Modeling a Complex Pole-Zero System in Terms of its
Low-Frequency/High-Frequency Cutoffs
Introduction
Even a simple amplifier circuit will tend to have multiple poles and zeros in its high-frequency
behavior, and if it is ac-coupled, in its low-frequency behavior as well. The amplifier frequency
response will act to alter a signal passing through it. When the transfer function is complicated, the exact
influence it will have on a signal becomes obscure. However, a good approximation to the way an
amplifier will alter a signal can be gotten from its high-frequency and low-frequency cutoffs.
High-Frequency Response
The high-frequency response of all amplifiers is limited, mainly by the internal capacitances of
the transistors, but also possibly by externally-applied load capacitances and the stray capacitance of
the wiring. These effects can be summarized in terms of the “high-frequency cutoff.” This is the
frequency at the high end of the frequency response plot, denoted by fH, at which the amplifier gain is
3 dB less than its midband value. The amount “3 dB” is somewhat arbitrarily chosen, mainly because it
results in a easier computation. If the exact transfer function is known, the exact high-frequency cutoff
can be calculated; however, the computation is rather tedious and requires finding the roots of a
polynomial. An approximation which can be used to accurately estimate fH when the i poles and zeros
of the high-frequency response function FH(s) are known explicitly is the following:
z H = 2of H j
1 2
z pi
Si
− 2S
i
1 2
z zi
− 12
(1)
Another approximation which can be used when the transfer function is not known is the method of
open-circuit time constants. This method requires only knowledge of which capacitors in the circuit act
to limit the high-frequency response. Assuming that there are k capacitors, the approximation is:
z H = 2of H j
Sk C k Rk
−1
(2)
In a well-designed amplifier, there tends to be one dominant pole controlling the high-frequency
behavior. In some cases, such as an oscilloscope vertical amplifier, the designer will take pains to
ensure that this is so. In many cases, the following approximation to the high-frequency response
function is a reasonable one:
F H (s )
j
1+
1
1
s
zH
(3)
In essence, a complicated high-frequency behavior is modeled by an equivalent single-time-constant
low-pass filter having its corner frequency at fH. Careful study of the effects of the basic
single-time-constant low-pass filter on a signal of interest can be used to predict what will happen to the
signal as it passes through an amplifier having the equivalent high-frequency cutoff.
Low-Frequency Response
The low-frequency response of some amplifiers is limited by the use of coupling capacitors.
Such amplifiers are said to be “ac-coupled.” Designs which avoid the use of coupling capacitors are
called “dc-coupled” and have no low-frequency response behavior (their midband extends down to
zero frequency). The effects of the coupling and bypass capacitors which may be present in an amplifier
can be summarized in terms of the “low-frequency cutoff.” This is the frequency at the low end of the
frequency response plot, denoted by fL, at which the amplifier gain is 3 dB less than its midband value.
If the exact transfer function is known, the exact low-frequency cutoff can be calculated; however, the
computation is rather tedious and requires finding the roots of a polynomial. An approximation which
can be used to accurately estimate fL when the i poles and zeros of the low-frequency response function
FL(s) are known explicitly is the following:
z L = 2of L j
Si z pi
2
− 2 S(z zi
)2
1
2
(4)
i
Another approximation which can be used when the transfer function is not known is the method of
short-circuit time constants. This method requires only knowledge of which capacitors in the circuit act
to limit the low-frequency response. Assuming that there are k capacitors, the approximation is:
z L = 2of L j
Sk C 1R
k k
(5)
In many cases, the following approximation to the low-frequency response function is a
reasonable one:
F L (s )
j
s
s + zL
(6)
In essence, a complicated low-frequency behavior is modeled by an equivalent single-time-constant
high-pass filter having its corner frequency at fL. Careful study of the effects of the basic
single-time-constant high-pass filter on a signal of interest can be used to predict what will happen to the
signal as it passes through an amplifier having the equivalent low-frequency cutoff.
Example
Consider an ac-coupled amplifier with the following transfer function.
