30), we obtain for the do component the
relation
(4.32)
and for its relative magnitude (with respect to Ymax)
(4.33)
This dependence is usually given graphically (see Fig. 73, the curve α0 = f()).
Fig. 73. The dependence of the relative magnitude of the output quantity αn = Yn/Ymax
on half the conduction angle , for n = 0, 1, 2, 3
By a similar procedure, the amplitudes of the fundamental and of the higher harmonic
components of the output signal can also be determined. For the amplitude of the
fundamental harmonic component Yl we get the relation
(4.34)
and for the relative magnitude of this output harmonic component
(4.35)
(the dependence αl = f() has also been plotted in Fig. 73).
In the general case, the amplitude of the k-th harmonic component of the output signal
(k > 1) will be given by the relation
,
(4.36)
and its relative magnitude
( 4.37)
(Fig. 73 gives the dependences α2() and α3()).
Fig. 74. The dependence of the relative quantities n = Yn/SX1 on half the conduction
angle  for n = 0, 1, 2, 3
The dc component and the amplitudes of harmonic components of the output signal can
also be related to the product SX1. Then for the k-th component we obtain the relative
quantity
(4.38)
The dependences x() for k = 0, l; 2 and 3 are given in Fig. 74.
From the expressions for Y0 , Yl and Yx it is evident that the magnitudes of the individual
components of the output signal depend on the quantity Ymax and on the angle . Taking into
account definitions (4.31) and (4.29), we can readily see that the magnitudes of the
component Yk depend on the one hand on the properties of the non-linear element or circuit
(i.e. on the values S and Xp), and on the other hand on the input signal (i.e. on the values X0
and X1).
As can be seen, the method of spectral analysis based on the approximation of the
characteristic by a piecewise linear function is simple and gives generalized results illustrated
by the curves in Figs 73 and 74. The actual characteristics of non-linear elements or circuits,
however, approach the chosen approximation only roughly. In practical applications we
therefore commit errors which, depending on circumstances, may amount to 10% to 20% or
even more (especially in the case of the higher harmonic components; this is because in the
calculation we start from a "more non-linear" characteristic than it actually is). Although the
above method of analysis is above all suitable for qualitative considerations and fundamental
orientation, it is frequently used in practice.
All the results have been derived on the assumption that apart from the do signal there is
only one single input harmonic signal acting on the non-linear element (circuit) (see Fig. 72).
The method cannot be extended by simple means to a case where several harmonic signals
act simultaneously on the non-linear element (circuit); this, of course, also means that the
backward action of the harmonic output signal on the circuit operation cannot be evaluated in
this way either. In spite of this, the method can be applied in cases where the backward effect
of the output signal on the current in the circuit is negligible (e.g. in circuits with multi-grid
valves). The method is also unsuitable when the harmonic input signal is small
(unadvantageous approximation). At times we approximate a characteristic by several fitted
straight-line segments. This approximation yields more accurate results but at the cost of
greater complexity of the whole task.
4.2.4 Analysis of signals when approximating the characteristic by a power polynomial
We shall again consider a simple input signal given by the sum of the do component and one
harmonic component as expressed by Eq. (4.27). The non-linear characteristic of the circuit
(element) y = f(x) is in this case approximated by a power polynomial, in the given case most
effectively by the Taylor polynomial, which best expresses the neighbourhood of the
quiescent operating point given by the do component X0:
(4.39)
where
f-k-(Xo) is the value of the k-th derivative of the function f(x) for the value x = X0 .
Substituting from Eq. (4.27) into Eq. (4.39), we obtain
(4.40)
For further calculation, the relations for the powers of cosine functions will be used:
where n = 1, 2, 3, ... .
Substituting from these formulae into Eq. (4.40) and recasting the individual terms
according to the multiples of argument α, we obtain a Fourier series in the form
(4.41)
where the dc component has the magnitude
(4.42)
and the individual harmonic components have the amplitudes
(4.43)
,
(4.44)
and generally
(4.45)
where m = (N - k)/2 for N - k even, while m = (N - k - 1)/2 for -N - k odd.
From the expressions for the individual components Y0, Y1 , Y2, ... , YN we can see that the
dc component Y0 and the amplitudes of the even harmonic components Y2 , Y4, Y6 , . . . only
depend on the even coefficients of polynomial (4.39) while the amplitudes of the odd
harmonic components Y1 , Y3 , Y5 , . . . only depend on the odd coefficients of this
polynomial. From an analysis of Eqs (4.42) through (4.45) it can further be found that as in
the preceding case the magnitudes of the individual components depend on the shape of the
non-linear characteristic (on the coefficients al , a2 , . ..), on the do component of the input
signal X0 (for which the coefficients al , a2 , . . . are defined), and on the amplitude X1 of the
input harmonic component.
Spectral analysis of output quantities in a non-linear circuit based on the approximation of the
non-linear characteristic by a power polynomial is suitable for both small and large input
signals.
4.2.5 Analysis of signals when approximating the characteristic by an exponential
function and an exponential polynomial
Let a non-linear characteristic y = f(x) be expressed by a simple exponential function
(4.46)
If the input signal x(α) is again given by relation (4.27), i.e. x = X0 + X1 cosα, then after
substituting into Eq. (4.46) we obtain the expression
(4.47)
From the theory of the Bessel functions [4] we know that
(4.48)
where Bk(bX1) are the modified Bessel functions of order k (with the argument bX1), i.e. the
Bessel functions of an imaginary argument, whose values can be found in tables (e.g. in
[32]). The plots of these functions are given in Fig. 75.
From Eqs (4.47) and (4.48) the output quantity is obtained in the form of the series
(4.49)
Fig. 75. The plots of the modified Bessel functions of order k (with the argument bX1)
The dc component has thus the magnitude
(4.50)
the k-th harmonic component (where k = 1, 2, 3, ...) has then the amplitude
(4.51)
It can be seen that also in this case the magnitudes of output components depend on the shape
of the non-linear characteristic (on the coefficients a, b) and on the parameters X0 and X1.
At times it is necessary to express the characteristic in a more complicated manner, by an
exponential polynomial. In this approximation, we obtain for the output quantity y an
expression which is only a little more complicated:
(4.52)
so that the dc component of the output signal has the magnitude M
(4.53)
and the harmonic components have the amplitudes
(4.54)
The above method is suitable for both small and large signals. Note further that when
approximating a characteristic by an exponential function we can also proceed by expanding
this function into a series. using the formula
(4.55)
This actually takes us to approximations by a power series.
4.6 POLYPHASE SYMMETRIZED SYSTEMS OF NON-LINEAR AND PARAMETRIC
CIRCUITS
Owing to their useful properties, systems are often employed in engineering practice which
consist of non-linear or parametric circuits in which both the acting input and the derived
output signals form symmetrical polyphase systems. These symetrized systems are known as
"push-pull", "symmetrical", "ring", "star", "balanced", "double-balanced", "bridge", or "fullwave" non-linear or parametric circuits (amplifiers, rectifiers, frequency multipliers, mixers,
modulators, and demodulators). Their common feature is the application of the compensation
of undesired current components so that a mutual electrical separation of the individual
sources of input signals and loads can be obtained, and filtration of desired spectral
components of the output signal facilitated.
4.6.1 Basic non-linear or parametric circuits
The basic circuit in which non-linear or parametric transformation of signals is realized is
usually simple. It consists of an active linear circuit (with one or more voltage or current
sources and with one or more loads) and a non-linear or parametric circuit element (resistor,
capacitor, inductor). It can be two-terminal (as in Fig. 82a) or multi-terminal; and it can be
controlled by one or more control signals. At the beginning of this chapter we examined the
transformation of spectra in circuits, where acting on a non-linear element with the characteristic expressed by several types of approximation function were either simple harmonic
or also more complicated polyharmonic signals. The situation is in fact yet more complicated.
Consider quite a simple non-linear system as indicated in Fig.82a. Replace the active linear
circuit by an equivalent source of voltage vi with internal impedance Zi (Fig. 82b), and in
place of the controlled non-linear element consider, for simplicity, a resistor R.
Fig. 82. Configuration of the basic non-linear and parametric circuits
The source of voltage vi(t) will cause a current i(t) in the circuit; owing to the
transformation in the non-linear parametric element, the current has a complicated spectrum
with many higher harmonic or combination components. This current i(t), however, flows
through the linear part of the circuit with the impedance Zi and causes a voltage drop across
it, the derived voltage v0(t). The latter is added to the voltage vi, and together they form the
voltage v(t), which acts on the non-linear parametric circuit. The derived voltage v0(t) or the
voltage v(t) across the non-linear element usually represents the starting point for obtaining
the circuit output voltage.
The spectrum composition of the voltage v0(t) (and thus also of v(t)) depends on the
frequency properties of the linear circuit. If it is a highly selective circuit, the derived voltage
is harmonic. At times, several harmonic components can be separated selectively, and the
voltage volt) is then periodic or quasi-periodic. The most complicated spectrum of a derived
voltage is in the case when the linear circuit is purely resistive (non-inertial).
For a given non-linear parametric circuit, a simple model as given in Fig. 83 can be
derived. Here, the individual symbols denote:
P0
- the do control quantity,
pl through pv
- the harmonic control quantities of frequencies l through v,
V0
- the do voltage,
vv+ 1 through vN
- the harmonic voltages of frequencies v+ 1 through
N, applied from external connected sources,
vN + 1_ through vN + M
- the harmonic voltages of frequencies N + 1 through
N+M originating during the circuit operation (derived
voltages).
In this model, the frequencies N+l through N+M of the derived voltages are from the set of
combination frequencies kll + k22 + ... + kNN, where kl , k2 , . . . , kN are integers.
In the above model, let the electrical parameters of a controlled non-linear element be
affected by the control quantity (for notation simplicity the complex symbolic notation will
be used)
(4.107)
Fig. 83. A generalized model of the non-linear parametric circuit
which is quasi-periodic with the frequency base {l, ..., v} and consists of the dc component
P0 and v harmonic components pn(t). In the circuit, this element is acted upon by the voltage
(4.108)
In addition to the do component V0 , this voltage contains on the one hand N - v harmonic
forced components coming from external sources of electric energy and having the
frequencies v+1, ... , N, and on the other hand M derived harmonic components originating
as voltage drops across the impedance of the linear part of the circuit and having the
combination frequencies
(4.109)
The combination coefficients cm,n are arbitrary integers satisfying the condition N+m > 0.
Since the frequencies N+m depend linearly on the frequencies 1 , ... , N, the voltage v(t) is
quasi-periodic with the frequency base { l , .. . , N}. Consequently, the current i(t) flowing
through the circuit when the quantities v(t) and p(t) are acting will also be quasi-periodic with
the frequency base l , . . . , N. It can therefore be expressed by the multidimensional
Fourier series (denoting t = αn)
(4.110)
in which the complex amplitudes are given by the Fourier coefficients
(4.111)
We shall now examine how these complex amplitudes change if the arguments an are
changed by a phase angle βn. Subjecting the expression for the current i(α1 + β1, α2 + β2, . . . ,
αN + βN) to spectral analysis according to Eq. (4.111), we obtain
(4.112)
where Jk1,k2,...,kN(0, 0, ..., 0) corresponds to the reference complex amplitude given by Eq.
(4.111) for β1 = β2 = ... = βN = 0.
It can be seen that a change in the initial phases of harmonic input quantities by the
angles β1 , β2 , . . . , βN will make the phase of the harmonic component of current
characterized by the combination coefficients k1, k2, ... , kN change by an angle
(4.113)
4.6.2 Polyphase systems
Symmetrical polyphase systems (a typical case of which is the three-phase voltage system of
the power supply network) are characterized by the fact that the composition of their spectra
makes it possible to suppress some undesired components by means of compensation, that is
to say without any necessity of removing them by means of a filter. Consider that a
symmetrical polyphase system of voltage sources and control quantities is available. Let Sn be
the number of phases of the n-th harmonic quantity (Sn ≥ 2 is a natural number). The
individual phases of this system will be denoted by the serial number sn = 1, 2, . . . , Sn . Then
the individual components of this Sn-phase system differ by the initial phase
(4.114)
Let there further be a set of identically arranged non-linear parametric circuits, in a
configuration as given in Fig. 83 but differing in the individual variants in that the phases of
the individual applied oscillations pn(t) or vn(t) are shifted by the angles
respectively (the
subscripts n and s„ assuming the values n = 1, 2, ... , N and sn = 1, 2, ... , Sn respectively). The
number H of these variants will obviously be given by the product H=S1S2...SN.
The complex amplitudes of the components of frequency
will be found for the individual variants by substituting, for the given sN, from Eq.
(4.114) into Eq. (4.112). We obtain
(4.115)
Discussing this relation, we find that the number of phases of the individual combination
components of current is equal to the number of variants H at the most, but generally
speaking it is less and depends on the combination coefficients k1, k2 , . . . , kN and on the
number of phases S1, S2, ..., SN. The component of frequency
has the number of
phases
,and it can be determined as the least common multiple of the denominators
in fractions kn/Sn, n=1, 2, ..., N rewritten in the basic form (i.e. properly cancelled).
If there is
, then the examined combination current components of
frequency
(and thus also their corresponding derived voltages in the circuit) are in
phase in certain groups of variants. The number of variants in these groups is given by the
quotient
.The forced voltages and control quantities and (with some exceptions)
the other harmonic current components in the given group are of different initial phases, and
thus they form polyphase symmetrical systems. This circumstance can be used for an easy
separation of useful in-phase spectral components which, when added up, give a one-phase
output signal, while all polyphase undesired spectral components compensate one another,
their sum being equal to zero.
Two-phase symmetrical systems, in which for all n = 1, 2, ... , N there is Sn = 2, form a
special case. The initial phases of the quantities pn or vn are in this case given by the relation
(4.116)
arid they assume only two values, βn,1 = 0 and βn,2 = ; consequently, they are either in the
same or in the opposite phase with respect to the original oscillations. For illustration, we
shall show what happens if in the circuit of Fig. 83 we change the polarity of the control
quantities pn, n = 1, ... , v, and of the applied voltages vn, n = v + 1, ... , N, respectively. In
that case, both the control quantities and the applied voltages form symmetrical two-phase
systems which, in addition to the original quantities pn or vn, are also made up of quantities
which are in opposite phase with them, i.e. the quantities -pn or -vn. Thus we obtain 2N
possible variants of the basic circuit which differ only in the signs (polarities) of the control
quantities and applied voltages. Denoting the polarity of the individual quantities pn or vn by
the familiar mathematical symbol for the sign such that
we can find by means of the spectral analysis of the current in the circuit that the individual
combination harmonic current components of frequencies kll + k22 + ... + kNN will differ
by the coefficient
(4.117)
Two important conclusions follow from this fact:
a) If in a non-linear parametric circuit there is a change in the polarity of an arbitrary
input voltage vn, then there is also a change in the polarity of the fundamental harmonic
component of the current supplied to the circuit by the source of this voltage; the same
conclusion also holds in the case that the control quantity pn of the parametrically controlled
element is a voltage or a current.
b) If in a non-linear parametric circuit the polarity of the input quantities is changed
arbitrarily, the phase of an arbitrary combination current component is always changed by a
whole multiple of , i.e. the given combination component is always either in phase or in
opposite phase with the corresponding combination component in the initial fundamental
circuit; this can be employed to separate the useful or to compensate the undesired
combination current components.
Fig. 84. Variants of the basic non-linear circuits in which two independent voltages v1, , v2 and one derived
voltage v0 are acting: a) through d) the basic series, e) through h) the complementary series for inversely
connected diodes
Fig. 84 gives a simple example. Acting in the circuit with a diode as indicated in Fig. 84a
are two independent voltages vl , v2 (i.e. v = 0, N = 2) with frequencies l and 2, and one
selectively separated derived voltage v0 of frequency Ω = l ± 2 (this corresponds to the
choice of the combination coefficients kl = 1, k2 = ± 1). Changing the polarity of the voltages
vl and v2 will yield a total of four variants, as indicated schematically in Figs 84a through d.
In the individual variants, the polarity of the derived voltages has been calculated on the basis
of Eq. (4.117). The variants given in Figs 84e through h, which differ by inversely connected
diodes, will be discussed later.
4.6.3 Structural synthesis of symmetrized systems
The above knowledge can be used with advantage in constructing symmetrized (balanced)
systems of non-linear circuits. The basic building element of symmetrized systems is the
joining circuit which joins two voltage sources such that they affect each other as little as
possible. Its connection is illustrated schematically in Fig. 85a. One source in the circuit acts
directly, the other through a symmetrizing (balanced, centre-tapped) transformer (in practice
usually a symmetrizing resonant circuit). The joining circuit has two output ports, a-0, b-0
(output a and output b, for short). The output voltage is
(4.118)
Fig. 85. Joining circuits
If two identical loads are connected to the outputs a and b, the currents ia and ib flow through
the circuit. It is apparent from Fig. 85 that the current component i2, of frequency cot
supplied from the source of the voltage v2 divides equally between the two halves of the
secondary winding so thát the current of frequency cot does not flow through the primary
winding of the symmetrizing transformer. The current component i1 of frequency 1
supplied to the loads from the source of the voltage vl through the symmetrizing transformer
does not flow through the branch containing the source of the voltage v2 either; this branch
forms a diagonal of the balanced bridge. In this joining circuit, the two voltage sources
(provided the loads are identical) do not affect each other at all. The circuit in Fig. 85a can be
replaced by an equivalent system of four independent sources, as indicated schematically in
Fig. 85b. This circuit, however, does not exhaust all the possible sign combinations of the
voltages vl and v2. For the purpose of an all-round compensation, we must therefore consider
a joining circuit as given in Fig. 85c,. which gives across the outputs a through d the voltages
respectively. The idea of a joining circuit with all-round compensation can be extended to
cover more voltage sources. For N sources in a chain, we can thus derive 2N chains of
independent sources.
When combining basic non-linear or parametric circuits to form symmetrized systems,
we apply the above knowledge and proceed as follows:
a) the joining of input voltage sources is chosen such that the sources affect one another
as little as possible;
b) applying Eq. (4.117), we find for each basic circuit used the polarity of the output
combination components of current, and thus determine the desired manner of connecting the
outputs of the individual basic circuits forming the system. Output signal components of
opposite phases are applied to the output through a symmetrizing transformer with centretapped primary winding.
A system of non-linear or parametric circuits designed in this fashion can further be
modified by re-arranging the sources, by choosing differently the common (usually
grounding) node, and possibly by using symmetrizing transformers with two or more
independent symmetrizing windings. Further favourable possibilities can be obtained when
employing complementary elements in the basic circuits. The complementary element to a
diode is an inverted diode. In Figs 84e through h we gave as examples circuits with inverted
diodes following up on the series given in Figs 84a through d. A simple reflection will reveal
that the case a corresponds completely to the case d taken with opposite polarity of all the
voltage sources.
Similar correspondence exists between the cases f - c, g - b and h - a. A similar extension of
the possibilities can also be obtained in symmetrized (balanced) systems of circuits
containing complementary transistors.
4.6.4 Some examples of symmetrized systems of non-linear circuits
A fu11-waverectifier is connected as shown in Fig. 86. It is a pair of diodes connected on the
input side to a symmetrizing circuit (transformer) and delivering on the output side a current
into the common load CR. The spectral components of the current flowing through the diode
D2 will differ from the corresponding spectral components of the current flowing through the
diode D1 (in keeping with expression (4.117) the current is reduced for v = 0 and N = 1) by
the coefficient
Here, k is the serial number of the spectral component (which is of frequency kl).
Fig. 86. Full-wave rectifier
Flowing through the load CR is the sum of currents through the diodes D1 and D2. Taking
into account Eq. (4.120), we can see that only the even spectral components (including the do
one) will show in the current flowing through the load, while the odd spectral components
(including the fundamental harmonic component) will compensate each other. The capacitor
C will function as a filter passing all ac current components and retaining the do component..
Across the load resistor R a voltage v0 will therefore develop with a pronounced do
component Yo0 ; it will be rippled a little owing to the effect of the remainder of even higher
harmonic current components present in the voltage (see also further considerations in
Chapter 7).
Frequency mu1tip1ier. The circuit given in Fig. 87, which is similar to the full-wave
rectifier but differs from it by the application of
Fig.87. Frequency multipliers with two diodes (the do voltage sources used to set the operating point of the
diodes are not indicated)
a selective resonant circuit CL at its output, operates as a frequency multiplier designed to
obtain even multiples of the fundamental frequency col. Eq. (4.120) holds for this circuit, too.
In the sum of currents flowing through the diodes D1 and D2, the odd spectral components are
again compensated; from the even spectral components the component of frequency l is
separated selectively, c being an even number.
Conversely, if the outputs from the diodes D1 and D2 are applied to the symmetrizing
resonant circuit (Fig. 87b), the even spectral components are compensated in it, and the odd
spectral components will show. The resonant circuit CL will separate selectively from them
only the component of frequency l, where c is an odd number. In keeping with the
preceding considerations, the inverted diode can be used in the circuit to replace the
multiplier of Fig. 87b by a symmetrized circuit as given in Fig. 87c. The parallel connection
of the diodes in this circuit has a characteristic which can be expressed by an odd function.
Thus a nonlinear element characterized by an odd function yields exclusively odd multiples
of fundamental frequency. This can be proved directly by the spectral analysis of the current
in a circuit containing a non-linear element with odd characteristic. Likewise, it can be
proved that an element with an even characteristic has exclusively even harmonic
components in the current spectrum.
Partiall y symmetrized mixer of two signals. A mixer of two signals of
frequencies l and 2 is usually tuned at its output to a frequency given by the sum or
difference of the frequencies l and 2 (Ω = | 1 ± 2 | ). In that case, a partially symmetrized
system made up of the circuits a and c as given in Fig. 84 and arranged as indicated in
Fig. 88, has these properties:
1. In the input (usually aerial) circuit, the currents flowing from the local oscillator are
compensated; a signal of the local oscillator frequency (in our case cot) will not reach the
input terminals and, consequently, it will not be radiated by the aerial.
2. In the output circuit (of intermediate frequency) the currents flowing from the local
oscillator are compensated, including the noise cur rents originating in this oscillator; this
results in a considerable reduction of the noise factor of the receiver.
3. There is, however, no compensation between the input and the output.
Fig. 88. A partially symmetrized mixer of two signals
The circuit given in Fig. 88 can also be employed for amplitude modulation of a highfrequency voltage of frequency 1 if the low-frequency modulation voltage has a frequency
2  l and if the resonant circuit is tuned to the frequency 1 (so that it also transfers the
components with frequencies l ± 2, i.e. the sideband spectral lines of the AM signal).
If in a given circuit we interchange the sources of the two signals, i.e. if the highfrequency signal is of frequency cot and the low-frequency signal is of frequency 1 < 2,
then at the output we obtain an AM signal with completely suppressed carrier oscillations.
Symmetrized systems with four diodes. A completely symmetrized mixer of two
signals of frequencies l and 2 whose output signal has a frequency Ω = | l ± 2 | can be
constructed in two ways. We either start from the basic set of circuits a through d (Fig. 84), or
we can replace a suitable pair from these four circuits by alternative variants from the set a
through h.
If by the above method we determine a completely symmetrized configuration of
independent sources, and if we connect the outputs of
Fig. 89. Completely symmetrized systems of non-linear circuits
the individual circuits to a symmetrizing output transformer (respecting the correct polarity of
output voltages), we obtain a circuit whose diagram is given in Fig. 89a. Comparing this
circuit with the preceding partially symmetrized mixer, we can see that a completely
symmetrized system is formed by two such partially symmetrized mixers which differ by the
polarity of the voltage v2 and have a common output (compare the upper and the lower part
of Fig. 89a). If such a completely symmetrized system is used for modulation, the spectrum
of the output current will lack the components with the frequencies of the two input signals;
we obtain an amplitude-modulated signal with completely suppressed carrier oscillations.
Rearranging the diodes and making use of transformers (or symmetrizing circuits) with
several symmetrizing windings, the system indicated in Fig. 89a can be changed to a system
illustrated in Fig. 89b, which has equivalent electrical properties and is known in engineering
practice as a "star" modulator.
When circuits with inverted diodes are also employed for structural synthesis, still
further variants of completely symmetrized systems will be obtained. To illustrate this point,
we give in Fig. 89c the diagram of a completely symmetrized system of circuits with diodes
which is made up of the variants a, b, e, and f given in Fig. 84. This system is generally
known as the "ring" modulator (but it can also be used as a mixer). This completely
symmetrized system is also characterized by the spectrum of the current output lacking the
components with the frequencies of the two input signals.
The principles of structural synthesis and compensation of undesired spectral
components of signals can also be made use of when designing symmetrized circuits
containing transistors or valves. Some applications are given in Chapters 6, 7, and 8.
Methods for analysing non-linear and
parametric circuits
The analysis of the phenomena taking place in linear circuits and systems does not usually
present any major difficulties. A description of these phenomena leads to linear equations and
systems of such equations, and for their solution there exist a number of efficient universal
procedures which are always successful.
There is, however, no such universal method to be used in analysing non-linear circuits.
From the physical point of view, the difficulties connected with the solution of the
phenomena in these circuits are in the first place due to the fact that in a non-linear circuit the
responses to individual stimuli do not simply add up; they affect one another considerably.
When examining the properties of non-linear circuits, we are often faced with the task of
solving a non-linear differential equation (or a system of such equations) describing the
phenomena in the circuit. The solution of such equations is usually very difficult and,
frequently, unknown in analytical form. Thus when analysing non-linear circuits, only
approximate methods are usually used; we must, of course, accept the fact that the calculation
result is always approximate only.
At present, there are a number of methods used in analysing non-linear circuits, both for
steady-state processes and for the examination of transient processes. All these methods can
be classed in three groups:
a) graphical methods,
b) analytical methods,
c) numerical methods (using computers).
Naturally, we also use methods which are based on a combination of the methods listed
above, not infrequently with the aid of computers.
5.1 GRAPHICAL METHODS
The graphical solution is applied especially to the study of phenomena in simple non-linear
circuits. Usually, it gives a clear idea of the problem in question, but it is laborious and
lengthy. Graphical methods do not exhibit great accuracy, and the solution is not obtained in
a general form. When there is a change in the parameters of circuit elements and acting
signals, the whole procedure must be repeated. With graphical methods, we start directly
from graphically presented characteristics of non-linear elements. In spite of their simplicity,
these methods make it in many cases possible to reveal typical phenomena in non-linear
circuits and their peculiarities. Thus they are very convenient, especially in the initial stage of
investigating circuits.
We shall first deal with the basic procedures of the graphical solution of simple nonlinear and parametric circuits of zero order, i.e. circuits made up of only resistive elements
and electrical energy sources. We shall proceed purposefully in that we shall replace the
linear part of the circuit by an equivalent voltage or current source (in accordance with
Thevenin's or Norton's theorem); the non-linear part of the circuit will gradually be simplified
to a single equivalent non-linear resistor, and only after this simplification will the circuit be
solved graphically. For this purpose, we shall give here the principles of simplifying series
and parallel combinations of non-linear resistors, possibly in connection with voltage and
current sources.
5.1.1 Resultant characteristics of one-ports consisting of several elements
Two non-linear resistors in series. Let there be a pair of non-linear resistors with the
characteristics vl(il) and v2(i2). When these elements are connected in series as indicated in
Fig. 90a, the resultant characteristic vs(is) of such a one-port is given by the relations
(5.1)
The graphical construction of the resultant characteristic vs(is) of the equivalent resistor in the
v - i plane is clear from Fig. 90b.
Two non-linear resistors in parallel. If two resistors with the characteristics il(vl)
and i2(v2) are connected in parallel as given in Fig. 91a, then the resultant characteristic ip(vp)
is given by
(5.2)
Fig. 90. a) Two series-connected resistors replaced by one equivalent resistor; b) graphical derivation of the
resultant volt-ampere characteristic vs(is)
Fig. 91. a) Two parallel-connected resistors replaced by one equivalent resistor; b) graphical derivation of the
resultant ampere-volt characteristics ip(up)
Fig. 91 b shows how to construct graphically in the i - v plane the resultant characteristic
ip(vp) of the equivalent resistor.
Note that we proceed analogously also in the graphical construction of the resultant
characteristic of several resistors connected in a more complicated manner (i.e. in series and
in parallel) or when constructing the resultant characteristics of non-linear capacitors or
inductors connected in series and in parallel.
Dc voltage source in series with a non -linear resistor. The ideal source of do
voltage V0 = const., connected in series with a non-linear resistor with the characteristic
i1 (v1), can be replaced by an equivalent resistor with the resultant characteristic i(v).
According to Fig.92a, it holds that
(5.3)
On connecting a non-linear resistor Rn, the quantities i and v are additionally governed by the
functional dependence
(5.4)
Solving now the two equations together, we obtain the two sought quantities v = vA and
i = iA . These equations are usually solved graphically as indicated in Fig.93b. Plotted in the
common graph will be both the characteristic i(v) of the given non-linear resistor and the load
line given by Eq: (5.4), which will be constructed with the aid of two points, (v, 0) and (0, ik),
where ik = v0/R0 is the short-circuited current. The operating point is given by the intersection
A of the two characteristics. The same current iA is flowing through the two resistors, with a
voltage drop vA appearing across the non-linear resistor. The co-ordinates of the operating
point A, i.e. vA, iA, give the sought solution.
Fig.93. The diagram of a circuit with a non-linear resistor; b) its graphical solution
Solving a circuit at ac voltage. If in the circuit given in Fig. 93, the voltage v0 changes
with time by an arbitrary dependence v0(t), and if the internal resistance Ro remains constant,
then the circuit is solved by the method indicated in Fig. 93b, successively for the appropriate
instants of time (usually with equidistant distribution along the time axis). An example of the
graphical determination of the waveform of the current in the circuit of a rectifier with noninertial load, supplied by a harmonic voltage, is given in Fig. 94. Knowing the waveform of
the current, we can determine on the one hand the do component of the current (so-called
rectified current) I0 , on the other the amplitude I1 of the fundamental harmonic component of
this current, and possibly also the amplitudes of the higher harmonic components. The
magnitude of the amplitudes of
Fig. 94. Graphical determination of the waveform of the current in the circuit of a rectifier with non-inertial load
current components will be found by means of the spectral analysis given above.
Graphical solution of a ci rcuit containing a controlled resisto r. If there is a
non-inertial circuit consisting of a linear part and one controlled resistor either linear or nonlinear, , then again the whole linear part of the circuit can be replaced, according to the rule
about equivalent source (i.e. Thevenin's theorem), by an equivalent voltage source with
internal voltage v0 and internal resistance R0.
Fig. 95. A general diagram of a non-inertial circuit with a controlled resistor; a) the circuit with a linear
controlled resistor; b) the circuit with a non-linear controlled resistor
This yields the equivalent circuit shown in Fig. 95; here, p is the control quantity of the
controlled resistor. In these circuits, the equivalent source v0, Ro itself exhibits a dependence
between the output voltage v and the current i which is described by Eq. (5.4). On connecting
the controlled resistor, the quantities v and i are additionally governed by its characteristic
(5.6)
in the case of a linear controlled resistor, or
(5.7)
in the case of a non-linear controlled resistor.
The system of Eqs (5.4), (5.6) or (5.4), (5.7) is solved either graphically or analytically
(especially in circuits containing a linear controlled resistor). In terms of geometry, we are
concerned here with determining a space line which originates as an intersection of a general
surface i(v, p) and the plane v = v0- R0i (independent of p). In engineering practice, however,
we usually work more frequently with the projections
Fig. 96. Graphical solution of a circuit with a controlled linear resistor; a) the solution in the plane of the
ampere-volt characteristics; b) and c) the operating (dynamic) characteristics i(E) and v(E); d) the diagram of a
circuit with a photoresistor
Fig. 97. The solution of a circuit with a semiconductor photodiode. a) The circuit diagram; b) graphical solution
for V0 = 40 V and R0 = 60 kS2; c) and d) the obtained dependences i(E) and v(E)
of this space line onto the planes v = 0 or i = 0, where for the given circuit configuration (i.e.
for v0, R0 and the given controlled element) we obtain the operating (dynamic)
characteristics i(p) or v(p).
Circuit with a linear controlled resistor . If there is a circuit as in Fig. 95a
consisting of a voltage source v0 Ro and a linear controlled resistor, e.g. a photoresistor,
whose characteristics are represented in the i - v plane by a family of straight lines with the
parameter p = E (E is illuminance in lux), see Fig. 96a, then the graphical solution follows
the method given in Fig. 93. The solution, however, must be repeated for all the necessary
values of the parameter p = E. In Fig. 96, the operating points corresponding to the individual
values of the parameter p = E are denoted by letters A through F. It is evident from Fig. 96
that for the parameter value E = E4, to which the operating point C corresponds, there
originates a -voltage v = vc across the photoresistor while across the resistor R0 there will be a
voltage drop vR = uRC = V0 - vc.
The graphical solution given in Fig. 96a makes it possible to determine (as indicated by the
dashed line) the operating, or dynamic, characteristics i(E) and v(E) given in Figs 96b and
96c. By means of these characteristics, the waveform of the current i(t) or the waveforms of
the voltages v(t) and vR(t) can be constructed for the given waveform of illuminance E(t).
Circuit with a non-linear controlled resistor . The circuit in Fig. 95b containing a
non-linear controlled resistor is usually solved graphically by the - method given in the
preceding section. In contrast to Fig.96, however, the ampere-volt characteristics of the
controlled resistor will be non-linear. Given as an example in Fig. 97 is the solution of a
circuit with a semiconductor photodiode, i.e. a controlled resistor, in which the current i
depends on the one hand on the voltage v, and on the other hand on the control quantity,
which is the illuminance E (given in lux).
Examples of the graphical solution of circuits containing electrically controlled nonlinear resistors (unipolar and bipolar transistors and valves) will be given in Chapter 6.
5.1.2 Graphical methods for solving first- and second-order non-linear circuits
Also in the analysis of first- and second-order non-linear circuits we often employ graphical
methods, which provide for a solution in the form of a graph. First-order differential
equations are usually solved by the method of isoclines, second-order differential equations
by the phase plane method. Both these graphical methods permit the examination of a
transient process including its settling.
The method of is oc1ines. A first-order differential equation can in general be written in the
form
(5.8)
This equation determines the dependence between the points with the co-ordinates x, y and
the slopes of the sought integral curve y(x). In geometrical terms; the derivative dy/dx is the
slope of the tangent to the integral curve y(x) at the point (x, y).
The substance of the method of isoclines can be expressed in the following way. If in the
differential equation (5.8) we put dy/dx = const. = k, then the algebraic equation obtained
(5.9)
Fig. 98. The principle of constructing the integral curve y(x) in a family of isoclines
gives the geometrical locus of the points through which the sought curve y(x) passes at a
given angle φ, whose tangent is equal to the quantity k. If, for example, k = 1, then the
integral curve will go through all the points f(y, x) = 1 at an angle φ = 45°. The curve
expressed by Eq. (5.9) is called the isocline (i.e. a line of the same slope).
