PROFESSOR’S NOTES
version 1.2
FEEDBACK AND STABILITY IN CIRCUIT APPLICATIONS
23-1 Generic nomenclature and definitions
Although feedback is essential for opamp applications, it also is a generic technique that applies to
any type of electrical system or circuit. In almost all forms it is a topology for which a sample of the
output is fed back to the input as a corrective contribution that pulls the circuit back into alignment.
We usually refer to this type of feedback as ‘negative feedback’ since the output must pull back on
the input to align it to the feedback loop. The concept is represented by figure 23.1-1
Figure 23.1-1: Generic context of feedback. Negative feedback is accomplished by insertion at the
subtractive input to the summation (= ‘inverting’ input if opamp).
Note that figure 23.1-1 implies that the result of the feedback will be a transfer ratio:
x0
1

1    
xS
A



1

if A = large
(23.1-1)
Typically x0 = v0 and xS = vS, although generically they can be any electrical quantity.
Notice that for A = large the transfer function is defined by the feedback factor . The feedback
factor can be an RLC construct. All RLC components are linear (i.e. i = Y x v ) so the transfer ratio is
then linear and the output will be an undistorted copy of the input. And so feedback is a good friend
to circuit performance.
If A is not large then we need to translate equation 23.1-1 into the form
x0
A

x S 1  A
(23.1-2)
which is mathematically the same except that it gives more acknowledgement to the gain of the signal
around the loop A. It is appropriate to assess the feedback condition in terms of the loop gain
L  A
 L e j
(23.1-3)
and be aware that loop gain has a phase shift as well as magnitude since A (and  ) should be
expected to have some sort of frequency character.
That brings up the caveat of what happens when the phase shift = 180o and |L| = 1. If so then the
denominator of equation (23.1-2) goes to zero and the transfer gain T(s) goes to impossible. When
the system approaches this condition it becomes unstable. And it will go into oscillation at the
frequency at which the denominator goes to zero.
This is more accurately interpreted by a plot in the complex plane, (also called a Nyquist plot).
Figure 23-1.2: Nyquist plot of feedback loop gain (conceptual). The point at which the circuit goes
to hell is located at (-1,0) (which is the same as |L| = 1 and = 180o ).
From the Nyquist plot we can identify the margins for stability. If at the condition |L| = 1 the phase
shift  is less than 180o, then the Nyquist trajectory will miss the (-1,0) singularity when it comes in
for a landing (converges toward zero). This margin is called the phase margin and is given by
  180 o   M
(also called the PM)
(23.1-4)
where M is the phase shift for which |L| = |A|=1. And that means we have to find the frequency fM
for which |L| = |A|=1. And that is usually a task. And that is why this dinky little equation for
phase margin gets an equation number of its very own.
There is also a gain margin, which is the margin in gain when = 180o . But not one ever uses it. So
we won’t either.
-----------------------------------------------------------------Example: A (bad quality) differential amplifier has gain characteristic A 
A0
1  s / p1 3
with A0 = 60 dB (same as A0 = 103). It is connected in a feedback loop for which  = .004. What is
the phase margin?
Solution: for the condition |A|=1 we have
1  .004 
3
1000
 1   2 p 2 
1


3
which gives
 1   2 p 2   4
1


and if we solve for  = M we get
and we have loop phase shift
 M   4 2 / 3  1  p1  1.233 p1


