Amplifier Design 705(20-4)An elegant form in which to express Eq. 20-1 is
obtained by letting S=jZ with Z equal to 2Sf and Z 0
equal to 2Sf 0. Eq. 20-1 after substitution and simplifica-
tion becomes
(20-5)Eq. 20-5 is the statement of Eq. 20-1 in the language
of the Laplace transform, which is really the basis for
transfer function analysis. It is worthwhile at this point to
note that Eqs. 20-1, 20-2, and 20-5 are alternative ways
of expressing the transfer function of the simple ampli-
fier under discussion. The form in Eq. 20-5 is that which
is used most often in practice because of its simplicity.
If S were allowed to assume any possible value,
whether it be real, imaginary, or complex, such that all
points in a two-dimensional complex plane were acces-
sible, there would be only one value of S in Eq. 20-5 for
which the denominator would become zero and A would
become infinite. That value of S is when. It
is said then that Eq. 20-5 has a single pole located at
. The pole order of a transfer function is deter-
mined by the power of S appearing in the denominator.
A two-pole amplifier would have an S 2 , a three-pole an
S^3 , etc., appearing in the denominator of the transfer
function. In the steady state as opposed to transient state
recall that S is restricted to the values and the
only accessible points lie on the positive imaginary axis
because the physical frequency values must be positive.
In the steady state even though the value of S never
coincides with the location of our example pole, the
pole location nevertheless influences the operation of
the amplifier. Changing the pole location in effect
changes the value of Z 0 and hence changes the value of
the transfer function at all frequencies other than zero
frequency.
A further study of the Laplace transform and the
inverse Laplace transform indicates that the transfer
function is a description also of the device’s impulse
response in the complex frequency plane while the
inverse Laplace transform of the transfer function is the
description of the device’s response to an impulse
described in the time domain—i.e., it is the device’s
transient response to an impulse expressed as a func-
tion of time. An important consequence of this is that in
order for a device to exhibit a transient response that
decays with increasing time, all of the poles of the
device’s transfer function must have negative real parts.
The amplifier under discussion satisfies this criterion
with a pole at Z 0 and hence its transient response
decays with time which allows the amplifier to exhibit a
stable steady-state response. If this were not true, the
device would not be useful as an amplifier.
The information contained in the amplifier’s transfer
function may be depicted in a variety of ways, the two
most popular of which are the Bode and Nyquist
diagrams. The Bode diagram displays Eqs. 20-3 and
20-4 in the form of a graph of plotted
versus and a graph of I plotted versus. Fig.
20-4 is the Bode diagram for the amplifier of the
example.An examination of the Bode diagram of Fig. 20-4
leads to the conclusion that this amplifier is in essence a
low-pass filter having a reference gain of 20 dB, a single
pole, and a half power point at Z=Z 0. The pole order is
deduced from the fact that even though the response is
low pass in nature its asymptotic slope is 20 dB per
decade or equivalently 6 dB per octave. A two-pole
low pass would produce 12 dB per octave, a three-pole
18 dB per octave, etc., in the asymptotic slope.
This same information is displayed in a different
form by means of a Nyquist diagram. A Nyquist
diagram is a graph in the complex plane of Eq. 20-1
plotted under the condition that Z is allowed to take on
all values from zero to infinity. Fig. 20-5 is the Nyquist
diagram for the amplifier of the example.I tan1–- f
f 0
= ----A10 Z 0
S+Z 0=---------------Sj–= Z 0S –= Z 0Sj= ZFigure 20-4. Gain and phase graphs for the example
amplifier.20 dBlogG
logZ logZ20 dB0.1 WO 1.0 WO 10 WO0 dBAsymptote at 20 dBAsymptote at
20 dB/octave0 o–45o–90o