The procedure of the graphical solution of a first-order non-linear differential equation
by the method of isoclines will be explained on the example of the equation
First we find the equation of isoclines. Putting dy/dx = k and solving the equation with
respect to the variable y, we obtain
which is the equation of a family of isoclines. With its aid, we plot for the known quantities b
and Q a number of isoclines corresponding to different values of k. Plotted in Fig. 98 is a
family of four isoclines for k = 1, 0.5, 0, -1, whose equation, when Q = 10 and b = 3, is in the
form
The isocline a corresponds to an angle of 45°, b to an angle of 26.6°, c to an angle of 0°, d to
an angle of 135°, and e (which is identical with the abscissa axis) to an angle of 90°.
Having constructed the family of isoclines through which the sought integral curve y(x)
passes at the given angles, and knowing the initial conditions, we can construct the integral
curve. The principle of the construction is also shown in Fig. 98. The initial conditions
determine the starting point A(x0, y0). We shall construct the integral curve by drawing two
line segments from the initial point A. The direction of the
1 segment is given by the angle
assigned to the isocline a, the direction of the segment
by the angle assigned to the
isocline b. Point B, lying on the isocline b approximately in the middle between points B1 and
B2 , is a point of the integral curve. From point B we again draw two line segments,
and
, and thus find point C, and similarly point D, etc. Connecting points A, B, C, D, ... we
obtain the desired integral curve y(x). Needless to say that with different initial conditions,
there will be a different integral curve. The family of isoclines, however, remains the same
for all possible initial conditions.
The accuracy with which the solution (i.e. the integral curve) of the given differential
equation is obtained is obviously limited. It increases with the density of the family of
isoclines. Simultaneously, however, the laboriousness of the solution increases. It is therefore
of advantage first to form a rough idea of the shape of the sought function y(x) and only then
in places where there are sudden changes in the shape of the integral curve to make the
family of isoclines denser, as need be.
The phase p1ane method. The phase plane method is applied when solving (usually
homogeneous) second-order non-linear differential equations. This method gives a
comparatively quick and clear idea of the behaviour and properties of the non-linear circuit
under investigation, especially in rough outlines. The method in itself does not contain any
limitations, so that it can be used in examining arbitrary autonomous circuits and systems of
the second order.
We know that the rectilinear motion of a certain point can be described uniquely by two
variables : its co-ordinate and the rate of change of the co-ordinate. A graphically expressed
idea of motion in the
Fig. 99. Representation of a harmonic signal in the phase plane; a) the waveform of the quantities x and x; b) the
phase trajectory; c) the phase portrait
rectangular system of co-ordinates gives a clear picture of the motion of the point and thus
also of the investigated phenomenon. The term phase plane is used to denote a plane with
rectangular co-ordinates in which plotted on the abscissa axis are the instantaneous values of
the investigated quantity x, while on the ordinate axis the derivative of this quantity dx/dt = x,
which is the rate of change of the quantity x with time. This plane is called the phase plane
because each point of this plane corresponds to a certain phase of the examined process.
Here, the term phase refers to the instantaneous operating state; it must not be mistaken for the
term which in electrical engineering is applied to denote the phase shift in the harmonic
waveforms of electrical quantities.
If for the instantaneous value
of the examined quantity we determine its derivative
x(t), then at each instant of time the position of the representative point M in the phase plane
with the co-ordinates x, x will be determined. Plotted in the phase plane in Fig. 99b is a point
Mn corresponding to the operating state at time tn. With varying time, the representative point
M moves along the curve x = f(x) called the phase trajectory, which is the graphical
representation of the dynamic properties of the circuit. The shape of this phase trajectory
Fig. 100. The waveforms and phase trajectories of damped and increasing oscillations
depends on the circuit configuration, on the parameters of its elements, and on the nature of
non-linearity of the non-linear element in the .circuit. The initial position of the representative
point in the phase plane is given by the values x = x(0) and
, i.e. by the values at
time t = 0.
If the process in the circuit is periodic, then the respective values x and are repeated
each period, and the phase trajectory forms a closed curve: If the process is non-periodic,
then the phase trajectory represents a non-closed curve (Fig. 100). In a number of cases, the
settling process of increasing or decreasing oscillations is such that their phase trajectory ends
in a closed curve; this curve is called the limit cycle.
As an illustration, the representation of harmonic oscillations in the phase plane will now
be given. The harmonic signal will be written in
the form
The rate of change of the quantity x, i.e. its derivative with respect to time, will then be equal
to
The waveforms of the two quantities x and x are illustrated in Fig. 99. From these data alone,
it would be possible to plot the phase trajectory of a harmonic signal. Eliminating from the
above pair of equations time t, we obtain the equation of phase trajectory in analytical form.
To do this, we divide the former equation by the amplitude X, the latter by the amplitude X,
then we raise the two equations to the second power and add them up. The phase trajectory
equation is then obtained in the form
It is the equation of an ellipse with the half-axes X and X. In a period T = 2/ the
representative point will circumscribe a closed curve, which is the phase trajectory of the
harmonic signal in the phase plane (Fig. 99b).
A set of phase trajectories representing possible processes in the given circuit under
different initial conditions or with different amplitudes is called the phase portrait of these
processes in the phase plane. In Fig. 99c, for example, the phase portrait is plotted of
harmonic oscillations differing by the amplitude X.
Consider now damped oscillations (Fig. 100a) changing by the time function
and express their derivative with respect to time:
We can again eliminate time t from these two equations in parametric form and work with
one equation. After rewriting, we obtain the phase trajectory equation in the form
which is the equation of an inward spiral illustrated in Fig. 100b.
Similarly, for increasing oscillations shown in Fig. 100c and changing by the time
function x = A eαt sin apt, we obtain a phase trajectory in the form of an outward spiral, see
Fig. 100d.
The above examples make clear the substance of the phase plane method. They show at
the same time that from the shape of the phase trajectories we can judge the nature of the
signal in the circuit and its behaviour.
The above considerations on the phase plane can be extended and generalized. Thus,
third-order differential equations can be examined both qualitatively and quantitatively in
three-dimensional representation by means of the phase space. In this case, the variable x is
plotted on one axis, its first derivative x = dx/dt (i.e. the rate of change) on the second axis,
and its second derivative x = d2x/dt2 (i.e. the change acceleration) on the third axis.
Solving second -order non-linear differential equations by the phase plane
method. The advantage of the phase plane method is much more pronounced in the
examination of non-linear circuits and system, when we usually do not know the exact
solution of non-linear differential equations. The phase trajectory is then with advantage
constructed by the method of isoclines.
In the first stage of the solution, the given second-order differential equation is rewritten
as a system of two first-order equations. The obtained first-order differential equation is then
solved by the method of isoclines, and from the obtained phase portrait we judge the nature
of the process or, possibly, we derive the desired time functions from the trajectories. This
procedure will be demonstrated in greater detail [9].
A second-order non-linear differential equation of an autonomous system can always be
rewritten in the general form
(5.10)
Substituting
we obtain Eq. (5.10) in the form
and hence
(5.12)
The second-order differential equation (5.10) has thus been replaced by a system of two
first-order differential equations, (5.11) and (5.12). Solving Eq. (5.12) by the method of
isoclines, we obtain the dependence y(x), i.e. the phase trajectory of the process under
investigation.
To do this, we substitute certain constant values of the derivative dy/dx = k into Eq.
(5.12) and solve it with respect to y. In this way we obtain the equation of isoclines in the
form
(5.13)
The most laborious part of the solution is constructing the isoclines and the phase
portrait. To evaluate the principal and most pronounced features of the process, it usually
suffices to construct only a few isoclines and an orientation plot of the phase trajectory. For
this reason, the phase plane method .is often regarded as a qualitative method. It has therefore
proved to be reasonable to introduce the concept of "principal isoclines", through the points
of which the trajectories pass at a slope of φ = 0°, 90°, 45°, and -45°. The values
corresponding to these angles are k = tan φ = 0,, 1, and -1. The equations of the principal
isoclines, derived from Eq. (5.13), will thus be in the form
(5.14)
These four principal isoclines permit us to form a quick idea of the basic features of the phase
trajectory or of the portrait of the examined circuit. A concrete example of the application of
this method is given in the chapter on oscillators.
Deriving the time function from the phase trajectory. The integral curve y(x)
is not, of course, the solution of a given differential equation; it is the phase representation of
the solution that is being sought. In many cases, the phase trajectory itself provides a
sufficient qualitative idea as to the nature of the phenomenon in the circuit and its properties.
The ultimate aim of solving the differential equation, however, is finding the time function
x(t). This waveform x(t) can be constructed from the phase trajectory. One of the methods
used in deriving the time function x(t) will now be given.
Fig. 102. Illustrating the derivation of the time function x(t) from the phase trajectory
Assume that a portion of the phase trajectory y(x) is of the shape given in Fig. 102. The
method of piecewise linearization of the phase trajectory wi11 be used; this means that the
curve y(x) will be replaced by straight-line segments. Then for the segment with serial
number n, the equation
holds, hence
The time in which the representative point traverses the segment from point An(xn, yn) to
point An+ 1(xn+ 1 , yn+ 1) will be denoted 0t„ and calculated from the relation
Since the slope of the line segment
can be changed to its final form
is bn = (yn+ 1 - yn)/(xn+ 1 - xn)- the equation for tn
(5.18)
The procedure adopted in . constructing the time function x(t) will therefore be as
follows. We shall start from a certain point, say A1 on the curve y(x), see Fig. 102. We
determine for the respective segment the values x2 - xl and y2 - yl and calculate the time tl.
The obtained values x1, x2, and tl will be used in constructing the first segment of the graph
x(t). Then we determine for the next segment the values x3 - x2 and y3 - y2, calculate the time
interval t2, and from the values x2, x3, and t2 we construct the next segment of the graph
x(t). Then we continue with the same procedure.
5.2 ANALYTICAL METHODS
In this section, some of the most widely applied analytical methods, which are most
frequently encountered in engineering practice, will be discussed. They are the method of
equivalent linearization, the piecewise solution method, the method of slowly changing
amplitudes, and the method of state equations.
5.2.1 The method of equivalent linearization
The method of equivalent linearization, sometimes called the quasi-linear method or the
method of harmonic linearization, is the most frequently applied method in engineering
calculations of second-order selective non-linear circuits operating in steady state. It is used
in those cases in which we are interested in the first harmonic components of currents and
voltages, i.e. in the analysis of circuits and systems containing selective filters which
markedly suppress the other harmonic components. It can, however, also be used to analyse
transient processes in these circuits. [31], [33].
The substance of the method of equivalent linearization consists in replacing a non-linear
element by a certain fictitious linear element which responds by its parameter to the
amplitude of the flowing current or acting voltage. In steady-state operation, i.e. with
constant amplitude, such a non-linear element behaves as a linear element with constant
parameter. In this case we introduce new, equivalent parameters (parameters for the first
harmonic component or also average parameters) defined by the ratio of the amplitudes of the
first harmonic components of electrical quantities.
The non-linear circuit is now described by linear equations which can be solved by
methods known from the theory of linear circuits. The circuit non-linearity shows in the nonlinear dependence of the equivalent parameter on the amplitude of the respective electrical
quantity.
Equivalent parameters of non -linear elements. The equivalent parameter of a
non-linear element (or circuit) is defined as the ratio of the amplitude Yl of the first harmonic
component of the dependent quantity y(t) to the amplitude X1 of the independent harmonic
quantity x(t):
(5.19)
The derivation of the equivalent parameter Pe and its geometrical significance follow from
Fig. 103. We shall consider a simple case of a non-linear element with a single-valued odd
characteristic y(x) as indicated in Fig. 103a. The cosine function yl(t) given in dashed line in
Fig. 103c represents the waveform of the first harmonic component of the output signal y(t).
The amplitude Yl of this harmonic component will be found by one of the methods of spectral
analysis.
Fig. 103a shows a straight line passing through points A and B, corresponding to the
characteristic yl(x) of an equivalent linear element which, when supplied with a harmonic
input signal x(t), would give the harmonic signal yl(t) at the output. We can see that the
equivalent parameter Pe = Y1 / X1 of the equivalent linear element under consideration is
proportional to the slope of the line segment
, i.e. Pe ~ tan β, and that it differs from the
differential parameter Pd(0) at the quiescent operating point. Recall that the differential
parameter Pd(0) is characterized by the slope of the tangent to the characteristic y = f(x) at the
quiescent operating point, that is to say in our case in the origin, Pd(0) - tan a. Since α β, it
also holds that Pd(0) - Pe(X1).
Fig. 103. Geometrical significance of the concept of the equivalent parameter
On the basis of Fig. 103, we can readily see that the magnitude of the equivalent
parameter Pe will depend on the amplitude X1 of the input signal x(t), and that with X1 → 0
the equivalent parameter Pe converges to Pd, i.e.
(5.20)
The non-linearity of the characteristic of an element thus entails not only the appearance of
new harmonic components but also the fact that the dependence between the amplitude Yl of
the first harmonic component of the output signal and the amplitude X1 of the harmonic input
signal is non-linear.
If we have a non-linear resistor with the characteristic
and if we supply it, as shown in Fig. 104, with a harmonic current i = I1 cos (t + φ) =
I1 cos3α, we obtain across the resistor a voltage
Fig. 104. The effect of harmonic current on a non-linear resistor
Hence the amplitude of the first harmonic component of voltage is
and the equivalent resistance for the first harmonic component is
The graphical representation of such a dependence is given in Fig. 104b.
There is yet another equivalence criterion which holds for the two resistors. It is the
principle of the equivalence of energy or, as it is sometimes termed, the principle of energy
balance. By this principle, the average power delivered by a current source into both the nonlinear and the equivalent linear resistor must be the same.
For non-linear capacitors, the equivalent parameter is the equivalent capacitance as
defined by the ratio Ce(V1) = Q1/V1 , and for non-linear inductors the equivalent inductance
Le(I1) = 1/I1. Here, Q1 is the amplitude of the first harmonic component of the charge, and
l the amplitude of the first harmonic component of the flux linkage.
Equivalent parameters can he both real (in circuits containing resistors only) and
complex (in circuits containing both resistors and storage elements), e.g. the equivalent
impedance Ze(I1)= V1/I1 or the equivalent transfer function Ke of a non-linear two-port.
The procedure in analysing a circuit by the method of equivalent linearization will be
illustrated by an example. Let there be a series resonant circuit RLC with a non-linear resistor
R and a non-linear inductor L, and with a linear capacitor C (Fig. 105).
Fig. 105. A series resonant circuit with a non-linear resistor and a non-linear inductor
At resonance, the current i(t) is approximately harmonic owing to the pronounced
selective properties of the non-linear resonant circuit. It is therefore of advantage to consider
for the description of the properties of non-linear elements the characteristics v(i) and -(i). Let
the characteristic of the resistor R be expressed by the function
(5.21)
and the characteristic of the inductor L by the function
(5.22)
First, we shall calculate the equivalent parameters Re and Le of the equivalent linear elements,
i.e.
(5.23)
Second, we shall write a linear equation for the circuit into which these parameters of
equivalent linear elements will be substituted; in complex symbolic notation we obtain
(5.24)
From this we calculate the amplitude of current
(5.25)
We can solve this equation with respect to the amplitude of the current I1 and obtain the
resonance curves I1() with the parameter V1.
5.2.2 The piecewise solution method
The substance of the piecewise solution method (sometimes also referred to as the method of
linearizing differential equations) consists in approximating the characteristic of a non-linear
element by a number of piecewise fitted linear segments. This replacement of a non-linear
characteristic by linear segments allows us to depart from the non-linear differential equation
and work with several linear differential equations which usually differ only in the values of
the coefficients. One linear differential equation holds for each segment of the characteristic.
The system of linear differential equations obtained in this way is solved by methods known
from the theory of linear circuits. At the points of fitting of the linear segments, the initial
conditions are chosen such that the individual solutions link up with each other (this method
is therefore sometimes referred to as the method of "piecewise fitted segments" [33]).
This method permits the investigation of both transient and steady-state processes in
circuits of first, second, and higher order containing sources of both do and harmonic voltage
or current. In solving a system of linear differential equations, it is of advantage to use
computers, since in more complicated circuits the quantities sought are usually calculated
from transcendental equations.
Fig. 106. Illustrating the solution of a non-linear circuit by the piecewise solution method
The basic steps of the calculation by the piecewise solution method will be illustrated by
the solution of the simple non-linear circuit of Fig. 106a. With the switch S turned on, the
capacitor C is charged from a source of do voltage V0 through the non-linear resistor Rn,
whose characteristic v(i) is given in Fig. 106b. We are to calculate the waveform of the
current i(t) in the circuit when the capacitor C is being charged.
The characteristic v(i) of the non-linear resistor is approximated by two linear segments
as indicated in Fig. 106b. On the first segment, from i = 0 to i = i1 it holds that v = R2i . On
the second segment, with i > i1 we have v = Vp + R1i . At any instant, vC + v = V0 . During the
charging of the capacitor C, the current i decreases from the maximum value I0 down to zero.
Consequently, the operating point on the volt-ampere characteristic will first move along the
first segment which is denoted (1) and defined by the currents I0 and il ; then it will move
along the second segment denoted (2), in which i  0, i1 . For the first segment,
(5.26)
and for the second segment,
(5.27)
Solving first Eq. (5.26) and subsequently Eq. (5.27), we obtain the desired dependence i(t).
For the first segment,
(5.28)
The current i reaches the value il at time t = tl . Substituting these values into Eq. (5.27), we
obtain for tl the equation.
For t > tl , we make use of the second segment of the characteristic so that the current
(5.29)
Signal amplification
Signal amplification is one of the most frequent and important tasks required from
electronic circuits. Basically, two principles can be applied in amplification. The first of them
is based on a controlled supply of energy from a dc voltage or current source, in which there
is no transposition of the spectrum of the signal that is being amplified during this process.
The other principle of signal amplification is, on the contrary, definitely associated with
a transposition of the frequency spectrum.
The most common and frequent amplification is that in circuits with an electrically controlled
non-linear resistor, so we shall first consider amplifier operation in these circuits.
6.1 THE PRINCIPLE OF AMPLIFIERS WITH CONTROLLED RESISTORS
An elementary circuit with a controlled resistor which, with a suitable choice
of the parameters of its circuit elements, permits the amplification of voltage, current, or
power, is arranged as indicated in Fig. 111.
Fig. 111. An elementary amplifier with an electrically controlled resistor
An independent source of dc voltage VS supplies a series combination of a non-linear resistor
R, controlled by either the voltage v1 or the current i1, and a loading two-terminal element ZL,
which can be, for example, a resistor, a resonant circuit, or some more complicated circuit.
The input signal v1 or i1 controls (with little power being consumed) the resistance R of the
controlled resistor R, which determines the supply of current i2 and, consequently, also the
power delivered from the voltage source VS to the load ZL.
Amp1ification factor. The amplification properties of such an amplifier are
characterized by the amplification factor. If a change of Δv1 in the input voltage v1 produces
a change of Δv2 in the output voltage v2 , the voltage amplification factor Av is defined by the
relation
(6.1)
Note that, depending on circumstances, a positive change Δv1 can cause either a positive or a
negative change Δv2. The amplifier amplifies a voltage if |Av| > 1. Depending on whether Av
> 0 or Av < 0, the amplifier does or does not change, during its operation, the polarity of the
amplified output signal as compared with the input signal. If Av < 0, we say that the amplifier
is inverting the signal; if Av > 0, the amplifier is non-inverting.
The current amplification factor Ai is defined analogously, that is as the relation
(6.2)
and, again, it can have either a positive or negative sign. What sometimes interests us about
an amplifier is the ratio of the average active powers P2 and P1 of the alternating signals at
the output and the input of the amplifier, respectively. The power - amplification factor is
then defined as
(6.3)
It is always a positive number, and it does not reveal whether the amplifier is or is not
inverting the voltage (the current).
Note that the output quantities need not always be the voltage v2 and the current i2, as
suggested in Fig. 111. The output voltage can be, for instance, a voltage drop across the load
ZL or a voltage derived from it via a suitable linear two-port which enables impedance
matching of the amplifier and its coupling with other circuits. The output current can be
represented, for example, by a current derived from the current i2 via a suitable coupling twoport. Some of these problems will be dealt with in the following text. The fundamental
principle of amplification in a circuit with a controlled resistor remains, however, unaltered.
6.2 TRANSISTOR AMPLIFIERS
A more concrete idea as to the properties of amplifiers with electrically controlled resistors
can be obtained from graphical and numerical analyses of valve and transistor amplifiers.
6.2.1 Transistor amplifier with a resistor load
An amplifier with a PNP transistor which operates in common-emitter connection is shown in
Fig. 113.
Fig. 113. The diagram of a transistor amplifier containing a PNP transistor in common-emitter connection
The configuration of this amplifier does not differ from the valve amplifier. However, the two
circuits function differently from one another in that the valve amplifier operates in most
cases without an input (grid) current, while the operation of a transistor amplifier is, on the
contrary, conditional on an input current being supplied to the transistor base. In a transistor
amplifier it is therefore of interest to follow not only the dependence of the output voltage vCE
on the voltage vBE but also the dependence of vCE on the voltage v1, to whose sources the base
is connected via the resistor R1.
Fig. 114. Graphical solution of the transistor amplifier from Fig. 113 with a KF517 transistor, for the given
values VS = - 28 V, R = 2 kΩ, and R1 = 5 kΩ
The graphical solution of a transistor amplifier is shown in Fig. 114. Since this is a PNP
transistor, all the characteristics are found in the third quadrant of the corresponding coordinate system. From the characteristics iC(vCE; iB), given in Fig. 114b, it is easy to derive
for the given values VS and R the dynamic characteristic iC(iB) plotted in Fig. 114a. The
dependence vCE(iB) could be derived in a similar fashion (it is not shown in Fig. 114), but it
does not tell us anything about the properties of the circuit as a voltage amplifier. To derive
the operating characteristics vCE(vBE) or vCE(v1), we must also know the input characteristic
vBE(iB), which is given in Fig. 114c. It can be seen from Fig. 114c how for a chosen voltage
v1 = -1 V (point A in Fig. 114c) we can, with the aid of a straight line corresponding to the
resistance R1 = 5 kΩ, derive point B which determines the input current iB. From this current,
the current iC (point C in Fig. 114a) and the output voltage vCE (point D in Fig. 114b) can then
be determined. This voltage vCE, together with the voltage vBE, gives us point E of the
characteristic v1(vCE), and with the voltage v1 point F of the characteristic v1(vCE). In this way,
we can successively derive the whole characteristics vBE(vCE) and v1(vCE) which are plotted in
Fig. 114d.
It can be seen that these characteristics have a similar shape to the analogous characteristics
of the valve amplifier. It is also obvious that the amplification decreases with increasing
internal resistance R1 of the source of the voltage v1 applied to the amplifier input. Since the
characteristics of a transistor strongly depend on temperature, care must always be taken lest
in consequence of the changes due to temperature the quiescent operating point of the
amplifier should be shifted undesirably towards the curved regions of the characteristics.
6.2.2 Emitter and cathode follower
A valve or a transistor need not be used in an amplifier solely in common-cathode
connection or in common-emitter connection. The valve can also operate as an amplifier in
common-anode connection or in common-grid connection, and, similarly, the transistor can
operate in common-collector connection or in common-base connection. The graphical
solution of amplifiers with a valve or a transistor connected in this manner does not differ in
principle from the preceding one, but it assumes the knowledge of the input and output
characteristics of the electronic element in the connection under consideration.
Manufacturers, however, do not usually publish such characteristics. We must therefore, even
in such cases, start with the characteristics available for valves in the common-cathode
connection or for transistors in the common-emitter connection, and by a suitable graphical
construction derive from them the desired dynamic characteristics of the amplifier.
We shall now show how to proceed in the graphical solution of an amplifier with
a transistor in the common-collector connection. In keeping with the preceding
considerations, the configuration of this amplifier should be as given in Fig. 115a, but far
more frequently it is connected as shown in Fig. 115b, in which, provided that R1 = 0 and
when the circuit quantities from Fig. 115a are applied, the input voltage is v1 = vBC + VS, and
the output voltage v2 = vEC + VS, i.e. the two voltages we are interested in are shifted by VS.
Fig. 115. An amplifier with an NPN transistor in common-collector connection
From the circuit shown in Fig. 115b, the following relations for the input and output voltages
result:
(6.4)
(6.5)
From the latter equation we determine the dependence of vCE on iC:
(6.6)
which is the equation of the load line of a source with the voltage VS - RiB and with internal
resistance R. If a set of output characteristics is available for the given transistor in the
common-emitter connection (Fig. 116b) the dynamic characteristic can be constructed which
corresponds to Eq. (6.6). For the individual values of the parameter iB, we plot the voltage
VS - RiB on the vCE axis, construct the load line corresponding to the resistance R, and
determine its intersection with the characteristic marked by this parameter iB. In this way, we
obtain a number of points forming the desired dynamic characteristic. It is shown in Fig.
116b by the dashed line, and it almost coincides with the solid line for vCE = VS - RiC. This is
because the voltage drop RiB « VS. With the help of this characteristic; another characteristic,
iC(iB), is derived as plotted in Fig. 116a. We must further know the transistor input
characteristic vBE(iB), which is given in Fig. 116c.
Now we can plot the dependence v1(v2) given by Eq. (6.4). To do this, we construct
some auxiliary lines: the curve i2(iB) = iC(iB) + iB in Fig. 116a (by adding up the ordinates of
the straight line iC = iB and of the curve iC(iB)), the straight line iC = vCE/R in Fig. 116b, and
the straight line v2 = v1 in Fig. 116d.
Fig. 116. Graphical solution of the amplifier from Fig. 115b with a KC508 silicon transistor, for the values
Vs = 20 V, R = 270 Ω, and R1 = 1 kΩ
Then we proceed as follows. For a chosen voltage R1iB + vBE (e.g. 1 V - see point A in Fig.
116c) and for a given resistance R1 = 1 kΩ we obtain the magnitude of the input current iB
(point B in Fig. 116c). By means of the current iB, we determine the current iC + iB (point C
lying on the characteristic iC(iB) given by the dashed line in Fig. 116a). This current flows
through the emitter resistor R, and we can determine from it the output voltage v2 = R(iC + iB)
- point D on the dash-and-dot auxiliary straight line in Fig.116b, and point E on the auxiliary
straight line in Fig. 116d. The ordinate v1 of point F of the desired dynamic characteristic
v1(v2) is found in Fig. 116d by adding to the established value v2 (in our case v2 ≈ 11.6 V) the
initial voltage R1iB + vBE, i.e. v1 ≈ 11.6 + 1 = 12.6 V (the distance between points E and F
corresponds to the voltage R1iB + vBE). In this way, the whole transfer characteristic v1(v2) can
be derived.
It can be seen from Fig. 116d that over a wide region the characteristic v1(v2) plots
almost as a straight line and can be used to amplify signals. Since, however, Δv1 > Δv2, the
amplification factor Av < 1. An amplifier with a transistor in the common-collector
connection has a voltage amplification close to one, and a certain change in the input voltage
will produce an almost identical change in the output voltage - the output voltage follows,
with only a slight difference, all the changes in the input voltage. Such an amplifier is
therefore called the emitter follower (its valve equivalent is the cathode follower). Although
the follower does not amplify the voltage, it amplifies the current and hence also the power.
For almost identical voltages v1 and v2, the currents flowing in the input circuit are of the
order of tenths of mA, while at the output there are currents of the order of tens of mA.
Fig. 117. Differential ampler and its current distribution characteristics
6.2.3 Differential amplifier
If we join the emitters of two transistor amplifiers operating in common-emitter connection,
and if we deliver to these joined transistors a common constant current IE (see Fig. 117a), this
current will be distributed between the two transistors, depending on the difference vd = v1 v2 in the voltages applied to the bases of the two transistors. The amplifier is therefore called
the differential amplifier. A simplified analysis follows.
Assume that the two transistors employed are identical, that their input ampere-volt
characteristics are approximately exponential,
or
, and that the
collector currents are directly proportional to the base currents, iC1 = βiB1 and iC2 = βiB2. For
β » 1 it is obvious that
(6.7)
(6.8)
Substituting for iBl and iB2 into Eq. (6.7) and taking into consideration that by Eq. (6.8)
vBE2 = vBE1 - vd, we obtain the relation
(6.9)
From this relation, we can immediately derive the coefficients α1 and α2 which govern the
distribution of the current IE between the transistors T1 and T2:
(6.10)
(6.11)
By means of these coefficients we can determine, for any differential voltage vd, the
magnitudes of the currents iC1 and iC2 or the voltages proportional to these currents, i.e.
vR1 = R1iC1 and vR2 = R2iC2. The distribution characteristics α1(vd) and α2(vd) are plotted in Fig.
117b. It can be seen that the inflexion point of the two curves is at vd = 0, and that in its
neighbourhood the curves are almost linear; in this operating region (bounded roughly by the
inequality |vd| < 1/(2b)), the circuit can be employed as an amplifier. The curves α1(vd) and
α2(vd) can in this region be replaced by tangents at the inflexion point, i.e.
(6.12)
It follows that the output currents iC1 and iC2 change in mutually opposite sense - the circuit
gives two mutually symmetrically changing output voltages.
From the discussion of the derived relations it follows that a positive change in the voltage v1
leads to a negative change in the collector voltage vC1 = VS - vRl of the transistor T1; by
contrast, a positive change in the voltage v2 yields a positive change in the voltage vC1. If we
draw an output voltage from the collector of the transistor T1, the base B1 shows as an
inverting amplifier input while the base B2 shows as a non-inverting input. From the
viewpoint of the transistor T2, the two inputs have opposite properties.
The transistor differential amplifier is an important building component since owing to its
symmetrical configuration it provides for a considerable compensation of the effects of
temperature changes in transistor parameters. It is therefore manufactured together with a
voltage-controlled transistor current source in integrated form (e.g. FAIRCHILD CA3000,
CA3005). The differential amplifier finds a wide range of application in the design of
operational amplifiers. The application of the non-linearity of its characteristics and its ability
to multiply two signals will be discussed later.
6.2.4 Types of transistor amplifier
Selective properties of amplifiers. The types of amplifier operating with a resistive
load that we have discussed are basically non-inertial circuits, and thus they can, theoretically
speaking, amplify signals with an unlimited spectrum range. The actual frequency bandwidth
in which such an amplifier can process signals without any linear distortion is given by the
inherent inertial phenomena entailed by the structure of the electronic element and the whole
circuit configuration. As we know, these phenomena can be modelled for the purpose of
analysis by inserting the respective capacitors into the circuit model. Even if this limitation is
taken into consideration, transistor and valve amplifiers with a resistor load can still be
characterized as wide-band amplifiers.
In many applications, however, we require that signals with a very narrow limited spectrum
should be amplified, or even that during the processing of a signal with a wide spectrum the
amplifier should pass only a narrow limited band to the output. For this purpose, selective and
band amplifiers are used in which the load is replaced by a resonant circuit or a system of
coupled resonant circuits forming a band-pass filter.
Amp1ifier coup1ing circuits. A general shortcoming of amplifiers is that they produce
an undesired shifting of the dc component of the signal. To remedy this, coupling circuits are
used which permit the connection of a load (or another amplifier) to the amplifier output such
that a required shifting of the dc voltage level may be obtained. Typical coupling circuits for
amplifiers are illustrated schematically in Fig. 118. Figs 118a, b give examples of coupling
circuits which also transfer the correspondingly shifted dc component of the signal and the
adjacent region of very low frequencies.
Fig. 118. Amplifier coupling circuits
In Fig. 118a it is assumed that the operation is practically without any current demand
on the output, and the voltage v3 is then reduced by a voltage drop RpI as compared with the
voltage v2. The operation of the circuit in Fig. 118b is similar, but thanks to the properties of
the voltage reference (Zener) diode, this circuit makes it possible to obtain across the output
the voltage v2 reduced by the stabilization voltage VZ even if a current i3 > -I is being drawn.
When solving amplifiers with these coupling elements, it is, of course, necessary to respect
the loading of the amplifier output by the current I or I - i3. In both the above coupling
elements the source of constant current is often replaced by a source of constant voltage with
a large internal resistance.
In amplifiers of ac signals, only dc quantities are used to set the operating point. In these
cases, the coupling circuits in Figs 118c, d are employed. The coupling element in Fig. 118c
has, at a frequency ω > 1/CR, a transfer factor equal to one. The coupling capacitor C is
charged to a voltage VC which is equal to the sum of the dc component of the voltage v2 and
the dc voltage V3. The output voltage v3 = v2 - VC has a dc component V3. An example of the
connection of the amplifier with a coupling element CR is given in Fig. 119a. When solving
this circuit, we must respect the fact that in determining the quiescent operating point P (Fig.
119b) we must consider the static load line which corresponds only to resistor RD.
Conversely, when determining the amplification properties of the circuit, we must start from
the dynamic load line with a slope corresponding to the resistance of a parallel combination
of resistors RD and R and passing through the statically determined quiescent operating point
P.
The transformer coupling shown in Fig. 118d fulfils two tasks at the same time: it
allows us to introduce dc voltages VS (supply voltage) and V3 into the input and output loops
of the coupling element and, at the same time, it makes it possible to transform (or match) the
resistance of the load attached to the output terminals of the coupling circuit to the primary
winding of the transformer (for the ideal transformer in the ratio of p2, where p = N1/N2, N1
and N2, being the numbers of turns of the primary and the secondary winding, respectively).
In the amplifier analysis, we must examine separately the dc voltages and currents
determining its quiescent operating point, and when solving the amplification properties of an
amplifier processing ac signals, we must take into account the resistance introduced into the
circuit through the capacitor C or transformer Tr. This circumstance is respected in the
solution of the amplifier with a transformer coupling shown in Fig. 119c. The slope of the
static load line (Fig: 119d) is in this case given only by the resistance of the primary winding
of the coupling transformer. The transformer coupling is very often used in selective
amplifiers where either the primary or the secondary winding of the transformer is by-passed
by a capacitor with which it forms a resonant circuit (tuned transformer).
Fig. 1I9. Class A amplifier with a coupling element CR and with transformer coupling
6.2.5 Amplifier operation classes
C1ass A amp1ifier. If an amplifier is to process both positive and negative changes
of the input signal (e.g. of an ac voltage) with a small non-linear distortion, the quiescent
operating point is set to the central part of the active region of the transfer characteristic.
A considerable quiescent current I2 flows constantly through the circuit and is so large that
even for the largest current change Δi2 superimposed on it (e.g. the largest amplitude of the
ac-current) the following requirement is met
(6.13)
Amplifiers satisfying this inequality operate in class A. When operating in this class, the
electronic element is conducting permanently. The most frequent amplifier operation is in
class A, since it is the easiest means of providing linear amplification. This is especially true
of amplifiers processing small signals (delivering small powers to the load).
In amplifiers delivering larger powers to the load (in the order of tenths to tens of watts) the
operation in class A is, from the energy point of view, of considerable disadvantage since the
amplifier draws permanently from the source of voltage VS a quiescent delivered power VSI2.
For instance, the emitter follower dealt with above (Figs 115 and 116) draws at a quiescent
input voltage v1 = V1 = 9 V a quiescent current I2 = 32 mA, and thus has at VS = 20 V a
quiescent delivered power P1 = 640 mW. At full excitation, the output voltage has the
amplitude V2 = 7.5 V, and with this amplitude it delivers to the loading resistor R with a
resistance R = 270 Ω an average power
.
The circuit efficiency is small.
In an effort to improve the amplifier efficiency, we often reduce its quiescent power, even at
the cost of non-linear amplification. Amplifiers of class B (or AB) and class C operate in this
way.