 M  3  tan 1 ( M / p1 )
 3  tan 1 (1.233 p1 p1 ) 152.8 0
From equation (23.1-4) the phase margin (PM) is then
= 180o - 152.8o = 27.2o
Note: If  is increased to .008 phase margin (PM) will go to zero (and the circuit would go into
oscillation.) You can check this out by replacing the .004 with .008 in the above example.
23.2 GAIN STABILIZATION AND INPUT/OUTPUT IMPEDANCE
The non-ideal opamp has finite input and output resistances (or impedances) for which feedback has
an effect of
Rif  Rin  1  A
ROf  RO 1  A
(23.2-2a)
(23.2-2b)
where Rif = input resistance with feedback and Rof = output resistance with feedback. Since loop gain
A is generally frequency dependent, then Rif  Z if and Rof  Z of .
Whether or not we employ an opamp, we commonly make use of feedback to condition the transfer
characteristics of a circuit. And feedback is a process for which signal is sampled at an output and
fed back to an input. If the signal being sampled is a voltage signal, it is sampled at a node. If the
signal being sampled is current, it does so using a loop since current requires a flow path.
If a voltage is fed back and inserted at the input, it (or actually its negative) must be added to the input
signal by means of a summing circuit. The summing of voltages is accomplished via the two inputs,
since voltages add in series.
However, if the feedback signal is fed to a single input node instead of two separate inputs, then
currents are the quantity summed, since currents sum at a node.
The context of this mouthful is that either voltage or current is sampled at the output and either
voltage or current is inserted back at the input. And how these sampled and inserted quantities are
accomplished is what defines the feedback topology. Different topologies stabilize different types of
gain, and have different effect on the I/O impedances.
A summary of the four different feedback topologies and their effect on the transfer characteristics is
given by table 23.2-1
Type of gain
stabilized
sampled
inserted
voltage
voltage
current
current
voltage
current
current
voltage
vO/vI
vO/iI
iO/iI
iO/vI
Effect on
impedances
Input (Rif)
Output (ROf)
Rin x (1+ A)
Rin / (1+ A)
Rin / (1+ A)
Rin x (1+ A)
Ro / (1+ A)
Ro / (1+ A)
Ro x (1+ A)
Ro x (1+ A)
Table 23.2-1: The feedback topologies and their consequences on (1) transfer gain Aof with
feedback and (2) on input and output impedances (simplified as Rif and Rof, respectively).
Note that there are four topologies associated with how voltage and current have to be sampled and
how they have to be inserted. And in each case the factor D = (1+ A) appears. The factor (1+ A)
is defined as the desensitivity factor because of its effect on the transfer gain.
The desensitivity factor is derived from the definition of sensitivity, namely, how sensitive a
performance factor (such as A) may be to a fractional change in some component. For the sensitivity
of the gain, the fractional sensitivity is dA/A. The sensitivity of gain with feedback is
dA f
Af
dA 1  A  AdA / 1  A

A 1  A
d A 1  A

A 1  A
d  A 1  A

A 1  A
dA 1  A

A 1  A
2

dA A
1  A

2
dA A
D
dA
 10% and the loop gain is such that A = 19, for which the
A
desensitivity factor D = (1 + 19), then the desensitized gain Af is only sensitive to 0.5% in variations
in the gain.
As an example, if gain sensitivity is
23.3 SIMPLEST MULTI-STATE PROFILE: BIQUADRATIC CIRCUITS
A transfer function is always identified in terms of the ratio of two polynomials in s = j.. The
numerator polynomial defines the zeros of the transfer function and the denominator defines the poles
of the transfer function. For second-order transfer functions, for which the numerator N(s) is
quadratic in s and the denominator D(s) is quadratic in s we call the transfer function a biquadratic,
which we cast in the form
T ( s) 
N ( s)
As 2  Bs  C
 2
D( s) s  s  0 Q   02
(23.3-1)
If the transfer function is a consequence of an RLC circuit, a natural resonance is expected since
energy will bounce back and forth between L and C with characteristic frequency f 0   0 / 2 and
quality factor Q for which the bandwidth is
f 
f0
Q
(23.3-2)
The denominator D(s) = s 2  s  0 Q   02 identifies these resonance characteristics. The numerator
N(s) defines the type of profile. Profile types are indicated by table 23.3-1. The profiles that have the
most circuit applications are the band-pass profile
T ( s) 
Bs
(23.3-3a)
s  s  0 Q   02
2
and the low-pass profile
T ( s) 
C
(23.3-3a)
s  s  0 Q   02
2
Strictly speaking, equation (23.3-2) only applies to the band-pass profile, but as represented by the
traces given by figure 23.3-2, the distinction is (essentially) lost when Q > 4.
Type
Transfer function
(a) Band-pass
T ( s) 
Bs
s  s  0 Q   02
2
(Q = 4 represented)
(b) Low-pass
T ( s) 
C
s  s  0 Q   02
2
(Q = 4 represented)
(c) High-pass
T ( s) 
As 2
s 2  s  0 Q   02
(Q = 4 represented)
Profile
(d) Band-stop
T (s) 
s 2  s  0 Q '   02
s 2  s  0 Q   02
(Q = 4 represented)
(e) All-pass
T ( s) 
s 2  s  0 Q   02
s 2  s  0 Q   02
(Q = 4 represented)
Table 23.3-1: Biquadratic profiles: The profile for (e) (all-pass) bears some explanation. It
is actually a phase-shift circuit, for which phase changes as a function of frequency but amplitude
does not.
Note: For the low-pass profile the quality factor Q also represents a factor of its magnitude profile at
resonance |T( = 0)| relative to its magnitude at zero frequency |T( = )|, as
Q
T (   0 )
T (  0)
(23-2.4)
This ratio is represented by figure 23.3-2.
Take note that there are several cases of interest associated with the low-pass profile and quality
factor Q. When Q = 0.5 (the poles are coincident, i.e. p1 = p2). Whrn Q = 1 2 the profile is
maximally flat. When Q =  (not shown) the circuit goes into oscillation (unstable).
Figure 23.3-2: Low-pass profile for several values of Q, i.e. Q = {0.5, 0.707, 2, 4}. The resonance
peak is approximately at f0 for Q > 4 .
Biquadratic transfer functions are realized (1) by RLC components in series or parallel or (2) by RC
circuits that are rendered mildly unstable by feedback.
For the situation in which gain component A has two intrinsic poles
A
A0
(1  s p1 )(1  s p 2 )
(23-2.5)
and is used in a feedback loop with (frequency independent) feedback factor , then
A
A0
A

