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1
BASIC CIRCUIT THEORY
Circuit and system design is the creative task of the design engineer, but not that of the
computer. However, the computer is an almost indispensable tool in the design process
as an analyser, a mathematical modeller, a simulator, and an optimiser of electric
networks. An essential part of many systems is the integrated circuit which today is ten
to ten thousand times more complex than the discrete circuit of just a few years ago.
Analysis, which is still the most important design step, must be systematic, accurate,
and economic if it is to be applied to such large circuits. The application of network
theory through matrix methods provides an approach which is suited to digital
computation and is also capable of considerable sophistication under the broad heading
of computer-aided design (CAD).
Many readers will be familiar with some of the network theory to be studied,
particularly in the first two chapters of this book. Chapter 1 briefly reviews basic circuit
theory, and Chapter 2 presents the basic methods of circuit analysis. These are expressed
in matrix form for the development in the remainder of the book of methods for dc, ac,
transient, sensitivity, and tolerance analysis. Chapter 2 should therefore be regarded as
important even if the work is familiar, as it orients analysis towards matrix and computer
methods.
1.1
LUMPED CIRCUITS
Electromagnetic theory tells us that energy is radiated whenever an electron is
accelerated. In any circuit, current varies with time, and energy is therefore radiated
and lost. However, the wavelength of the generated electromagnetic wave is usually
large in comparison with the physical dimensions of the circuit and the energy loss is
therefore negligible.
Circuits in which radiated energy may be assumed negligible are called lumped
circuits, and they are obtained by interconnecting lumped circuit elements.
Of course, lumped elements and circuits are idealised models of physical elements
and circuits. A lumped circuit is assumed to have negligible physical dimensions, and
may be analysed exactly. The corresponding physical circuit, depending on its frequency
of operation, may or may not behave as predicted by the analysis. The circuit analysed,
therefore, is not the physical circuit but a model drawn in the form of a circuit diagram
and subject to a number of assumptions. Within these assumptions, the analysis is
simplified at the expense of accuracy in modelling the behaviour of physical circuits.
1
2
APPLIED CIRCUIT THEORY
The most important assumption throughout this book is that circuits are lumped and
therefore obey Kirchhoff’s laws.
Circuits which are not lumped are called distributed.
Usually the dimensions of a distributed circuit are specifically designed to utilise the
electromagnetic wave properties at their frequency of operation. A television antenna,
for example, is designed to receive some of the energy radiated from another antenna.
The shape and size of the receiving antenna are designed to selectively receive the
required band of frequencies from a particular direction. A less obvious example of a
distributed circuit is the electricity supply system. The wavelength corresponding to
50 Hz is 6000 kM. The supply system is of comparable size, and must therefore be
considered as a distributed circuit.
The most common circuit elements such as resistors, capacitors, inductors, transistors,
voltage sources, should be familiar to the student from first courses in electric circuits
and electronics. The lumped versions of these elements are assumed to have zero physical
dimensions, and therefore interact with the circuit voltage or current at a point. Each
element has two or more terminals by which it is connected to other elements to form a
circuit.
A point in a circuit where two or more terminals are connected is called a node.
A two-terminal element connected between two nodes in a circuit is called a
branch.
The circuit shown in Fig. 1.1(a) has four nodes numbered (0), (1), (2), and (3), and
six branches labelled i1, i2, ... i6 with arrows to show current direction.
The branch voltage is the voltage across a branch, and the branch current is the
current through a branch.
Voltage polarity is illustrated in Figs. 1.1(b) and 1.1(c), where the branch voltage is,
by convention, in a direction opposite to the branch current. Indication of polarity,
either by means of an arrow or by + and – signs, is redundant on the circuit diagram,
since the polarity is fixed by the choice of current direction and by the sign convention.
In circuits with sinusoidal or other time-dependent sources polarity signifies the direction
of positive variation.
i1
i5
i2
2
1
i3
i3
3
V3
E
i6
i4
0
(a)
Fig. 1.1.
+
i3
–
(b)
(c)
(a) A bridged-T circuit; (b), (c) The branch sign convention.
Not all textbooks utilise the same convention for indicating voltage polarity, and
often arrows similar to the one in Fig. 1.1(b) refer to voltage drops. The convention
used here ensures that measured voltages and currents in actual circuits have the same
3
BASIC CIRCUIT THEORY
directions as those on the circuit diagram. Another consequence of the convention is
that the product, branch voltage × branch current, represents the power delivered to the
branch element.
