Download Section H2: Preliminary Material

Section H2: Preliminary Material
In this section, we are going to briefly review topics in s-domain analysis
and transfer functions, as well as discussing the frequency characteristics of
capacitive components.
s-Domain Analysis
Most of our work will be performed in the complex frequency domain, or
s-domain, where s=jω and j = − 1 . In s-domain, capacitors are replaced
by either their admittance, given by YC=sC, or their impedance, given
ZC=1/sC. Inductors are replaced by an admittance of YL=1/sL or an
impedance of ZL=sL. All circuit analysis techniques are still utilized and all
circuit parameters and behaviors may be defined in terms of s-domain
impedances. For example, in the s-domain, the frequency dependent voltage
gain may be expressed as a transfer function, T(s), as
T (s) =
Vo (s)
.
Vi (s)
Note: The discussion that follows holds for all transfer functions. The
transfer function notation, T(s), is used above instead of our usual
Av(s) to keep stuff as general as possible.
Once the transfer function has been determined, it may be evaluated for
physical frequencies by replacing s with jω. The resulting complex transfer
function T(jω) may be defined in terms of a magnitude and an angle for all
values of ω. The magnitude, |T(jω)|, provides the magnitude of the amplifier
response with respect to frequency and the angle of T(jω) gives the phase
response as a function of frequency.
In general, we are going to be dealing with transfer functions of the form
T (s) =
am s m + am−1 s m−1 + K a1 s 1 + a0
s n + bn −1 s n −1 + K b1 s 1 + b0
, where
(1)
¾ the order of the numerator (m) is less than or equal to the order of the
denominator (n), and
¾ the coefficients (all a’s and b’s) are real numbers.
In addition, if all the roots of the denominator have negative real parts, the
system is stable… very important and we’ll get to it shortly.
The transfer function of Equation 1 may also be expressed in the form:
T (s) =
am (s − Z1 )(s − Z 2 ) L (s − Z m )
, where
(s − P1 )(s − P2 ) L (s − Pn )
(2)
¾ Z1, Z2,…,Zm are the roots of the numerator polynomial, called the zeros
of the transfer function,
¾ P1, P2,…,Pn are the roots of the denominator polynomial, called the poles
of the transfer function, and
¾ am is the multiplicative constant (the coefficient of sm in the original
numerator polynomial).
A transfer function is completely defined in terms of its poles, zeros and the
multiplicative constant. Poles and zeros may be real or complex numbers.
If complex, poles (or zeros) must occur in complex conjugate pairs. For
example, if a root of the denominator (pole) occurs at 6+j3, there is
automatically another pole at 6-j3 (note that the real part of a complex
conjugate pair may be equal to zero; i.e., for this example with a zero real
part the pair would consist of +j3,-j3).
Since we are focused on linear amplification, the amplifiers we are interested
in may be analyzed as linear systems where the complete frequency
response is given by the magnitude and phase shift for all input frequencies.
We will be concentrating on the Bode plot method, which will allow us to
analyze and generate frequency plots almost by inspection after we define
the poles and zeros of the system transfer function as shown in Equation 2
above.
Capacitive Considerations
Before we start talking about specific frequency characteristics and
responses, let’s look at what actually happens to the frequency dependent
components – specifically external capacitances - that we have previously
considered as ideal. We have used bypass and coupling capacitors to couple
stages, provide dc isolation, and reduce or eliminate the effects of certain
resistors. In our studies to date, the capacitor was considered an open to dc
and
frequencies
below
the
operational range and a short to
frequencies in the operational
range. This allowed complete
blocking of dc bias voltages and
any propagating dc signal while
allowing
all
desired
signal
components to pass without attenuation, for a frequency response as
illustrated in the figure to the right.
Realistically, capacitors do not instantaneously switch from a short circuit
condition to an open circuit condition. Recall from circuit theory that
capacitive impedance is given by
ZC =
1
1
=
,
jωC
sC
where ω is the radian frequency (in radians per second) and is related to the
cyclical frequency, f (in cycles per second or Hertz) by ω=2πf. This means
that the coupling and bypass (external) capacitances cannot respond as
ideal, and that they gradually take on open circuit characteristics as the
frequency gets smaller. Also, in addition to any capacitances that are
intentionally placed in a circuit, unintentional capacitances also exist.
Remember that a simple parallel plate capacitor is composed of two
conducting plates with a non-conducting (dielectric) medium between them.
Using this concept, we can see that internal capacitances exist within
semiconductors in the junctions and depletion regions, between contacts,
and between conductors in IC layouts. We’re going to be talking more about
these internal capacitances when we discuss high frequency behavior but,
usually, their overall effect is to cause a decrease in gain at high frequencies
by effectively shorting the output signal.
The frequency dependent behavior of capacitive impedance, regardless of
the source of capacitance, forces a modification to the ideal gain versus
frequency plot presented
above. A generic gain
versus
frequency
response plot for an RC
coupled
amplifier
is
presented to the right
(Figure 10.1 of your
text, slightly modified).