C1ass Bamp1ifier. For amplifiers operating in class B, we set the quiescent operating
point in the region in which current ceases to flow through the electronic element (the
element is biased to cut off). For example, in the case of the emitter-follower with a silicon
transistor we should choose the quiescent input voltage v1 = Vl ≈ 0.5 V; this means we should
place the quiescent operating point on the lower knee of the characteristic v1(v2) given in
Fig.116d. The quiescent current of the amplifier and thus also the quiescent delivered power
are approximately zero. The amplifier processes signals of only one polarity, the signals of
opposite polarity make the current cease flowing through the electronic element, and the
amplifier therefore does not amplify in this region. The output signal is considerably
distorted. The emitter-follower with an NPN transistor, operating in class B, would, when
excited by a harmonic input signal, process only its positive halfwaves, the negative
halfwaves being suppressed.
If linear amplification with a small distortion but a good efficiency is required, even in this
case there is a comparatively easy remedy. In parallel with a class B amplifier,
a complementary class B amplifier is connected which processes signals of opposite polarity
to that of the first amplifier. The two distorted signals are summed at the output, while at the
load they form a signal similar to the input signal. Such a pair of class B amplifiers is usually
regarded as a single amplifier, and it is referred to as the push-pull amplifier operating in
class B. A transistor push-pull amplifier of class B is shown in Fig. 120a. The resistor
dividers R1, R2 and R3, R4 serve as dc coupling elements (their principle is as given in Fig.
118a), and their task is to move the quiescent operating point of the transistors to the limit of
their conducting state, i.e. for v1 = 0 we have vp = vn ≈ 0.5 V. Fig. 120b shows the transfer
characteristic v1(v2) of this amplifier, formed in the first quadrant by the operation of the
transistor T1, and in the third quadrant by the operation of the transistor T2.
For comparison, we shall now calculate the efficiency of such a push-pull amplifier at full
excitation. At VS = 20 V, the attainable amplitude of the output voltage is about V2 = 16 V
and an average power
is delivered to the load
RL with the resistance RL = 270 Ω. Through either of the transistors, a pulsating current with
a peak value Imax = V2/R = 16/270 = 0.0593 A = 59.3 mA is flowing. The dc value of this
pulsating current is I2 = Imax/π (it is calculated on the basis of Eq. (4.3)). Thus for the two
transistors the dc delivered power at full excitation amounts to P1 = 2VSI2 = 2 x 20 x
59.3/3.14 = 755 mW. The efficiency of this push-pull emitter-follower operating in class B is
η = P2/P1 = 470/755 = 0.62 and is thus considerably better than in the preceding case.
Fig. 120. A push-pull amplifier wig complementary transistors in common collector connection, operating
in class
A somewhat larger non-linear distortion of the push-pull amplifier operating in class B (the
characteristic v1(v2) shown in Fig. 120b has a marked curvature in the neighbourhood of the
origin of the co-ordinate system) is usually removed by cascading before it an amplifier
operating in class A and introducing a suitable negative feedback. At the cost of reducing the
total amplification, the feedback suppresses the undesired components appearing in the
amplifier due to non-linear transformation.
C1ass AB amp1ifier. A class AB amplifier operates like a class B amplifier. Its quiescent
operating point; however, is shifted a little (it lies between the operating points corresponding
to classes A and B), and a. small quiescent current flows through it (smaller than in a class A
amplifier). Small signals are thus processed by this amplifier in class A, while larger signals
are partly processed in class B. To give an undistorted signal transfer, these class AB
amplifiers must therefore also be of the push-pull type.
C1ass C amp1ifier. With amplifiers operating in class C, the amplification is again of a
markedly non-linear nature. The quiescent operating point is placed not on the boundary
between the conducting and the non-conducting state of the electronic element, but well into
the non-conducting state region. It is used for power amplification of harmonic signals for
which the current flowing through the electronic element is of the nature of periodically
repeated sinusoidal segments. At the output a harmonic voltage is recovered from these
pulses by selective filtration with the aid of a resonant circuit tuned to the frequency of the
fundamental harmonic component of this pulse train.
Both valve and transistor amplifiers can operate- in class C. Since in amplifiers for very large
powers (in the order of kilowatts to hundreds of kilowatts) valves still predominate, we shall
choose as an example the valve variant of class C amplifiers. Its simplified diagram is given
in Fig. 121a. The resistor R is used to model on the one hand the supply of power to the load,
and on the other hand the losses in the resonant LC circuit itself, which is tuned to the
frequency ωl of the harmonic input voltage vl. The amplitude of the harmonic output voltage
v2 will be proportional to the amplitude IA1 of the first harmonic component of the anode
current iA. Therefore, when examining the dependence of the output voltage on the input
voltage, we must know the dependence of IA1 on the amplitude Vl of the output voltage v1.
This dependence IA1(Vl) will be called the oscillation characteristic of the amplifier. The
oscillation characteristic obtained at a constant anode voltage (vA = const.) is static. If in the
amplifier we take into consideration the backward action of the ac component of the anode
voltage (vA = VS - v2), we obtain the dynamic oscillation characteristic.
Fig. 121. A selective valve amplifier, operating in class C
At times, this backward action can be neglected (e.g. in tetrodes or in pentodes, whose
amplitude of the output voltage V2 < VS), and the dynamic oscillation characteristic almost
merges with the static one. In these circumstances, it is possible to start the analysis from the
static transfer characteristic of the valve iA(vG), valid for vA = const. If we approximate this
characteristic by a piecewise linear line, then in agreement with the conclusions of Section
4.23 (Eq. (4.30)) we obtain the dynamic characteristic
(6.14)
Here, S is the slope of the characteristic iA(vG) in the conducting region of the valve
characteristic, cos Θ = (Vp - VG)/V1, and Vp is the voltage of the breakpoint of the
approximated characteristic.
Fig. 121b shows the waveform of the grid voltage vG(t) = VG + V1 sin (ω1t) together with the
corresponding waveforms of the anode voltage vA(t); the respective pulses of the anode
current iA(t) are shown in Fig. 121c. The curves denoted (1) hold for a valve operating with
a zero load (R = 0, vA = VS), and they can therefore be derived from the static characteristic of
the valve. In the case of the curves denoted (2), we have R > 0, and the anode voltage vA(t)
has an ac component with an amplitude
. When the valve is conducting, the voltage
VA(t) is smaller than VS, and the pulses of the anode current are therefore somewhat smaller
(curve (2) in Fig. 121c). The curves (3) correspond to a special, hitherto unconsidered
operating state. If the resistance R is increased, as compared with the preceding case, the
amplitude of the voltage v2 also increases, and if the resistance R is sufficiently large, the
amplitude V2(3) will be larger than the supply voltage VS. For one portion of the period, the
voltage vA(t) will therefore be negative. At vA < 0, the valve will obviously be nonconducting, and the current pulse will therefore exhibit a deep valley. A valley already
develops in the current pulses at small positive anode voltages. This operating state of the
amplifier is referred to as overexcited. Because of this valley, the dynamic oscillation
characteristic of the valve amplifier is as given in Fig. 121d. It can be seen that in the
underexcited state the dependence IA1(Vl) initially increases steeply, but beginning with
a certain value of Vl = Vlc, at which the overexcited state occurs (V2 ≈ VS, IAl = V2/R ≈ VS/R),
this increase stops suddenly. A valley in the anode current pulse is evidenced by the
amplitude IA1 increasing only slightly with increasing V1. The boundary between the
underexcited and the overexcited state is referred to as the critical state.
The efficiency of a selective power amplifier is given by the ratio of the active power 0.5
V2IA1 delivered to the load R and the dc delivered power VSIA0. Since by Eqs (4.35) and
(4.33) we have IA1 = Imaxα1 and IA0 = Imaxα0, the efficiency is
(6.15)
The efficiency of a class C amplifier will be the greater, the greater the voltage V2. The
magnitude of V2 is optimal when the amplifier operates in critical state, when V2 ≈ VS.
Fig. 122. The dependence of α1/(2α0) on half the conduction angle Θ
The efficiency further depends on the coefficient α1(2α0); its dependence on half the angle of
conduction Θ is shown in Fig. 122. As can be seen, the efficiency of a class A amplifier
cannot exceed η = 0.5. For class B we have η = 0.78. In class C, i.e. for Θ < 90°, the
efficiency continues to increase and e.g. at Θ = 60° we have η ≈ 0.9. However, with the angle
of conduction decreasing, it is necessary to increase the amplitude of the input voltage Vl and,
in proportion to this, also the power delivered to the input. Increasing the efficiency beyond
a certain limit given by the design of the valve is no longer of advantage. For most valves,
this limit lies at η = 0.76 to 0.83. The total efficiency of a valve amplifier of class C is usually
still lower, owing to losses in the resonant circuit and the power taken by the heater or
filament or by other grids of the valve.
In class C amplifiers, the coupling circuits previously described are also often used. For
safety reasons, the resonant circuit is often required not to be connected to a high voltage VS
but to a grounded common node. In that case the anode is usually supplied from a source of
anode voltage via an auxiliary inductor (choke), and the resonant circuit is coupled to the
anode by an isolating capacitor.
6.3 AMPLIFIERS WITH NEGATIVE RESISTANCE
For amplification, circuits can also be applied which include non-controlled twoterminal resistors whose ampere-volt characteristics have a shape with a descending portion
and thus a negative differential resistance (conductance).
6.3.1 The principle of amplification
The amplification principle will be made clear on the basis of the elementary idea illustrated
in Fig. 123a. Acting on the resistor divider, which consists of a linear resistor R and a nonlinear resistor Rn with the N-type ampere-volt characteristic as given in Fig. 123b, is the
voltage v(t) = VS + v1(t).
Fig. 123. A resistor with negative differential resistance employed for amplification
A current i(t) is flowing through the circuit, and across the output we have a voltage v(t). The
voltage and the resistance are chosen such that with v1 = 0 the quiescent operating point
A lies approximately in the middle of the descending portion of the ampere-volt characteristic
of the resistor Rn. If there is a comparatively small ac voltage v1(t) acting across the input, the
operating point moves along the characteristic in the interval between points B and C (see
Fig. 123b). It is obvious from the graphical derivation that the output voltage
vo(t) = VA + v2(t) has an ac component v2(t) which is larger than the input voltage v1(t). The
voltage amplification factor of the amplifier is Av = V2/V1, where V1 and V2 are the amplitudes
of the input and output harmonic voltages, respectively.
If the portion of the ampere-volt characteristic between points A and B can be approximated
by a straight line, then the task can be linearized for this case. The non-linear resistor Rn is
replaced for this region by a linear one with a differential resistance Rn = (voB - voC)/(iB – iC),
where (voB, iB) and (voC, iC) are the co-ordinates of points B and C, respectively, in the plane
vo - i. Since voB > voC while iB < iC, this resistance is negative, Rn < 0. The solution of the dc
operating conditions is in this case not considered, which is admissible in a linear system (the
principle of superposition holds for it). The voltage amplification factor Av is given by the
division ratio of the resistor divider
(6.16)
Fig. 124. The dependence of the voltage amplification factor A- on the resistance ratio R/Rn of the resistor
divider
The hyperbolic dependence of Av on R/Rn is plotted in Fig. 124. It can be seen that with both
resistances positive, the resistive divider attenuates. If Rn < 0, then Av > 1, and if R/Rn
approximates the value -1 from the right, Av increases rapidly, and with R/Rn = -1 it increases
beyond all bounds. In the region of
the output voltage is in phase with the
input voltage, and the circuit operates as a non-inverting amplifier. In the region
characterized by the inequality R/Rn < -1 the amplifier reverses the phase, it is inverting, and
the absolute value of amplification |Av| increases if the ratio R/Rn approaches the value -1. If
we want to specify in which of the regions considered an amplifier with a resistor of the N–
type characteristic can operate, we must go back to the non-linear model. As can be
determined from the slope of the segment
and from the slope of the load lines in Fig.
123b, the amplification phenomenon considered has been derived for the case of R/Rn > -1.
The dash-and-dot line in Fig. 123b gives an example corresponding to the value of R/Rn < -1.
As can be seen, the load line intersects the N-type characteristic at three points, of which only
two, namely D and E, are stable (see the discussion on oscillators with negative resistance in
Chapter 9). However, when the operating point is moved into the neighbourhood of these
points, the circuit obviously does not amplify; it attenuates. The situation is reversed in
a circuit containing a resistor with the S-type characteristic. Such a circuit can operate only in
the region of R/Rn ( -2, -1), that is as an inverting amplifier. In the region of R/Rn ( -1, 0),
the load line intersects the S-type characteristic again at three points, the consequences being
the same as in the preceding case.
Fig. 125. Amplifier with a tunnel diode
6.3.2 Selective amplifier with a tunnel diode
Atypical representative of amplifiers making use of the negative resistance is the amplifier
with a tunnel diode. Its diagram is given in Fig. 125. The quiescent operating point of the
tunnel diode is set by the dc voltage supplied through the resistive divider R3, R4 and through
the inductor L. For ac current, the capacitors C1, C2, and C3 represent a short circuit, so that
from the viewpoint of ac quantities there are in this circuit in parallel connection the source of
the voltage v0 with internal resistance R1, the parallel resonant circuit LC (closed via the
capacitor C2), the tunnel diode TD, and finally the loading resistor R2 with resistance R2. If
the resonant circuit LC is tuned to a frequency ω1 of the input voltage v0, only the real
resistances R1, R2 and the resonant resistance of the resonant circuit Rr are applicable. By
Thévenin's theorem, the whole linear part of the amplifier can be replaced by a single
equivalent voltage source with internal resistance R = 1/(1/R1 + 1/R2 + 1/Rr) and internal
voltage vl = v0R/R1. This brings us to the equivalent circuit in Fig. 123a, whose operation has
already been analysed (the resistor Rn is here replaced by a tunnel diode TD). Amplifiers with
tunnel diodes have the advantage of operating at frequencies in the order of up to tens of
gigahertz, and they have good noise properties.
Signal rectification and shaping
In this chapter we shall focus our attention on the wide range of circuits processing
a single input signal which can be of harmonic or any other waveform. We shall include in
this group rectifiers, signal shaping circuits, functional converters, and also such specialized
signal shaping circuits as frequency multipliers and dividers. In the discussion we shall avail
ourselves of both conventional principles and some more advanced views based on the
systems approach.
7.1 RECTIFIERS
Rectifiers are circuits converting ac quantities (voltage or current) to dc quantities.
Typical of rectifiers is the application of electronic elements with unidirectional conductance,
especially semiconductor diodes and transistors and also controlled switching elements such
as thyristors. Rectifiers containing valves have recently lost significance.
Depending on the circumstances, the output quantity of a rectifier (rectified voltage or
current) can be more or less rippled, because of the insufficient removal of the harmonic
components originating during rectification. To suppress the undesired harmonic
components, a low-pass filter is frequently connected to the rectifier output which, in turn,
usually affects the rectification process so that the dynamic properties of the filter must be
included in the rectifier analysis. Since rectifiers with non-linear signal transformation have
qualitatively different properties from those of parametric rectifiers, we shall deal with each
group separately.
7.1.1 Non-linear rectifiers
Characteristic of non-linear rectifiers is the application of two-terminal or even multiterminal
elements with a pronounced unidirectional conductance, which is large in the forward
direction (infinite in the ideal case) and small in the reverse direction (down to zero).
Whether the element conducts or does not conduct depends on the voltage (current) which
acts on it in the rectifier.
Fig. 132. Diode rectifiers with a resistive load; a) series and c) parallel half-wave rectifiers, b) and d) the
corresponding voltage waveforms
Diode rectifiers with a resistive load. The simplest type of rectifier is the diode with a
non-inertial, i.e. resistive, load. In the series diode rectifier, shown in Fig. 132a, all the circuit
elements (a source of ac voltage v1(t), the diode D, and the load represented by the resistor R)
are series-connected in a closed loop. The rectifier proper is delineated in Fig. 132a by the
dashed line; it is a simple degenerate non-linear two-port with the diode D in the longitudinal
branch. With the diode polarity marked, only a positive current i(t) > 0 can always flow
through the circuit. Assuming that the diode D is ideal, the operation of this rectifier can be
described by the relation
(7.1)
If the voltage v1(t) is harmonic (Fig. 132b), then at the output we obtain a pulsating voltage
v2(t), which is referred to as half-wave rectified voltage. It is obvious that this is a strongly
rippled voltage where the fundamental harmonic component with frequency f = 1/T is
manifested markedly. A spectral analysis of the half-wave rectified voltage shows that the
magnitude of its dc component is
(7.2)
There are no losses in the ideal diode, and the theoretical efficiency of this idealized rectifier
is therefore η = 1. In the power delivered to the load R by the half-wave rectified voltage
v2(t), the dc component V20 is a mere 41% (to be more exact, (4/π2) x 100%); the rest is the
power delivered to the load by all the remaining spectral components of the voltage v2(t).
A dual analogy to the series rectifier is the parallel rectifier (Fig. 132cá, whose three circuit
elements (a source of ac current i1(t), the diode D, and the load R) are connected in parallel.
In this case, too, the rectifier proper (shown by the dashed line) is a degenerate two-port, but
this time with the diode in the transverse branch. When the ideal diode is polarized as
indicated in Fig. 132c, for the given rectifier we have
(7.3)
If the current i1(t) is harmonic, then the half-wave rectified voltage v2(t) illustrated in Fig.
132d will be a complete analogy to the voltage v2(t) in Fig. 132b. However, its maximum
value will be RI1 and its dc component
(7.4)
The considerations on the efficiency of the parallel rectifier are analogous to those for the
series rectifier. In rectifiers containing real diodes and a non-zero source resistance, we must
take into account the voltage drops originating in the circuit. With silicon diodes, for
example, we must reckon with a voltage drop of about 0.6 V when they are forward-biased.
Interchanging the polarity of the diodes in the two rectifiers considered so far, we transfer the
negative half-waves to the output and suppress the positive half-waves. The dc component of
the output voltage will be of opposite polarity.
If we connect in parallel the outputs of two series rectifiers which are supplied on their inputs
with two voltages in opposite phases (i.e. v1(t) and - v1(t)), we obtain a full-wave rectifier as
given in Fig. 133a.
Fig. 133. Full-wave diode rectifiers with a resistive load
The diode D1 conducts if v1(t) > - v1(t); in the contrary case it is the diode D2 that conducts.
When considering ideal diodes, there appears across the load R a pulsating voltage v2(t) (Fig.
133b) developed by positive half-waves of the two input voltages; it can be described by the
relation
(7.5)
The period of the voltage v2(t) is half the period of v1(t), and its fundamental harmonic
component will therefore have a frequency f2 = 2/T. The average value of a full-wave
rectified voltage,
(7.6)
is twice that of the half-wave rectified voltage. The theoretical efficiency of a full-wave
rectifier with ideal diodes is again η = 1, but the dc component's share in producing the
average output power is as much as 81% , i.e. twice as much as in the half-wave rectifier.
Input ac voltages of opposite phases are easy to obtain from a transformer with a symmetrical
output (the secondary winding is centre-tapped) or, if need be, a voltage of opposite
polarity can be obtained with the help of an invertor (an amplifier with the amplification
factor A = -1).
With inversely polarized diodes (see, for example, the diodes D3 and D4 in Fig. 133c), there
appears across the load (R2) a full-wave rectified voltage -v2(t) of opposite polarity. The
double full-wave rectifier in Fig. 133c is, however, interesting also from another point of
view. If diode D1 conducts, then at the same time diode D4 also conducts. In the subsequent
half-period, the remaining two diodes D2 and D3 conduct. If we assume that the resistors R1
and R2 have the same resistance, then owing to the symmetry of the circuit there is no current
flowing through the branch between the nodes A and B. This branch can hence be left out,
and the two sources as well as the two loads can be merged. This gives rise to the bridge fullwave rectifier shown in Fig. 133d, which is usually called the Graetz rectifier, after its
inventor. In comparison with Fig. 133c, for this rectifier v1(t) = 2v’1(t) and R = 2R1. For this
type of rectifier, we must bear in mind that a voltage drop across two simultaneously
conducting diodes is always present.
Interesting possibilities are offered by polyphase diode rectifiers. In an N-phase rectifier there
are N simple rectifiers operating with a common load which are supplied with respective
input voltages
(7.7)
which form a symmetrical N-phase system. Fig. 134a shows a three-phase rectifier which can
be supplied via, for example, a suitable transformer from a three-phase distribution network.
Fig. 134. A three-phase rectifier with a resistive load
It can be seen from Fig. 134b that for a chosen polarity of the diodes only one of them is
always conducting, namely the one which is supplied from the source which at the given time
supplies a larger voltage than the two remaining sources. Within one period, all the three
diodes conduct in turn, each for a time T/3. The output voltage
is rippled
comparatively little, and its dc component
(7.8)
is 1.17 times larger than the effective value of the input voltage.
From a three-phase distribution network we can easily obtain a six-phase symmetrical system
by means of a transformer with symmetrical outputs for each phase. The six-phase rectifier
has an even smaller ripple than the three-phase one,
, and the dc
component of the output voltage has a magnitude of 0.955 V1.
Diode rectifiers with a storage capacitor. Complementing the preceding elementary
rectifier with the simplest filter element - the capacitor, which with regard to its function will
be termed the storage capacitor - we obtain a rectifier with qualitatively different properties.
The kernel of the series half-wave rectifier arranged as given in Fig. 135a is a non-linear twoport with the diode D in the longitudinal branch and the capacitor C in the transverse branch.
The voltage across the capacitor is at the same time the output voltage v2(t). When supplying
a harmonic input voltage v1(t) to the circuit in Fig. 135b, the capacitor is periodically charged
via the diode D and discharged via the resistor R. If, for simplicity, we consider that the diode
D is ideal, then when the diode is conducting, the output voltage will be v2(t) = v1(t). Assume,
in keeping with Fig. 135b, that at time t1 (with t1 ≠ T/4) the period of charging just terminates,
i.e. the diode ceases to conduct and the capacitor C starts to discharge, following an
exponential curve with the time constant τ = RC. The discharge curve links up with the curve
of the input voltage v1(t) = V1 sin (ωt) at point A1 with the co-ordinates (t1, V1 sin (ωt1)). The
discharge curve will therefore be given by the relation
. The
two curves must link up smoothly at point A1, and their derivative at this point must be the
same: ωV1 cos (ωt1) = -V1 sin (ωt1)/τ, hence t1 = arctan (- ωτ)/ω. During the discharge time
the output voltage is v2(t) = vd(t), and across the diode there is a voltage v1(t) v2(t) < 0. The diode therefore remains non-conducting until time t2, when v1(t2) = vd(t2).
Beginning with this instant, the diode starts to conduct, and the output voltage in the region
between points A2 and A3 equals the input voltage. Point A3 is only a periodic repetition of
point A1.
Fig. 135. A series diode rectifier with a storage capacitor
In this half-wave rectification, the voltage v2(t) is periodic with period T. Its ripple depends
on the discharge rate of the capacitor C through the resistor R. Fig. 135b shows the waveform
v2(t) corresponding to the time constant τ = T. The larger the time constant τ, the smaller the
ripple, and vice versa. For illustration, a set of discharge curves is shown in Fig. 135c for the
given voltage v1(t) and for different values of τ. It can be seen that with τ/T < 0.05 we are
approaching the earlier considered case with non-inertial load. The dc component can be
determined by Fourier analysis; however, it is obvious at first sight that
(7.9)
and that the larger the ripple, the smaller the voltage V20 will be. Thus the voltage V20 is
dependent on the magnitude of the current that is being drawn; the rectifier behaves as a socalled soft source. As for the. rectification with non-inertial load, for τ  0 we obtain V20 =
V1/π. For the majority of practical applications, the rectified voltage is required to have as
little ripple as possible. Consequently, we usually choose τ > 50T so that the dc voltage
V20 ≈ V1. A rectifier operating in this mode is referred to as the peak rectifier. When real
semiconductor diodes are used, this voltage will be smaller, approximately by the residual
voltage across the conducting diode, i.e. by about 0.6 to 1 V. The theoretical efficiency of the
rectifier with an ideal diode is again η = 1, but the dc component V20 is almost 100 % of the
total output power.
The full-wave rectifier containing a storage capacitor differs from the schematic circuit in
Fig. 133a only by the storage capacitor added in parallel with the resistor R. The discharge
time of capacitor C is reduced to less than a half, the ripple is smaller, and its period is T/2.
This is clear from Fig. 135c, where the dashed part of the sinusoid -v1(t) corresponds to the
charging line in full-wave rectification. Connecting a capacitor C in parallel with the resistor
R in the circuit of Fig. 133d gives the bridge-type Graetz rectifier with storage capacitor.
In the parallel diode rectifier with a storage capacitor we arrange the source of ac current
i1(t), the storage capacitor C, the diode D, and the load R in parallel. For practical
applications, however, it is of advantage to replace the parallel combination of the current
source and the capacitor by an equivalent series combination of the source of ac voltage and
the capacitor. A parallel rectifier modified in this fashion is shown in Fig. 136a. In the nonlinear two-port shown, the capacitor is placed in the longitudinal branch and the diode D in
the transverse branch. The output voltage v2(t) across the load R is simultaneously the diode
voltage drop.
Fig. 136. A parallel diode rectifier with á storage capacitor
Unlike the preceding case, here the voltage v2(t) is considerably rippled (see Fig. 136b)
because when the diode is non-conducting, the input voltage v1(t) acts via the capacitor C
directly on the output. At time t1 = T/4, the diode D is conducting and the output voltage v2(t)
is therefore of zero level; at this instant, the capacitor C is charged to a voltage vc(t) = V1.
Over the interval
the diode D is non-conducting, and the capacitor discharges via
the resistor R and the source of voltage v1(t). In the region between points A1 and A2, the
curve vc(t) is formed by superimposing the discharge exponential function with the time
constant τ = RC and the shifted sinusoidal function with period T and an amplitude which is
inversely dependent on the ratio τ/T. Fig. 136b gives the waveforms of v2(t) and vc(t),
corresponding to the ratio τ/T = 1. At time t2, the voltage v2(t) assumes zero value; the diode
conducts and the voltage across the capacitor follows the input voltage v1(t) in the region
between points A2 and A3. Beginning with time t1 + T the whole process is repeated.
The dc component V20 of the voltage v2(t) can be determined by spectral analysis. For large
values of τ/T there will be
(7.10)
The ripple will be harmonic with the amplitude V1. The parallel rectifier with a diode will
again have a theoretical efficiency of η = 1. The dc component V20 will, however, participate
in the total average power delivered by the voltage v2(t) into the load R by only 67%. To
remove excessive ripple of the rectified voltage, a filter is usually connected after the parallel
rectifier (e.g. a low-pass filter RC).
Changing the diode polarity in the parallel rectifier circuit, we obtain across the output
a rectified voltage of opposite polarity to that in the case above.
C1amping circuits. A parallel rectifier can also be applied as a clamping circuit. In
engineering practice, the term clamping circuit is used to denote a circuit which complements
an ac signal by a dc component of such magnitude that the peak values (positive or negative)
of this newly obtained signal are of a certain prescribed level. In this sense, a rectifier as
shown in Fig. 136a can be regarded as a clamping circuit with zero clamping level. If
a different clamping level is required, a source of dc voltage of corresponding magnitude and
polarity must be connected in series with the diode D. In Fig. 137 we give an example of
processing a composite video-signal by means of a clamping circuit. Such a circuit is often
referred to as the dc restorer or level shifter.
Fig. 137. A diode clamping circuit
Voltage doubters and multipliers. Two series half-wave rectifiers connected as
indicated in Fig. 138a can deliver the voltage v2(t) to the common load R. With a sufficiently
large time constant τ = C1C2R/(C1 + C2), e.g. τ > 50T, the dc component of the output voltage
will be
(7.11)
that is to say twice as much as with a simple rectifier. This rectifying voltage doubter is called
the Greinacher voltage doubter, after its inventor.
Another possibility of doubling an output dc voltage is provided by the cascade of a parallel
and a series rectifier as indicated in Fig. 138b. As we know, during the operation of a parallel
rectifier, a strongly rippled rectified voltage develops across the diode D1, with a peak value
of almost 2V1. The cascade-connected series rectifier will process this voltage such that the
capacitor C2 will be charged to as much as the above-mentioned peak voltage. With the time
constants C1R and C2R sufficiently large, the output voltage will be rippled only a little, and
its dc component will be of the magnitude
(7.12)
This doubter was invented by Delon. On the above principle, he also developed the voltage
multiplier illustrated in Fig. 138c.
Fig. 138. Rectifying voltage doubters and multipliers
In this multiplier, an ac voltage from a source acts on each of the diodes, via a chain of
capacitors which were charged to a certain dc voltage during the previous operation. When in
operation, the capacitor C1 is charged to a voltage close to V1, the remaining capacitors to
voltages close to 2V1. Thus in a multiplier with four diodes it is possible to obtain across the
output a dc voltage
(7.13)
The chain of rectifiers can be extended still further. With N diodes (N > 2), we obtain
a rectified voltage V20 ≈ NV1.
Transistor rectifi ers. The non-linearities of the characteristics of multiterminal electronic
elements, especially transistors and grid valves, can also be employed for rectification. For
the given purpose, any characteristics are suitable which exhibit or cause unidirectional
conductance. While in diode rectifiers the entire output power was drawn from the input
voltage source, rectifiers with multiterminal electronic elements provide, in certain
circumstances, for operation with a negligible loading of the input voltage source. In these
cases, the entire output power of the rectifier is delivered at the expense of an auxiliary
supply source. Typical representatives of this group of rectifiers are class B non-linear
amplifiers, which were discussed above. In this connection, however, it must be remembered
that the rectified voltage may be superimposed on a quiescent dc output voltage.
Valve rectifiers, such as the grid rectifier (applying for the rectification the unidirectional
conductance between the grid and the cathode), anode rectifier, and cathode rectifier
(basically a class B amplifier), now belong to history. As for transistor rectifiers, we shall
give a simple example, namely the emitter rectifier, shown in Fig. 139.
Fig. 139. Emitter rectifier with a storage capacitor
The charging and discharging of the capacitor are the same as in the series diode rectifier,
with the difference that the charging current i(t) is supplied to the capacitor not via the diode
from an input voltage source but through the transistor T from a source of voltage VS. In this
case, the source of input voltage v1(t) delivers to the circuit only the current coming to the
transistor base, il(t) ≈ i(t)/(β + 1), where β is the current amplification factor of the transistor.
For the ripple, for the magnitude of the dc component of the output voltage, and for the
dimensioning of the time constant τ = RC, the same principles are valid as in the case of the
series rectifier with a diode. The voltage VS must be chosen with VS > Vl. The rectifier is not
suitable for rectifying larger voltages since the base-to-emitter junction is at peak values
exposed to a voltage of 2V1 , which must be less than the maximum reverse voltage of the
transistor permitted by the manufacturers. For a rectifier with opposite polarity of the output
voltage, we shall employ a complementary transistor supplied from a source delivering
a voltage of opposite polarity.
Operational rectifiers. In all the rectifiers considered up to now, the imperfect electrical
properties of real non-linear elements appear to an undesired degree. For these rectifiers, we
must take into account voltage drops across diodes; and, when smaller voltages are being
rectified, a deterioration arises which is due to the insufficient curvature of the characteristics,
etc. For measuring purposes, however, a direct proportionality between the input voltage
amplitude and the dc component magnitude of the output voltage is required. A solution to
this problem can be found in operational rectifiers [17], [27].
Fig. 140. Operational rectifier
Basically, these are operational amplifiers complemented in the output by a non-linear circuit
permitting the decomposition of the output voltage according to its polarity. For each polarity
of the output voltage, a separate feedback path is provided, leading to the summing point of
the operational amplifier. An operational rectifier is shown in Fig. 140. Used as decomposing
elements are the diodes D1 and D2. If there is a positive voltage v1(t) across the input,
a negative voltage appears across the output of the inverting operational amplifier OA. It cuts
off the diode D1, and an approximately zero voltage will appear across the output 2 - 0. Then
the diode D2 conducts and a negative voltage develops across the output 3 - 0 such that the
voltage at the summing point A will be close to zero. For this region of input voltages it
therefore holds that
(7.14)
Conversely, in the region of negative input voltage v1(t), the diode D1 conducts and the diode
D2 becomes non-conducting. Under these conditions,
(7.15)
The boundary between these regions is very sharp, and the transfer characteristics v2(vl) and
v3(vl) approximate with high accuracy the ideal characteristic which is of piecewise linear
shape. These favourable properties make it possible that unlike other rectifiers, the
operational rectifier can even process very small voltages with amplitudes of the order of
a fraction of a millivolt.
Phase-sensitive rectifiers. Interesting possibilities arise if we apply to the rectifier input
a sum of two voltages (see Fig. 141a) of the same frequency ω but mutually shifted in phase
by an angle φ.
Fig. 141. Phase-sensitive rectifiers
If these voltages are v0(t) = V0 sin (ωt) and vl(t) = V1 sin (ωt + φ), then their sum gives
a harmonic voltage v(t) of frequency ω and having, in accordance with Fig. 141b, the
amplitude
(7.16)
where a = V0/V1. It can be seen that the amplitude V depends not only on the amplitudes V1
and V0, but also on the angle φ. After rectification in a diode rectifier with a storage capacitor
with a large capacitance, we obtain across the output a negligible rippled voltage v2(t) with
the dc component V20 ≈ V, which is thus a function of the quantities φ, V1, and V0. If V1 = V0,
i.e. a = 1, the dependence of the voltage V20 on the angle φ is simple: V20≈ 2 V1 cos (φ/2).
Frequently, it is desirable for the dependence V20 (φ) to be linear if possible. In that case it is
of advantage to employ two rectifiers with inverted diodes, one of them processing the sum
and the other the difference of the voltages v0 and v1. Such a rectifier is shown in Fig. 141c,
and the composing of amplitudes in Fig. 141d. The output voltage v2(t) is given by the sum of
the voltages across the capacitors C1 and C2, which for the given polarity of the diodes have
opposite signs. The dc output voltage
(7.17)
is thus a function of the quantities φ, V1 and V0. In Fig. 142, a set of curves V20 (φ, a)/ V1; is
plotted, from which it can be seen that closest to the linear dependence is the curve with the
parameter a = 1 (i.e. V1 = V0), and that all the curves have their inflexion point at the
argument φ = π/2. Note that for a = 1, Eq. (7.17) can be simplified to the form of V20 ≈ 2V1
(cos (φ/2) - sin (φ/2)).
Fig. 142. The characteristics V20 (φ; a)/ V1 for the rectifier from Fig. 141c
7.1.2 Parametric rectifiers
Parametric rectifiers use, in place of non-linear elements, controlled elements whose
conductance is a periodic function of time and the operation of which is only little affected by
circuit voltages or currents. Of most frequent application in this respect is a periodically
operating switching element which is ON for half a period (large conductance) and OFF for
the other half-period (small conductance). A mixed case is that of elements periodically
switched on by a control quantity but disconnected by the transition into the non-conducting
region, i.e. non-linearly.
Synchronous rectifier. A parametric variant of the diode rectifier is the synchronous
rectifier, whose series version is shown in Fig. 143a; the parallel version is shown in Fig.
143b.
