5. Frequency Response Considerations
5.1
Introduction
Up to now, consideration has been confined to purely resistive
feedback circuits and the op-amps have been largely taken as ideal apart
from the treatment of a finite CMRR. Another important practical
limitation of op-amps is their finite bandwidth and the associated
frequency response. Internally, the bandwidth of the op-amp is limited by
the effects of frequency dependent elements of the transistors. Parasitic
capacitances present limit the frequency of operation of individual gain
stages.
5.2
Frequency Response of a Single Inverting Gain Stage
The simplified, equivalent, small-signal model of a MOS transistorbased inverting amplifier stage is shown in Fig. 1. The input resistance is
assumed to be very high and is therefore omitted from the model. The
parasitic capacitive elements have been grouped together and appear as
single equivalent capacitors at the input, Ci, the output, Co, and as a
feedback capacitance, Cf, between output and input of the stage. The
amplification is shown as a current source producing an output current
controlled by the small-signal transconductance, gm.
Cf
i1
i2
Ci
Vi
Vgs1
gm1Vgs1
i3
i4
rO
CO
Fig. 1 Equivalent Circuit of a MOS Transistor Amplifier Stage
VO
To determine the frequency dependent gain, Kirchhoff’s Current
Law can be applied to the output node taking Vgs1 = Vi. Then:
i1  i 2  i 3  i 4
Vi - VO jω Cf
 g m1Vi 
jω Cf Vi  g m1Vi 
VO
 VO jω CO
rO
VO
 jω CO VO  jω Cf VO
rO
1

Vi jω Cf  g m1   VO   jω CO  jω Cf 
 rO

jω Cf  g m1 
VO

1
Vi
 jω CO  jω Cf
rO

C 
 g m1  1  jω f 
g m1 
VO


Vi  1

 r  jω CO  Cf 
 O


C 
 1  jω f 
g m1 
VO

 g m1rO
1  jω CO  Cf  rO 
Vi
The low frequency gain is
VO
Vi
ω 0
 g m1rO
and indicates an inverting amplification stage.
2
which is dimensionless
The response is of the general form:

ω 
K 1  j 
ωZ 
VO (ω)
 
Vi (ω)


1  j ω 

ωp 

with K  g m1 rO
g m1
ω

The numerator has a value of Z
Cf
1
ω

p
The denominator has a value of
CO  Cf  rO
In pole and zero terms, the negative coefficient in the numerator indicates
that the zero is located in the right half plane while the positive coefficient
in the denominator indicates that the pole is located in the left half plane.
Consider the numerator frequency dependent term:
2
 ω 
 ω 
ω
 Tan 1  

1 j
 M Z φ Z  1  
ωZ
ω
ω
Z 
 Z

ω0
MZ 1
φ Z  0
ω  ωZ
ω
M Z  2  3dB
φ Z  tan 1  1  45o
MZ  
φ Z  tan 1    90o
The magnitude and phase responses for the numerator term of the
frequency response are shown plotted in Bode diagram form as functions
of frequency in Fig. 2 below.
3
MZ
magnitude
slope
20dB/dec
+3dB
ω
ωZ
log frequency scale
0.1ωZ
ωZ
10ωZ
ω
- 45o
- 45o / dec
phase
- 90o
 Z
Fig. 2 Magnitude and Phase of Numerator Term of Frequency Response
4
Consider the denominator frequency dependent term:
ω
ω
ω
1 j
ωp
ωp
ωp
1
1
x