1  A (1  s p1 )(1  s p 2 )  A0
A0
1
1  A0


 1
1

1  s

 p1 p 2
1
s2 
 

 D p1 p 2 D 
(23-2.6)
for which D = 1+A0, is the desensitivity factor, same as identified by section 23-2..
If we compare denominator of (23-2.6) to the one for the normalized quadratic denominator
D( s )  1 
s 1 s2

 0 Q  02
 02  p1 p 2 D  p1 p 2 (1  A0 )
we then see that
and
Q
(23.3-7a)
0
(23.3-7b)
p1  p 2
These resonance characteristics can also be identified relative to the stability of the feedback loop, as
defined by the condition |L| = 1. When this criterion is applied to equation (23.3-5) we have
1  A 
A0
(1  s p1 )(1  s p 2 )

1  
2
p1 p 2

A0
2
  2 1 p1  1 p 2 
(23.3-8)
2
for which we have a quartic equation in 
 


 4   2 p12  p 22  A0 2  1 p12 p 22  0
(23.3-9)
In order to assess the loop for stability, equation (23.3-9) must be solved for frequency  = M at
which we can define a phase shift M and the phase margin PM = 180o - M. And equation (23.3-9)
is reasonably tractable, with solution
1

 p
 M2   p12  p 22 
2
2
1
 p 22

2



2
 4 A0   1 p12 p 22 

(23.3-10)
Equation (23.3-10) simplifies considerably if we make the assumption that one pole is considerably
larger than the other (which is usually the case). If we assume p2 >> p1, then (23.3-10) becomes
 M2 

p 22 
p2
 1  4 A0 2  1 12  1
2 
p2




(23.3-11)
Equation (23.3-11) becomes even more compact if the gain factor A0 is large (as is typical of an
opamp) and the feedback factor  is sufficient to meet the condition
A0 
p2
2 p1
(23.3-12)
in which case (23.3-11) reduces to
 M2  p1 p 2 A0
(23.3-13)
and if we compare (23.3-13) to (23.3-7a)
M  0
(23.3-14)
Actually, this should be no great surprise, since it is to be expected that a resonance should be
strongly linked to stability.
The relationship between the frequency profile and the phase margin is more evident if we regard
systems for which  = constant. Since A = L = |L|ejt then
A
1

1
L

L e j
(23.3-15a)
At the phase margin frequency  = M, for which |L| =1 we then have
A( M ) 
1
e j

(23.3-15b)
Therefore the transfer function with feedback as defined by (23.1-2) becomes
AF ( M ) 

A( M ) 1
e j m
 
1 L
 1  e j m
1


e j M 2
e  j M 2  e j M
2

1


e j M 2
2 cos( M / 2)
If we look at the magnitude and also acknowledge that
1

cos M 2  cos 180  PM   sin PM 2
2


then
AF ( M ) 
1


1
2 sin PM / 2
which can be rewritten as
(23.3-16a)
(23.3-16)
AF ( M )
AF (0)