Fig. 143. Synchronous rectifiers
The switching element can be realized mechanically (a relay) or, more often, by an electronic
switch. A diode switch is formed by a bridge with four diodes (Fig. 143c), which is supplied
transversely by a square-wave current is(t) = Is(1 + sign sin (ωt))/2. The switch is ON at
periodically recurring intervals
, where T = 2π/ω and where n is
integer. For the switch to operate correctly, the switching current of the diodes Is must always
be larger than the peak switched current i2(t) ≈ i1(t), i.e. Is > max |i2(t)|. A switch with
a field-effect transistor (Fig. 143d) can be controlled practically without any current, but
a residual resistance of the order of tens to hundreds of ohms must be reckoned with.
A switch with a bipolar switching transistor as in Fig. 143e makes it possible to switch only
voltages of currents of one polarity. For signals of both polarities a complementary pair of
switches must be used.
The operation of a synchronous rectifier can be described by the mathematical model
(7.18)
where s(t) = (1 + sign sin (ωt))/2 is the switching function with period T = 2π/ω. If v1(t) is an
arbitrary harmonic voltage, i.e. v1(t) = V1 sin (ω1t + φl), and if the function s(t) is expressed
by a Fourier series, Eq. (7.18) changes to the form
(7.19)
Since the system {sin nx} is orthogonal the dc component of the output voltage will be
(7.20)
It follows from the discussion of this relation that the parametric rectifier is selective: it gives
a non-zero dc output voltage only for mutually synchronized signals such that their
frequencies satisfy the condition ω1 = (2k + 1) ω, i.e. ω1/ω = 1, 3, 5, 7, . . . ; hence the name
synchronous rectifier. In addition, the voltage V20 depends on the initial phase of the input
voltage v1(t) - the synchronous rectifier is at the same time phase-sensitive. Of all the possible
cases, the voltage V20 will be maximum for k = 0 and φ1 = 0. For
we have
V20 > 0, for
we have, on the contrary, V20 < 0.
Note: A parallel synchronous rectifier can be used with advantage as a keyed clamping
circuit. The switching function is in this case a periodic sequence of short pulses which
activate the switching element. Unlike the diode clamping circuit, the keyed clamping circuit
is insensitive to any disturbing voltages during the gaps between the switching pulses.
Contro1led rectifiers. In engineering practice, it is often required that the magnitude of
the dc voltage from a rectifier be controlled continuously. This purpose can be served by
controlled rectifiers, which today usually employ thyristors (also known as silicon controlled
rectifiers). Basically, they are of the same configuration as diode rectifiers, but the diodes are
replaced by thyristors provided with appropriate control circuits.
Fig. 144. The diagram of a full-wave controlled thyristor rectifier
Fig. 144a is an example of a controlled full-wave thyristor rectifier. The control circuit CC
supplies the gates of the thyristors T1 and T2 with control currents derived from the input
voltage v1(t) and suitably shaped so that the thyristors will always start to conduct at the
prescribed time.
Fig. 144b gives the waveforms of the rectifier's input and output voltages for the so-called
phase control of the rectifier. The thyristors do not conduct immediately the voltage v1(t) or v1(t) crosses zero, but later, at times t1, t1 + T/2, etc. If we change the phase φ1 = ωt1 at which
the thyristor starts to conduct, then the time also changes for which the thyristor remains
conducting. For this type of control, the magnitude of the dc component of the output voltage
will be
(7.21)
The control circuits for the phase control of a thyristor rectifier can be conceived in various
ways, ranging from simple adjustable resistors through more complicated phase-shift
networks to complicated transistor delay-circuits [25].
A disadvantage of phase-controlled rectifiers consists in switching at an instant when the
voltage v1(t) is fairly large. Abrupt voltage and current jumps appear in the circuit, and this
results in undesired phenomena. Following the development of digital techniques,
a controlled rectifier with more or less periodically omitted whole half-periods has emerged.
The rectifier circuit remains unchanged, but the control circuit CC is changed substantially.
The latter is in this case realized by a complicated logic system which is set to a required
value of the output voltage. it calculates the order of the omitted half-periods, and supplies
a control pulse train to the thyristors T1 and T2. Fig. 144c shows the waveform of v2(t) in
which two out of every seven half-periods are left out. The magnitude of the dc voltage is in
this case V20 ≈ 5 V1/(7π).
7.2 WAVE-SHAPING CIRCUITS
Any signal of sufficient magnitude is deformed when passing through a non-linear network.
Denoted as wave-shaping circuits are such non-linear circuits which in a prescribed manner
deform, or rather shape, determined (e.g. periodic) or even stochastic signals of known
parameters [23], [25], [42]. The great majority of wave-shaping circuits are formed by
circuits made up of only resistors and sources of dc voltage or current, that is to say noninertial circuits; we shall demonstrate, by an example, that even non-linear storage elements
can be employed for signal shaping.
7.2.1 Clippers
Fundamenta1 notions. One typical class of wave-shaping circuits is formed by circuits
which clip the signal at a certain level of voltage (current). Such circuits are termed clippers.
The peak clipper with its operating characteristic v2(vl) in the shape of a piecewise linear line
in Fig. 145a clips the "top" of the signal v1(t); that is, it replaces all the voltages v1(t) > VH, by
the voltage VH . Expressed analytically,
(7.22)
Of opposite operation is the base clipper' with its characteristic v2(vl) as given in Fig. 145b.
This circuit will clip the "bottom" of the signal v1(t) at a clipping level VL, which can be
expressed by the relation
(7.23)
A base clipper is in this sense also the quite common diode rectifier with a resistive load as
shown in Fig. 132, where, of course, VL = 0. This holds analogously also for the peak clipper
(with a reverse-biased diode and a voltage VH = 0). The characteristic v2(vl) with two
breakpoints as shown in Fig. 145c pertains to the two-level clipper with the top clipping level
VH and the bottom clipping level VL . Its operation can be described by the equation
(7.24)
Fig. 145. The operating characteristics and signal shaping in signal clippers
The characteristic v2(vl) with two breakpoints which pertains to the clip-out circuit is of
another type (Fig. 145d). It is a circuit which clips a band between the levels VL and VH out of
the signal v1(t) and shifts the remaining "clippings" to the time axis. The properties of the
clip-out circuit are rendered by the relation
(7.25)
The clipper is often required to give at its output a signal clipped at a certain level of the input
signal but shifted so that the clipping takes place at another voltage level. This is indicated in
Fig. 146 for the peak clipper. If an input signal v1(t) is to be clipped at the level VH1 and
shifted to the level VH2 ≠ VH1, then the characteristic v2(vl) must be of the shape shown in Fig.
146. In contrast to Figs 145a, c, the slanting portion of the characteristic v2(vl) does not go
through the origin (it does not even head towards it, as in Fig. 145b). The required shifting of
the characteristic is easy to obtain by inserting a suitable source of shifting voltage at an
appropriate point of the clipper.
Fig. 146. Signal clipping with subsequent shifting of the clipped signal
At present, signal clippers are realized with semiconductor diodes and transistors; where the
level of clipping must be maintained accurately, feedback networks with operational
amplifiers are used.
Diode c1ippers. The configuration of the peak diode clipper is as shown in Fig. 147. The
diode D is non-conducting if v1(t) < VH. On the assumption that i2 = 0, there appears in this
case no voltage drop across the resistor R and therefore v2(t) = v1(t).
Fig. 147. Peak diode clippers and their characteristics v2(vl)
If v1(t) > VH, the diode D supplied through the resistor R conducts, a voltage vd appears
across it (its magnitude depends on the current flowing through the diode), and a voltage
VH + vd is obtained across the output. The characteristic v2(vl) given for this clipper in Fig.
147b differs therefore from the ideal characteristic shown by the dashed line, especially in its
horizontal portion. This deviation must be reckoned with in applications (in the case of
silicon diodes the voltage vd amounts to as much as 0.6 V), and one must also take into
account its temperature dependence which is due to the considerable effect of temperature on
the ampere-volt characteristics of semiconductor diodes.
Another variant of the peak diode clipper is shown in Fig. 147c. If v2(vl) > VH, the diode will
be non-conducting, and since i2 = 0, a voltage v2(t) = VH will originate across the output. If
v1(t) < VH, the diode will start to conduct, and a voltage drop vd will appear across it. Across
the output, we shall obtain the voltage v2(t) = v1(t) + vd. The characteristic v2(vl) is of the
shape shown in Fig. 147d, and it differs from the ideal characteristic especially in its slanting
portion (indicated by the dashed line).
By merely interchanging the polarity of the diodes in the preceding two circuits, two variants
of the base diode clipper are obtained, as shown in Figs 148a, b.
Fig. 148. Base diode clippers
Connecting in parallel the peak and the base clippers of Figs 147a and 148a, and replacing
the parallel combination of resistors by a single resistor, we obtain a two-level diode clipper,
as shown in Fig. 149.
Fig. 149. A two-level clipper
The circuit operates correctly when VH > VL. In applications, we must again take note of the
voltage drop across the diodes D1 and D2. Drawing the output voltage from the resistor R
would yield a diode clip-out circuit, but we shall not go into details here.
Transistor c1ippers. In the examination of the properties of transistor amplifiers we
derived in Fig. 1144 the characteristic v1(vCE), which exhibits two breakpoints fitting the
regions of almost constant voltage corresponding to a full conduction (saturation) and full
non-conduction of the transistor. An amplifier with such a characteristic can then operate
with a sufficiently large input signal as a peak, base, or two-level clipper, depending on the
choice of the quiescent operating point. The clipping of a signal is always associated with
a voltage shift of the output as compared with the input signal (the same as in the case of
simple transistor amplifiers).
A very efficient transistor clipper makes use of the distribution of a constant current by two
transistors. In substance, it is identical with the differential amplifier, and it is shown in Fig.
150a. For v1(t) = 0, approximately half the current I0 flows through each of the transistors T1
and T2. The node E will, relative to the common conductor (node 0), have a voltage of about
-0.6 V. When the input voltage is increased to v1(t) > 0.2 V, the voltage of the node E will be
larger (more positive) than -0.4 V, and the transistor T2 will therefore cease to conduct and
the entire current to will flow through the transistor T1. A voltage drop R1I0 will appear across
the resistor R1, so that in this situation v3(t)= VS - R1I0. At this instant, there is no current
flowing through the resistor R2, thus v2(t) = VS. If, on the contrary, v1(t) < -0.2 V, the entire
current I0 will be taken over by the transistor T2, so that v2(t) = VS – R2I0 and v3(t) = VH. In the
region of -0.2 V < v1(t) < 0.2 V there is a smooth transition between the two voltage levels v2
and v3 mentioned above. Fig. 150 shows the curves of the characteristics v2(vl) and v2(vl)
together with an orientation scale on the axis v1. The characteristics have been plotted for R1
< R2. For small voltages v1 (max | v1| « 0.2 V) the circuit can serve as a voltage amplifier;
larger voltages v1 (max |v1| > 0.2 V) will be amplified and at the same time clipped at both
ends. An advantage is that during the clipper operation the transistors will not reach the
saturation region.
Fig. 150. A transistor clipper operating with the distribution of the constant current to by two transistors
This type of clipper is often used as the basic component of the amplitude limiter, which
processes ac voltages of different (say stochastically changing) amplitudes such that a voltage
of the same frequency but with a constant amplitude is obtained across the output. It is used
for the reconstruction of phase- or frequency-modulated signals which, during transmission
by a communication system, are distorted by an undesired amplitude modulation and
additional interference.
Fig. 151. A transistor amplitude limiter
The clipper of an amplitude limiter (Fig. 151a) operates identically with the preceding case.
The transistors T1, T2, and a transistor source of constant current form a single integrated
circuit (such as CA3006). At its output, the clipper is connected via a coupling capacitor Cc
with the resonant circuit CL, which is tuned to the frequency of carrier oscillations of the
useful signal. This filter CL will pass to the output only the first harmonic component of the
signal. It can be seen from Fig. 151b that the signal v1(t) with a linearly increasing amplitude
V1(t) will produce a response v2(t) whose amplitude V2(t) will cease increasing if V1(t)
exceeds a level of about 0.3 V.
Clippers with an operational amplifier. A disadvantage of all the clippers dealt with
so far is that they exhibit voltage drops across the diodes and an undesired curvature of the
characteristics. An almost ideal clipping of the signal can be obtained by means of clippers
containing an operational amplifier. In the section on rectifiers we described the operational
rectifier. If an additional dc current is introduced to its summing point, its characteristic can
be shifted along the horizontal axis as need be (for further details on this problem see also
Section 10.1).
Fig. 152. A single-level inverting clipper with an operational amplifier (the characteristics are plotted for
R1 = R2 = R3)
An inverting clipper with an operational amplifier and two diodes which decompose its
output signal according to polarity is shown in Fig. 152a. If the output voltage v2(t) is used,
the circuit behaves as a base clipper, with the characteristic v2(vl) as in Fig. 152b.
Simultaneously with the clipping of the signal, however, the input signal. is inverted and the
clipping level is shifted to zero voltage. When the output voltage v3(t) is used, the circuit
exhibits the properties of the peak inverting clipper, with the characteristic v3(vl) as in
Fig. 152c. The voltage v3 is also shifted in its clipping level to zero voltage. The actual
characteristics approximate the ideal ones (with an error of a few millivolts at the most). If
we choose R2 > R1 or R3 > R1, the clipped signal can be amplified simultaneously.
With the aid of operational amplifiers and decomposition diodes, two-level clippers and clipout circuits can also be realized [17], [27].
7.2.2 Functional converters
Converters with logarithmic and exponential characteristics. In engineering
practice, analogue circuits find wide application which make use of the advantage of
simplified calculation by means of logarithms. For this purpose, functional converters are
necessary which on the one hand make it possible to take the logarithm of the input signal
and, when the necessary analogue calculating operations have been performed (summation,
subtraction, multiplication, or division by a constant, etc.), to return from the logarithm back
to the original signal [31], [54].
A typical element for modelling a logarithmic function or its inverse exponential function is
the semiconductor PN junction. Fig. 153a shows the ampere-volt characteristic of a silicon
diode with PN junction, shown in semilogarithmic representation.
Fig. 153. Converters with exponential and logarithmic characteristics
Over a wide range of several decades of current, this characteristic plots as a straight line, and
between points, A and B it can be approximated with great accuracy by the exponential
function
(7.26)
where I0 [A] and b [V-1] are constants characterizing the given diode, while VA, VB are the
boundary voltages of the exponential portion of the characteristic.
A converter with an exponential characteristic is easy to construct if we complement the
diode with a current-controlled voltage source. If an operational amplifier is used for this
purpose, we obtain the circuit of Fig. 153b. Since vi ≈ 0 and i1 = id ≈ - i2, the converter
properties can be characterized by the relation
(7.27)
If the converter is to operate correctly, it is required that the voltage
.
Interchanging the elements R and D in the feedback network of the preceding circuit, we
obtain a converter with a logarithmic characteristic, shown in Fig. 153c. In this converter,
the volt-ampere characteristic of the diode is employed, derived from Eq. (7.26) in the form
(7.28)
where
and
. Since it holds for this circuit that v1 ≈ 0, i1 ≈ - i2 = id and v2
≈ -vd, we can describe its properties by the equation
(7.29)
For correct operation of this inverting converter we must ensure tha
t.
In place of diodes, transistors in diode connection can be used for the same purpose (the base
is connected to the collector). A substantial extension of the applicable exponential region of
the characteristics can be obtained with transistors operating in common-base connection as
current repeaters. Semiconductor elements have a marked temperature dependence. The
changes caused by temperature can to a great extent be removed by compensation circuits
[17], [54].
Modelling functional dependences by piecewise linear characteristics. On
principles similar to those of the preceding two cases, functional converters can be designed
which make use of the characteristics of other non-linear elements: varistors, field-effect
transistors, etc. Sometimes, however, the "natural" characteristics of electronic non-linear
elements are not sufficient to meet our requirements. In this case, we can model the required
function (or its inverse function) by piecewise linear characteristics. For this purpose we use
diode or transistor functional converters.
The basic idea of a diode functional converter will be illustrated on the basis of Fig. 154.
Fig. 154. A diode functional converter
In Fig. 154a, four parallel branches connected to the basic resistor R0 are formed by a series
combination of a resistor, a diode, and a voltage source. The voltages of these sources are
chosen such that V4 < V3 < V2 < V1 (see also Fig. 154b). The diodes D1 and D2 are of
opposite polarity to the diodes D3 and D4. If the voltage is v
, all the diodes are
non-conducting, and there is a current i = i0 = v/R0 flowing through the circuit. If the voltage v
exceeds the value V2 , the diode D2 conducts and the current through the circuit is i = i0 + i2,
and hence
(7.30)
Over the interval (V2, Vl ), the characteristic i(v) will thus have a slope corresponding to the
resistance R0R2/(R0 + R2), and it will head towards a point marked on the current axis by the
ordinate - V2/R2. Similarly, for the remaining portion of the characteristic we can derive the
equations
(7.31)
(7.32)
(7.33)
With a suitable choice of the number of parallel branches with diodes, the individual shifting
voltages and slopes of the linear portions delimited by these voltages, we can model with
comparatively great accuracy the given non-linear characteristic. The actual characteristic
differs from the ideal piecewise linear shape because the voltage drops across real
semiconductor diodes will also affect the performance of the circuit. This circumstance must
be respected in the design of the circuit. With a correctly designed diode functional converter;
very satisfactory results can be obtained. For example, a diode converter converting
triangular oscillations to sinusoidal ones, whose characteristic has 13 portions, gives at its
output sinusoidal oscillations with a distortion of less than 0.5% if properly dimensioned and
adjusted.
In place of diodes, transistors can also be used for switching. Diode (or transistor) functional
converters are often used in feedback networks of operational amplifiers [27].
7.2.3 Pulse shaping by means of a transformer
Non-linear storage elements can also be employed to shape signals, as will be shown by the
example of shaping needle pulses in a transformer with closed ferromagnetic core whose
material has an almost rectangular magnetization curve. This circumstance is expressed in
Fig. 155 by the characteristic Ф(i1) which gives the dependence between the magnetic flux Ф
and the magnetization current i1 flowing through the primary winding of the transformer. For
a harmonic excitation current i1(t), the flux Ф(t) will have an almost trapezoidal waveform.
On the secondary winding with N2 turns, a voltage is induced which is given by the
differentiation of the flux linkage with respect to time:
(7.34)
It can be seen from Fig. 155 that the output voltage will be in the form of narrow, periodically
repeated positive and negative pulses.
Fig. 155. Pulses originating in a shaping transformer
7.3 FREQUENCY MULTIPLIERS
The frequency multiplier is. a circuit which, when excited by a harmonic input quantity of
frequency ω1, gives at its output a harmonic quantity of frequency ωk, which is a k-th
multiple of the frequency ω1 (k > 1 is a natural number). In a more general respect, the
signals can also be non-sinusoidal and differing in shape; only their fundamental frequencies
must be in the ratio ωk / ω1 = k.
There are two basic methods for multiplying a frequency. The first of them is based on
a pronounced distortion of the input signal flowing through a non-linear two-port. As a result,
the spectrum of the output signal is rich in higher harmonic components, and for the required
purpose the respective spectral component can be separated by a selective filter (most
frequently a resonant circuit). Operating on a similar principle is the multiplier containing
a parametrically controlled linear element. In this case, both resistive and storage elements
can be employed for the transformation of the spectrum.
In the second method, the signal-shaping circuit has an operating characteristic such that,
when the input signal is acting, the output of the shaping circuit gives the desired signal of
multiple frequency. Thus, this method can dispense with filtering of the useful signal.
However, greater demands are made on the non-linear circuit used.
7.3.1 Frequency multipliers with non-linear two-terminal elements
Depending on the manner in which the non-linear two-terminal element is connected to the
signal source and to the load, two basic variants of frequency multipliers can be
distinguished. Fig. 156a is the basic diagram of the series connection of a frequency
multiplier. A source G, a non-linear resistor Rn (such as a diode), and a load R are seriesconnected in a closed loop. A filter F1, represented by a parallel resonant circuit C1L1, is
tuned to the frequency of the input signal, i.e.
, and represents practically
a short circuit for all the components of the current spectrum, with the exception of the
component of frequency ω1. A filter F2 with the resonant circuit CkLk is tuned to the k-th
multiple of the frequency ω1, i.e.
and has a negligibly small impedance
for all spectral components of the current, excepting the desired component of frequency kω1.
A harmonic voltage of frequency ω1 develops across the filter F1, while across the filter F2
there is a harmonic voltage of frequency kω1. Acting on the non-linear element Rn is
therefore a periodic voltage given by the sum of both these harmonic voltages and having the
fundamental frequency ω1. A non-sinusoidal current with higher harmonic components based
on the fundamental frequency ω1 flows through the element Rn. The component with
frequency kω1 goes through the load R and produces across it the useful harmonic output
voltage.
Fig. 156. Frequency multipliers with two-terminal non-linear elements
The parallel connection of the frequency multiplier of Fig. 156b operates in dual manner.
The sum of two harmonic currents of frequencies ω1 and kω1 (to which the filters F1 and F2
are tuned, respectively) acts on the non-linear element Rn. From the polyharmonic spectrum
of the voltage across the element Rn, only the harmonic component of frequency kcal reaches
the load via the filter F2.
Since sufficiently large signals usually act on the non-linear element, the operation of
a frequency multiplier can in many cases be analysed by approximating the characteristic of
the non-linear resistor by a piecewise linear line. Such an analysis has been given in Chapter
4. Either from Fig. 73 or by establishing the maxima of the functions α(Θ) from Eq. (4.37),
the optimum value of half the conduction angle can be determined:
(7.35)
To multiply a frequency by two, the optimum value of half the conduction angle Θ2 = 60°, to
multiply by three, Θ3 = 40°, etc.
General power relations for active powers in non-linear resistors (see Chapter 4) show that
frequency multipliers containing non-linear two-terminal resistors can have a maximum
efficiency
(7.36)
i.e. when multiplying the frequency by two, the theoretically attainable efficiency is η = 1 /4,
when multiplying by three, it is η = 1 /9. Of considerably greater advantage from the energy
point of view are, therefore, frequency multipliers with a non-linear lossless capacitor or
inductor for which a theoretical efficiency of η = 1 was derived in Chapter 4.
Figs 156c, d show series and parallel variants of a frequency multiplier with a non-linear
capacitor Cn (say a capacitive diode). The source of dc voltage V0 is used to set the quiescent
operating point on the farad-volt characteristic of the capacitor Cn. The coupling capacitor Cc
prevents the dc current from flowing through the load R. In the circuit of Fig. 156d, the
resistor R0 prevents the capacitor Cn from being by-passed directly by the source of voltage
V0. Its resistance R0 must be considerably larger than the reactance of the capacitor Cn at
frequency ω1.
7.3.2 Frequency multipliers with non-linear three-terminal resistors
Three-terminal resistors are often employed in frequency multipliers. They may be valves, or
unipolar or bipolar transistors, in which the non-linearity of their transfer characteristic is
used. The basic diagram of such a frequency multiplier is shown in Fig. 157.
Fig. 157. The basic diagram of a frequency multiplier with a three-terminal non-linear resistor
A harmonic voltage v1(t) of frequency ω1 is applied to the control electrode of a threeterminal non-linear resistor Rn. In the output circuit of this resistor, a parallel resonant circuit
CL is connected, which is tuned to the k-th harmonic component of the non-sinusoidal
current i2(t), i.e.
. A harmonic voltage v2(t) of frequency kω1 develops across
this circuit. Because of the selective properties of the circuit CL, the other spectral
components are ineffective on the output. For optimum operation of the circuit, the amplitude
V1 of the input voltage and the quiescent bias V0 must be chosen such that half the conduction
angle of the element Rn satisfies the requirement of Eq. (7.35). In this type of frequency
multiplier, the output power can be even larger than the power delivered to the input, i.e.
simultaneously with the non-linear conversion of the spectrum, a power amplification takes
place; the necessary energy is drawn from the source of supply voltage VS.
7.3.3 Parametric frequency multipliers
The basic element of a parametric frequency multiplier is a linear controlled element of the
resistive or storage type. To simplify our considerations, we shall give the operation principle
of a frequency multiplier containing a linear controlled resistor, as shown in Fig. 158.
Fig. 158. The basic diagram of a parametric frequency multiplier
Let the resistance R(t) of a controlled resistor R vary, owing to the input harmonic signal
s1(t), sinusoidally in time
(7.37)
where m = R1/R0 < 1, R1 being the amplitude of the change in resistance. On the assumption
that max | v2(t)| « VS, the current i(t) equals
(7.38)
Spectral analysis of this current on the basis of expanding the function 1/(1 - x) into the
power series 1/(1 - x) = 1 + x + x2 + x3 + + . . . , yields
(7.39)
We can , see that in a parametric circuit there also appear higher harmonic components of
current with frequencies 2ω1, 3 ω1, 4 ω1, ...; their amplitudes depend on the magnitude of the
dc voltage VS and on the relative resistance change m. Using a filter CL, dimensioned such
that
, we obtain a harmonic voltage v2(t) of frequency kω1 across the load RL.
7.3.4 Frequency multiplication by means of wave-shaping circuits
The application of shaping circuits for multiplication will first be illustrated by a simpler
example of frequency multiplication of triangular oscillations. Fig. 159a shows triangular
oscillations v1(t) with period T1 = 2π/ ω1, and oscillations of the same waveform but with
period T5 = T1/5; that is to say, oscillations with five-fold the basic frequency ω5 = 5ω1. From
the waveforms we can derive, by the familiar method, the shape of the required characteristic
v2(vl) of the functional converter. The characteristic is easy to construct because it consists of
straight line segments. To multiply a frequency by five, there are five such segments; in
general, to obtain a k-fold frequency, we must apply k segments. A piecewise linear
characteristic v2(vl) of the shape given in Fig. 159a can be realized by a functional converter
with operational amplifiers [27].
Fig. 159. The generation of oscillations of multiple frequency by means of wave-shaping circuits
Somewhat more complicated is the synthesis of the characteristic v2(vl) for a frequency
multiplier which, when excited by a harmonic voltage v1(t) of frequency ω1, gives a harmonic
voltage v2(t) across the output but of a multiple frequency kω1. The situation for k = 3 is
plotted in Fig. 159b. Let there be an input signal v1(t) = V1 sin (ω1t). For the chosen value of
v1, which will be the abscissa of a point of the characteristic v2(vl), we shall find the
corresponding argument
(7.40)
Across the output, there will be a voltage v2(t) = V2 sin (k ω1t) which together with the
argument from Eq. (7.40) will give us the equation of the desired characteristic
(7.41)
Denoting v1/V1 = x and v2/V2 = y, we can formulate Eq. (7.41) for the individual k in the form
of the Chebyshev polynomials with the negative sign [1], [4]:
(7.42)
7.4 FREQUENCY DIVIDERS
The operation in which we obtain periodic oscillations of a fundamental frequency k-times
lower than the fundamental frequency of the input periodic oscillations is called frequency
division. The non-linear and parametric circuits by means of which this operation is realized
are frequency dividers. Both the input and the output oscillations of a frequency divider can
be harmonic.
7.4.1 Concept of frequency dividers
There are two concepts of the frequency divider. According to one of them, the supply of
energy to the output circuit of the frequency divider is controlled by a time filter which
periodically opens and closes with period kT, where T is the period of input oscillations. The
time filter can operate with analogue signals and employ for the counting of time various
inertial phenomena (integration of charges in a capacitor, free oscillations of a resonant
circuit tuned to a frequency 2π/(kT), etc.). For the counting of periods, however, digital
techniques can be used with advantage (a counter which after k periods returns to its original
state and repeats its operation). These digital frequency dividers are of special advantage in
the realization of large division ratios (with k in the range of several decades), and they can
simultaneously be employed .for shaping the output signal. However, they are beyond the
scope of this book, and we shall not deal with them in any detail. The other concept of the
frequency divider is based on a more complicated system in which use is made of frequency
multiplication, mixing, and filtration by a resonant circuit to obtain oscillations of a lower
frequency.
A characteristic property of frequency dividers is that output oscillations can be produced in k
variants which do not differ in waveform but are mutually shifted by time T, i.e. by the period
of the input signal. Which of these variants will exist in the circuit is determined by the
conditions under which the circuit starts operating. The choice of individual variants can also
be affected by external intervention into the circuit operation (e.g. in the logic circuit called
the parametron this possibility of changing the initial phase is used to express binary
information).
7.4.2 Frequency dividers with a time filter
Fig. 160a is the block diagram of a frequency divider in which the time filter is realized by an
integrator. A periodic (say harmonic) input voltage v1(t) (see Fig. 160b) of fundamental
frequency ω1 is clipped by a half-wave rectifier with a non-inertial load. The negative pulses
of the rectified voltage vr(t) are delivered to the input of an inverting integrator (an
operational amplifier with a capacitor in the feedback network). Across its output, we obtain
the integral of this voltage in the form of an increasing staircase function v2(t). As soon as the
voltage v2(t) reaches the value Vref (in the example given it is always every five periods), the
comparison circuit (voltage comparator) VC responds and sends a zero-setting pulse to the
integrator, thus restoring the initial conditions of the integration process. It can be seen that
the output voltage v2(t) of this frequency divider resembles the rippled sawtooth oscillations
whose repetition frequency is one fifth of the frequency of the input voltage v1(t).
Fig. 160. A frequency divider with an integrator
Another method of counting the periods is used in the frequency divider shown in Fig. 161.
Fig. 161. A frequency divider with a feedback time filter
The circuit resembles a class C selective amplifier (Fig. 121), even as regards the function.
The resonant circuit CL is tuned to the frequency ω1/k, i.e.
. A non-linear threeterminal resistor Rn with the characteristic i2(v) as given in Fig. 162 performs simultaneously
the function of a time filter. The information as to when further energy is to be supplied to the
resonant circuit LC is provided by the output voltage v2(t) itself or by a voltage v’2(t) derived
from it, which is applied back to the input circuit. The input of the element Rn is thus acted on
by the sum of two harmonic voltages, namely v1(t) of frequency ω1, and v’2(t) of frequency
ω1/k. It can be seen from Fig. 162 that the voltage v(t) exhibits a pronounced positive peak
after every k periods of the voltage v1(t). If the threshold voltage Vp on the characteristic of
the element Rn is chosen such that only short pulses show up in its output current, which
corresponds to the above peaks of the voltage v(t), we obtain across the resonant circuit CL a
harmonic voltage v2(t) of the required frequency ω1/k. The operating conditions of this
frequency divider must be chosen with some caution. If the quiescent operating point and the
degree of positive feedback are chosen unsuitably, the circuit may start oscillating
spontaneously, even if no voltage v1 is present across the input. These problems are dealt with
in greater detail in Section 9.2.2.
Fig. 162. Illustrating the operation of a frequency divider with a feedback filter
The operation of the time filter need not always be so apparent in all frequency dividers as in
the preceding two cases. This is so in, for example, the frequency divider of the order of 1/2
realized by a resonant circuit with a non-linear capacitor (whose properties will be discussed
in Chapter 10).
7.4.3 Feedback frequency divider
The configuration of a frequency divider of this kind is obvious from the block diagram of
Fig. 163a. The circuit consists of two two-ports indicated by the dashed line, an amplifier
two-port (containing a selective amplifier A with a filter F1), and a feedback two-port (with
a frequency multiplier FM, a filter F2, and a mixer M).
Assume that the circuit is in steady state, i.e. it generates oscillations v2(t) of frequency ω1/k.
By means of the frequency multiplier FM provided with a filter F2, we derive from this
output voltage a harmonic voltage vp of frequency ωp = ω1(k - 1)/k, which is a (k -1)-th
multiple of the frequency ω1/k. The voltage vp(t) and the input voltage v1(t) of frequency ω1
are delivered to the mixer M. From the spectrum of the voltage v’1(t) produced by non-linear
transformation in the mixer, we select and amplify by means of the selective amplifier A with
the filter F1 the harmonic component of the difference frequency, i.e.
(7.43)
Across the amplifier output, a harmonic voltage v2(t) is obtained which by Eq. (7.43) has the
frequency ω1/k.
For the circuit to operate correctly, the component of the difference frequency ω1 – ωp,
contained in the spectrum of the voltage v’1(t) and delivered to the input of the amplifier A,
must have a sufficient amplitude and a suitable phase. For the self-excitation and settling of
oscillations, the same conditions are valid as in the autonomous circuits called feedback
oscillators, which will be dealt with in Chapter 9. The feedback frequency divider is not
autonomous, though. If no input voltage v1(t) is supplied to the mixer M, no spectral
component with frequency ω1 – ωp = ω1/k will be contained in the spectrum of the voltage
v’1(t), and the circuit will therefore not develop self-oscillations.
Fig. 163. A frequency divider with a mixer and a frequency multiplier in the feedback branch
A particular connection of such a frequency divider is given in Fig. 163b. The mixer M is
realized by a diode D1 with coupling elements L3, C4, and the selective amplifier by
a transistor T with a resonant circuit C2C3L2. The frequency multiplier FM is provided with
a diode D2 supplied by a voltage from the tap of the capacitive divider C2C3 , and its filter F2
is formed by a resonant circuit C1L1 connected in series with the source of input voltage v1(t).
A resistor R1 completes the circuit of dc current for the diode D2.
Signal mixing, modulation, and
demodulation
8.1 FREQUENCY MIXERS AND CONVERTERS
The term frequency mixer will be applied to two-ports which are used to shift (to transpose)
the frequency spectrum of a given input signal along the frequency axis by a fixed frequency
difference. An idea as to the manner in which to process a signal si(t) = cos (ωit) such that it
yields on the frequency converter output a signal s0(t) = cos (Ωt), where Ω = ωi ± ωh (the
sign + holds for so-called sum frequency, the sign - for so-called difference frequency) is
provided by the familiar trigonometric relation
(8.1)
It can be seen from this relation that in addition to a signal of frequency ωi there must be an
auxiliary harmonic signal of frequency ωh and a device capable of multiplying the two
signals. The product of two functions, together with other components, is obtained in the
transformation of the spectrum of two signals in a non-linear or parametric circuit which in
this case is referred to as the mixer of these two signals. A signal obtained by such
transformation has in general a very rich spectrum; the useful spectral components can be
separated from it by a selective filter. Every frequency converter (FC in Fig. 164a) therefore
always consists of three functional blocks: a mixer M, an oscillator O supplying harmonic
oscillations of frequency ωh = |ωi ± Ω|, and a filter F passing only the desired spectral
components to the output.
Fig. 164. The block diagram of a frequency converter and the spectra of the signals processed by the
converter
Applying to the frequency converter input a modulated signal which carries some
information, we require that the frequency conversion take place without any loss of
information, i.e. without any distortion of the spectrum of the modulated signal. It must
therefore be ensured that during the operation of a frequency converter the amplitudes of all
frequency-shifted components of the signal spectrum always change in a constant ratio (Fig.
164b).
Depending on whether the mixing is realized in a non-linear or in a parametric circuit, we
distinguish non-linear and parametric mixers.
8.1.1 Non-linear (additive) mixers
Fundamental considerations. In the case of a non-linear mixer, we deliver to a nonlinear element NE (Fig. 165), which is usually a two- or three-terminal resistor, the sum of
two signals - the input (say amplitude-modulated) signal si(t) and an auxiliary harmonic
signal h(t), i.e. the signal
(8.2)
Since acting on the non-linear element is the sum of both signals, this type of mixing is
sometimes referred to as additive mixing.