j
2
2
2
ω
ω
 ω 
 ω 
 ω 
1 j
1 j



1
1
1
ωp
ωp
ω 
ω 
ω 
 p
 p
 p
1 j
Mp 
Mp 
ω0
Mp 1
φ p  0
 ω
1
ω
 p
 ω
1
ω
 p








2
2
;
 ω
φ p  tan 1  
 ω
p





;
 ω
φ p  tan 1 
ω
 p




1
 ω
1
ω
 p




2
ω  ωp
Mp 
ω
1
 3dB
2
φ p   tan 1  1   45o
Mp  0
φ p    90o
Bode plots of the magnitude and phase of the denominator term of the
frequency response are shown in Fig. 3 below.
5
ωp
ω
-3dB
slope
-20dB/dec
Mp
log frequency scale
magnitude
0.1ωp
ωp
10ωp
ω
- 45o
- 45o / dec
phase
- 90o
p
Fig. 3 Magnitude and Phase of Denominator of Frequency Response
6
The numerator expression gives a break frequency at ωz and it can
be seen that the gain rises at a rate of 20dB/dec above this frequency, being
+3dB when ω = ωz. The phase on the other hand is negative going between
0 and -90o being -45o when ω = ωz. It should be noted that in the Bode
approximations the gain only begins to change at the break frequency
while the effect of the phase begins at a decade below the break frequency
and levels off at a decade above the break frequency. Note also that the
negative phase contributed by the numerator term comes from the negative
coefficient of the imaginary term. This in turn comes from the feedback
from output to input via Cf.
The denominator expression gives a break frequency at ωp and it can
be seen that the gain falls off at a rate of -20dB/dec above this frequency,
being -3dB when ω = ωp. The phase response on the other hand is identical
to that of the numerator term going negative between 0 and -90o being -45o
when ω = ωp. The negative phase in this instance is due to the fact that it is
contributed by the denominator inverse term which gives the negative
coefficient of the imaginary term when rationalised.
For the typical values of stray capacitances present in a MOS
transistor gain stage used in an op-amp the associated values of dc gain and
the frequencies of the pole and zero are obtained as:
g m 1rO  100  40dB
;
f p  5.3MHz
;
f z  3.2GHz
The magnitude and phase plots for the frequency response of the amplifier
stage having these characteristics are shown in Fig. 4 below, where it can
be seen that the pole occurs at a much lower frequency than the zero.
7
Gain
(dB)
gmro 40
30
20
-20dB/dec
10
fz
fp
0
500k
5M
50M
0.1fp
fp
10fp
f (Hz)
5G log scale
500M
Cf -10
Co  Cf
-20
0.1fz
fz
10fz f
0
-90
-180
-270
-45o /dec
-45o /dec
-360
Phase o
Fig. 4
Gain and Phase Response for MOS Transistor Single Gain Stage
8
As the frequency is increased, the first noticeable effect is the phase
shift due to the first pole which begins at a decade below the frequency at
which this pole is located. Once the frequency of the pole is reached the
magnitude begins to fall off at a rate of -20dB/decade and the phase reaches
-225O, i.e. -180O-45O = -225O with the inclusion of the inversion in the
amplifier. When the frequency reaches ten times that of the pole the phase
levels off at – 270O, but the magnitude continues to fall. At a frequency of
ten times that at which the zero is located, the phase begins to fall further
due to the effect of the zero. At the frequency of the zero, the phase reaches
– 315O, while the magnitude levels off as the rising effect of the zero cancels
the falling effect of the pole. At a frequency of ten times that of the zero the
phase reaches its lowest point of – 360O and levels off here. The overall
phase is asymptotic to -360O.
5.3
Feedback and Stability
Consider a feedback amplifier for which the closed loop frequencydependent gain is given as:
V
A(ω )
AV  O 
Vi
1  A(ω )β
If A(ω) β  1 ,
VO
1
 and the closed loop gain is independent of A(ω).
Vi
β
From a stability point of view it is the loop gain A(ω)β which is important.
If A(ω)β =  10 o or 1180 o then:
AV 
V0 A(ω ) A(ω )



Vi
11
0
This gives rise to instability in that the amplifier becomes self-sustaining
requiring no input to produce an output. In practice it starts to oscillate.
If β is real and fractional then the amplitude response of the loop
A(ω)β is simply that of A(ω) scaled down by a factor of β. The phase
response of the loop is identical to that of A(ω).
In order to ensure stability of the amplifier the feedback loop must
be prevented from reaching the condition above. To do this, it must be
ensured that the magnitude of the loop gain A(ω )β is less than unity when
the total loop phase A(ω )β (including the inversion through the
amplifier) is 0o or 360o. If the amplifier is inverting this applies to a
feedback loop phase shift of 180 o.
9
5.4
Two-Stage Non-Inverting Amplifier
Consider two stages similar to the previous single stage amplifier
placed in cascade so that the combined gain is non-inverting. This has a
frequency response the form:

ω 
ω
1  j
A 0 1  j
ω Z1 
ω Z2
VO (ω)


Vi (ω)