1
2 sin PM / 2
(23.3-16b)
since |AF(0)| = 1/, if we make the not unreasonable assumption that A0 = large
Assuming  M   0 , equation (23.3-16b) relates directly to equation (23.3-4) for which we have
sin PM / 2 
1
2Q
(23.3-17)
This equation identifies that resonance and stability are close cousins, if not siblings.
23.4 STABILITY AND FREQUENCY PROFILES
Section 23.4 shows that it is not necessary to use complementary energy storage components (i.e. L
and C) to achieve a resonance condition. Resonances can be achieved by phase shift and instability,
provided that a gain element is included in the topology. Stability characteristic of feedback loops are
associated with phase shift. Resonance is associated with Q. And equation 23.4-17 shows that they
can be directly related.
Therefore any second-order circuit with a gain element in its topology can inspire the necessary phase
shift and gain to create a resonance and produce a biquadratic profile. Many options exist. One of
the more useful topologies is that of the Sallen-Key circuit shown by figure 23.4-1, which has the
merit of being a tuneable-Q topology.
Figure 23.4-1. Sallen-Key single-amplifier biquad (= low-pass biquadratic transfer function)
The Sallen-Key topology relates to two RC time constants, 1 = R1C1 and 2 = R2C2. By electing
these time constants to be equal the transfer function resolves into a concise low-pass biquadratic
form. For the sake of flexibility in the outcome a ‘tapering factor’ C2 = aC1 and R2 = R1/a can be
applied for which parity 1 =2 is maintained.
Analysis shows the merit of these choices. Nodal analysis at nodes vx and vI give
(1)
(2)
v x G1  G2  sC1   v S G1  vO sC1  0
v I G2  sC 2   v x G2  0
If the tapering factor option C2 = aC1 and R2 = R1/a is applied then
v x G1 (1  a)  sC 1   v S G1  Kv I sC 1  0
v I G1  sC1   v x G1  0
(1’)
(2’)
where the simplification vO = KvI is also applied. Eliminating vx between (1’) and (2’) gives
vI 
G12
vS
s 2 C12  sG1C1 2  a   K   G12
This result is cleaner if we divide both numerator and denominator by C12 , for which
vI
G12 C12
 2
v S s  s G1 C1 2  a   K   G12 C12

 02
s 2  s 0 2  a   K    02
Note that we can use  0  G1 C1  1 R1C1 . Since vO = KvI the transfer function is then
T ( s) 
vO
K 02
 2
v I s  s 0 2  a   K    02
(23.4-1)
This equation is of the form of a low-pass biquadratic with 0  1 R1C1 and a Q of
Q
1
2aK
(23.4-2)
The Sallen-Key circuit is also called a tuneable-Q circuit.
A reasonable choice for gain element K is an opamp configured in the non-inverting mode (figure
23.4-2) for which
R
K  1 b
Ra
Figure 23.4-2. Sallen-Key biquad with non-inverting opamp topology for gain element
The tuneable-Q relationship (23.4-2) then becomes
Q
1
1  a  Rb Ra
(23.4-3)
Simulation of the circuit of figure 23.4-2 for Q = 0.707, 2.0, and 5.0, respectively is shown by figure
23.4-3. The value of Q = 0.707 (= 1 / 2 ) is of particular interest since for this value, the low-pass
profile is maximally flat .
Figure 23.4-3. Simulation results of figure 23.4-2 for a Sallen-Key single amplifier biquad
with tapering factor a = 1 and values of Q = {0.5, 0.707, 2, and 5}
The use of phase shifts and loop instability is more evident in the class of circuits called twointegrator loops. Since the transfer function of an integrator is of the form
T ( s)  

1
 0
sRC
s
(23.4-4)
for which successive stages will change the profile to another biquadratic state.
This is evident in the case of the two integrator loop shown by figure 23.4-4, for which one of the
integrators is a lossy integrator. This topology is also called the Tow-Thomas or Ring-of-three
biquad. The necessary phase shift to approach loop instability is accomplished by the use of the two
integrators plus an inverter, as represented by the figure.
Figure 23.4-4. Tow-Thomas Biquad
The input to the lossy integrator also serves as a summing point. Therefore its output (= v2 ) is
v2  
G3
G4
vS 
v1
G1  sC1
G1  sC1
(23.4-5)
and since
v1  v3  
G2
v2
sC2
(23.4-6)
then
v2 
s C C
 sG3 C 2
2
1
2
 sC 2 G1  G2 G4
v
S

s
 sG3 C1
2
 s G1 C1  G2 G4 C1C 2
v
S
(23.4-7)
which is of the form of a bandpass. Using equation (23.4-6), we also obtain
v1 
s
 G3 G2 C1C 2
2
 s G1 C1  G2 G4 C1C 2
v
S
(23.4-8)
which is of the form of a lowpass.
The benefit of the Tow-Thomas biquad is not just its simplicity, but because it is graced with a good
tuning algorithm, namely
(1) Let R2 = R4 = R and C1 = C2 =C. Then we can define a characteristic frequency
0=1/RC and equation (23.4-7) gives transfer ratio
 sG3 G
v2
sk
 2
 2
2
vS
s  s 0 G1 G    0
s  s  0 Q   02


(2) Choose Q by means of Q = G/G1 = R1/R
(i.e. the value of R1 defines Q)
(3) Choose the amplitude factor by k = G3/G1=R1/R3
(i.e. the value of R3 defines k)