If we approximate the non-linear element characteristic y(x) in its operating region by the
power polynomial
(8.3)
we obtain the output signal y(t) = y(x(t)) of the non-linear element NE in the form
(8.4)
Contributing from this polynomial to the generation of components with frequencies ωi ± ωh
is only the term 2a2sih, whose appearance is due to the quadratic term a2x2 of the
characteristic y(x). The amplitudes of these components will be found from the relation
(8.5)
Along with the useful components, there will be at the mixer output a number of further
combination components with frequencies k1ωi + k2ωh (k1, k2 = ... , -1, 0, 1, 2, ...), which must
be removed by a selective filter (F in Fig. 165).
Fig. 165. The block diagram of a non-linear frequency converter
This selective filter, usually realized by a parallel resonant circuit tuned to the difference (or
sum) frequency Ω = ωi ± ωh, will pass to the frequency mixer output only the harmonic
component
(8.6)
To facilitate the filtration of the useful signal out of the rich spectrum of non-linear
transformation products in mixers, the elimination of undesired spectral components in
symmetrized (balanced) mixers is often employed. The principle and basic properties of
symmetrized systems of non-linear circuits were described in Section 4.6.
Three basic facts can be concluded from the above analysis of the operation of an additive
mixer: a) The useful product of mixing with difference (or sum) frequency appears because
of the presence of the quadratic term in the approximation function y(x) describing the characteristic of the employed non-linear element. An element with the characteristic y = ax2 is
therefore best suited for mixing. b) The amplitude So(t) ~ a2HSi(t) of the output signal is
directly proportional to the product of the amplitudes of both input signals. When mixing
small signals si(t), it is therefore of advantage to employ auxiliary oscillations h(t) with large
amplitude. c) The envelope curve So(t) of the modulated output signal resembles (in the
geometrical sense of the word) the envelope curve Si(t) of the input signal. Thus during the
conversion of the frequency, no distortion of the information transmitted can be observed.
The theoretical efficiency of a mixer containing a non-linear two-terminal resistor,
established on the basis of general relations for active powers in such circuits (these relations
were discussed in Section 4.5.2), amounts to η = 0.5 at the most. Mixers with three- or
multiterminal non-linear resistor can at the same time amplify, at the expense of the energy
supplied to the circuit by a source of dc supply voltage.
Mixers with lossless non-linear capacitive or inductive elements have a total theoretical
efficiency of η = 1; according to the general relations for active powers in non-linear storage
elements (see Section 4.5.3), the distribution of power in these mixers depends on the
frequency structure of the circuit. With suitably chosen frequencies, they can, simultaneously
with frequency conversion, amplify the power of the input signal, at the expense of the
energy supplied at frequency ωh by the local oscillator; they are therefore usually included
among amplifiers (parametric amplifiers, as discussed in Section 6.4). We shall therefore
concentrate on the analysis of mixers containing non-linear resistors.
Fig. 166. The diagram of a diode mixer
Diode mixers. A frequency mixer with a diode is shown in Fig. 166. Acting on the mixing
diode D is the voltage
(8.7)
The voltage v1(t) ≡ si(t) corresponds to the input signal, the voltage v2(t) ≡ h(t) represents
oscillations supplied by the local oscillator, and finally the voltage v3(t) ≡ vo(t) represents the
frequency-transposed output signal and is given by the voltage drop across the resonant
circuit C3L3 tuned to the frequency ωr ≈ Ω = ω1 ± ω2. Assume that the tuning is not quite
accurate, and that for the current component with frequency Ω = ω1 ± ω2 the resonant circuit
C3L3 has an impedance Z = Z ejφ; for the other components of the current i(t), let this resonant
circuit have a negligibly small impedance. Let us now calculate the amplitude V3(t) and the
initial phase Ф of the output signal, on the assumption that the ampere-volt characteristic of
the diode in the operating region can be approximated with sufficient accuracy by the power
trinomial
.
(8.8)
Substituting for v = v(t) from Eq. (8.7) and performing the necessary mathematical
operations, we find that the harmonic current component with frequency Ω = ω1 ± ω2 is
given by the relation
(8.9)
or in complex symbolic notation
(8.10)
where V3(t) = V3(t) ejФ. On the above assumption, a non-zero voltage will be produced across
the resonant circuit C3L3 only by the current component iΩ(t); this voltage will have the
magnitude
(8.11)
Substituting for iΩ(t) in this equation from Eq. (8.10), we obtain the relation
(8.12)
from which we can determine the complex amplitude of the voltage across the resonant
circuit C3L3
(8.13)
It can be seen that in a frequency mixer containing a diode, the relation between V3(t) and
V1(t) is characterized by a transfer coefficient Kc, and it depends on the shape of the amperevolt characteristic of the diode (as expressed by the coefficients a1 and a2), on the amplitude
V2 of the voltage from the local oscillator, and on the impedance Z(Ω) of the output resonant
circuit C3L3 (including the transformed load).
Typical of diode mixers is their combination into symmetrized systems, which owing to the
symmetrical structure make it possible to avoid interaction between sources and loads (for
details and some examples see Section 4.6).
Three-terminal resistor as a mixer. A frequency converter with a non-linear threeterminal resistor Rn is shown in Fig. 167. The sum vi(t) of two voltages v1(t) and v2(t) with
frequencies, ω1 and ω2 is applied to the input 1- 3 of the mixing element Rn. A current io(t)
flows through its output 2 - 3 from a source of voltage VS. Owing to the non-linear
transformation, this current has a rich spectrum and also contains a useful component with
frequency Ω = ω1 ± ω2, which on flowing through the resonant circuit CL tuned to the
frequency Ω, gives the derived output voltage v3(t). Various non-linearities of the threeterminal element can be employed for mixing, for example the curvature of its input
characteristic ii(vi) or the curvature of the transfer characteristic io(vi), which are the most
frequent.
Fig. 167. The basic diagram of a frequency converter with a non-linear three-terminal resistor Rn
A typical example of a mixer operating on the basis of the non-linearity of the input
characteristic is the mixer with a bipolar transistor, whose application in a frequency
converter is shown in Fig. 168a. The quasi-periodic voltage
(8.14)
Fig. 168. Two basic types of frequency converter with a non-linear three-terminal resistor a) and c), and
their models b) and d)
produces on the approximately exponential characteristic iB(vBE) of the transistor a quasiperiodic current iB(t), whose spectrum also contains components with frequency Ω = ω1 ± ω2.
In a bipolar transistor, the transfer characteristic iC(iB) is almost a straight line and is
characterized by the current amplification coefficient β = iC/iB. Thus, a quasi-periodic current
flows through the collector circuit which (with the exception of the different dc component)
is a β-fold of the base current, i.e. iC(t) ≈ βiB(t). If a resonant circuit CL tuned to a frequency
Ω is inserted in the path of this current, the desired output voltage v3(t) will develop across it.
On the assumption that the output voltage v3(t) does not affect the base circuit, the circuit of a
frequency converter with a transistor mixer can be replaced by the model shown in Fig. 168c.
A non-linear transformation takes place in the diode De, whose characteristic is used to model
the input ampere-volt characteristic of the transistor iB(vBE). In contrast to the frequency
converter with a diode, there is no resonant circuit tuned to a frequency Ω in the diode circuit;
such a circuit is in this case connected in the transistor output circuit, in which a current iC(t)
≈ βiB(t) is flowing. Across the load R there appears a harmonic voltage of frequency Ω
because the circuit CL represents roughly a short circuit for the other current components.
In some cases it is not possible to neglect the backward effect of the output voltage v3(t) on
the operation of the mixer. In that case it is sufficient to add to the chain of sources in the
model in Fig. 168c another source of voltage Dv3(t) of frequency Ω, and to solve the circuit in
the same manner as for the diode frequency converter. The quantity D, the penetrance, is then
the measure of the voltage v3(t) penetrating from the output to the input.
Fig. 168b is a simplified diagram of a frequency converter with a field-effect transistor,
which is a typical representative of mixers applying for non-linear transformation the
curvature of the transfer characteristic iD(vGS). The field-effect transistor is especially suited
for mixing because its transfer characteristic is approximately quadratic. For this frequency
converter, a model can be made with a non-linearly controlled source of current iD = iD(vGS) in
the configuration shown in Fig. 168d. Also, in this case, the backward effect of the voltage
v3(t) on the mixing process can be allowed for by introducing penetrance D.
Non-linear frequency converter with an autoparametric mixer. In a circuit
containing a non-linear mixer, the voltage supplied by the local oscillator is often much larger
than the voltage of the useful signal, V2 » V1. In that case, it is of advantage to analyse this
frequency mixer as an autoparametric circuit. These problems were treated in general terms
in Section 4.4.2.
Let there be a real element whose non-linear characteristic, which is employed for mixing
(hence also the transfer characteristic), can be approximated by a function i(v). We shall use a
procedure starting with Eq. (4.68) (for x, y, Pd we substitute v, i, S, respectively). Assume that
two voltages are acting in the circuit, namely v1(t) = Vl cos(ω1t) and v2(t) = V2 cos(ω2t), with
V2 » V1. On rewriting with the newly introduced quantities, we obtain the resultant Eq. (4.76)
in the form
(8.15)
It can be seen that the spectrum of the current Δi also contains the desired current
components with frequencies Ω = ω1 ± ω2,
(8.16)
The quantity Sc = S1/2 = IΩ/V1, given in siemens or in A/V, is the conversion slope.
8.1.2 Parametric (multiplicative) mixers
Fundamental considerations. The problem of transforming signal spectra in parametric
circuits was discussed in Section 4.4. It follows from Eq. (4.66) that a parametric circuit
supplies, among other components, also the desired harmonic components with frequency Ω
= ω1 ± ω2 (corresponding to k1 = 1, k2 = ± 1). If both the input voltage and the parameter of
the controlled element (e.g. the slope) change harmonically; e.g. v1(t) = V1 cos (ω1t) and S(t)
= S0 + S1 cos (ω2t), the process of mixing will be characterized by the simple relation
(8.17)
Thus, for optimum operation of the parametric mixer it is suitable that the parameter S(t)
should vary harmonically or, to put it differently, it is of advantage if the relation between the
harmonic control quantity p(t) and the parameter S(t) is linear, S(t) = kp(t), where k is
a constant.
The parametric element could be controlled with the same effect by an arbitrary control
quantity; in practice, however, only . electrically controlled elements are used (mostly
voltage-controlled elements). For this purpose, special four-terminal or even multiterminal
resistors have been developed (tetrodes, pentodes, etc.). Formerly, they were mixing valves
with several grids [34]; today, they are unipolar transistors with two separate gates [25].
A considerable advantage of these mixers is that they provide, practically without any
problems, for mutual electrical separation of all sources and all loads.
Since in Eq. (8.17), describing the operation of the parametric mixer, there appears the
product v1(t) S(t), these mixers are often called multiplicatiue mixers.
Fig. 169. Simplified diagrams of a parametric converter with a hexode and an FET transistor with two
gates
Frequency converter with a mixer valve. Although a valve with two grids, i.e.
a tetrode, would suffice for parametric mixing so far as the function is concerned, further
(screen or suppressor) grids have been added to the mixing valve to give better mutual
separation of the sources of signals, and thus the most common mixing valves came into
existence - the hexod- and the heptode. Fig. 169a is a simplified diagram of a frequency
converter with a hexode. An input voltage v1(t) of frequency ω1 is applied to the first grid of
the hexode, while a voltage v2(t) coming from the local oscillator operating at frequency ω2
acts on the third grid. A source of dc voltage V0 provides a suitable negative bias for the two
grids. The second and the fourth grid are screen grids. The slope S(t) of the transfer
characteristic iA(v1) is changed periodically by the voltage v2(t). The useful component of the
anode current iA(t) = v1(t) S(t) produces across the resonant circuit CL a harmonic output
voltage v3(t) of frequency Ω = ω1 ± ω2 .
Frequency converter with a semiconductor tetrode. The semiconductor element
most suitable for parametric mixing is the %eldeffect transistor provided with two gates, i.e.
the semiconductor tetrode with MOS structure. A frequency converter with this mixing
element is shown in Fig. 169b; it can be seen that it is an analogy to the preceding case. The
slope of the transfer characteristic iD(v1) is markedly dependent on the voltage v2, the
dependence being almost linear in a certain operating region. The operation of a mixer with
a semiconductor tetrode can therefore be described approximately by Eq. (8.17).
Parametric mixer with bipolar transistors. This mixer is connected as shown in Fig.
170.
Fig. 170. A transistor mixer with a parametrically controlled current distribution
It makes use of the differential transistor unit whose principle was explained in Section 6.2.4.
However, in place of the constant current IE flowing through the transistor unit, we have
a time-varying current IC3(t) from the current source transistor T3. Neglecting the base current
of this transistor,
(8.18)
where VBE3 is the (approximately constant) voltage between the base and the emitter of the
transistor T3. Taking into consideration Eqs (6.10) through (6.12), in which by Eq. (8.18) we
substitute for the current IE the current iC3(t), and for the difference voltage vd the voltage v1(t)
across the symmetrizing resonant circuit C1L1, we can formulate the basic relations
(8.19)
and similarly
(8.20)
The last expressions in Eqs (8.19) and (8.20) are valid for only a small difference voltage v1
(max |v1 | < 20 mV, see the explanation of the differential amplifier).
On the assumption that the voltages v1(t) as well as v2(t) are harmonic, there is a quasiperiodic current flowing through the resonant circuit C2L2:
(8.21)
which also contains the desired harmonic component iΩ(t) with frequency Ω = ω1 ± ω2. This
component will produce across the circuit C2L2, tuned to the frequency Ω, the desired
harmonic output voltage v3(t). These mixers are also manufactured in integrated form (under
the name differential amplifier, e.g. CA3005).
8.1.3 Self-oscillating mixers
Self-oscillating mixers (frequency converters) combine a mixer and a local oscillator.
A single multiterminal resistor (transistor, valve) performs two functions simultaneously: it
generates harmonic oscillations, and it is used for the non-linear parametric transformation of
the signal spectrum. A typical self-oscillating mixer of earlier days was represented by
a heptode (pentagrid) or an octode. In these multigrid valves, the system of the first two grids
and the cathode was employed as a triode which in conjunction with a feedback selective
system (see Chapter 9) generates harmonic oscillations. This modulated the space charge
which formed the virtual cathode for the other grids and the anode, and periodically changed
the slope of the octode. Hence the mixer operáted on the principle of parametric mixing.
Fig. 171 shows a frequency converter in which a bipolar transistor T is used as the selfoscillating mixer. Together with the resonant circuit C3L1, a feedback coil L2 and a coupling
network C2R3, the transistor forms an oscillator (for details on feedback oscillators refer to
Section 9.2.2) oscillating at frequency ω2. Simultaneously, an input voltage v1(t) of frequency
ω2 is delivered via the coupling capacitor C1 to the transistor base. Between the base and the
emitter of the transistor there is a quasi-periodic voltage vBE(t) = v1(t) – v2(t). Consequently,
the spectrum of the collector current iC(t) will also contain harmonic components with
frequency Ω = ω1 ± ω2. Across the resonant circuit C4L3, tuned to the frequency Ω, we shall
therefore obtain the desired harmonic voltage v3(t).
Fig. 171. A frequency converter with a self-oscillating transistor mixer
Self-oscillating mixers, which are used quite frequently, are simple and economical.
However, they are not suited for more demanding applications, because of the comparatively
small frequency stability of the generated oscillations and also because the optimum
operating conditions for the oscillator and the mixer are different.
8.2 MODULATORS
8.2.1 Modulated signals
In communications engineering, signals carrying information are modified to signals suitable
for transmission; this is known as modulation. It allows manifold utilization of the wide-band
transmission medium for a simultaneous transmission of several signals. Each signal is
shifted by modulation to a frequency band reserved for it, and all the modulated signals are
transmitted simultaneously. In the receiver the desired modulated signal is separated by
frequency selection, and the original modulating signal is recovered from it by demodulation.
The most common modulation is that of harmonic oscillations. In the course of modulation,
we affect by a modulating signal s(t) one of the three parameters characterizing harmonic
oscillations:
(8.22)
i.e. the amplitude V, or the frequency ω, or the initial phase φ. Oscillations without
modulation are referred to as carrier oscillations, their frequency ω being equal to the carrier
frequency ωc , which usually lies in the middle of the frequency band used for transmission.
When, in modulation, the amplitude V = V(t) = V(s(t)) varies with the modulating signal s(t),
we have amplitude modulation of harmonic oscillations. If there is to be no distortion, the
modulation characteristic V(s) must be linear:
(8.23)
Here, Vc is the amplitude of the carrier oscillations alone (for s ≡ 0), and kAM is a coefficient
determining the extent to which the modulating signal s affects the amplitude V during
modulation. It is sometimes desirable that no carrier oscillations shall be contained in the
spectrum of amplitude-modulated oscillations. In that case, Vc = 0, and the modulation
characteristic is given by the direct proportionality V(s) = kAMs.
If, in modulation of the signal s(t), the frequency ω = ω(t) = ω(s(t)) changes, we have
frequency modulation of harmonic oscillations. The modulation characteristic ω(s) must
again be linear, and without modulation (s ≡ 0) it must give the carrier frequency ωc. It
therefore obeys the equation
(8.24)
in which the coefficient kFM determines the extent to which the modulating signal s affects the
frequency ω during modulation.
The third possibility is phase modulation of harmonic oscillations, in which their initial phase
φ = φ(t) = φ(s(t)) is affected' by the modulating signal s(t). Usually, the initial phase of the
carrier oscillations alone is φc= 0, so that the linear modulation characteristic passes through
the origin:
(8.25)
Here the coefficient kPM is the coefficient of the proportionality between the value s and the
corresponding change in the initial phase φ of the phase-modulated harmonic oscillations.
There is nothing to prevent a similar modulation of analogous parameters of other, nonsinusoidal oscillations. Rectangular carrier oscillations are often applied. Specific
possibilities are provided by pulse modulation, where the carrier oscillations are formed by
a train of short, periodically repeated pulses whose height, width, repetition frequency, or
position (phase) can be affected by modulation. Such modulated pulses can be
advantageously interspaced, so that between two adjacent impulses of a pulse train we
successively insert impulses pertaining to the other modulated pulse trains in a determined
sequence. After transmission, the individual modulated pulse trains are separated by time
selection (by means of periodically switched time filters).
Note that for modulation, the requirement of the Shannon - Kotelnikov theorem must also be
fulfilled (the sampling theorem - see e.g. -26-, -51-), by which the waveform of a signal with
a limited spectrum is determined completely if we know its values sampled periodically with
a frequency larger than twice the highest frequency of the spectrum of this signal. For
modulation, this means that the frequency ωc of carrier oscillations must be at least twice the
highest frequency of the spectrum of the modulating signal. For instance, for the speech
signal of an upper limit frequency of 3400 Hz, carrier oscillations of a frequency of 8000 Hz
are used. For other reasons, the carrier frequency is usually considerably higher.
8.2.2 Modulators for amplitude modulation
Fundamenta1 considerations. It follows from the above considerations that a generally
formulated relation holds for the amplitude modulation of harmonic oscillations, namely
(8.26)
Consider first the elementary case of amplitude modulation by a single harmonic signal s(t) =
Vmcos(ω2t). For simplicity, we choose φ1 = 0. With the modulation characteristic V(s)
expressed by Eq. (8:23), we obtain
(8.27)
where m = kAMVm/Vc is the modulation index or modulation factor. From Eq. (8.27), we
calculate
(8.28)
It can be seen that the spectrum of harmonic oscillations which are amplitude-modulated by
one harmonic signal consists of three harmonic components with frequencies ω1, ω1 - ω2, and
ω1 + ω2, i.e. the carrier oscillations and two side oscillations. The waveforms of oscillations
described by Eq. (8.27) are shown in Fig. 172x; Fig. 172b shows on the one hand the spectra
of the two signals participating in the modulation, and on the other hand the spectrum of the
amplitude-modulated signal corresponding to Eq. (8.28). Fig. 172a shows that the
information carried by the modulating signal s(t) is contained in the envelope V(s(t)) of the
modulated oscillations v(t).
Fig: 172. Amplitude-modulated harmonic oscillations and their spectra
In the general case, the modulating signal s(t) is non-sinusoidal. Its spectrum covers acertain
frequency band, and this is also seen in the composition of the spectrum of the modulated
signal. In Fig. 172c, this is indicated for a modulating signal s(t) whose spectrum falls within
the band bounded by the upper limit frequency Ω. It can be seen that the spectrum of the
modulated signal contains, in addition to the carrier oscillations, two sidebands carrying
information as to the waveform of the modulating signal s(t).
In this kind of modulation, which constantly maintains the carrier oscillations of amplitude Vc
even in the absence of the modulating signal and is therefore of little advantage as regards
energy, the modulation index is usually m ≤ 1, because a signal modulated in this manner can
be demodulated by simple demodulators.
If for the generation of amplitude-modulated oscillations we employ a simpler modulation
characteristic V(s) = kAMs, and again consider the simplest case of the harmonic modulating
signal s(t) = Vm cos(ω2t), Eq. (8.26) will change (assuming again φ1 = 0) to the form
(8.29)
Fig. 173a shows that the envelope V(s(t)) of oscillations v(t) follows the modulation function
s(t) into the region of negative values.
Fig. 173. Amplitude modulation with completely suppressed carrier oscillations
The passage of the envelope through zero is accompanied by a typical jump in the initial
phase of oscillations v(t) by an angle π, which corresponds to a change in the sign of the
function V(s(t)). Without modulation (s(t) ≡ 0) we also have v(t) ≡ 0. It follows from Eq.
(8.29) that the spectrum of a signal modulated in this way contains only the two side
oscillations (see Fig. 173b); this is amplitude modulation with totally suppressed carrier.
When modulating by a non-sinusoidal signal s(t) with the spectrum bounded by the upper
limit frequency Ω(Fig. 173c), we obtain the two sidebands, again without the carrier
oscillations. However, each of these sidebands carries complete information as to the
waveform of the modulating signal s(t). For transmission efficiency (decreasing the necessary
bandwidth) either the lower or the upper sideband is removed (by frequency filtration or by
compensation in polyphase modulation systems), and only one sideband is transmitted; this is
amplitude modulation with a single sideband.
It is evident from Eq. (8.26) and from the derived equations (8.27) and (8.28) that, leaving
aside various multipliers which affect the phenomena quantitatively but not qualitatively, the
process of amplitude modulation consists in multiplying two time functions, namely the
cosine function cos(ω1t) characterizing the applied carrier oscillations, and the modulation
function s(t). Thus, we are solving an identical task as in the case of mixers, and we can in
fact solve it by the same means, i.e. by the transformation of spectra in non-linear or
parametric circuits, which in this case are called amplitude modulators. These modulators
differ from mixers, in principle, only by the manner of choosing selectively the spectral
components forming the output modulated signal.
Non-linear amplitude modulators. Diode modulators are in principle arranged in the
same way as diode mixers. The output circuit, however, is dimensioned such that it will pass
to the load the complete spectrum of an amplitude-modulated signal, i.e. both the carrier
oscillations and the two sidebands. A simplified diagram of a simple diode modulator is
shown in Fig. 174a. Acting in the circuit are the source of harmonic carrier oscillations v1(t)
of frequency ω1, and the source of the modulating signal v2(t) with limited spectrum. We
assume for the analysis that the signal v2(t) is harmonic and has a frequency ω2 « ω1. As
a result, the current in the circuit will be quasi-periodic with the frequency base { ω1, ω2},
and its spectrum will contain harmonic components with combination frequencies k1 ω1 + k2
ω2 > 0 (k1, k2 = ... , -1, 0, 1, 2, ...).
The first line of Fig. 174c gives the spectra of both acting voltages, and the second line the
spectrum of the current ia(t) excited by these voltages in the circuit of Fig. 174x. It can be
seen that in the neighbourhood of the carrier oscillations of frequency ω1 and of the two
desired side oscillations of frequencies ω1 + ω2 there are still some undesired components
with frequencies ω1 ± 2 ω2, ω1 ± 3 ω2, etc. Considerations similar to those used for mixers
lead to the conclusion that optimum properties from the viewpoint of suppressing these
undesired spectral components will be exhibited by a modulator with a quadratic ampere-volt
characteristic. The desired modulated oscillations consisting of the carrier oscillations and the
two sidebands will be obtained by means of a band-pass filter connected to the modulator
output (in a simple case, by applying, for example, a simple resonant circuit as indicated by
the dashed line in Fig. 174a).
To facilitate the filtration of the useful signal, diode modulators are often arranged in
symmetrical systems whose general properties and possibilities of structural synthesis were
discussed in Section 4.6. A "balanced" modulator, as shown in Fig. 174b, is a partially
symmetrized system from which it can be found, on the basis of Eq. (4.117), that in the
spectrum of the output current ib = ib1 - ib2 all the spectral components in the neighbourhood
of the even multiples of the frequency ω1 compensate one another (including the
neighbourhood of zero frequency), and only the spectral components from the neighbourhood
of the odd multiples of the frequency ω1 will remain. The spectrum of the current ib(t) is
illustrated graphically in the third line of Fig. 174c. If we add to this circuit another two
diodes of opposite polarity (D3 and D4 in Fig. 174d), we obtain a completely symmetrized
system, the "ring modulator". An analysis based on the idea advanced in Section 4.6 shows
that in addition to the spectral components given for the balanced modulator, in the ring
modulator all the components characterized by the combination frequencies mω1 ± nω2,
where m = 1, 3, 5, . . . and n = 0, 2, 4, . . . , are also mutually compensated (the spectrum is
illustrated in the fourth line of Fig. 174d). Thus, the carrier oscillations also disappear; after
selecting the desired components by means of a band-pass filter, we obtain an amplitudemodulated signal with the carrier oscillations completely suppressed.
Fig. 174. Diode modulators
Modulators with a three-terminal resistor have the same connection as the corresponding
mixer (Fig. 167), differing only in the dimensioning of the selective resonant circuit CL.
Analogously to mixers, various non-linearities of the three-terminal element Rn can be used
for the
non-linear transformation of signals in such modulators; here we work most frequently with
the input characteristic ii(vi) and the transfer characteristic io(vi).
An example of employing the input characteristic ii(vi) for modulation is supplied by the
transistor modulator, which is an analogy to the mixer shown in Fig. 168a. In this modulator,
carrier oscillations of frequency ω1 and a modulating signal of frequency ω2 « ω1 are acting
in the base circuit. Also, in this case, the model given in Fig. 168b can be used for
a simplified analysis.
A typical example of a modulator based on the utilization of the non-linearity of the transfer
characteristic of a valve is the grid modulator. Its simplified diagram is shown in Fig. 175a.
The voltage vG(t) consisting of the components v1(t) (carrier oscillations), v2(t) (modulating
signal), and VG (grid bias), acts on the grid of the valve. The waveform of the voltage vG(t)
produced by this superposition is plotted in the bottom left part of Fig. 175b. In the anode
circuit, a current iA(t) will flow, whose waveform will be determined by the shape of the
dynamic transfer characteristic iA(vG). A modulated output voltage v3(t) will originate across
the resonant circuit CL. The envelope V3(t) of this output voltage is somewhat distorted as
compared with the modulating signal v2(t).
Fig. 175. Grid modulator
The anode modulator operates on the basis of the non-linearity of the transfer characteristic
and simultaneously on the basis of the non-linearities of anode characteristics. If we compare
its diagram in Fig. 176 with the diagram of the class C amplifier from Fig. 121a, we can see
that the anode modulator differs only in that the supply anode voltage Vs has the ac
component v2(t) superimposed on it corresponding to the modulating signal. An example of
oscillation characteristics Ia1 (Vi; Vs) with the parameter Vs is given in Fig. 176b. It is obvious
that in the region of overexcitation the oscillation characteristics are spaced approximately
uniformly. From Fig. 176b, we can derive the modulation characteristics Ia1 (Vs; Vi) which are
plotted in Fig. 176c. It is evident that for V1 = 30 V, the modulation characteristic over the
whole range of the supply voltage under consideration is almost linear. The quiescent supply
voltage of the modulator must be chosen approximately in the middle of the linear portion of
the modulation characteristic, i.e. in the given case e.g. Vs = 300 V; the amplitude V2 must be
adapted in such a way that the instantaneous voltage Vs + v2(t) does not lie beyond the
boundary of this linear region (e.g. V2 = 200 V).
Fig. 176. Anode modulator
These modulators can also be combined to form symmetrized systems, the aim being to
separate by compensation the undesired spectral components [34].
Parametric amplitude modulators. Perhaps the simplest example of parametric
modulators can be seen in circuits with a periodically operating switching element (Fig. 177),
which are a complete analogy to the circuits in Fig. 143. For the circuit of Fig. 177a,
(8.30)
Fig. 177. The series and the parallel parametric modulators with a switching element
where, as in Eq. (7.18), s(t) is the switching function, s(t) = (1 + sign sin(ω1t))/2. Expressing
the switching function by the Fourier series and substituting for v2(t) = V2 sin(ω2t), we obtain
(8.31)
Discussion of this equation reveals that the spectrum of modulated rectangular oscillations
will contain components with frequencies (2k + 1) ω1 ± ω2. As well as electromechanical
switching elements, which can be used only for low frequencies (up to 400 Hz), these
modulators can be provided with various electronic switching elements such as those shown
in Figs 143c through e.
The transistor parametric amplitude modulator with a differential transistor unit has the same
configuration as the corresponding mixer (see Fig. 170). The carrier oscillations are used to
control the current source with transistor T3, and the modulating signal is applied to the
differential input of the circuit. Because of the circuit symmetry, the carrier oscillations in the
spectrum are suppressed.
Multiterminal resistors (tetrodes, pentodes, etc.) can also be employed for parametric
modulation, basically on the same principles as in the case of mixers.
Circuits for single sideband amplitude modulation deserve special attention. At first sight, it
may seem simple to suppress one of the sidebands in the modulated signal with suppressed
carrier oscillations and thus obtain the desired single sideband. In fact it is a complicated
problem because it is difficult, especially for higher frequencies, to build band-pass filters
with the necessary slope of the attenuation characteristic. Nevertheless, this method of
filtering one sideband is used but in the form of a multiple modulation system. An example of
such a system is given by the block diagram in Fig. 178a. After passing through the balanced
modulator BM1, the modulating signal s1(t) changes to a modulated signal s2(t) with both
sidebands but without the carrier oscillations, which have a comparatively low frequency ω1
(in telephony for instance f1 of up to 60 kHz). The band-pass filter BP1 will separate from the
signal s2(t) only one (say the upper) sideband which gives a signal s3(t). The spectral
components of this modulation as well as the attenuation characteristic of the filter BP 1 are
indicated by the first line of Fig. 178b. The whole process is then repeated in the next cascade
BM2, BP2. We can see from the second line in Fig. 178b that the two sidebands
corresponding to the signal s4(t) on the output of the balanced modulator BM2 are at a
considerable distance from each other, and the desired sideband (say the upper one) can be
separated by the comparatively simple band-pass filter BP2, on whose attenuation
characteristic no special demands are made. (Note: If we did not stick to tradition, we should
have to denote the system BM1,2, BP1,2, and O1,2 not as a modulation system but simply as
a frequency converter - see Fig. 164.)
Fig. 178. Modulators for amplitude modulation with a single sideband and completely suppressed carrier
oscillations ,
There is, however, yet another method which permits us to suppress the undesired sideband, a
method based on the application of polyphase symmetrical systems of non-linear or
parametric circuits. We have seen that in a completely symmetrized system built on the basis
of two-phase symmetrical systems (e.g. with a ring modulator) we can suppress the two
acting signals but that we still have the two sidebands to cope with. If we want to suppress
a further component, we must apply symmetrical systems with more than two phases. It can
be done with systems of only three phases [60], but in practice four-phase symmetrical
systems are mostly used, arranged as shown in Fig. 178c. Carrier oscillations of frequency ω1
are applied on the one hand as sine oscillations to a ring modulator RM1 (we know that sine
oscillations of opposite signs are acting here on diodes of opposite polarity), on the other
hand as cosine oscillations to a modulator RM2 (in which they act even with opposite
polarity). Similarly, a sine and a cosine modulating signal of frequency ω2 is applied to the
modulators RM1 and RM2, respectively. Thus the two systems are four-phase systems, their
components are mutually shifted by 90°, and they are given by four functions, namely sin,
cos, -sin, and -cos. It is easy to deduce that on the outputs of the modulators RM1 and RM2
we obtain the difference or the sum of two components with frequencies ω1 - ω2 and ω1 + ω2,
respectively. Adding (substracting) these components, we obtain a single harmonic (cosine)
component of the desired frequency ω1 - ω2 (or ω1 + ω2).
To operate also with non-sinusoidal (e.g. speech) modulating signals, special wide-band
phase-shift networks have been developed which give at the output two signals whose
corresponding spectral components are mutually shifted in phase by 90° with an error of
several degrees at the most.
8.2.3 Modulators for frequency modulation
Fundamenta1 considerations. A general relation for frequency-modulated harmonic
oscillations is obtained using Eq. (8.24) in the form
(8.32)
We substitute for the modulating signal s(t) a dimensionless function
dividing it by the maximum value Smax = max |s(t)|,
by
(8.33)
Here, Δ ω = kFMSmax is the frequency deviation which is the maximum frequency difference
caused by the modulating signal (for x = 1). Eq. (8.32) can now be rewritten in the form
(assuming φ1 = 0)
(8.34)
where a = Δ ω/ ω1 is the relative frequency deviation, also called the frequency modulation
index.
Frequency modulation is realized in non-linear or parametric circuits which are called
frequency modulators. The basic (even if not the only) method for obtaining frequencymodulated signals consists in affecting directly the frequency of harmonic oscillations in the
process of their generation. Thus in most cases the frequency modulator will be realized by
an oscillator (see Chapter 9) in which the parameters of the elements determining the
frequency are affected by a control quantity (the modulating signal) such that the dependence
of frequency on the control quantity is as linear as possible [12].
In oscillators with a resonant circuit, the frequency of generated oscillations is determined
especially by the resonant frequency
. For frequency modulation, we can then
employ either control of the capacitance C(x(t)) or control of the inductance L(x(t)). We shall
now examine the dependence C(x) corresponding to a linear dependence of the frequency ωr
on the control quantity x:
(8.35)
We calculate C(x) and expand it into the series
(8.36)
This relation shows that to obtain a linear modulation characteristic, the capacitance C(x)
determining the frequency of the oscillations generated by the oscillator must be controlled
non-linearly, in inverse proportion to the expression (1 + ax)2. Usually, we require linear
control, and in that case we must choose the relative frequency deviation a = Δω/ω1 « 1 so
that in the expansion on the right-hand side of Eq. (8.36) all the powers of more than the first
degree may be neglected. A similar requirement also follows for the modulation by means of
a controlled inductor with inductance L(x). The simplest possible realization of this kind of
frequency modulation is the LC oscillator to whose resonant circuit a capacitive microphone
is connected in parallel; its capacitance is controlled by the incident sound waves. We usually
employ for this purpose, however, either an autoparametrically operating non-linear storage
element, say a capacitive diode, or synthetically constructed storage elements, usually
controlled by voltage.