1.2
KIRCHHOFF’S LAWS
Kirchhoff’s current law (KCL) and voltage law (KVL) are the two fundamental laws
which govern the voltages and currents in a circuit. They may be derived from Maxwell’s
equations under the lumped circuit assumption.
Kirchhoff’s current law states that “the sum of the currents entering a lumped
circuit element is zero at all times.”
Kirchhoff’s voltage law states that the sum of the voltages round a closed path in a
lumped circuit is equal to zero at all times.
KCL may be generalised by regarding a node as a circuit element, so that the sum of
the currents entering a circuit node is zero. Also a circuit may be divided into two
halves, each half being regarded as single elements with a larger number of terminals
than usual. The sum of currents entering one half of the divided circuit is zero. KCL
applied to a circuit element, circuit node, and a divided circuit is illustrated by Fig. 1.2.
In each case KCL states that
i1(t) + i2(t) – i3(t) = 0.
i1(t)
i1(t)
i1(t)
i2(t)
i2(t)
i3(t)
i3(t)
Fig. 1.2.
i5(t)
i2(t)
E
i6(t)
i4(t)
i3(t)
Kirchhoff ’s current law.
The circuit division shown in Fig. 1.2 as a broken line will later be defined precisely
and called a cut.
The node equations for the circuit in Fig. 1.2 are obtained by application of KCL at
each node, giving
–i1 – i5 – i6 = 0
i5 – i2 – i4 = 0
i1 + i2 – i3 = 0
i3 + i4 + i6 = 0.
The loop equations for the circuit are obtained by application of KVL round the
closed paths, or loops, as they are usually called. Using the three obvious loops and the
outer loop we obtain
E – v5 – v4 = 0
–v1 + v2 + v5 = 0
4
APPLIED CIRCUIT THEORY
v4 – v2 – v3 = 0
E – v1 – v3 = 0.
Three further equations can be derived for the more complicated loops in the circuit.
These two sets of equations are linear; they express linear constraints on the currents
and voltages in a circuit which exist solely as a result of the circuit connections. The
elements themselves have not yet been considered except for the assumption that they
are lumped. Both the node and loop equation sets may by observed to be linearly
dependent since it is possible to express anyone equation as a linear combination of the
remaining three.
1.3
TELLEGEN’S THEOREM
Tellegen’s theorem applies to any network which obeys Kirchhoff’s laws, it is essentially
a statement “that energy is conserved in a lumped network.” Consider a network
with n branches with branch currents i1, i2, ... in and corresponding branch voltages
v1, v2, ... vn.
Tellegen’s theorem states that the sum Σ nk = 1 v k i k equals zero at all times.
We have already noted that the product of branch voltage and branch current gives
the power delivered to the branch. The total power delivered to the circuit by a branch
containing a source must therefore be absorbed by the remaining branches so that energy
is conserved.
Proof
Consider the branch in a network connected between node r and node s . Introduce
a reference point in the circuit so that node voltages er and es may be defined relative to
it as shown in Fig. 1.3.
ers
irs
s
r
er
es
Reference point
Fig. 1.3.
Tellegen’s theorem.
Then
ers = er – es,
and the power delivered to the branch is given
ersirs = erirs – esirs.
The total delivered branch power for all the branches in the network is therefore
given as the sum
Σ nr = 1 Σ ns = 1 ersirs = Σ nr = 1 Σ ns = 1 (erirs – esirs).
This may be rewritten as
Σ nr = 1 er Σ ns = 1 irs – Σ ns = 1 es Σ rn = 1 irs .
5
BASIC CIRCUIT THEORY
The term Σ ns = 1 irs can be recognised as the sum of all currents entering node r ,
which equals zero by KCL. Similarly Σ nr = 1 irs can be recognised as the sum of all
currents leaving node s , and is therefore also zero. The total of power delivered to the
network branches is therefore equal to zero.
The only assumption made is that the circuit is lumped and obeys Kirchhoff’s laws.
No reference has been made to specific/element values or types, and it is necessary
only that the branch currents obey KCL and the branch voltages obey KVL. The actual
values are arbitrary.
Example 1.1. In the bridge circuit (Fig. 1.4) let the branch voltages and currents
take arbitrary values, and check Tellegen’s theorem.
