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Chapter 10
Feedback
EE 3120 Microelectronics II
Suketu Naik
Operational Amplifier Circuit Components
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1. Ch 7: Current Mirrors and Biasing
2. Ch 9: Frequency Response
3. Ch 8: Active-Loaded Differential Pair
4. Ch 10: Feedback
5. Ch 11: Output Stages
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Feedback
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Two Stage
Op Amp
(MOSFET)
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Stability
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10.10 The Stability Problem
 In a feedback amplifier, the open loop gain (A) is generally
a function of frequency.
 It is called open-loop transfer function A(s).
 Question: What happens to gain at higher frequencies?
 This has huge implications on stability of the amplifier.
EE 3120 Microelectronics II
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10.4.1 The Ideal Case
A  s
(10.81) closed-loop gain t-function: A f  s  
1  A  s β  s 
A  j 
(10.82) closed-loop gain t-function: A f  j  
1  A  j  β  j 
(10.83) loop-gain: L  j   A  j  β  j   A  j  β  j  e



angle

j φ w 
magnitude of gain
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10.4.2 Nyquist Plot (Loop Gain with Varying Freq)
Figure 10.34: The Nyquist plot of an
unstable amplifier
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1) At ω=ω180 , the feedback
becomes positive
2) If the loop gain at ω=ω180
crosses the x-axis to the left of (1,0), the amplifier will be unstable
because Aβ < -1: oscillations will
grow with nonlinearity
3) If the loop gain at ω=ω180
crosses the x-axis exactly at (-1,0),
the amplifier will be unstable
because Aβ = -1: sustained
oscillations
4) If the loop gain at ω=ω180
crosses the x-axis to the right of (1,0), the amplifier will be stable
5) If the Nyquist plot encircles (1,0), then the amplifier will be
unstable
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10.4.1 The Ideal Case
(10.84) instantaneous voltage: v  t   e 0t ent  ent   2e 0t cos nt 
(10.85) feedback-ampflier pole constraint: 1  A  s  β  s   0
A0
(10.86) open-loop transfer function: A  s  
1  s / P
(10.87) closed-loop transfer function: A f  s  
A0 / 1  A0  
1  s / P 1  A0  
(10.88) pole: Pf  P 1  A0  
A0P
(10.89) closed-loop transfer function: A f  s  
 A  s
s
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10.11. Effect of Feedback on the Amplifier Poles
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Figure 10.35: Relationship
between pole location and
transient response.
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10.11 Effect of Feedback on the Amplifier Poles
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Figure 10.36: Effect of feedback on (a) the pole location and (b) the frequency
response of an amplifier having a single-pole, open-loop response.
EE 3120 Microelectronics II
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10.12 Stability Study Using Bode Plots
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Since the open-loop gain A(s)*β = 1
at low frequencies, we define
A(s)*β= 1ejθ, where
1)β= feedback factor at low
frequencies
2) θ=180-phase margin (PM)
At low frequencies closed-loop
gain=(1/β)
At phase margin=45, closed-loop
gain=1.3(1/β)
At phase margin=70, closed-loop
gain=0.87(1/β)
1
PM 
BW
The stability of the feedback
amplifier reduces as the phase
margin reduces
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10.12 Stability Study Using Bode Plots
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The stability of the feedback
amplifier can be determined
directly from the plot of A(s)
(open-loop gain frequency
response)
After plotting A(s), we look at
the phase at 1/β
phase margin (PM) = 180-phase
If the phase < -180deg:
amplifier will be unstable
If the phase is very small:
amplifier will be stable but the
BW will be small
If the phase is about 110-120
deg: stable with acceptable BW
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10.13 Miller Compensation and Pole Spitting
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Problem: open-loop response
A(s) shows instabilty
Solution: shift the response to
the left so that the phase
angle is positive and lies
between 110-120 deg
-While shifting, we end up
reducing the BW and desired
DC gain.
-We can shift the pole at the
intersection of 1/β and A(s)
curves to the right by
introducing compensation
capacitor.
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10.13 Miller Compensation and Pole Spitting
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C1 and C2 include the Miller component due to Cμ
R1 and C1 = total resistance and capacitance at the input
R2 and C2 = total resistance and capacitance at the output
Cf = compensation capacitor
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Compensation Capacitor in Two-stage BJT Op-amp
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Suketu Naik
Compensation Capacitor in Two-stage CMOS Op-amp
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Suketu Naik
List of Problems
Feedback and Stability
p10.82: stability of op amp with feedback
p10.92: phase margin of op amp
p10.99: Miller capacitance compensation
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Suketu Naik
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Summary
 Negative feedback is employed to make the amplifier gain less
sensitive to component variations; to control input and output
impedances; to extend bandwidth; to reduce nonlinear distortion;
and to enhance signal-to-interference ratio
 The advantages above are obtained at the expense of a reduction in
gain and at the risk of the amplifier becoming unstable (that is,
oscillating). The latter problem is solved by careful design
 For each of the four basic types of amplifier, there is an appropriate
feedback topology. The four topologies, together with their
analysis procedures, are summarized in Table 10.1.
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Summary
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 The key feedback parameter are the loop gain (A), which for
negative feedback must be a positive dimensionless number, and the
amount of feedback (1+A). The latter directly determines gain
reduction, gain desensitivity, bandwidth extension, and changes in
input and output resistances
 Since A and  are in general frequency dependent, the poles of the
feedback amplifier are obtained by solving the characteristic equation
1+A(s)(s) = 0
 For the feedback amplifier to be stable, its poles must all be in the lefthand side of the s-plane.
EE 3120 Microelectronics II
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Summary
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 Stability is guaranteed if at the frequency for which the phase angle
of A is 180O, |A| is less than unity; the amount by which it is less
than unity, expressed in decibels, is the gain margin. Alternatively,
the amplifier is stable if, at the frequency at which |A| = 1, the phase
angle is less than 180O, the difference ifs the phase margin
 The stability of a feedback amplifier can be analyzed by constructing
a Bode plot for |A| and superimposing it on a plot for 1/||. Stability
is guaranteed if the two plots intersect with a difference in slope no
greater than 6dB/decade.
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Summary
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 To make a given amplifier stable for a given feedback factor, the
open-loop frequency response is suitably modified by a process
known as frequency compensation.
 A popular method for frequency compensation involves connecting
a feedback capacitor across an inverting stage in the amplifier.
This causes the pole formed at the input of the amplifier stage to
shift to a lower frequency and thus become dominant, while the
pole formed at the output of the amplifier stage is moved to a very
high frequency and thus becomes unimportant. This process is
known as pole splitting.
EE 3120 Microelectronics II
Suketu Naik