Non-sinusoidal oscillations can also be frequency-modulated. It follows from the
considerations in Section 9.3.1 that the frequency of rectangular oscillations generated by
a relaxation generator (shown in Fig. 207) is by Eq. (9.40) a function of both the currents I1
and I2 which charge and discharge the capacitor. If the circuit is dimensioned such that I1 = I2
= I/ 2, then on the basis of Eq. (9.40) we can express the dependence of the repetition
frequency of originating rectangular oscillations in the form
(8.37)
If the two current sources are controlled by modulating voltage, we obtain a linear
dependence of frequency on this voltage. Frequency modulators of this type have been
brought to considerable perfection; their linearity error can be less than 0.1%, and in
measurement techniques they are used as voltage frequency converters.
Fig. 179. A frequency modulator with a capacitive diode
Modulator with a capacitive diode. In Fig. 179a is a simplified diagram of the
frequency modulator with a capacitive diode CD, in which the CL oscillator is indicated only
blockwise. Through the resistor Rc with a large resistance Rc (the resistor Rc can be replaced
by an hf choke) a voltage vD(t) = V0 + v2(t) formed by the sum of the dc voltage V0 and the
modulating signal v2(t) (e.g. a harmonic voltage of frequency ω2) is applied to the diode CD.
By means of a coupling capacitor Cc which operates as a high-pass filter the capacitive diode
is connected in parallel with the resonant circuit CL of the oscillator. Owing to changes in the
modulating voltage v2(t), the differential capacitance Cd(t) of the capacitive diode changes
(see Fig. 179b) and so affects the frequency of the generated oscillations.
Modulators with controlled synthesized storage elements. Controlled capacitors
or inductors can also be realized synthetically by means of controlled sources of voltage or
current. Their simple variants are known as "reactance valves" and "reactance transistors".
We shall now give the general principles of such controlled synthesized storage elements.
Fig. 180a is a simplified diagram of a synthesized controlled capacitor with the current source
CS controlled on the one hand by the current ic(t), on the other by the modulating voltage
v2(t). The output current of the source CS is proportional to the product ic(t) v2(t). As we have
seen, the product of two time functions can be realized both in a non-linear circuit (with a
quadratic characteristic) and in a parametric circuit. In the case of the source CS, we are
concerned with a multiplier of two signals with current output. For the circuit it holds that
(8.38)
where the equivalent capacitance Ce of the synthesized capacitor is a linear function of the
modulation function, i.e. Ce(v2) = C(1 + Kav2). In the simplest case, the controlled source CS
can be realized by a single transistor (e.g. by an FET) or by a valve [12], [25]; in the optimum
case, the transfer characteristic of these controlled resistors should be quadratic.
Fig. 180. The basic principles of controlled synthesized storage elements
A different synthesized capacitor is connected as in Fig. 180b. The voltage source VS is
controlled by two voltages, an input hf voltage v1(t) and a modulating voltage v2(t). The
output voltage of the source is proportional to the product v1(t) v2(t). The situation in the
circuit can be described by the relation
(8.39)
It can be seen that in this case also the equivalent capacitance Ce(v2) = C(1 + Kbv2) is a linear
function of the modulating voltage v2. (Note: A capacitance transformed in this way was
found in valve amplifiers, and it is often called the Miller capacitance, after its discoverer.)
Controlled synthesized inductors can also be designed on similar principles [12], [25].
8.2.4 Modulators for phase modulation
Fundamental considerations. In this kind of modulation, the modulating signal s(t)
affects the initial phase of the carrier oscillations. Applying Eq. (8.25), we obtain for the
phase-modulated harmonic oscillations the relation
(8.40)
where the relative modulating signal x(t) = s(t)/Smax is of the same significance as in Eq.
(8.33) and where Δφ = kPMSmax is the phase deviation.
Modulators for phase modulation are basically phase-shift networks in which we alter the
phase shift of the output signal (the modulated signal) relative to the input signal by means of
parametric or autoparametric circuit elements controlled by the modulating signal. The block.
diagram of such a modulator is given in Fig. 181a.
Fig. 181. Phase modulators
Another kind of phase modulation is based on the application of pulse-position modulation
(see below). It is a modulation which is a direct analogy of the phase modulation of harmonic
oscillations and has the advantage that the respective modulators are comparatively simple. A
phase-modulated harmonic signal is separated from the spectrum of a position-modulated
pulse train by frequency filtration and, if need be, it is transferred into the higher frequency
region by multiplying the frequency (the phase deviation is also multiplied) or by mixing (the
phase deviation is preserved).
Phase modulator with a controlled two -port RC. A simple realization of the phase
modulator is arranged as in Fig. 181b. Basically, it is a two-port RC in which we control, by
the voltage v2(t), the capacitance C(v2(t)) of the capacitor C and thus change, at a constant
frequency ω1, the phase shift of the voltage v3(t) relative to v1(t). A considerable disadvantage
is the necessarily related amplitude modulation of the signal v3(t).
It is easy to derive for this simple element RC that at the frequency ω1 it will produce a phase
shift
(8.41)
An analysis of this expression reveals that the modulation will be linear only for small values
of φ, i.e. for ω1RC(v2) « 1.
Phase modulator with a detuned resonant circuit. The modulator is again
a comparatively simple two-port RCL (Fig. 181c) in which by detuning the resonant circuit
CL we control the phase shift of the modulated voltage v3(t) with respect to the input voltage
v1(t) of constant frequency ω1. In complex symbolic notation, the transfer of this two-port is
(8.42)
Since C = C(v2), the transfer factor K is also a function of the voltage v2 . Now we can
establish the phase shift
(8.43)
It is obvious that if the capacitance C is changed linearly by the modulating voltage v2; the
modulation characteristic φ(v2) will be linear only for small values of φ. The vector diagram
corresponding to this circuit is shown in Fig. 181d. It can be seen that in this case also the
phase modulation is accompanied by undesired amplitude modulation.
8.2.5 Modulators for pulse modulation
Modulators for pulse-amplitude modulation. In principle, all amplitude modulators
are suitable for this purpose, but they process carrier oscillations in the form of a periodic
train of short pulses. This pulse train has a rather broad spectrum, and this must be taken into
consideration in the design of the modulator.
A simple method for obtaining pulse-amplitude modulated signals is the application of a time
filter in which an electronic switch, opened for short periods, allows periodically taken
samples of the modulating signal to reach the output. Thus at the time filter output we obtain
the desired amplitude-modulated pulses.
Fig. 182. Pulse-width and pulse-position modulation
Modulators for pulse-width modulation (also called pulse-duration modulation or
pulse-length modulation). A simple modulator for obtaining a pulse-width modulated signal
is shown in block diagram form in Fig. 182a; the waveforms in this modulator are shown in
Fig. 182b. Added to the modulating voltage v2(t) (say a harmonic voltage of frequency ω2)
are triangular oscillations v1(t) with the repetition frequency ω1 equal to the repetition
frequency of the carrier pulse train, and their sum vi(t) is applied to the input of the two-level
clipper TLC. Here, the signal is clipped at both ends, and at the clipper output we obtain
a trapezoidal pulse-width modulated voltage v3(t). If the clipping levels are very close to each
other. the output signal will be very close to the rectangular pulse-width modulated signal
shown by the second line of Fig. 182b.
Modulators for pulse-position modulation (also called pulse-phase modulation). The
problem of pulse-position modulation is most frequently solved by means of a converter
which converts a pulse-width modulated signal to a pulse-position modulated signal. In
pulse-width modulation, the information content of the modulated signal is obviously hidden
in the position of the leading or the trailing edge of modulated
pulses. It is therefore sufficient to transfer information about the position of these edges. This
purpose is served by a converter arranged as in Fig. 182c, with the signal waveforms as in
Fig. 182d. The pulse-width modulated voltage v3(t) from the output of the PWM modulator is
applied to the differentiator D. At its output, we obtain an alternating train of positive and
negative pulses v4(t). The positive pulses are the response of the differentiator to the leading
edges of width-modulated pulses, the negative pulses are the response to their trailing edges.
Pulses of one polarity only are allowed to be transmitted, since they carry complete
information about the modulating signal. For this purpose, a rectifier R with non-inertial load
is inserted in the circuit, operating as a base clipper with zero voltage level of clipping. At its
output, we obtain the desired pulse-position modulated voltage v5(t).
8.3 DEMODULATORS
8.3.1 Demodulation of modulated signals
Demodulation is the opposite process to modulation. Its task is to recover from the modulated
signal the original modulating signal carrying information. The circuits in which
demodulation is realized are called demodulators. Depending on which type of the modulated
signal they are designed for, we distinguish demodulators for amplitude-, frequency-, phase-,
and pulse-modulated signals.
A number of principles already discussed are used for demodulation. We shall therefore
sometimes refer to circuits whose operation we have already discussed in other contexts.
However, it can generally be said that where the transformation of spectrum in non-linear or
parametric circuits was used for modulation, the same principles must be used for
demodulation.
8.3.2 Demodulators of amplitude-modulated signals
Fundamenta1 considerations. The demodulation of amplitude-modulated harmonic
oscillations is performed in non-linear or parametric circuits. The demodulators used will
depend on whether the modulated signal contains carrier oscillations and whether it has both
sidebands or only one sideband.
In the case of an amplitude-modulated signal consisting of carrier oscillations and both
sidebands, the modulating signal, as we know, is carried directly by the envelope of these
modulated oscillations. For demodulation, therefore, a simple circuit which follows this
envelope is sufficient, and, if necessary, separates the added constant component
(corresponding to the amplitude of unmodulated carrier oscillations). These demodulators are
also called detectors.
In a modulated signal with completely suppressed carrier oscillations, the modulating signal
is again carried by the envelope, but its waveform also extends into the negative region.
Luckily, it suffices to add, during demodulation, carrier oscillations to a signal so modulated,
and we have again the preceding case with the possibility of simple detection of the envelope
of modulated oscillations.
The single sideband amplitude-modulated signal (with completely suppressed carrier
oscillations) is in fact the result of transposing the spectrum of the modulating signal by the
carrier frequency. For demodulation we must again transpose the spectrum of the modulated
signal back to the original position (to the baseband) with the help of a frequency converter.
Rectifiers used as demodulators of amplitude -modulated osci11ations. One
very efficient and simple means of demodulating amplitude-modulated oscillations is the
rectifier. Any rectifier can be used for this purpose in which the dc component of the rectified
voltage is directly proportional to the amplitude of the input harmonic voltage. If we apply an
amplitude-modulated voltage to the rectifier input, the average value of the rectified voltage
will, tó a certain extent, correspond to the envelope of the amplitudes of the modulated
voltage and thus (with the exception of the added dc component) also to the waveform of the
modulating signal.
It is of advantage to use for this purpose a rectifier with inertial load CR, which was
discussed in Section 7.1.1. If it is used to detect the envelope of amplitude-modulated
oscillations, care must be taken that the time constant of the discharge of the storage capacitor
should be dimensioned suitably. The discharge must be rapid enough to follow the envelope
of modulated oscillations even in their most steeply decreasing portions. If this requirement is
not fulfilled, the demodulated signal is distorted [13], [34].
Distortion also occurs if the modulated voltage which is being processed is too small. In that
case, the curvature of the diode characteristic has an undesired effect (e.g. near zero the
semiconductor diode practically does not conduct at all) showing up in a distortion (here we
are concerned with quadratic detection [34].
To demodulate amplitude-modulated oscillations, a parametric circuit can also be used,
namely the synchronous rectifier. The rectifier must be provided at its output with a filter to
remove the undesired hf components accompanying the demodulated signal. The
demodulator based on the synchronous rectifier has, moreover, outstanding selective
properties (see Section 7.1.2).
Demodulation of a double sideband ampli tude-modulated signal with
suppressed carrier oscillations. We have already said that this type of modulated signal
can be adapted for demodulation based on detecting the envelope of amplitude-modulated
oscillations by adding to it, during demodulation, the carrier oscillations from the local
oscillator. The frequency of the added carrier oscillations must satisfy severe requirements.
For instance, for a comparatively simple transmission of a speech signal the frequency
deviation of the carrier oscillations must not exceed 5 Hz if unacceptable distortion is to be
avoided. At times, the modulated signal is therefore transmitted with partially suppressed
carrier oscillations which at the receiver are used as a reference signal to control the
frequency of the oscillator generating additional carrier oscillations.
Demodulation of single sideband amplitude -modulated osci11ations. Also in
this case, demodulation can be performed subsequent to adding a carrier frequency voltage
but with some limitation since, on being summed, two harmonic oscillations of frequencies
ω1 and ω1 + ω2 produce beats, and their envelope is in general not harmonic; thus by
rectifying the beats, we obtain a distorted signal. The distortion is maximum if the amplitudes
of the two oscillations are the same (this can be seen, for example, in Fig. 173a). To ensure an
acceptable level of distortion, the amplitude of the added carrier oscillations must be chosen
substantially larger than the amplitude of the modulated oscillations.
However, demodulation can also be realized in a circuit analogous to the one in which the
modulation was performed. If we apply to the modulator input a signal of frequency ω1 + ω2
(that is, the upper sideband), then together with the carrier oscillations of frequency ω1 we
obtain at the modulator output two components with frequencies ω1 + 2ω2 and ω2. The latter
obviously represents the original modulating signal. Thus, a ring modulator can also serve as
a demodulator.
8.3.3 Demodulators of frequency-modulated signals
Fundamenta1 considerations. In frequency modulation, the modulating signal is
converted to the corresponding frequency change. For demodulation, the demodulator must
be in the form of a frequency-to-voltage converter. In measurement techniques, such circuits
are known as analogue frequency meters. However, their principle is not very suitable for
processing signals from higher frequency bands. For this reason, we use indirect methods to
demodulate frequency-modulated signals. A frequency-modulated signal is first converted in
a modulation converter to an amplitude-modulated signal, and then it is demodulated by an
arbitrary demodulator for amplitude-modulated signals. Another method is based on
converting the frequency modulation to phase modulation, and the demodulation is then
realized by means of a phase-sensitive rectifier. Note that these demodulators are often
referred to as discriminators.
Demodulators employing the conversion of frequency modulation to
amplitude modulation. An elementary frequency-to-amplitude modulation converter is
the inductor. Its reactance XL = ωL changes in proportion to the changes in frequency. When
supplied with frequency-modulated current, the inductor develops an ac voltage whose
amplitude will be proportional to the frequency ω(s(t)). More usual is the case where
a resonant circuit is used whose resonant frequency is set such that the frequency of carrier
oscillations without modulation falls on the side of the resonance curve. Since the resonance
curve of a simple resonant circuit is curved laterally, the conversion of frequency modulation
to amplitude modulation is accompanied by distortion.
To remove this distortion, two resonant circuits (C1L1 and C2L2 in Fig. 183a) are used in the
amplitude discriminator. One of them is tuned to the resonant frequency
, and the other to the resonant frequency
; here,
ω1 is the frequency of the carrier oscillations. The resonance curves of these two resonant
circuits are shown by the dashed line in Fig. 183b. After demodulation by means of the
demodulators with the diodes D1 and D2, the two output signals (i.e. the voltages across R1
and R2) are subtracted from each other; the resultant characteristic v2(ω) is then of the shape
indicated by the heavy line in Fig. 183b.
Fig. 183. Amplitude discriminator
The approximately linear portion of this characteristic in the neighbourhood of the frequency
of carrier oscillations is amenable to demodulation.
Demodulators using the conversion of frequency modu 1ation to phase
modu1ation. In the converter of frequency modulation to phase modulation we make use of
the frequency dependence of the argument of the complex transfer factor of coupled resonant
circuits. The demodulator, usually referred to as the phase discriminator, is arranged as in
Fig. 184.
Fig. 184. Phase discriminator
When the resonant circuit C1L1 tuned to the frequency ω1 of the carrier oscillations is
supplied with a frequency-modulated current, a voltage v1(t) originates which is
simultaneously transferred through the capacitors C4 and C5 to the inductor (choke) L3. The
primary resonant circuit C1L1 is inductively coupled with the resonant circuit C2L2, which is
also tuned to the frequency ω1. It is known from the theory of linear circuits that at resonance
there appears across the circuit C2L2 the voltage v2(t) which is phase-shifted against the
voltage v1(t) by exactly the angle φ = 90°. If the frequency of the modulated current i1(t)
changes, then the angle φ changes with it. Acting on the rectifier with the diode D1 is thus the
sum v1 + v2/2 of two phase-shifted voltages, while their difference is acting on the rectifier
with the diode D2. We have obtained the phase-sensitive rectifier whose properties were
discussed in Section 7.1.1. It can be seen from Fig. 142 that the characteristic v3(φ) of this
rectifier is in the neighbourhood of the phase shift by φ = 90°, suitable for the demodulation
of phase-modulated oscillations.
8.3.4 Demodulators of phase-modulated signals
For the demodulation of phase-modulated oscillations it is possible to employ the phasesensitive rectifier discussed in the preceding paragraph. However, in addition to the phasemodulated voltage, there must be un-modulated carrier oscillations with an initial phase of
90°. It is not easy to generate such oscillations at the receiver.
8.3.5 Demodulators of pulse-modulated signals
Signals with pulse-amplitude or pulse-width modulation are very easy to demodulate. In
addition to a number of other components, their spectrum contains also the baseband
spectrum of the original modulating signal. A frequency low-pass filter is therefore sufficient
to separate the useful signal. This method, however, has a drawback in that the power of the
original baseband spectrum in the spectrum of the modulated signal is very small. It is
therefore of advantage in pulse-amplitude modulation to use the approximate restoration of
the original modulating signal from the samples by a simple interpolation of zero order with
the help of a pulse-stretcher. The latter contains a memory element (a capacitor) which is
charged to the peak value of the amplitude-modulated pulse and remains so charged until the
arrival of the next pulse which charges (or discharges) it to its peak value, etc. The simplest
(but imperfect) stretcher is the common diode rectifier with a storage capacitor which is used
for the demodulation of amplitude-modulated signals. Pulse stretchers of better performance
use synchronous pulse-controlled electronic switches of analogue signals which ensure
a proper charging or discharging of the storage capacitor at prescribed instants (here we have
to do with sample-and-hold circuits).
In pulse-width modulation, a converter is used which converts the width-modulated
rectangular pulse to an amplitude-modulated pulse (usually applying the linear charging of
the capacitor for the duration of the width-modulated pulse). This amplitude-modulated
signal is then demodulated with the help of a demodulator with pulse-stretcher.
The pulse-position modulated signal is first converted to a pulse-width modulated signal, and
this in turn is demodulated by the method given above. The conversion of a positionmodulated pulse to a width-modulated pulse can be performed, for example, in such a manner
that using a position-modulated pulse we open a switch supplying current to the load, and
then we close it again by a periodically repeated pulse derived from a synchronizing
unmodulated pulse train. At the output we thus obtain width-modulated pulses with a varying
leading edge position and a periodically repeated trailing edge position.
Generation of oscillations
9.1 GENERAL CHARACTERISTICS OF GENERATORS OF ELECTRICAL
OSCILLATIONS
An electrical circuit generating periodic undamped electrical oscillations without any outside
periodic excitation is called a generator. It is an autonomous circuit which, from the energy
viewpoint, represents a converter of dc voltage to ac voltage.
Depending on the waveform of generated oscillations (of voltage or current), generators are
classified into two basic groups: a) generators of harmonic oscillations, generally called
oscillators; b) generators of shaped oscillations (relaxation oscillations), for instance,
sawtooth, triangular, or rectangular oscillations or pulses.
Periodic oscillations are excited in circuits containing one or more storage elements serving
as energy storage. For undamped oscillations to be excited in such a circuit, electrical energy
must be supplied in such amounts and at such times that the oscillations may be sustained
permanently. Thus between the energy source and the circuit with storage elements a device
must be connected which automatically controls the energy supply at a required pace to the
circuit where energy oscillation occurs. The energy supply must, of course, also provide the
energy supplied to the connected load.
Fig. 185. The basic component parts of oscillators
Oscillators consist of three basic parts (Fig. 185): a source of dc voltage (or current),
a resonant circuit with storage elements and with a connected load, and an automatic
regulator of the delivered quantity of energy controlled by a suitable feedback [11]. The
function of the automatic regulator is usually performed by a controlled (three-terminal) nonlinear resistor (transistor, electron valve) or a non-controlled nonlinear resistor with an Ntype or S-type ampere-volt characteristic. In controlled non-linear elements, the feedback
controlling their performance is apparent (shown by the solid line in Fig. 185); in the case of
non-controlled elements with the N-type or the S-type characteristic this feedback is nonapparent (shown by the dashed line in Fig. 185).
9.2 GENERATORS OF HARMONIC OSCILLATIONS - OSCILLATORS
Depending on the elements used in the resonant circuit, there are two qualitatively different
classes of generators of harmonic oscillations: LC oscillators and RC oscillators. The two
classes will be treated separately.
9.2.1 LC oscillators with negative resistance
Both the physical basis and the analysis of the operation of LC oscillators can be explained
clearly and comparatively simply by the example of oscillators with a non-linear resistor
whose ampere-volt characteristic is of the N- or S-type, which are known as oscillators with
negative resistance.
Let there be a simple resonant circuit as shown in Fig. 186a. If a certain amount of electrical
energy is applied to the circuit (e.g. in such a way that by switching over the switch S for a
short time, the capacitor C is charged to a voltage V0), free oscillations appear in the circuit.
For the resonant circuit under consideration the following differential equation holds:
(9.1)
with the solution
(9.2)
where I0 is the initial amplitude of current i in the circuit depending on the magnitude of the
energy supplied to the circuit, α = R/2L is the circuit damping factor,
is the
resonant frequency of the circuit, and
is the natural frequency of free
oscillations in the circuit.
Fig. 186. Free oscillations in a simple resonant circuit
Depending on whether the damping factor α is positive, negative, or zero, the free oscillations
in the circuit can be of three possible types, as indicated in Figs 186b (increasing oscillations
for α < 0), 186c (damped oscillations for α > 0), and 186d (undamped oscillations with a
constant amplitude when α = 0). A similar (dual) consideration also holds for the parallel
resonant circuit..
In an actual resonant circuit there are always losses, and, moreover, the circuit is required to
deliver permanently a certain average power to the load. If the oscillator is to generate
undamped oscillations, the damping factor of the applied resonant circuit must be zero. This
can be achieved by connecting to the resonant circuit LCR a resistor with a descending
portion on the ampere-volt characteristic, i.e. a resistor which has negative differential
resistance in a certain region of its characteristic.
Oscillator with a parallel resonant circuit. Connect to a parallel resonant circuit
CLRr of Fig. 187a a resistor Rn characterized by its equivalent negative conductance
Gn(V1) < 0, which is a function of the amplitude V1 of the first harmonic component of the
voltage v (Fig. 187b). Denoting the resonant conductance of the circuit by Gr = 1/Rr, the total
equivalent conductance in the oscillator circuit will be
.
(9.3)
This conductance is also a function of the amplitude V1, and in Fig. 187b it is shown by
a heavy line. In the sense of the above considerations, a steady harmonic voltage appears in
the circuit, with the amplitude V1s at which the condition Gt(V1s) = 0 is fulfilled.
If we do not want the oscillations in the oscillator to increase beyond all bounds, damping of
oscillations must be provided for the voltage amplitude
, which means that the total
equivalent conductance in the circuit must be positive,
. The shape of Gt(V1) given
in Fig. 187b meets this condition. It is easy to verify (e.g. by derivation from the ampere-volt
characteristic) that it is a shape typical of a resistor with an N-type ampere-volt characteristic.
A resistor with an S-type characteristic cannot be used for an oscillator with a parallel
resonant circuit.
Fig. 187. An oscillator with a parallel resonant circuit
Dynamic stabilit y of the oscillator. If the quiescent operating point of the applied
resistor with an N-type characteristic is chosen differently, we can obtain a somewhat
different shape of the dependence Gn(V1), as shown in Fig. 187c. The dependence Gt(V1),
derived on the basis of Eq. (9.3), has then a shape satisfying the condition Gt(V1) = 0 at two
points characterized by the amplitudes V1s, and V1u. It would seem that in this circuit there
can appear steady oscillations either with the amplitude V1s or with the amplitude V1u. In fact,
however, only oscillations with the amplitude V1s are sustained permanently in the oscillator.
To show why this is the case, we will introduce the concept of dynamic stability of
oscillations. Oscillations will be denoted as stable when there is a tendency for the circuit to
maintain a steady (constant) amplitude of oscillations. Assume that owing to some external
cause the amplitude of oscillations changes. If subsequent to the removal of the disturbing
action the system returns to the original amplitude, the system is dynamically stable in the
mode considered. If, however, the amplitude of oscillations changes spontaneously in the
same sense even when the external disturbance has ceased, the system is said to be
dynamically unstable. Let us apply these considerations to the case indicated in Fig. 187c.
Any reduction in the amplitude V1 below the value V1u results in a situation where Gt(V1) > 0,
i.e. the oscillations are decaying. Conversely, an increase in the amplitude V1 above the value
V1u results in an increase in the oscillations (due to Gt(V1) < 0) until they become steady at the
amplitude V1s. With the amplitude V1 decreasing below the value V1s we have Gt(V1) > 0 and,
consequently, the oscillations will increase until their amplitude becomes steady at V1s. On
the contrary, with the amplitude increasing above the value V1s we have Gt(V1) > 0, and the
oscillations are damped gradually until they again reach the steady amplitude V1s. Thus in the
case of oscillations with amplitude V1u the oscillator under consideration is dynamically
unstable, while oscillations with amplitude V1s are steady, and the oscillator operates in
a dynamically stable mode.
Soft and hard self-excitation of the oscillator. On connecting an oscillator
characterized by the curves in Fig. 187b the oscillator will start oscillating spontaneously at
the amplitude V1s without any external causes. This is because for all the voltages with the
amplitude
the total equivalent conductance in the resonant circuit is negative,
and even a negligible cause, such as thermal noise in the circuit, is sufficient to set the
oscillator into oscillation. In this respect we speak of the soft self-excitation of the oscillator.
In contrast to this, an oscillator with the properties characterized in Fig. 187c must be given a
certain initiating impulse that will make it oscillate at the amplitude V1 > V1u, since
subsequent to this the voltage amplitude increases spontaneously up to the value V1s
corresponding to steady oscillation. Thus in this case we are concerned with an oscillator in
hard self-excitation. Note that an oscillator with soft self-excitation can be designed even
with a resistor whose characteristic Gn(V1) is as given in Fig. 187c. It is sufficient to employ
a resonant circuit with a smaller conductance , as indicated by the dashed line in Fig. 187c.
The curve of the total initial conductance
is then below the curve Gt(V1) such that its
initial value is
Fig. 188. An oscillator with a series resonant circuit
Oscillator with a series resonant circuit. Dual to the preceding oscillator is the
oscillator with a series resonant circuit CLRs (Fig. 188a) with a connected resistor Rn
characterized by its equivalent resistance Rn(I1) < 0 which is a function of the amplitude I1 of
the first harmonic component of the current i flowing through the circuit. The total equivalent
resistance in the circuit
(9.4)
is a function of the amplitude I1; in Fig. 188b this dependence is shown by the heavy solid
curve. Originating in the resonant circuit will be steady harmonic oscillations of current with
the amplitude I1s satisfying the condition Rt(I1s) = 0. Analogously to the preceding case, the
oscillations will be damped for any
, and thus the plot of the dependence Rt(I1) must
be as given in Fig. 188b. This corresponds to a non-linear resistor with an S-type
characteristic. The N-type characteristic is not suitable for an oscillator with a series resonant
circuit.
By analogy, all the considerations on dynamic stability and on soft and hard self-excitation
(see also Fig. 188c) also hold for the oscillator with a series resonant circuit.
Note that, basically, it makes no difference by what mechanism the N-type or S-type
characteristics are obtained, whether we are concerned here with a direct application of the
laws of motion of charge carriers in electronic circuit elements (the tunnel effect, the
dynatron effect, effects in multilayer semiconductor structures), or whether we have to do
with the application of more or less complicated positive feedbacks in amplifiers. This will be
seen in the following treatment of feedback oscillators.
9.2.2 Feedback LC oscillators
For different types of LC oscillators with three-terminal controlled resistors and external
feedback, the general principles of their theory, the physical basis of their operation, and the
differential equations and methods of their solution are similar. It is therefore of advantage to
study in the first place the properties of a simple feedback LC oscillator and only then to
apply the knowledge gained to other types of this oscillator. In our case we shall analyse the
properties of the oscillator with transformer feedback.
When investigating an oscillator, we are interested in these questions: a) Is the oscillator able
to get self-excited from quiescent state? b) Which is the steady amplitude of its oscillations?
c) How do oscillations increase with time in the oscillator? d) What stability or what accuracy
does the frequency of the generated oscillations exhibit? These are the problems that we shall
briefly deal with.
Condition of spontaneous self -excitation of the oscil lator. Let there be an
oscillator as shown in Fig. 189. The three basic parts as discussed in connection with the
block diagram in Fig. 185 are easy to distinguish. A source of dc supply voltage VS delivers
energy to a parallel resonant circuit CLR through a regulator formed by a transistor T (a fieldeffect transistor - MOSFET) and a transformer made up of coils L1 and L with mutual
inductance M. An external feedback (FB) delivers voltage v from the resonant circuit directly
to the gate of the transistor.
Fig. 189. The basic configuration of an LC oscillator with transformer feedback
The differential equation valid for this circuit is
(9.5)
The expression on the right-hand side of the equation gives the voltage induced in the
resonant circuit by the current iD flowing through the coupling coil L1. In the transistor T,
however, the current iD is controlled by the gate-source voltage v, and the differential relation
holding for it is diD = S(v) dv, where S(v) is the slope of the transfer characteristic of the
transistor at the operating point given by the voltage v. The current i can be expressed by the
relation i = C dv/dt. Taking into account these circumstances, Eq. (9.5) can be expressed in
the form
and after rewriting,
(9.6)
where
We shall now examine whether oscillations, if any, can originate in this circuit without an
external impulse. Since at the start of oscillations the originating voltage will be very small, v
≈ 0, it is sufficient for our purpose to consider the constant slope at the quiescent operating
point, S0 = S(0). In this way the non-linear differential equation (9.6) changes to the familiar
differential equation of the type
(9.7)
with the solution
,
where
. This equation is a complete analogy to Eq. (9.2). The damping factor
of oscillations in the circuit,
(9.8)
has in this case two components. The first, R/2L, is the same as in the resonant circuit without
applied feedback; the other,
, expresses the effect of feedback on the damping of
oscillations in the circuit.
Spontaneously increasing oscillations will originate in the oscillator (owing to an arbitrarily
small impulse such as noise voltage) when the damping factor is α < 0. From this, the
condition for the soft self-excitation of the oscillator from quiescent condition can be
formulated e.g. in the form
(9.9)
(since the quality factor of the resonant circuit Q = ω0L/R). It can be seen that an oscillator
with a given resonant circuit and a given transistor is set to soft self-excitation if a sufficiently
strong positive feedback is introduced in the circuit by choosing a sufficiently large mutual
inductance M between the coils L1 and L.
The condition of α < 0 for the self-excitation of an oscillator with transformer feedback can
also be interpreted in another way. Let us write the damping factor as the ratio of the total
resistance Rt in the circuit to twice the inductance L, i.e. α = Rt/2L. From Eq. (9.8) we
determine
It can be seen that the feedback introduces into the resonant circuit a negative resistance
whose magnitude can be affected by the choice of mutual inductance M. The oscillator will
be excited to self-oscillations if the total resistance in the circuit is Rt < 0, i.e. R < - Rn, or
(9.10)
Steady-state oscillations in oscillator. To examine the amplitude of the oscillations
generated by the feedback LC oscillator of Fig. 189, the method of equivalent linearization
can be used with advantage. We shall assume that when exciting the circuit CLR by feedback
introduced through the coupling transformer L1, L it is only the fundamental harmonic
component of the current iD with amplitude ID1 that will, owing to the selective properties of
the circuit, play a significant role. Under this assumption, all the quantities in the oscillator
circuit are harmonic, and a complex symbolic method can be used to describe the phenomena
occurring in it. For the resonant circuit of the oscillator we thus obtain the relation
(9.11)
Introducing into this equation the complex amplitude V of the gate-source voltage (which is
coupled with the complex amplitude I by the relation I = jωCV) and multiplying
simultaneously its two sides by the expression
, we obtain
(9.12)
Assume for simplicity that the transistor T is a purely resistive element, i.e. ID1 and V are in
phase. Its equivalent slope
(9.13)
is then a real quantity, and it is a function of the .amplitude of the voltage V. Eq. (9.12) can
now be rewritten in the form
.
(9.14)
We have obtained a complex quadratic equation for the frequency ω. By solving it, we find
that the frequency of steady-state oscillations of the oscillator
(9.15)
is conditional on satisfying the condition (R - MSe(V)/C) = 0, which can with great advantage
be formulated in the form
(9.16)
Solving the non-linear equation (9.16), we can find the amplitude of steady-state oscillations
of the oscillator. To do this, we must first derive the dependence Se(V) of the given transistor,
which we shall do by using Eq. (9.13), in which we shall find ID1 by some of the methods of
spectral analysis. In doing so, we can obtain two typical qualitatively different shapes of
Se(V);.which are shown in Figs 190a, b; they depend on the shape of the transistor transfer
characteristic and on the choice of the quiescent operating point on this characteristic.
Fig. 190. Illustrating the derivation of the amplitude of steady-state oscillations in a feedback oscillator
Eq. (9.16) will be solved graphically. Plotted in the graph in Fig. 190a will be both the lefthand side of the equation, i.e. the function Se(V), and the right-hand side of the equation,
RC/M = const., which does not depend on V and is thus represented by a straight line running
parallel with the V axis. Its distance from the V axis is inversely proportional to the
magnitude of the feedback (as expressed by the mutual inductance M), hence it will be
referred to as the feedback line. If the feedback line is above the curve Se(V), as shown by the
dashed line in Fig. 190a, Eq. (9.16) is not satisfied, and no oscillations will originate.
Increasing the feedback (by increasing the mutual inductance from the value M' to M) will
lower the feedback line, which will intersect the curve Se(V) at point A, the abscissa of which
gives the amplitude Vs of steady-state oscillations in the feedback oscillator. A further
increase in the feedback will move the feedback line still lower, and point A will be shifted to
the right; the amplitude of oscillations will increase. Conversely, reducing the feedback will
move point A to the left until it reaches the ordinate axis. The amplitude of oscillations will
be reduced to zero, and the equivalent slope will coincide with the differential slope at the
quiescent operating point, Se(0) = S0. A feedback oscillator with the characteristic Se(V) as
shown in Fig. 190a will always exhibit soft self-excitation of oscillations.
The case of an oscillator with the Se(V) characteristic as indicated in Fig. 190b is different. So
long as condition (9.10) is satisfied, i.e. RC/M < S0, the oscillations in this oscillator are also
the result of soft self-excitation. However, this oscillator can also oscillate at RC/M > S0, as
indicated in Fig. 190b by the solid feedback line intersecting the curve Se(V) at two points, A
and B. To evaluate the dynamic stability of oscillations at these points, let us examine the
case when Eq. (9.16) is not satisfied. If Se > RC/M, the feedback system delivers energy to the
resonant circuit in excess of the energy lost in it; if Se < RC/M, then the dissipation of energy
in the circuit is greater than its supply. At point A, the system is dynamically unstable, since
any reduction of the amplitude V below the value Vu results in the oscillations decaying,
while increasing the amplitude above the value Vu is accompanied by further amplitude
increase. At point B, the system is dynamically stable; the oscillator will generate steady-state
oscillations with amplitude Vs. Such an oscillator exhibits hard self-excitation. To obtain soft
self-excitation, it suffices to increase the feedback (the dashed feedback line in Fig. 190b,
with the point of intersection C whose amplitude of steady-state oscillations is V´s).