As previously mentioned,
high frequency behaviors
are usually determined
by
the
internal
capacitances, while the
external capacitances (coupling and bypass) are responsible for lowfrequency response.
The frequency axis in the figure above indicates the two possible labeling
schemes; i.e., linear frequencies in Hz (cycles per second) or radian
frequencies in radians per second. Don’t let this confuse you, but be sure
that you know which frequency representation is being used! Remember,
ω=2πf and f=ω/2π.
Note that there are three distinct regions defined in Figure 10.1. In the midfrequency range (also called midband) the gain is maximum and the
frequency response is (ideally) flat. The corner frequencies, which are also
called break frequencies, half-power frequencies, -3dB frequencies,
or cutoff frequencies, are indicated in the figure above by fL and fH (or ωL
and ωH) and define the lower and upper limits of the midband range.
Whatever name is used to refer to these points, they are by definition the
frequencies at which the current or voltage gain drops to 0.707 ( = 1 2 ), or
(
)
–3dB ( = 20 log 1 2 ), of the midrange value. All frequencies below fL (ωL)
are considered to be in the low frequency range, while those above fH (ωH)
are in the high frequency range.
Rather than try to include all the effects of all possible capacitances at one
time, which is virtually impossible to solve without a computer, we are going
to break up the analysis into separate steps involving the low and high
frequency regions and input and output circuits. For the low to midfrequency ranges, we can use the transistor models we have been using so
far. High frequency analysis will involve the development of transistor
models that include the capacitances between each pair of transistor
terminals…we’ll get to that later!
Before we go further, let’s have a definition review in an attempt to
reduce the number of parentheses in the following discussion:
¾ pole: root of the denominator of the system transfer function or,
equivalently, a frequency that causes the denominator of the system
transfer function to go to zero.
¾ zero: root of the numerator of the system transfer function or,
equivalently, a frequency that causes the numerator of the system
transfer function to go to zero.
¾ decade: a factor of ten difference in frequency. For example, a decade
above 500Hz would be 5kHz and a decade below 500Hz would be 50Hz.
¾ octave: a factor of two difference in frequency. For example, an octave
above 500Hz would be 1kHz and an octave below 500Hz would be 250Hz.
¾ dominant pole: a pole that is separated by at least a decade (your
author states two or three decades while other sources define a
separation of two octaves as sufficient) from its nearest neighbor. A
dominant pole for the low frequency response would be at least a decade
above its nearest neighbor, while a dominant pole for the high frequency
response would be at least a decade below its nearest neighbor.
To perform hand analysis (and design) of frequency dependent amplifiers,
we are going to want to simplify things as much as possible while still
maintaining a reasonable level of realism. Specifically, we will be looking at
the location of the poles of the transfer function. If we can define a
dominant pole, any other poles would have little or no effect on the
frequency response of interest and the frequency of the dominant pole would
be the cutoff frequency (or whatever other name you may want to call it).
However, if the poles are not widely separated and we cannot define a single
dominant pole, or if the circuit is complex and it is not obvious that a
dominant pole exists, we can determine the cutoff frequency through the
following procedure:
1. Find the time constant associated with each capacitor. First, set all
independent sources to zero (short voltage sources and open current
sources), then select a capacitor and eliminate all other capacitors in the
circuit. Now…how you accomplish this depends on whether you are
performing low or high frequency analysis.
¾ For low-frequency analysis, external capacitances are dominant.
Capacitances (other than the one of interest) are set to infinity; i.e.,
short circuits. All internal capacitances are considered opens.
¾ For high-frequency analysis, internal capacitances are dominant.
Capacitances (other than the one of interest) are set to zero; i.e.,
open circuits. All external capacitances are considered shorts.
For each capacitance Ci, we are left with a circuit that contains one
capacitor and a combination of resistances. The equivalent resistance,
Reqi, across the capacitor terminals is calculated and the product of this
equivalent resistance and the capacitance yields the time constant
associated with the particular capacitance (τi=CiReqi).
2. Determine the cutoff frequencies from the time constants. The
reciprocal of each time constant represents the frequency associated with
the individual pole (ωi=1/τi). If the frequencies are widely separated, a
dominant pole may be defined. If not, the poles will interact and the
cutoff may be crudely approximated by
ωL ≅
1
∑τ
i
i
=
i
∑C R
i
i
eqi
for the low frequency cutoff, and
ωH
⎡
⎤
≅ ⎢∑ τ i ⎥
⎣ i
⎦
−1
=
1
for the high frequency cutoff,
∑ C i Reqi
i
where each summation is the superposition of the contributions of all
relevant capacitances.
If poles (and zeroes if necessary) are known, can be determined, or if the
crude approximation is not sufficient, a more accurate approximation for
the cutoff frequencies may be found through
ω L ≈ ω P 1 + ω P22 + L − 2(ω Z2 1 + ω Z2 2 + L)
2
ωH ≅
.
1
1
ω P21
+
1
ω P22
+
1
ω P23
⎞
⎛ 1
1
1
+ K − 2⎜⎜ 2 + 2 + 2 + K⎟⎟
⎠
⎝ ω Z1 ω Z 2 ω Z 3