2
T(s )
=
[1 ]
s
1+ 1e6
s(s+20 )
(s+10 )(s+100 )
s
1+ 100K
s
1+ 100K
(7)
The transfer function has been arranged into the standard form which separates midband gain, FL(s),
and FH(s). The poles and zeros of both FL(s) and FH(s) are explicitly known because the expressions
have been given in factored form. Note that FL(s) has a dominant pole at -100 /s (although it is barely
dominant because the nearest singularity, the zero at -20 /s, is just a bit more than 2 octaves removed).
FH(s) has a double pole at -100K /s, so it cannot be said to have a dominant pole at all. Equation (1) is
used to estimate ωΗ = 71.1 K/s, and (4) is used to estimate ωL = 96.4 /s. Based on these estimates,
the actual transfer function (7) can be replaced by the approximate transfer function:
T(s )
j
[1 ]
s
(s+96.4 )
1
s
1+ 71.1K
(8)
Both the actual transfer function (7) and the approximate transfer function (8) were simulated
using PSpice. There is a convenient device called “LAPLACE” which allows the user to simply type in
the s-domain polynomial representing a transfer function. This device acts as an ideal voltage amplifier
having that transfer function. PSpice was used to produce the frequency response magnitude, the pulse
response for a lengthy pulse, and the pulse response for a fast (short) pulse. The schematic used and
these three response plots follow.
Comments on the Frequency Response
The two traces on the frequency response magnitude plot represent the actual transfer function
(VDB(R1:2)) and the approximate transfer function (VDB(R2:2)). Note that the syntax
“VDB” results in the decibel-equivalent of the voltage, with respect to a 1-V reference. It can be seen
that the actual and approximate magnitudes track well through the midband and somewhat past the lowand high-frequency cutoffs. The actual transfer function shows its more-complicated behavior in the
low- and high-frequency regions. However, because the gain is much reduced in these regions,
accuracy here is often not needed so that the approximation is justified.
Comments on the Long-Pulse Response
A time-domain simulation of the response of the two transfer functions to a 10-ms 1-V pulse
shows that a system having limited low-frequency response cannot faithfully reproduce a long pulse.
When the 1-V step change first occurs, it can be seen that the ac-coupled system will initially show a
1-V change at its output, but that the response will “sag” back towards zero. The amount of sag,
normalized with respect to the initial step height, is given by:
DS
S
= 1 − exp(−T z L )
(9)
In (9), T is the width of the pulse and ωL is the low-frequency cutoff (in radian measure). The sag
predicted by (9) is 61.9% (based on ωL = 96.4 /s); the sag observed from the actual system is 57.4%.
3
The agreement between the simplified system and the actual with regard to sag is very good for
engineering purposes. It can be seen from (9) that a long pulse can only be reproduced faithfully by a
system having good low-frequency response.
Comments on the Rise/Fall Times
A time-domain simulation of the responses of the two systems to a 10-µs 1-V pulse shows that
a system having limited high-frequency response cannot reproduce a fast-changing signal. While the
1-V change in the input signal seems to take place instantly on this time scale, the output exhibits “rise
time” as it catches up with the step change. Rise time is usually defined as the time needed for the
output to transition from 10% to 90% of its ultimate change. For a linear system, the rise and fall times
are identical. The rise time is given by:
tR =
2.2
zH
=
0.35
fH
(10)
Here ωH is the high-frequency cutoff. The rise/fall time predicted by (10) is 30.9 µs, based on
ωH = 71.1 K/s. The rise/fall time observed from the actual system is 33.0 µs. The agreement between
the simplified system and the actual with regard to rise time is also very good for engineering purposes.
It can be seen from (10) that a step change can only be reproduced faithfully by a system having good
high-frequency response.
It should be noted that the long-pulse waveform previously discussed also exhibits a 33-µs rise
time. It is just not visible on the time scale used to display the long pulse.
R. King
October 2001
4
2
1
VDB
V1
1V
+
s*(s+20)
(1+s/1e6)
(s+10)*(s+100)
(1+s/100e3)*(1+s/100e3)
R1
-
LF
Note: The boxes are the "Laplace"
device - just type in the s-domain
functions!
1k
HF
Complex System
0
0
VDB
s
B
1
B
(1+s/71.1e3)
(s+96.4)
R2
LF
1k
HF
Approximate Equivalent
based on
lf and hf cutoff frequencies
0
A
A
Electronics II
EECS 3420
-
2
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V(V1:+)
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V(R2:2)
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