Module and argument (phase) conditions of excitation of steady -state
osci11ations. The oscillator under discussion can be viewed also from another angle. If in
the circuit of Fig. 189 the connection to the gate of the transistor T is interrupted, we obtain
a selective amplifier with a transistor whose collector circuit includes a transformer-coupled
resonant circuit CLR. In this case, the whole output voltage of this amplifier is returned back
to its input on applying the feedback (100 % positive feedback). Generally, however, a linear
feedback network is inserted between the output of the amplifier and its input. Thus we get
a general feedback system as shown in Fig. 191, consisting of an amplifier with the
equivalent amplification factor A(V1, ω) = V2/V1 (which is a function of the amplitude of
input voltage V1 and the frequency ω) and also of a feedback network, whose transfer factor
K(ω) = V3/V2 is a function of the frequency ω but does not depend on the amplitude of the
given voltage. The total transfer factor of the cascade is given by the relation
(9.17)
The system is assumed to be selective, so that at the frequency of the generated oscillations
the output voltage of the cascade is harmonic. It is only on this assumption that the complex
symbolic notation of the circuit quantities can be used.
Fig. 191. Oscillator as an amplifier with a feedback network
On applying the feedback FB from the output to the input of the cascade as indicated by the
dashed, line in Fig. 191, it is obvious that V3 = V1. Steady-state oscillations with amplitude
V1s and frequency ω1 will originate in the circuit and by Eq. (9.17), the following condition
applies:
(9.18)
Expressing the complex quantities in Eq. (9.18) in exponential terms as
, the above condition can be decomposed into two, namely
and
(9.19)
(9.20)
The former is the module condition which means that the total transfer factor of an oscillator
generating steady-state oscillations with amplitude V1s always equals one. Since the
frequency ω1 is usually known (it is equal or very close to the natural frequency
of the employed resonant circuit), the amplitude V1s of the generated oscillations can
be determined by solving the non-linear equation (9.19).
Eq. (9.20) is the argument (phase) condition from which it follows that the total phase shift in
the feedback loop must be either zero or an integer multiple of 2π. Thus if the amplifier shifts
the phase by π (inverting amplifier), the feedback network must shift the phase either by π or
by - π to satisfy condition (9.20).
The time dependence of increasing oscillations in the osci11ator. When
switching on the-oscillator, a transient process originates in the course of which the
oscillations increase and, after some time, achieve stable amplitude and frequency. We shall
first demonstrate the principles of the increase of oscillations in a feedback oscillator while
solving the non-linear differential equation for oscillator in the phase plane. Later on, this
knowledge will be extended by conclusions obtained by the method of slowly changing
amplitudes.
Let us again consider the oscillator of Fig. 189. For this oscillator, we have written the
differential equation
(9.21)
Approximate the non-linear transfer characteristic by an incomplete power polynomial of the
third degree,
where a1 > 0, a3 < 0, from which we determine the slope
(9.22)
After substituting for S(v) in Eq. (9.21), we obtain the non-linear equation for the oscillator in
the form
(9.23)
where
is the initial damping factor of the resonant circuit respecting
the effect of feedback, and
To facilitate further calculations, it is convenient to introduce normalized time τ = ω0t. Then
Substituting these expressions into Eq. (9.23) and dividing the whole equation by the factor
, we obtain
This non-linear second-order differential equation can be solved in the phase plane by the
method of isoclines. A comparison with Eq. (5.10) reveals that in the given case
We substitute these expressions into Eq. (5.13), and the resulting equation of isoclines is in
the form
(9.24)
From the known quantities α0, β, ω0 we calculate a number of isoclines with suitably chosen
parameters k. The isoclines are plotted into the phase
plane, and in the field of isoclines we draw the phase trajectory from the given initial
conditions
(9.25)
Fig. 192. An example of the phase trajectory of an oscillator
Fig. 192 shows an example of the phase trajectory of an oscillator for the case α0 / ω0 = 0.15 and β/ω0 = 0.13. The family of isoclines has been constructed for k = 4, 2, 1, 0.5, 0.3, 0.1, - 0.5, -1, - 2, - 4. Starting from point A, which corresponds to the initial conditions, the
phase trajectory is constructed in the familiar way. For the given oscillator, the phase
trajectory is in the shape of a spiral developing from point A and changing gradually into
a closed curve, the stable limit cycle. It is easy to find also that trajectories starting from any
point outside the limit cycle curve gradually approximate this curve until they coincide with
it. The intersection of the limit cycle curve with the axis a corresponds to the amplitude Vs of
steady-state oscillations. It can be seen from the course of the-trajectory that we are
concerned here with a system with soft self-excitation.
An oscillator with hard self-excitation has two limit cycles, a stable cycle and an unstable
cycle. This case is illustrated in Fig. 193. For small oscillation amplitudes, the damping of
this oscillator is positive, and the phase trajectories convolve spirally towards the origin of
co-ordinates. The region of positive damping is enclosed by the limit cycle, i.e. by a closed
curve (the dashed line in Fig. 193) corresponding to the zero damping of oscillations in the
circuit. This limit cycle, however, is unstable. The phase trajectories do not end on this cycle,
they originate here; they either convolve towards the origin of co-ordinates or they develop
outside this limit cycle. Outside the unstable limit cycle is the region of negative damping,
and there the phase trajectory of oscillations develops until it reaches the next limit cycle with
which it gradually coincides. This limit cycle is stable. Again, outside this stable cycle is the
region with positive damping in which the phase trajectory of decaying oscillations
approximates the stable limit cycle, ultimately coinciding with it.
Fig. 193. The phase portrait of an oscillator with hard self-excitation
To find out more details of the waveforms of oscillations increasing with time in an
oscillator, we shall use in the solution of this process the method of slowly changing
amplitudes. The principles of this method were dealt with in Section 5.2.3. As in the
preceding case, consider an oscillator whose properties are described by the non-linear
differential equation (9.23):
(9.26)
With regard to the selective properties of the resonant circuit of the oscillator, we assume the
solution in the form
(9.27)
with V(t) being the waveform of the envelope of originating oscillations. Applying the same
reasoning as that in Section 5.2.3, i.e. leaving out all the components of any but the
fundamental frequency and neglecting all the components that are not of decisive significance
in the solution, we obtain two shortened non-linear differential equations
(9.28)
These shortened equations provide on the one hand information on steady-state oscillations in
the oscillator, and on the other hand information on the process of developing oscillations.
The steady-state oscillations have an amplitude V(t) = Vs, hence dV/dt = 0. The first of the
two equations gives us immediately the frequency of steady-state oscillations
(9.29)
From the second equation we determine their amplitude
(9.30)
The waveform of the envelope of increasing oscillations is found from the second of Eqs
(9.28). Multiplying this equation by the quantity V and taking into account that 2VdV = d(V2),
we obtain
We separate the variables:
(9.31)
and we ascertain the dependence V(t) by integration. The left-hand side of Eq. (9.31) yields
the integral
so that integrating Eq. (9.31) gives
(9.32)
We take the antilogarithm, and after rewriting we obtain
With regard to Eq. (9.30) we can now calculate
(9.33)
where
is a constant derived from the integration constant K in Eq. (9.32)
and given by the initial conditions of oscillator operation.
The settling process of oscillations (i.e. the waveform of the envelopes of originating
oscillations) for different initial conditions characterized by the constant D is illustrated in
Fig. 194 in normalized representation V/Vs = f(-2α0t). Note, however, that the given curves
hold only for an oscillator whose non-linear element has the transfer characteristic given by
a third-degree polynomial. The course of this transfer characteristic is usually even more
complicated, so that the curves V(t) can be of a slightly different shape; in spite of this,
however, their nature remains roughly the same.
Fig. 194. The settling process of oscillator oscillations for different initial conditions
9.2.3 Basic types of LC oscillators
LC oscillators can be classified in several ways. We have already distinguished two basic
oscillator classes, namely oscillators with negative differential resistance and feedback
oscillators. Other important criteria include the choice of the electronic non-linear element
and the method of energizing this element, and in the case of feedback oscillators the
configuration of the feedback network. These problems will now be considered.
Kinds of feedback LC osci1lator. In feedback oscillators, the same three-terminal
electronic elements are used as in amplifiers. Thus we have oscillators with a bipolar
transistor, with a field-effect transistor (JFET or MOSFET), and with a valve. A threeterminal non-linear resistor in combination with the respective supply sources forms an
amplifier which is connected to a feedback two-port always at three points. In Fig. 195a is an
oscillator with an inverting amplifier (e.g. a transistor in common-emitter connection, a valve
in common-cathode connection), so that the feedback network must also invert the generated
oscillations to satisfy in the circuit the argument (phase) condition (9.20).
Fig. 195. The three-point connection of the amplifier and the feedback two-port FB in an oscillator
Leaving aside, for the time being, the problem of connecting the supply sources, we can
simplify the circuit of Fig. 195b and deal in the following with only the way in which the
three-terminal (controlled) resistor TR and the feedback two-port FB are connected. In place
of the TR we can use any of the electronic elements given in Figs 196a through d; we shall
use as the feedback two-port FB (with the input 2 - 3 and the output 1- 3) one of the circuits
in Figs 197a through h.
Fig. 196. Electronic realizations of the three-terminal resistor in feedback oscillators
We are already familiar with the oscillator with a feedback network as in Fig. 197a. It is the
oscillator with transformer coupling which was shown in Fig. 189 in connection with the
analysis of the feedback oscillator. The resonant circuit LC is connected to the amplifier
input, and the coupling coil L1 is energized from its output. The frequency of generated
oscillations is given roughly by the resonant frequency
. The magnitude of the
feedback is determined by the value of mutual inductance M between the coils L and L1.
Another oscillator with transformer coupling is obtained when using a feedback network as
shown in Fig. 197b. It differs from the preceding one in that the resonant circuit LC is
connected to the amplifier output, and the coupling coil L1 is connected to its input. The other
features are the same as in the preceding case.
In the oscillator with autotransformer coupling, the feedback two-port is as shown in Fig.
197c. The resonant circuit LC has the resonant frequency
, and it is
approximately at this frequency that the originating oscillations are established. The coil L
with a tap represents an autotransformer, and the magnitude of feedback will be given by the
transfer factor of this autotransformer which in turn is given by the ratio of the number of
turns p = N31/N23, where N31 is the number of turns between the terminals 3 - 1, and N23 the
number of turns between the terminals 2 - 3. This is the Hartley oscillator, named after its
originator.
Fig. 197. Feedback two-ports used in feedback oscillators
Another type is the Colpitts oscillator, which operates in conjunction with a capacitor divider
(Fig. 107d). The oscillator oscillates at approximately the resonant frequency
,
where Cs is the capacitance of the series combination of capacitors C1 and C2, i.e. Cs =
1/(1/C1 + 1/C2). The total voltage across the resonant circuit is divided by the chain of
capacitors C1 and C2 in such a way that the voltages across the capacitors will be inversely
proportional to their capacitances. Thus the transfer factor of the feedback network and,
consequently, also the magnitude of the feedback, are given.
The oscillator with inductor divider operates on a similar principle. Although its feedback
network as indicated in Fig. 197e resembles the circuit of Fig. 197c, it differs qualitatively in
that the inductors L1 and L2 are not mutually coupled by common magnetic flux, i.e. M = 0.
The resonant frequency determining the frequency of the generated oscillations is
. The total voltage across the resonant circuit is divided in such a way that the
voltages across the inductors L1 and L2 are proportional to their inductances L1 and L2,
respectively. This gives the magnitude of the feedback.
In oscillators operating at higher frequencies, the input, output, and transfer capacitances of
the amplifier are manifested. Thus the preceding oscillator type changes to an oscillator with
a more complex structure of the feedback two-port, as illustrated in Fig. 197f. Since this is a
reactance network with two inductors and three capacitors, it will exhibit several resonant
frequencies. We are interested only in the resonant frequency for which both the resonant
circuit C1L1 and the resonant circuit C2L2 are of inductive nature, i.e.
,
. In that case the circuits C1L1 and C2L2 appear as
inductors with equivalent inductances Le1 = X1/ω0 and Le2 = X2/ω0, respectively, and we
obtain the case given in Fig. 197e. Observe that the magnitude of inductances Le1 and Le2 and,
consequently, also the magnitude of the feedback, can in the feedback network of Fig. 197f
be adjusted by tuning the resonant circuits C1L1 and C2L2.
In oscillators in which greater demands are made on the frequency stability of the generated
oscillations we usually require that the connection of the amplifier has the least possible
effect on the parameters of the resonant circuit. In such cases the coupling of the amplifier
with the resonant circuit is made as loose as possible, but such that the conditions of setting
the oscillator into oscillation are still satisfied. This can be done, for example, as indicated in
Fig. 197g. This circuit differs from that in Fig. 197d in that a chain of three capacitors C 1, C2,
and C3 is used. The frequency of generated oscillations again corresponds to the resonant
frequency
where Cs is the total capacitance of the chain Cs = 1/(1/C1 + 1/C2 +
1/C3). In the chain we choose C1 « C2, C1 « C3 (usually C2 = C3). The input and the output of
the amplifier are by-passed by the capacitors C2 and C3 , which at the frequency ω0 have a
low reactance, and all the changes in amplifier parameters are transformed into the resonant
circuit in attenuated form. An oscillator with this type of connection of the resonant circuit to
the amplifier is called the Clapp oscillator, after its originator. There are a number of
frequency-stable oscillators based on similar considerations. Very severe requirements on the
stability and accuracy of the frequency can be met by an oscillator with a crystal resonator,
whose feedback network is as shown in Fig. 197h. In principle, it is an analogy to the system
in Fig. 197g in which, however, the series resonant circuit C1L is replaced by the
electromechanical resonator CR, whose basic component is a quartz mechanical resonator
made to oscillate piezoelectrically.
Kinds of oscillator power suppl y. The electronic element of the oscillator operates in
the required operating region only on condition that it is supplied on both the input and the
output side with the required voltages or currents. So far we have assumed the transistor to
operate in common-emitter (the valve in common-cathode) connection. This is to say that the
dc supply for the input circuit of an electronic element (in order to set the quiescent operating
point - the bias) is introduced through the terminal pair 1 - 3 (see Fig. 195), and the dc supply
for its output circuit through the terminal pair 2 - 3. The conductor connected to terminal 3
represents the common conductor (the common node) of the circuit.
A voltage source can be inserted at any point of the dc loop (which can, in addition to simple
connections, contain also inductors). However, the source is usually required to be connected
by one terminal to the common node, and the point of inserting the voltage source is chosen
accordingly. The source is frequently located outside the oscillator proper, and it is therefore
usually by-passed in the oscillator circuit by a capacitor whose reactance at the operating
frequency of the oscillator is so small that it represents a short circuit for the ac current
generated. Since in the given case the voltage source, the feedback network, and the
electronic element are connected in the dc circuit in series, we refer to it as the seriesconnected power supply. For instance, the transistor in the oscillator of Fig. 189 has a seriesconnected power supply.
The oscillator can also be supplied immediately at the terminals 1 - 3 or 2 - 3, the supply
coming from a dc source. In that case, of course, the current source must be prevented from
being short-circuited by the dc circuit. If this occurs, we must interrupt the dc current circuit
by inserting a capacitor, whose reactance at the operating frequency of the oscillator is very
small and does not therefore prevent the passage of ac current through the oscillator circuits.
In practice, however, the source of dc current is often replaced by a source of dc voltage with
large internal resistance (realized by a resistor in series with the source), or a source with
large internal reactance is used (realized by a choke in series with the source). The current
source, the electronic element, and the feedback two-port are in this case connected in
parallel; this is the parallel-connected power supply. Examples will be given in the following
section.
There is, however, yet another method of providing a dc voltage source which is typical of
oscillators. It is by rectifying the generated voltage and applying the rectified voltage to the
amplifier input as a bias for automatically setting the operating point of the non-linear
electronic element. With increasing amplitude of the generated oscillations the average
operating point is shifted into the region of lower slope on the transfer characteristic. This
will substantially affect the amplitude stability of ac voltage in the oscillator.
Examples of connecting feedback LC oscillators. Fig. 198 gives the diagrams of
several oscillators with different types of electronic element, feedback, and power supply.
Fig. 198. Examples of feedback LC oscillators
Fig. 198a illustrates the connection of a valve oscillator with transformer feedback. The
resonant circuit L2C4 is connected to the anode circuit of the valve (i.e. it corresponds to the
configuration given in Fig. 197b). In the anode circuit, the parallel-connected power supply is
used. The voltage source Vs is by-passed by a capacitor C1, and the anode current is supplied
through a resistor R1. A blocking capacitor C2 is inserted between the anode of the valve and
the resonant circuit L2C4. In the grid circuit, a bias is used which results from rectifying the
ac voltage supplied by the coupling coil L1 of the feedback transformer. In the rectifier circuit
there is a diode formed by the grid and the cathode of the valve; together with a resistor R2
and a capacitor C3 this diode forms the parallel rectifier circuit. When setting the oscillator
into oscillation, the negative bias of the valve increases, the equivalent slope of the valve
gradually decreases, and this process continues until steady-state oscillations are generated.
The transistor oscillator of Fig. 198b operates with autotransformer feedback (corresponding
to Fig. 197c). The transistor T is series-supplied from the voltage source Vs through the coil
L. The circuit formed by the resistors R1, R2, R3 , and the transistor T, stabilizes the collector
current due to the negative current feedback caused by inserting the resistor R3 into the
emitter connection of the transistor. To prevent this negative feedback from affecting also
alternating currents, the resistor R3 is by-passed by the capacitor C3. The resonant circuit LC2
is connected to the transistor base through the blocking capacitor C1.
Another autotransformer oscillator with a MOSFET transistor is given in Fig. 198c.
However, it differs in principle from the preceding oscillators in that the transistor operates in
common-drain connection. This type of amplifier does not invert the signal which is being
amplified. The feedback network must therefore be also modified so as not to invert the
phase. In fact, the voltage across the resonant circuit LC1 (applied via the capacitor C2 to the
gate of the transistor T) is in phase with the voltage between the tap of the winding of the coil
L and the common node (the source is connected to the tap). The transistor bias is delivered
from the resistor divider R1, R2. Obviously, the transistor has a series-connected power
supply.
Parallel-connected power supply through the choke L2 is found in the oscillator with a JFET
transistor shown in Fig. 198d. The feedback is introduced by the capacitor divider which
forms an integral part of the resonant circuit L1, C1, C2. The automatic bias circuit formed by
the capacitor C3, the resistor R, and the input port of the transistor T, contributes towards the
settling of the oscillation amplitude in the oscillator.
Types of LC oscillator with negative differential resis tance. In Section 9.2.1 we
discussed the basic properties of oscillators with negative resistance. We saw that an
electronic element with an S-type ampere-volt characteristic can generate oscillations only
when connected to a series resonant circuit, while an element with an N-type ampere-volt
characteristic can form an oscillator only in connection with a parallel resonant circuit.
The S-type ampere-volt characteristic is found, for example, in the four-layer diode or in the
avalanche diode, the N-type characteristic in the tunnel diode or in the tetrode valve. The
descending portion of the input or output ampere-volt characteristic can also be found in twostage amplifiers (non-inverting) in which positive feedback is introduced by connecting the
input to the output (either directly or via a blocking capacitor). In elementary cases, these
amplifiers with positive feedback are reduced to a simple combination of two transistors
complemented with one or several resistors and energized respectively. Two such circuits are
shown in Fig. 199.
Fig. 199. Transistor two-terminal elements with negative resistance
In the circuit of Fig. 199a, the transistor T1 represents the -amplifying element whose output
current iC ≈ i is controlled by the base current iB ≈ iR = (V0 - v + vBE)/R, which is a function of
the voltage v. Applying these relations we can, for the given values V0 and R, determine the
ampere-volt characteristic i(v) in the family of collector characteristics of the transistor T1.
This is shown in Fig. 199b for the transistors KF507 (T1) and KF517 (T2) and for V0 = 7 V
and R = 6.8 kΩ. It can be seen that this circuit has an N-type ampere-volt characteristic.
In the other circuit (Fig. 199c), the current iB controls the voltage vC on the collector of the
transistor T2. The voltage vC is used to control the output voltage v of the emitter follower
containing the transistor T1. In this circuit, i ≈ iB and v = vC - vBE. The voltage vC can best be
found in the set of collector characteristics of the transistor T2, where for the given values of
V0 and R the respective load line is plotted (in Fig: 199b this was done for V0 = 7 V and
R = 100 Ω, the dashed line). For the individual values of i = iB the magnitude of the voltage
vC is determined, after which vBE = vBE(i) is subtracted, and thus the corresponding values of v
are obtained. On the basis of Fig. 199b and the manufacturer's catalogue characteristic vBE(i)
the ampere-volt characteristic i(v) is derived for the circuit containing the transistors KF517
(T1) and KF507 (T2). It is plotted in 'Fig. 199d and is of the S-type (its upper branch with
positive slope lies in the region of the currents i > 5 mA, i.e. outside- the format of our
figure).
Fig. 200. Illustrating the type of supply for oscillators with negative resistance
The power supply of oscillators containing negative differential resistance requires special
consideration. If the oscillator is to generate stable oscillations, its quiescent operating point
must be made to lie on the descending portion of the characteristic and be single-valued. This
is to say that an electronic element with an S-type characteristic must be supplied from a
source of constant current (IS in Fig. 200a) or at least from a very soft voltage source (with
the load line shown dashed). For the power supply from a hard voltage source, the load line
(shown as a dash-and-.dot line) intersects the S-type characteristic at -three points, and the
quiescent operating point is shifted into one of the dynamically stable points (A or B), i.e.
outside the region of the negative differential resistance.
For an element with an N-type ampere-volt characteristic the dual consideration holds. Such
an element must be supplied from a hard voltage source (the solid or the dashed load line in
Fig.200b). An. excessively soft voltage source (the dash-and-dot load line) will make the
quiescent operating point shift outside the region of negative differential resistance (i.e. into
point A or B).
Examples of LC oscillators containing negative resistance. The circuits of a few
simple oscillators will be given.
Fig. 201a shows the connection of an oscillator with a four-layer diode. The oscillator
oscillates at the resonant frequency of the series circuit LC in which, of course, both the
capacitance and the inductance of the diode itself must be included. By reducing the
inductance L of the external inductor, the oscillation frequency of the oscillator can be
increased; in the extreme case, this inductor can be left out altogether (Fig.201b). The
oscillator is supplied from a source of dc voltage VS through the resistor RS. The resistor RL
represents the load. Fig. 201c shows another oscillator, containing an avalanche transit diode,
which is used for the microwave frequency band, for instance for f ≈ 10 GHz. In this case the
resonant circuit is realized by means of microwave techniques (by the cavity resonator).
Fig. 201. Oscillators with four-layer and avalanche diodes
Fig. 202 is an example of the connection of an LC oscillator with a tunnel diode which
represents the resistor with the N-type ampere-volt characteristic. The oscillator oscillates at
the resonant frequency of the resonant circuit LC in which the capacitance and the inductance
of the tunnel diode must again be included. RL is the resistance of the load; the resistors Ra,
Rb form the voltage divider for the supply of the tunnel diode; Cb is the by-pass capacitor
through which the ac component of the diode current flows. By reducing the capacitance C,
the frequency of the generated oscillations can be increased; when the capacitance disappears
altogether, we have
, where CD is the capacitance of the tunnel diode.
Fig. 202. Oscillators with a tunnel diode
9.2.4 RC oscillators
In principle, RC oscillators operate in the same way as feedback LC oscillators. Their basic
arrangement corresponds completely to the block diagram in Fig. 195a, and for steady-state
oscillations there again hold module and argument (phase) conditions (9.19) and (9.20),
respectively. They differ principally in that the feedback network of an RC oscillator does not
contain a resonant circuit consisting of two dual reactance elements the capacitor and the
inductor; instead, it is made up of two or several reactance elements of the same kind capacitors, which together with two or several resistors form transfer two-ports with
a frequency-dependent transfer function. These feedback transfer two-ports can be of the
nature of low-pass, high-pass, and band-pass filters or band-rejection filters. Their
characteristics do not exhibit resonance curves which are typical of two-ports with resonant
circuits LC; they are comparatively flat (e.g. the module characteristic of the band-pass filter
RC corresponds roughly to the resonance characteristic of an LC circuit which has a very
small quality factor, Q < 1). The factor determining the frequency of an RC oscillator is the
phase shift in the feedback loop which, at the frequency ω0of the generated oscillations, must
satisfy the argument (phase) condition (9.20). Depending on whether an inverting or noninverting amplifier is used in the oscillator design, the feedback network RC must at
frequency ω0 shift the phase by either ± π or 0 (or ± 2π).
Fig. 203. Typical feedback networks of RC oscillators
Typical feedback circuits of RC oscillators are shown in Fig. 203. The ladder low-pass filter
with three RC sections as indicated in Fig. 203a has, on the assumption that R1 = R2 = R3 = R
and C1 = C2 = C3 = C, the transfer function
(9.34)
For further derivation we shall start from the assumption that the amplification factor of the
amplifier used is a real quantity, i.e. A = A, and that its output . resistance is very small,
Rout « R (if this condition is not satisfied, the resistance R1 must be reduced to R – Rout). The
frequency ω0 is then determined from the requirement that the transfer function of the ladder
network should also be real, i.e. Im K(ω0) = 0. This will be true for the frequency
. At this frequency we have K(ω0) = -1/29, which means that the amplifier used
must have an amplification factor of A = - 29 (hence it must be inverting). For the given
purpose, low-pass ladder filters with several RC sections are also used.
In Fig. 203b, a ladder high-pass filter with three CR sections is given. If C1 = C2 = C3 = C
and R1 = R2 = R3 = R, its transfer function is
(9.35)
If the applied amplifier meets the same conditions as in the preceding case, we can again
calculate the frequency ω0 from the equation Im K(ω0) = 0; we obtain
. For
this frequency there will be K(ω0) = -1/29 so that in. this case also the amplifier used will
have an amplification factor of A = -29 and a small output resistance. A ladder high-pass
filter can contain even more than three CR sections.
The feedback network of Fig.203c has two sections, a low-pass filter R1C1 and a high-pass
filter C2R2, whose phase shifts at a certain frequency ω0 just compensate each other. On the
assumption that C1 = C2 = C and R1 = R2 = R, the following relation holds for the transfer of
this band-pass filter:
(9.36)
From the condition Im K(ω0) = 0 we determine the frequency of generated oscillations ω0 =
1/(CR). Since for this frequency we have K(ω0) = 1/3, it is necessary that a non-inverting
amplifier with an amplification factor of A = 3 should be used in the oscillator with this
network.
In the band-pass filter of Fig.203d, the high-pass filter C1R1 and the low-pass filter R2C2 are
arranged in cascade in opposite sequence to that of the preceding case. For R1 = R2 = R and
C1 = C2 = C, its transfer function is again given by Eq. (9.36). The same conclusions
therefore also hold for this feedback network as for the network of Fig. 203c.
The following feedback two-port (Fig. 203e) takes advantage of the transfer properties of the
Wien network. If C1 = C2 = C and R1 = R2 = R, then also in this case the transfer function is
expressed by relation (9.36), with all the other consequences that ensue from it.
The last of the circuits given (Fig. 203f) has a twin-T network as its basis. If in this two-port
we choose R1 = R2 = 2R3 = R and C1 = C2 = C3/2 = C, its transfer function will be given by
the relation.
(9.37)
For the frequency ω0= 1/(CR) we have K(ω0) = 0, and this means that the twin-T network
behaves as a band-rejection filter. Its argument (phase) characteristic in close neighbourhood
of ω0 changes stepwise from - π/2 to + π/2 and passes through zero while doing so, i.e. arg
K(ω0) = 0. Unlike the preceding cases, we are concerned here with a band-rejection filter
(which from the viewpoint of the transfer properties is of a similar selective performance as
that of the transfer two-port with a series resonant circuit). and this must be reflected in the
arrangement of the oscillator. The amplifier is provided with a permanent-acting frequencyindependent positive feedback and complemented with a negative feedback loop with a twinT network. The negative feedback counteracts the effect of the positive feedback at all
frequencies with the exception of the frequency ω0, at which it is zero. The oscillator
therefore oscillates at frequency ω0.
Fig.204. The voltage-dependent negative feedback as a means of limiting the amplitude of oscillations of
RC oscillators
Feedback RC networks have not such pronounced selective properties as LC elements, and if
oscillators containing them are to generate harmonic oscillations without distortion, we try to
curb as much as possible the amplifier non-linearity, which gives rise to higher harmonic
components. This requirement, however, contradicts the requirement of maintaining a
constant amplitude of oscillations which in LC oscillators is established exactly owing to the
amplifier non-linearity. To limit the amplitude of oscillations, a special voltage-dependent
negative feedback is therefore used in RC oscillators, as indicated in Fig. 204. To make the
introduction of the feedback possible, a differential amplifier OA with an inverting and a
non-inverting input is employed. Through the resistor divider R1, R2 such part of the output
voltage is applied to the inverting input that the amplification from the non-inverting input to
the output just equals the required value A {e.g. A = 3 for the feedback networks of Figs 203c
through e). In the divider, either the resistor R1 or the resistor R2 is thermally inertial, and its
resistance depends on the amplitude of the flowing harmonic current (which causes the
heating of the resistor). If the amplitude of the output voltage increases, we require the
negative feedback to increase, i.e. the division ratio of the attenuator R1, R2 to be reduced.
This will be the case if either the resistance R1 is reduced with increasing current and thus
with the resistor R1 heated (a thermistor with a negative temperature coefficient of
resistance), or if the resistance R2 is increased with increasing current (an incandescent lamp
with a positive temperature coefficient of filament resistance). On other occasions, instead of
the resistor R2 a controlled resistor is used (e.g. a field-effect transistor) whose resistance is
controlled by the rectified output voltage of the amplifier. The controlled resistor must again
be connected so that its resistance increases with the increasing amplitude of the oscillator
output voltage.
Examp1es of RC osci11ators. Fig. 205 shows three typical connections of oscillators
with feedback RC networks. In Fig. 205a is the circuit of an oscillator containing a ladder
high-pass filter. For the filter output not to be loaded excessively by the input resistance of
the connected amplifier, the Darlington combination of transistors T1 and T2 is used in the
amplifier which operates with the load resistor RC. Its resistance RC is dimensioned so that the
amplification factor A ≈ -29. The resistor RB provides for the dc supply of the transistor base.
The oscillator shown in Fig. 205b has two branches in the feedback network of the
operational amplifier OA - a selective RC band-pass filter determining the frequency, and
a voltage-dependent divider R1, R2 with thermistor whose purpose is to maintain a constant
amplitude of oscillations.
The oscillator with the Wien two-port shown in Fig.205c has the negative feedback branch
formed by a chain consisting of R1, R2, and T. The field-effect transistor T serves as a linear
controlled resistor. The resistors R3, R4 complemented with a blocking capacitor C3 form a
network of auxiliary negative feedback to linearize its ampere-volt characteristic. The control
voltage is supplied from a rectifier (a voltage doubter) with the diodes D1, D2, storage
capacitors C1, C2, and discharge resistor R5.
Fig. 205. RC oscillators
9.3 GENERATORS OF NON-SINUSOIDAL OSCILLATIONS
So far we have examined the properties and connections of oscillators generating more or less
harmonic oscillations. In engineering practice, however, periodic oscillations are also used
which by their waveform differ basically from harmonic oscillations. They are usually
characterized by more or less steep steps or abrupt changes in the slope of waveform. Such
oscillations are referred to as relaxation oscillations. Typical relaxation oscillations are
triangular, rectangular, and sawtooth waveforms and the square pulse wave (a periodically
repeated sequence of rectangular pulses). These waveforms are shown in Fig. 206. It can be
seen that the waveform vb(t) is the result of differentiating the waveform va(t) with respect to
time and, similarly, the waveform vd(t) also represents a derivative of the waveform vc(t) with
respect to time.
While oscillators generating harmonic oscillations always represent a system of at least the
second order (with two storage elements, i.e. capacitors or inductors), a system of the first
order with a single storage element will often suffice when generating relaxation oscillations.
The mechanism of periodic operation in generators of non-sinusoidal oscillations is, of
course, other than that in oscillators.
Fig. 206. Typical waveforms of relaxation oscillations
9.3.1 Explanation of the operation of the relaxation generator
The storage elements in relaxation generators are almost exclusively capacitors, so that all the
following considerations will be focused on them. Note, however, that there are no
fundamental objections to the dual analogy of these generators in which inductors would be
used as storage elements.
Fig. 207. A general diagram of the relaxation generator with a capacitor
Basic configuration of relaxation generator. The operation of the generator
containing a capacitor can be explained on the basis of the elementary circuit configuration
presented in Fig. 207. During the operation of the generator there is a periodic alternation of
intervals when the capacitor C is supplied with the positive current iC = I1 and with the
negative current iC = - I2. The transition from one source to the other is made possible by the
(electronic) switch S, whose operation is controlled by the levels of the voltage vc(t) across
the capacitor C. If the switch S is in position 1, the positive current iC = I1 flows into the
capacitor C and the voltage across the switch increases linearly with time:
(9.38)
where VCL is some initial voltage across the capacitor (e.g. its lower level). The voltage vc(t)
acts on the input of the circuit VC, the voltage comparator. Its task consists in that when
a certain upper voltage limit is reached on the capacitor (let us denote it VCH) its output
voltage vK(t) changes from one constant value, e.g. VKL, to another, e.g. VKH (these two levels
can represent the logical value 0 and the logical value 1, respectively). This step change in the
voltage vK(t) makes (possibly after processing in the logic circuit LC) the switch S change to
position 2. The time T1 required to charge the capacitor C from the voltage VCL to the voltage
VCH is, by Eq. (9.38), T1 = (VCH - VCL) C/I1.
In the second stage of the process, the capacitor C is being supplied with the negative current
iC = - I2. Its charge decreases, the voltage
(9.39)
decreases linearly with time, falling sometimes down to negative values. If the voltage vc(t)
reaches a certain lower limit VCL, the comparator VC responds again in such a way that its
output voltage changes from the level VKH again to the initial level VKL (i.e. from the logic
value 1 to the logic value 0). As a result, the switch S changes (with the aid of the LC) to
position 1, and the whole process is repeated. Since the discharge time of the capacitor C
from the voltage VCH to VCL is T2 = (VCH - VCL)C/I2 , we can conclude that the whole process
is periodically repeated with period
(9.40)
It is obvious that with the two currents equal, I1 = I2 , we obtain a triangular voltage vC(t)
with the waveform shown in Fig. 206a, and along with it a rectangular voltage vK(t), whose
waveform corresponds to Fig. 206b. If I2 > I1, the generator will generate a sawtooth voltage
vC(t) shown in Fig. 206c and also, a square pulse train vK(t) as shown in Fig. 206d.
The above elementary idea must be elaborated. In some relaxation generators, several
capacitors and several comparators with logic circuits and switches are used. The
comparators usually respond to the voltage level of a capacitor other than that whose
charging or discharging they control. The system forms a closed chain, periodically
generating the prescribed time sequence of the given time functions. .