Assigning arbitrary branch voltages and currents which obey Kirchhoff’s laws we
obtain,
v1 = –12
i1v1 = – 24
i1 = 2
v2 = 2
i2v2 = 2
i2 = 1
v3 = 13
i3v3 = 13
i3 = 1
v4 = 1
i4v4 = 3
i4 = 3
v5 = –11
i5v5 = – 44
i5 = 4
v6 = 10
i6v6 = 50
i6 = 5
Σ 6k = 1 ik vk = 0
The sets of branch voltages and currents should be checked to see that Kirchhoff’s
laws are satisfied, and other sets of values should be tried to check the validity of
Tellegen’s theorem.
i1
i2
i3
i5
i4
i6
Fig. 1.4.
1.4
Bridge circuit.
TIME-INVARIANT CIRCUITS
Kirchhoff’s laws and Tellegen’s theorem apply to any lumped circuit at any instant of
time. In most circuits the elements are normally fixed in value and vary only over a
long period because of temperature changes and ageing.
A circuit with an excitation voltage or current x and an output voltage of current y
with the characteristic y = f(x) is time invariant if the function f is independent of
time.
6
APPLIED CIRCUIT THEORY
Note that the function f depends on the circuit element values such as resistance and
capacitance, and it is these values which must remain constant with time. The circuit
variables x and y are of course expected to vary with time. All circuits considered in this
book are assumed to be time-invariant.
1.5
LINEAR CIRCUITS
A circuit with an excitation voltage or current x and an output voltage or current y is
linear if y is proportional to x; that is, if y = f(x) then ky = f(kx).
Circuit elements are included within this definition of linearity since they may be
regarded as circuits. Resistors, capacitors, and inductors are normally close to linear
within the range of operation specified by the manufacturers. In general, however, all
physical components are non-linear owing to effects such as internal heating, voltage
breakdown, and magnetic core saturation, which alter the component value and depend
on the applied voltage.
As we shall see, the linearity assumption greatly simplifies circuit analysis. It is
important to make this assumption even if, circuits are only approximately linear.
Analytical techniques, some of which exclusively apply to linear circuits, may then be
used to gain insight into circuit behaviour even though approximation is involved in
assuming linearity. The disadvantage in using non-linear analysis, which for accuracy is
normally implemented as a computer programme, is that this human insight is then lost.
The linearity assumption has several important consequences. If we consider a change
Δx in the excitation x, a corresponding change Δy is produced in y. In a linear circuit
these are proportional, therefore
Δy Δx
=
=k
...(1.1)
y
x
and
Δy = f (Δx).
The changes in the circuit voltages and currents produced by variation of the fixed
sources are therefore independent of the nominal values which exist in the circuit. Also,
since
Δy
Δx
=1+
= 1 + k,
...(1.2)
y
x
(y + Δy) = f (x + Δx),
the effects of source variation are additive to the nominal values.
If the variations Δx and Δy are considered small compared with the nominal values x
and y, the analysis for variation only is called small signal analysis. It is performed
separately from the large signal analysis used to determine nominal values. Usually,
large signal analysis is meant to imply non-linear analysis since, as we have seen, large
signals and small signals have proportional effects in a linear circuit.
1+
Superposition
The principle of superposition states that a linear circuit excited by n sources x1, x2, ...
xn , which when applied individually give outputs y1 , y2 ... yn, will give the output Σ nk = 1 y k
when the sources are applied collectively.
7
BASIC CIRCUIT THEORY
Superposition extends the properties of Eqs. (1.1) and (1.2) to linear circuits with
more than one source by stating that the sources do not interact.
Application of this principle means that one complete analysis consists of separate
analyses for each source in the circuit. All other independent voltage sources are short
circuit, and all other independent current sources are open circuit, but any dependent
sources, for example transistor gm, must be left operative.
Example 1.2. Determine the voltage across the resistor R2 in the circuit shown in
Fig. 1.5.
R1
Fig. 1.5.
v 2 = E1
Superposition example.
FG R R IJ FG R + R R IJ
HR +R K H R +R K
2
3
2
3
1
2
=
E2
R2
V2
E1
With E2 short circuit,
R3
V
3
2
3
E1 R2 R3
.