1  j ω 1  j ω 

ω p1 
ω p2 

where A0 >>1



and ωp1 < ωp2 < ωz1 < ωz2
Fig. 5 shows the open-loop response and the feedback-loop response
for this amplifier where A0 = 105, β = 0.1 and AV = 10. The pole and zero
pair of each individual stage are separated by almost 3 decades, but the
zero of the first stage is close to the pole of the second stage. This has the
effect of arresting the 40dB/dec fall in the gain while increasing the slope of
the phase characteristic. The effect of this is that a loop phase of -180o is
reached before the magnitude of the loop gain has been reduced to unity,
thereby allowing instability to occur. This effect is due primarily to the
existence of the negative-coefficient zeros in the right half plane.
Consequently, high gain amplifiers generally require to be
compensated, either to ensure that the gain is reduced to unity prior to a
phase of -180o being reached, or to lift the phase characteristic so that the
phase of -180o occurs at a higher frequency where the loop gain has fallen
below unity. A very simple approach to this is often adopted in low and
medium grade operational amplifiers by including a dominant pole at a
very low frequency to guarantee adequate gain and phase margins. The
gain and phase responses of such a compensated amplifier are shown in
Fig. 6 below. The gain and phase margins are measures of how far away
the amplifier response is from the boundary of instability and are formally
defined below.
10
Gain
dB
100
Amplifier Open-Loop A( ω)
-20dB/dec
Feedback Loop βA( ω)
80
60
-40dB/dec
40
Gain @ 180o Loop Phase
-20dB/dec
20
10ωz1
0
0.1ωp
-20
10ωp1
0.1ωp2
ωp1
0.1ωz1ωp2
ωz2
.1ωz2
f
ωz1 10ωp2
-40
0.1ωp
0.1ωp2
10ωp1 0.1ωz1 ωp2
ωp1
ωz2
ωz1 10ωp2 10ωz1 .1ωz2
f
0
log scale
45
-45o /dec
90
135
-180o Loop Phase
180
-90o /dec
-225
-270
Amplifier Open-Loop A( ω)
and Feedback Loop A( ω)
-315
-45o /dec
-45o /dec
-360
Phase o
Fig. 5 Gain and Phase Responses of 2-Stage Negative Feedback Amplifier
11
Gain
dB
100
Amplifier Open-Loop A( ω)
80
Feedback Loop
60
A( ω)
-20dB/dec
40
-40dB/dec
20
ωz1
0
log scale
ω
ωp1
ωpc
-20
Gain Margin
-20dB/dec
-40
-60
0
0.1ωpc
ωpc
10ωpc
0.1ωp1
ωp1
.1ωz1 10ωp1
ωz1
10ωz1 ω
-45
-90
-135
-180
-45o /dec
-45o /dec
Phase Margin
-90o /dec
-225
-270
Phase o
Amplifier Open-Loop A( ω)
and Feedback Loop A( ω)
-45o /dec
Fig. 6 Gain and Phase Responses of a Compensated Two-Stage Amplifier
12
Gain Margin: is defined as the difference between the magnitude of the loop
gain A(ω)β and unity at the frequency at which the loop phase is -180o
(neglecting the inversion of the amplifier).
Gain Margin  l / Aω   180o  0  Aωβ dB
  180o
The design recommendation is that this should be at least 6dBs (a factor of
2) to ensure adequate stability.
Phase Margin: is defined as the difference between the loop phase shift
φ Aβ and -180o at the frequency at which the loop gain A(ω)β is unity.
Phase Margin  180o  φ Aβ
Aβ  1
where φ Aβ -ive
This is usually quoted as a positive quantity and the design
recommendation is that this should be at least 30o and is safer when greater
than 45o to guarantee adequate stability.
Often, manufacturers list op-amps as being unity-gain stable. This means
that full feedback with β = 1 can be applied around the amplifier and
stability maintained. Note that under these conditions A( ω)β  A( ω) , the
open loop gain of the op-amp. Sometimes, however, this condition cannot
be satisfied and manufacturers will specify a minimum closed loop gain for
the op-amp. This is really a way of specifying the maximum value of β
which can be tolerated while maintaining stability. This is necessary if the
open-loop gain A( ω) of the amplifier itself is above unity when the phase
A( ω) is 180o. The feedback network itself can also be made frequencydependent and given a phase shift β( ω) to correct the overall phase
response.
13