Types of current delivery to capacitors. In the above consideration we have assumed
the capacitor C to be supplied from sources of constant current. A current source with a very
small internal conductance can be realized by means of a transistor. In the wide range of
collector voltages with iB = const. (or with vBE = const.) its ampere-volt characteristics are
almost horizontal. That is to say, in the above operating region an almost constant current iC
flows through the transistor, increasing only a little with increasing voltage vCE. This
transistor property can be substantially enhanced by introducing a current negative feedback
which will appear when a resistor RE is inserted into the emitter connection (Fig. 208a). A
current source designed in this way will deliver a current iC ≈ iE = (VB - vBE)/RE (on the
assumption that the current amplification factor of the transistor h21E » 1). For a current of
opposite sense, an analogous circuit with a complementary transistor must be designed.
Another way of delivering constant current to the capacitor is shown in Fig. 208b. In
principle, it is an integrator with an operational amplifier. The input voltage of the operational
amplifier vi ≈ 0, and thus a constant current iR = VB/R flows through the resistor R. The
capacitor C in the feedback branch is supplied with constant current iC = - iR so that the
output voltage vC changes linearly with time (if VB > 0, then vC(t) is decreasing; if, however,
VB < 0, then the voltage vC(t) is increasing).
Fig. 208. The possibilities of supplying the capacitor a) from a transistor current source, b) in the integrator
circuit with operational amplifier,
There is, however, an often used and simpler method of charging or discharging the capacitor
through a resistor connected to the voltage source (Fig. 208c). The charging (discharging) is
known to take place in exponential time dependence:
(9.41)
where V0 is the initial voltage across the capacitor. From this equation we can also determine
the time T01 which is necessary to change the voltage vC(t) from the initial value vC(0) = V0 to
the final value vC(T01) = V1. We obtain for it the relation
(9.42)
From this, we can calculate the period of oscillations generated by the relaxation generator in
which the capacitor is charged and discharged from voltage sources via a resistor.
Vo1tage comparators. We have characterized voltage comparators as circuits responding
to a certain input voltage level being reached by a step change in the output voltage. A
comparator can be any amplifier having an upper and a lower region of saturation on the
transfer characteristic.
The simplest comparator is the simple transistor amplifier of Fig. 209a. Its transfer
characteristic is given by the solid line in Fig. 209b. It terminates at the value v1 ≈ 1 V, at
which the permissible base current is not exceeded. The fact that the input ampere-volt
characteristic of this comparator corresponds to the characteristic of a semiconductor diode is
in some applications detrimental, but elsewhere it can - be turned to advantage (see the
following section on multivibrators). At times, the input current magnitude is limited by
inserting a resistor R2 into the base connection; in this case the voltage v1 can be increased by
the voltage drop across this resistor (indicated in Fig. 209b by the dashed line).
Fig. 209. A simple transistor comparator
Of greater advantage are the properties of the comparator with the differential amplifier,
because it permits us to shift the level of the reference voltage as necessary. The principle is
clear from Fig.210a.
Fig. 210. Comparators with -a differential operational amplifier
The comparator is an operational amplifier OA (e.g. an integrated monolithic amplifier) with
differential input. Its transfer characteristic v2(vi) is plotted in Fig. 210b. It can be seen that if
the input voltage v1 is lower than the reference voltage Vref we have vi = v1 - Vref < 0, so that at
the output there is a positive saturated voltage v2 = V2p. If v1 > Vref we have vi > 0, and across
the output there is a negative saturated voltage v2 = V2n. The transition from V2p to V2n is very
steep, as can be seen from the orientation scales on the transfer characteristic axes.
We have postulated that the comparator must respond by two opposite step changes to two
different levels. This can be done by introducing in the differential amplifier a frequencyindependent positive feedback. The connection is obvious from Fig. 210c. The voltage
vf = v2R2/(R1 + R2) = pv2 acts on the non-inverting input of the operational amplifier.
Depending on whether there is a positive or a negative saturated voltage at the output, we
apply to the non-inverting input either the positive reference voltage Vfp = pV2p or the
negative reference voltage Vfn = pV2n. The voltage V2p appears at the output of the operational
amplifier only if vi < 0 i.e. if v1 < Vfp. If the voltage v1 approaches the reference voltage Vfp,
the amplifier will pass from the saturation region to the descending portion of the transfer
characteristic v2(v1) shown in Fig. 210b. A negative change in the output voltage v2 entails
immediately an also negative change in the reference voltage vf. Consequently, there is
a positive voltage change vi = v1 - vf which causes another negative change v2, and the whole
process continues avalanche-wise until the output voltage reaches the negative saturated
value V2n (see also the ampere-volt characteristic in Fig. 210d - but observe that unlike Fig.
210b, the scales on the two axes are in this case identical). At the same time, the reference
voltage will change automatically to the value Vfn. The circuit will remain in this state so long
as vi > 0, that is to say if v1 > Vfn. If the voltage v1 decreases almost to Vfn, the output voltage
v2 will change by a similar avalanche process supported by positive feedback, and will jump
from the value V2n to V2p. Simultaneously; the reference voltage will also change, jumping to
Vfp. Thus the comparator transfer characteristic is in the form of a hysteresis loop, and, as
expected, the comparator responds by step changes to two voltages Vfp and Vfn. If necessary,
the whole hysteresis loop can be shifted along the v1 axis, for instance by adding a dc voltage
source in series with the resistor R2.
The simplest case of a feedback comparator is the one-stage differential amplifier of
Fig.211a, with the feedback introduced by the resistor divider R3, R4. It is known as the
Schmitt trigger, and its transfer characteristic v2(v1) exhibits the hysteresis phenomenon (Fig.
211 b).
Fig. 211. The Schmitt trigger
Another type of comparator, responding, however, to two voltage levels by step changes in
the current, is represented by the non-linear resistor with an S-type ampere-volt characteristic.
It can be seen from Fig. 212 that if the voltage v increases from zero up to the value VH, a
comparatively small current i < I2 flows through the resistor. When the voltage VH is reached,
the operating point jumps from the lower branch of the characteristic to the upper branch
(from point 2 to point 3), and a large current i > I4 is now flowing through the resistor. With
the voltage v decreasing, the operating point moves along the upper branch of the
characteristic towards point 4. With the voltage v reaching the value VL, the operating point
changes stepwise to the lower branch of the characteristic (from point 4 to point 1).
Consequently, the current i decreases stepwise from the value I4 to I1. The whole process can
now be repeated. It follows that a resistor with an S-type characteristic behaves as an
imperfect switch, responding to two voltage levels VL and VH. This can be applied in
controlled periodic charging and discharging of a capacitor.
Fig. 212. The resistor with S-type ampere-volt characteristic used as comparator
9.3.2 Examples of relaxation generators
We shall give a number of simpler examples of relaxation generators.
Generator of sawtooth oscillations with a four -layer diode. Fig. 213a is
a simplified diagram of the relaxation generator with a four-layer diode FLD. The amperevolt characteristic of the diode corresponds to Fig. 212. The current 1 supplied from
a transistor current source is constant and its magnitude is chosen such that I2 < 1 < I4 (where
I2 and I4 are the characteristic diode currents, see Fig.212). The capacitor C is supplied with
current iC = I - i. This current can be either positive or negative, depending on whether i < I
or i > I.
Fig. 213. Generator of sawtooth oscillations with a four-layer diode
On connecting the current source to the circuit, i < I, the current iC is positive, and the voltage
v across the capacitor and across the four-layer diode increases, as shown in Fig. 213b. As
soon as the voltage v reaches the level VH, the current i changes stepwise to the value I3 > I.
Consequently, the current iC is now negative, and the capacitor C discharges until the voltage
across it decreases below the value VL, at which the current i decreases stepwise to the value
I1. The whole cycle is repeated periodically, and the circuit generates sawtooth oscillations as
indicated in Fig. 213b. The waveform is curved because neither the charging nor the
discharging capacitor current is constant. The frequency of the generated oscillations is
mainly given by the capacitance C of the applied capacitor, but it is also affected by the shape
of the ampere-volt characteristic of the four-layer diode. ,
Function (waveform) generator. A simplified diagram of the generator of triangular and
rectangular oscillations is given in Fig. 214. The comparator is an operational amplifier OA;
its output voltage is adjusted to the prescribed magnitude by a two-level clipper consisting of
a resistor R3 and a pair of regulator (Zener) diodes ZD1 and ZD2. In the comparator, the
divider R1, R2 introduces a positive feedback with which the comparator has the required
hysteresis transfer characteristic v2(v1). There are two sources of constant current in the
circuit. They are dimensioned so that I1 > I2 . While the I2, current source is connected
permanently to the capacitor C, the I1 current source is connected commutatively by means of
a periodically operating logic circuit with the diodes D1 and D2. If there is a positive saturated
voltage V2p > v1 across the output, the diode D2 is non-conducting, and the current I1 flows
through the diode D1. The capacitor C is thus supplied with the current iC = I1 – I2 > 0. The
voltage v1 across the capacitor increases linearly, and as soon as it attains the level
Vfp = V2pR2/(R1 + R2), the comparator output voltage changes stepwise to the negative
saturated - voltage V2n. As a consequence, the diode D2 starts conducting, and all the current
I1 flows through it to the comparator output (where it joins the current flowing through the
diodes ZD1 and ZD2). As a result, the diode D1 ceases to conduct and a constant current iC =
-I2 flows from the capacitor C. The voltage v1 decreases linearly with time until the level
Vfn = V2nR2/(R1 + R2) is reached; the output voltage returns stepwise to the value V2p, and the
whole cycle can be repeated.
Fig. 214. A simplified diagram of the function (waveform) generator
If a voltage v1 is to be drawn from the generator, the charging and discharging processes must
not be affected. A buffer amplifier must therefore be used which provides for a current-free
drawing of this voltage from the capacitor C.
The shape of generated periodic oscillations will be affected by the ratio of the currents I1 and
I2 . If I1 = 2 I2 , symmetrical triangular oscillations v1(t) are generated. If the current I1 < 2 I2 ,
the charging current of the capacitor will be smaller than the discharging current, and
sawtooth oscillations v1(t) sloping to the right will originate. Conversely, for I1 > 2I2 the
originating sawtooth oscillations will be sloping to the left. Needless to say, simultaneously
with the triangular or sawtooth oscillations v1(t) the corresponding rectangular oscillations
v2(t) are also generated whose shape (but not the location of levels) is similar to the derivative
of the time function v1(t).
Since the capacitor charging time T1 = C(Vfp - Vfn)/( I1 – I2) and its discharging time
T2 = C(Vfp - Vfn)/ I2, the period of generated oscillations is
(9.43)
Mu1tivibrators. One class of relaxation generators with two capacitors is formed by
autonomous flip-flop circuits called multivibrators. Their valve version was invented by
Abraham and Bloch. The diagram of a transistor multivibrator is given in Fig. 215a, and the
corresponding waveforms of generated voltages in Fig. 215b. The transistors T1 and T2
operate as both comparators and switches simultaneously.
Fig. 215. Transistor multivibrator
Assume that in consequence of the preceding operation the transistor T1 is at time t = 0
conducting until saturation (v1 ≈ - 0.2 V, v4 ≈ 0.7 V), that the transistor T2 is not conducting,
and that the charging of the capacitor C2 has ended (i.e. v3 ≈ Vs, v6 = VS - v4 ≈ VS - 0.7 V). At
time t = 0, the voltage v2 is negative but increasing with time, because the capacitor C1 is
supplied positive current through R2. At time t1 the voltage v2 is already positive and so large
that the transistor T2 conducts slightly. The small negative change so produced in the voltage
v3 is transferred through C2 to the base of the transistor T1 which passes from the saturated
state to the region of active operation, and on its collector a small positive change in the
voltage v1 appears which is transferred immediately through C1 to the base of the transistor T2
and contributes to its further opening. Owing to this positive feedback action, a very fast
avalanche process of circuit triggering takes place, resulting in T1 ceasing to conduct and T2
conducting until saturation.
In the next stage, two processes take place simultaneously. The capacitor C1 is charged
comparatively quickly from the source VS through R1 and the diode formed by the baseemitter junction of the transistor T2. Considerably slower is the change of the voltage v4
originating in the charging capacitor C2 due to the positive current flowing through the
resistor R4. At time t2 the voltage v4 reaches such a positive value that the transistor T1
conducts slightly, which analogously to the preceding case causes a rapid avalanche
triggering of the circuit. After its termination, T1 conducts until saturation, and T2 is nonconducting. The capacitor C2 is charged comparatively quickly through R3 and the diode
formed by the base-emitter junction of T1 to the voltage v6 ≈ VS - 0.7 V. Simultaneously,
there is a slower increase in the voltage v2 until at time t3 it reaches a positive value to which
the comparator T2 responds by triggering the circuit. Thus at time t3 the situation is the same
as at time t1, and the whole process can be repeated periodically with period T = t3 – t1.
It is obvious that the period and thus also the frequency of the generated oscillations will be
decisively affected by the time constants C1R2 and C2R4 and also by the parameters VBEsat and
VCEsat of the transistors operating as comparators and switches, and also by the supply voltage
VS. This follows from the application of Eq. (9.42) to the charging processes in the
multivibrator.
B1ocking osci11ator. The blocking oscillator is used as the source of a periodic train of
very short pulses (with a duration of hundredths of a microsecond to tens of microseconds}.
Its circuit is given in Fig. 216a. Again, the transistor T operates as a comparator and
simultaneously as a switching element. In the circuit, a positive feedback is introduced from
the collector to the base of the transistor T by means of a pulse transformer Tr, which is
characterized by a close coupling and a small magnetic flux leakage.
The waveforms of the generated oscillations are shown in Fig. 216b. Assume that at time t =
0 the voltage v1 ≈ v2 is negative, i.e. the transistor T is non-conducting so that v3 = VS. The
capacitor C is charged through the resistor R, and the voltage v1 increases with time until at
time t1 it reaches such a positive value that the transistor T conducts slightly. Consequently,
the voltage v3 decreases slightly, the change is transferred with opposite polarity to the
secondary winding of the transformer Tr, where the change is added to the voltage v1 and the
resulting positive change in the voltage v2 will produce greater conduction through the
transistor; the whole process develops avalanche-wise until the transistor T is saturated (i.e.
v3 ≈ 0.2 V). In this manner, the negative front of the pulse of voltage is being shaped.
Now follows the stage in which the capacitor C is charged from the secondary winding of the
transformer Tr through the diode formed by the base-emitter junction of the transistor T. This
stage is of only short duration, since the charging current decreases very quickly to such a
value that at time t2 the transistor T passes from saturation state to active operation. A small
positive change in the collector voltage v3 originates and, owing to the positive feedback,
causes an avalanche-like process of cutting off the transistor T which shapes the positive
trailing edge of the voltage v3 pulse. Since the magnetization flux cannot decay immediately,
a positive voltage peak of v3 appears which exceeds considerably the level of VS. Since this
voltage peak could destroy the transistor, a protective diode D is usually connected to the
primary winding of the transformer Tr; the diode yields the waveforms v2(t) and v3(t) given in
Fig. 216b by the dashed line.
The period T = t3 – t1 of the generated oscillations is determined mainly by the time constant
CR, by the parameters VBEsat and VCEsat, of the transistor, and by the turns ratio of the
transformer Tr. The width (duration) of the generated pulses is determined mainly by the
dynamic properties of the transformer Tr [18], [44].
Fig. 216. Blocking oscillator
Non-linear feedback and resonance
phenomena
In this chapter we shall set forth some general views on non-linear transformations in systems
based on the utilization of feedback in circuits containing amplifiers. We shall further deal
with problems of resonance phenomena in non-linear and parametric resonance circuits.
10.1 NON-LINEAR FEEDBACK SYSTEMS
Non-linear circuits are often used in which the properties of negative feedback are exploited
to obtain prescribed transfer properties. Such non-linear feedback systems make it possible to
realize a whole range of useful circuits that are suitable for non-linear transformation of
signals. Their significance has increased especially with mass introduction of general-purpose
amplification units - integrated operational amplifiers.
In principle, any arbitrary non-linear or parametric elements can be applied in a non-linear
feedback system. For simplicity, however, we shall restrict our discussion to resistive nonlinear (parametric) elements only. The non-linear feedback systems under consideration are
usually used in two basic configurations, namely as a) (linear) amplifiers with nonlinearities
in the feedback network, b) non-linear amplifiers with a linear feedback network. The basic
properties of both these groups will be discussed.
10.1.1 Amplifiers with a non-linear feedback network
For an amplifier with non-linear feedback to possess the required dependence between the
input and the output voltage, the feedback network must include non-linear elements whose
ampere-volt or transfer characteristics correspond, with the required accuracy, to the desired
characteristic (defined, for example, on the basis of the theoretical solution of the problem).
In many cases, these requirements will be satisfied by "natural" characteristics of electronic
elements which are basically given by the principles of the motion of electric charge carriers
in electric or magnetic fields and various media. In some cases, however, simple elements
will not be sufficient to perform this task, and we must resort to combinations of such
elements. A very effective method can be seen in modelling the required characteristics by
piecewise linear characteristics in circuits made up of linear resistors, diodes, and constant
voltage sources (these were discussed in Sections 7.2.2 and 3.5).
Non-linear resistor in a feedback network. In the inverting amplifier (which is the
most frequently employed connection in operational amplifiers) there are three possibilities
of inserting a non-linear (either non-controlled or controlled) resistor in the feedback
network: in the input branch, or in the feedback branch, or in both of these branches
simultaneously.
Consider first the case when a non-controlled non-linear resistor defined by the ampere-volt
characteristic iA = fA(vA), is used in the input branch of the feedback network of an inverting
amplifier (Fig. 217a). If we start the solution from the usual simplifying conditions of the
operation of the operational amplifier (vi ≈ 0, ii ≈ 0), the following relations will hold in the
circuit: v1 ≈ vA, v2 ≈ vB, il ≡ iA ≈ - iB. This means that iA ≈ fA(v1) and that v2 ≈ RBiB ≈ -RBiA.
Thus, the output voltage will be of the magnitude
(10.1)
In this way, we can derive from the characteristic iA = fA(vA) the transfer characteristic v2(v1)
(perhaps graphically, point by point). Fig. 217c shows the graphical solution. We start from
the fact that the currents iA and iB flowing through the two elements which form the feedback
network are opposite, iA ≈ - iB. The resultant transfer characteristic is plotted in Fig. 217d. It
can be seen from the derivation arid from the graphical solution that the characteristic v2(v1)
has been formed from the characteristic iA = fA(vA) of the applied non-linear resistor by
a transformation of the reflection type about the horizontal axis (see Table 2 in Section 3.7)
accompanied, moreover, by a scaling in the vertical axis direction. From the physical point of
view, we are concerned here with the transformation from the ampere-volt plane iA - vA into
the volt-volt plane v2 - v1 (into the voltage transfer plane).
Fig. 217. Inverting amplifier with a non-linear resistor in the input branch of the feedback network
In the circuit shown in Fig. 218a, the positions of the linear and non-linear elements in the
feedback network are interchanged. To simplify the notation, assume that the electrical
properties of the non-linear resistor are described by its volt-ampere characteristic vB = gB(iB),
which is inverse to the ampere-volt characteristic iB = fB(vB). The current iB flowing through
the feedback branch determines the voltage across the non-linear resistor, and so the output
voltage is
(10.2)
The graphical solution of this relation is illustrated in Fig. 218c. Since we usually know the
ampere-volt characteristic of the non-linear element iB = fB(vB) (Fig. 218b), it is convenient to
start from it in the graphical solution. It follows from a comparison of the ampere-volt
characteristic iB = fB(vB) and the resultant transfer characteristic v2(v1) (Fig. 218d) that in this
case we are concerned with a transformation of the rotation type by an angle of + 90° around
the origin, which is, moreover, accompanied by scaling.
In the special case it is possible to leave out the resistor R1(R1 = 0) and to excite the circuit
directly by the source of current i1. The output voltage is then a function of the input current:
.
(10.3)
This is again a transformation of the rotation type, but this time without any scaling, and the
ampere-volt plane iB - vB of the non-linear resistor is transformed into another (transfer)
ampere-volt plane i1 - v2.
Fig. 218. Inverting amplifier with a non-linear resistor in the feedback branch of the feedback network
A general configuration of the inverting amplifier with a non-linear feedback network is
plotted in Fig. 219a.
Fig. 219. Inverting amplifier with a non-linear feedback network
The solution is similar to that in the preceding cases, noting, of course, that iA = fA(vA) and vB
= gB(iB) (an inverse characteristic to iB = fB(vB)). In this case the output voltage is
(10.4)
The graphical solution is indicated in Fig.219c, starting from the characteristic plotted in Fig.
219b. The resultant transfer characteristic v2(v1) is shown in Fig. 219d.
It follows that an inverting amplifier with non-linear feedback has a very important property.
If in the feedback amplifier with the transfer characteristic v2 = f(v1) we interchange the two
elements of the feedback network, we obtain an amplifier with the transfer characteristic v2 =
g(v1), which is inverse to the characteristic f(v1). This property makes it possible that by
merely interchanging the elements in the input and the output branches of the feedback
network of an inverting amplifier we can model, for example, x2 and x1/2, ln x and ex, sin x
and arcsin x, etc.
In linear applications, the possibility is often exploited of delivering input current to the
summing point by several input branches to form a summing inverting amplifier. This can
also be made use of in the non-linear variant, as shown schematically in Fig. 220.
Fig. 220. Summing amplifier with a non-linear feedback network
If all the elements of the feedback network are non-linear, the output voltage will be
a function of the sum of currents flowing through all the input branches, that is to say
a composed function of all N input voltages
(10.5)
Inserting into the feedback branch a linear resistor with resistance RB, the output voltage will
be
.
(10.6)
If in all the input branches we use the linear resistors with resistances R1, R2, ..., RN, and in the
feedback branch we keep the non-linear resistor, we have
(10.7)
The summing amplifier can be used with advantage to shift -the transfer characteristic v2(v1).
For the output voltage in the circuit of Fig. 221a, we obtain the relation
(10.8)
As can be seen from Fig. 221b, the transfer characteristic v2(v1) can be shifted in the vertical
axis direction by changing the voltage v0 . Different are the properties of the circuit in Fig.
221c for the output voltage of which we can. on the basis of Eq. (10.7), derive the relation
(10.9)
In this case, a change in the voltage v0 permits the shifting of the transfer characteristic in the
horizontal axis direction (see Fig. 221d).
Fig. 221. Employing the summing amplifier to shift the transfer characteristic
To shift the transfer characteristic v2(v1) to the respective operating position, yet another
possibility can be used which is based on the application of the differential amplifier with
a non-linear feedback network. The circuit is shown in Fig. 222x. Applying the usual
simplifying assumptions (vi ≈ 0, ii ≈ 0), in this circuit vA ≈ v1 – v0, iB ≈ - iA ≈ - fA(v1 – v0) so
that the output voltage is
(10.10)
It follows that the original transfer characteristic v2 ≈ gB(-fA(v1)) (given by Eq. (10.4) and
corresponding to the value v0 = 0) will, owing to the voltage acting on the non-inverting
terminal of the amplifier, be shifted by a value of v0 in both the horizontal axis direction and
the vertical axis direction (see Fig. 222b).
The above methods can thus be used in an arbitrary configuration of the amplifier with a nonlinear feedback network to shift the transfer characteristic v2(v1) to an arbitrary place in the v2
- v1 plane.
Fig. 222. Shifting the transfer characteristics in a differential amplifier with a non-linear feedback network
Controlled non-linear resistor in a feedback network. The preceding
considerations can be extended to cover the use of controlled non-linear resistors in the
feedback network.
Fig. 223. A controlled non-linear resistor in the input branch of an inverting amplifier
If a controlled non-linear element with the characteristic iA = fA(vA, p) is connected in the
input branch of a feedback network (see Fig. 223a), the output voltage is given by the relation
(10.11)
which is an analogy to Eq. (10.1). The non-linear dependence between the input and the
output voltage will obviously depend on the control quantity p. A family of the ampere-volt
characteristics of a controlled linear resistor is plotted in Fig. 223b, while Fig. 223c gives the
set of the transfer characteristics v2(v1, p) derived from it.
Interchanging the elements in the feedback network, the output voltage will (analogously to
Eq. (10.2)) be given by the relation
(10.12)
If in the general case there is a controlled non-linear resistor with the characteristic iA = fA(vA,
pA) connected in the input branch of the feedback network while in the feedback branch there
is a controlled nonlinear resistor with the characteristic vB = gB(iB, pB), then the output
voltage of such a circuit will be
(10.13)
It is obvious that the solution of circuits with controlled non-linear elements in the feedback
network of an inverting operational amplifier is similar to that of amplifiers with noncontrolled elements. It is only necessary to include in the resultant relations the dependence
on the control quantity p (or on pA, pB).
Non-linear three-terminal resistor in a feedback network. If a three-terminal nonlinear resistor is suitably connected in the feedback network of an amplifier, we obtain
a number of technically interesting results. Inserting a three-terminal resistor into the input
branch of the feedback network of an inverting amplifier, we obtain the situation illustrated
schematically in Fig. 224a.
Fig. 224. A non-linear three-terminal resistor in the input and in the output branch of an inverting amplifier
It is obvious from the figure that the terminal Z of the given three-terminal resistor is
approximately at the potential of the common node. This is because vZ ≡ vi ≈ 0. In the solution
of this circuit, it is convenient to distinguish between the case when a voltage source, and the
case when a current source, is connected to the terminals X - X' or Y - Y'.
a) If the sources of voltage vX, vY are connected to the terminals X - X' and the terminals Y Y', respectively, it is of advantage to start the analysis from knowledge of the characteristic
iZ(vX, vY), since from the latter we can easily determine the output voltage
(10.14)
b) If the source of voltage vX is connected to the terminals X - X' while the source of current
iY is connected to the terminals Y - Y', it is more convenient to apply the hybrid characteristic
iZ(vX, iY). Then the output voltage is
(10.15)
c) If, on the other hand, the current source is connected to the terminals X - X' and the voltage
source to the terminals Y - Y', the output. voltage is
(10.16)
It can be seen that in all the three cases the resultant transfer characteristics v2(vX, vY), v2(vX,
iY), v2(iX, vY) are similar to the corresponding characteristics of a non-linear three-terminal
resistor. The fourth case, with the sources of currents iX and iY, is of no practical importance
since the currents flowing into a three-terminal resistor are governed by Kirchhoff s current
law, i.e. iX + iY + iZ = 0, and the properties of a three-terminal resistor are in this case of no
effect on the magnitude of the current iZ .
In the circuit of Fig. 224b, the three-terminal non-linear resistor is inserted into the feedback
branch of the feedback network. The terminal Z is again connected to the amplifier summing
point, and so vZ ≡ vi ≈ 0. The solution will differ, depending on whether the voltage source or
the current source is connected to the terminals Y - Y'.
a) If the source of voltage vY is connected to the terminals Y - Y', it is convenient to apply the
hybrid characteristics vX(iX, vY) and iY(iX, vY). On the basis of the generalized Kirchhoff's
current law we can construct for the given three-terminal resistor the implicit equation
(10.17)
This equation can be solved, graphically or numerically. For the known values of vY and iZ,
we find the magnitude of the current iX, and this in turn serves as the basis for determining
the magnitude of the output voltage
(10.18)
b) If the source of current iY is connected to the terminals Y - Y', we can calculate the output
voltage v2 ≡ vX from the implicit equation
(10.19)
From this equation, we calculate for the known values of iY and iZ directly the voltage vX.
A summing amplifier can also be realized with three-terminal non-linear resistors. It is
sufficient to simply take into consideration all the currents flowing through the input
branches to the summing point of the inverting amplifier. In this case, too, the resultant
characteristic can be shifted by introducing an additional current into the summing point of
the amplifier or by an additional voltage acting on its non-inverting input terminal.
10.1.2 Feedback networks with a non-linear amplifier
In Chapter 6 we explained non-linear amplification in class B, AB, and C amplifiers which,
from the point of view of energy efficiency, is of great advantage. We also said that even
with non-linear class B amplifiers we can obtain linear amplification such that in two
complementary class B amplifiers the positive and the negative parts of the input signal are
amplified separately and then, after amplification, they are summed again on the output. We
also mentioned that major non-linear distortions of so arranged class B push-pull amplifiers
are usually removed with the help of negative feedback.
In. contrast to this, the following text will focus on problems of separate processing of the
positive and the negative parts of the signal to be amplified in complementary feedback
structures. They form the basis for creating circuits known as operational rectifiers, absolute
value amplifiers, and the signal-shaping circuits derived from them.
Fig. 225. Non-linear operational amplifier with two complementary class B amplifiers at the output: a)
general configuration, b) amplifier with complementary emitter followers, c) amplifier with divider diodes, d)
the transfer characteristics
Non-linear operational amplifier. At present, the simplest method for realizing a class
B amplifier with two complementary outputs is to employ a general-purpose integrated
operational amplifier and to complement it at the output with two complementary simple
class B amplifiers as shown schematically in Fig. 225a. These amplifiers need not amplify
voltage; they can operate, for example, as complementary emitter followers as indicated in
Fig. 225b, or they can even degenerate to passive unidirectional conducting elements diodes, as shown in Fig. 225c. Fig. 225d shows the transfer characteristics of a currently used
operational amplifier to which silicon diodes have been connected to divide the signal into
the positive and the negative part. For brevity, the operational amplifier with two separate
complementary outputs will be referred to as the non-linear operational amplifier.
As in the case of the operational amplifier, a feedback network can also be connected to the
non-linear operational amplifier but with the difference that for each polarity of the signal a
separate feedback path must be realized. Because of the large amplification of the preconnected amplifier there is a sharp decomposition of the output voltage (or current) after
polarity. The non-linear operational amplifier with a connected feedback network has
therefore two transfer characteristics in the form of a piecewise linear line with a very sharp
change in the slope, which cannot be achieved by conventional non-linear circuits. The above
circuit configuration with feedback makes it possible to reduce to a negligible level the
unfavourable properties of the circuits or elements used for decomposition after polarity.
When connecting a feedback network to the non-linear operational amplifier, we shall
assume that class B amplifiers do not invert signal polarity; in the contrary case, the inverting
and non-inverting input terminals of the operational amplifier must be interchanged. In the
analysis of the properties of feedback structures with a non-linear operational amplifier we
shall start from the assumptions that the input voltage of the amplifier is approximately zero,
vi ≈ 0, that there is no current flowing to the input terminals, ii ≈ 0, and that the voltage which
is decisive for switching on or cutting off class B amplifiers is zero.
Inverting connection of the non -linear operational amplifier. The parallel manner
of applying linear negative feedback to the non-linear operational amplifier is obvious from
Fig. 226a.
Fig. 226. Inverting non-linear amplifier with a linear feedback network and the corresponding transfer
characteristics
The circuit operates on the principle of the inverting amplifier. If a positive polarity voltage,
v1 > 0, is applied to the input, a negative voltage develops across the output of the operational
amplifier OA. This voltage is processed by the amplifier BN so that the feedback loop is
completed through the resistor RN. The amplifier BP is now inoperative, and there is a zero
voltage, v2 = 0, at its output. For the node S it holds that iA + iP + iN + ii = 0 and since ii ≈ 0
and iP ≈ vP/RP ≈ 0, it simply holds that iA ≈ - iN. The output voltage is therefore v3 ≈ v1RN/RA.
If a negative voltage v1 < 0 is applied to the circuit input terminals, a positive voltage arises at
the output of the amplifier OA. This voltage is processed by the amplifier BP and the
feedback loop is in this case completed through the resistor RP. The amplifier BP is cut off,
and there is a zero voltage, v3 = 0, across its output. Since iA ≈ -iP, the output voltage is
v2 ≈ v1RP/RA.
Thus the circuit operation can be described analytically as follows:
(10.20)
(10.21)
The transfer characteristics v2(v1) and v v3(v1) are in the shape of a piecewise linear line with
the breakpoint in the origin of co-ordinates, as can be seen from Fig. 226b. It is obvious that
the properties of the amplifiers BP and BN (e.g. the difference voltages vBE between the input
and the output of the emitter followers in the circuit of Fig. 225b) do not show in the resultant
relations. As will be seen later, these results are valid on the assumption that only passive
resistive loads are connected to the circuit.
Note that the output resistance of the circuit of Fig. 226a changes in dependence on the
input voltage polarity:
(10.22)
(10.23)
Here, RoP and RoN denote the output resistances of the amplifiers BP and BN, respectively, in a
region where their amplifications are zero (i.e. their active elements are cut off).
The shape of the transfer characteristics of the inverting non-linear amplifier provided with
linear feedback predetermines its application. It can be applied as a precision clipper, as a
half-wave rectifier, or as a basic building block of more complicated circuits such as fullwave rectifiers and signal-shaping circuits.
Inverting non -linear amplifier with active loads. If an active load is connected to
the output terminals of an inverting non-linear amplifier, the performance of the circuit will
be affected considerably. This situation is encountered in cases when, connected to the output
terminals of the amplifier, is an arbitrary active circuit (with a voltage or a current source) or
a circuit with storage properties which, by the principle of compensation, can be replaced by
such an active circuit. For analysis, we can consider that connected to the output terminals of
the inverting non-linear amplifier are two active loads (Fig. 227a) in which the effect of the
output resistances RoP and RoN of the amplifiers BP and BN; in the cut-off state has been
included.
To determine the behaviour of the circuit of Fig. 227a, we shall first assume that the two
feedback loops are interrupted at points marked X, and that the voltage vi ≈ 0. All the circuit
quantities will be denoted for this case by the additional subscript X. On the above
assumptions, the voltages across the output terminals 2 - 2' and 3 - 3' will be
(10.24)
If v2X v3X, we can determine which of the two amplifiers BP and BN; is in the active mode
on the basis of this consideration. Let us determine the (fictitious) current
(10.25)
which in the given conditions would flow to the input terminal of the amplifier OA. From its
polarity, we can determine the polarity of the output voltage of the amplifier OA and,
consequently, which of the amplifiers BP and BN will amplify. From Eq. (10.20 we derive
that
(10.26)
where
(10.27)
If we now connect the two feedback loops to the non-linear operational amplifier, then for v1
> Vr the amplifier BN will operate while the amplifier BP will be cut off; conversely, for v1 <
Vr the amplifier BP will be in operation and the amplifier BN will not amplify. In the special
case, when v1 = Vr, both amplifiers are cut off and the output voltages v2 = v2X and v3 = v3X.
The behaviour of the inverting non-linear amplifier with active loads, for which v2X v3X,
can be described by the transfer characteristics
(10.28)
(10.29)
An example of the transfer characteristics is given in Fig. 227b.
The situation is different in the case which is characterized by the condition v2X < v3X, where
the voltages v2X and v3X are given by Eq. (10.24). In a circuit dimensioned like this it occurs
that over the interval characterized on the output side by the condition
the
two amplifiers BP and BN are always operative at the same time. This disturbs the correct
operation of the linear feedback in this region, and the output characteristics exhibit a curved
knee in this region, as can be seen from Fig. 227c. The intersections of the two characteristics
lie on the straight line v2,3 = v1R2R3/(R1R2 + R1R3).
Fig. 227. Inverting non-linear amplifier with a linear feedback network and active loads; the corresponding
transfer characteristics