( R1 R2 + R1 R3 + R2 R3 )
With E1 short circuit,
v2 =
E 2 R1 R2
,
( R1 R2 + R1 R3 + R2 R3 )
R2 R3 E1 + R1 R2 E 2
.
R1 R2 + R1 R3 + R2 R3
It is instructive to obtain this result by mesh analysis, since the linear form of the mesh
equations and solution by Cramer’s rule demonstrates that a solution must be obtained
with separate factors for each of the sources present.
When the sources are sinusoidal, the principle of superposition implies that sinusoids
are added in a linear circuit. Product terms, which give rise to harmonics and distortion,
cannot arise. The property of linearity is therefore essential for sinusoidal steady-state
analysis to be valid.
Finally, it should perhaps be noted that fixed sources are non-linear by our definition
of linearity, because their voltage and current are independent. Also, it is reasonable to
expect that a circuit is linear if, it is composed only of inter-connected linear circuit
elements.
therefore,
1.6
V=
SINUSOIDAL STEADY-STATE ANALYSIS
The characteristic equation relating voltage and current is given for inductors by
v(t) = Ldi(t)/dt.
...(1.3)
8
APPLIED CIRCUIT THEORY
and for capacitors by
i(t) = Cdv(t)/dt.
...(1.4)
Both equations involve differentiations with respect to time. Their use in analysis
therefore produces a differential equation which can be solved to give both the transient
and steady-state solutions. These are added to obtain the total response as a function of
time. When the circuit input is sinusoidal and when the circuit is assumed to have
reached the steady state, analysis for transient response may be replaced by sinusoidal
steady-state analysis for frequency response. Here we will briefly recall the main steps
in the transition from analysis in the time domain to analysis in the frequency domain
through rotating vectors, phasors, and complex numbers.
A rotating vector of length V is shown in Fig.
1.6, rotating about the origin of rectangular
coordinate axes at a rate of ω, radians/second in the
w rad./sec.
anticlockwise direction. The projection on to the
horizontal axis represents its instantaneous value
V
as a function of time.
In a linear circuit excited by sinusoidally varying
wt
sources, all voltages and currents may be
V cos wt
represented by rotating vectors. However, linearity
Fig. 1.6. A rotating vector.
ensures that all vectors rotate at the same angular
frequency ω, and it is more convenient to consider the vector stationary with the axes
rotating at ω rad/sec in the opposite direction. The vectors are then called phasors and
the diagram on which they are plotted is called a phasor diagram. A phasor V is plotted
as a straight line of length |V| at an angle θ to the horizontal, where | V | and θ represent
magnitude and phase of the phasor respectively. A phasor diagram portrays the relative
amplitudes and phases of the circuit voltages and currents at a given frequency. Since
sufficient information about a network in the steady state can be obtained from the
phasor diagram, projection of the phasors on to the rotating axes is no longer necessary.
When required, however, the rotational term can be reinserted, giving v(t) = | V | cos
(ωt + θ) for the phasor V.
Analysis proceeds by addition, subtraction, integration, and differentiation of phasors,
and is limited in practice because the technique, either graphically or algebraically, is
cumbersome, the more so as networks become bigger and have large phase shifts. Also,
a phasor diagram relates only to one frequency, whereas in most cases a range of
frequencies are of interest and a frequency response is required.
Another representation of a phasor is derived through Euler’s identity
...(1.5)
e jθ = cos θ + j sin θ,
Imaginary
where, j = –1 .
e jθ may be plotted on the Argand diagram as a
vector of unit length at an angle θ to the horizontal
axis, as in Fig. 1.7. An Argand diagram may be
identified with a phasor diagram, and the phasor
V = V ∠θ may be represented by the complex
quantity Ve jθ since it has magnitude V and
phase θ.
sin q
1
q
Real
cos q
Fig. 1.7.
Argand diagram of e jθ.
9
BASIC CIRCUIT THEORY
If we excite a network by a current i(t) = I cos (ωt + φ) and obtain a response v(t) =
V cos (ωt + θ) we can represent the phasors I and V as Ie jφ and Ve jθ respectively. The
rotational term e jωt is omitted from these expressions because phasors are stationary,
and it is applied to the rotating axes of the phasor diagram instead. To recover the time
domain behaviour the rotation term e jωt must be reinserted, and then the real part taken
to recover the cosine; that is,
i(t) = Re{| I | e jφ e jωt} = I cos (ωt + φ)
v(t) = Re{| V | e jθ e jωt} = V cos (ωt + θ).
It is this cancellation of e jωt which simplifies analysis in the sinusoidal steady state
by removal of trigonometric terms and conversion of differential equations to algebraic
equations.
Substitution of the excitation current I cos (ωt + φ) into the inductor characteristic.
Eq. (1.3) yields
d
{I cos (ωt + φ)}
dt
= ωL I cos (ωt + φ + π/2).
v(t) = L
In terms of phasors,
V = ωL I e jφ e jπ/2
= jωL I.
A similar result can be obtained for capacitors as
...(1.6)
1
I.
...(1.7)
jωC
The quantities jωL and 1/jωC are interpreted as complex impedance. Mutual inductors
may be similarly treated and terminal voltage and current phasors thus shown to be
proportional for all linear circuit elements. Kirchhoff’s laws therefore apply to complex
phasors, and sinusoidal steady-state analysis proceeds in the same way as linear d.c.
analysis. Three forms of steady-state analysis are illustrated by the following example.
Example 1.3. Determine the output voltage of the filter circuit shown in Fig. 1.8.
V=
R1 = 1 W
2V
1 rad/sec
Vin
L = Ö2 H
i1
C = Ö2 F
Fig. 1.8.
i2
R2 = 1 W
Low-pass filter.
The impedance of the two reactive components can be written in three forms.
Component
Numerical
Function of ω
Function of ω and L or C
C
1/ 2 j
1/jω 2
1/jωC
jω 2
jωL
L
2j
10
APPLIED CIRCUIT THEORY
Each of these forms can be used in analysis, giving three solutions of increasing
usefulness in the steady state.
If we use the numerical form of impedance, the mesh equations for the filter are
(1 – 0.7J) i1 + 0.7J i2 = 2
0.7J i1 + (l + 0.7J)i2 = 0.
Solving for i2 gives i2 = – 0.7J,
hence
vOUT = 0.7 ∠ –90°.
In order to analyse the circuit at other frequencies the mesh equations must be
re-formed and re-solved at each frequency required.
If impedance is a complex function of ω the mesh equations become
(1 + 0.7/Jω)i1 + 0.7/Jω)i2 = 2
– (0.7/Jω)i1 + (1 + 1.4 Jω + 0.7/Jω)i2 = 0
Solution for i2 gives i2 = l/{(Jω)2 + 1.4Jω + 1}, and the voltage gain can be written
vOUT
0.5
=
.
...(1.8)
2
vIN
(Jω ) + 14
. ( Jω ) + 1
We can now substitute for any value of input voltage and any value of frequency to
obtain vOUT without re-solving the mesh equations. The frequency response is quickly
evaluated by substituting a range of values for ω into the gain expression. For comparison
with the previous result, substitution of ω = 1 and vIN = 2 yields vOUT = 1/1.4J = –0.7J
as before. The analysis can be taken one stage further, using the symbols L, C, R1, and
R2 in place of numerical values. The mesh equations become
(R1 + 1/JωC)i1 – i2/JωC = vIN
(–1/JωC)i1 + (1/JωC + JωL + R2)i2 = 0.
This can be solved, noting that vOUT = i2R2, giving
vOUT
R2
=
.
...(1.9)
2
vIN
( Jω ) R1 LC + Jω( L + CR2 ) + (1 + R2 )
Numerical values for R1, R2, L, and C can be substituted, giving gain as a function of
ω identical to the previous results in Eq. (1.8) above. The additional advantage of this
symbolic solution is that circuits with different component values may be analysed
without re-solving the mesh equations. Also, the expression for gain can be differentiated
with respect to the circuit components to predict the variation in gain as component
values are varied.
The two expressions (1.8) and (1.9) are called transfer functions. In general, they
take the form of a ratio of two polynomials in the variable Jω. In order to investigate the
properties of transfer functions we must generalise the frequency variable Jω to the
complex frequency variables s = σ + Jω. Where Jω has appeared we now have s. For
example, the transfer function in Eq. (1.9) becomes
F(s) =
R2
.
R1 LCs 2 + ( L + CR2 ) s + (1 + R2 )
and the frequency response is evaluated by the substitution s = Jω. The impedance of
the inductor and capacitor become sL and 1/sC respectively. Analysis to obtain transfer
functions with numerical coefficients is called transfer function analysis or s-domain