EENG400_12d_week11.pdf

Chapter 12 Feedback
(d)
➢ 12.1 General Considerations
➢ 12.2 Properties of Negative Feedback
➢ 12.3 Types of Amplifiers
➢ 12.4 Sense and Return Techniques
➢ 12.5 Polarity of Feedback
➢ 12.6 Feedback Topologies
➢ 12.7 Effect of Finite I/O Impedances
➢ 12.8 Stability in Feedback Systems
1
Textbook: Fundamentals of Microelectronics, 2nd Edition, Behzad Razavi, Wiley, 2014

LIU- Prof. Adnan Harb
Outline
➢ Feedback is an integral part of our lives.
EENG400 CH 12 Feedback 2

12.8 Stability in Feedback Systems
➢ Negative feedback: many important benefits.
➢ Unfortunately, if designed poorly, negative-feedback
amplifiers:
– may behave “badly”
– or even oscillate.
– We say the system is marginally stable or simply unstable.
➢ In this section, we re-examine our understanding of
feedback so as to define the meaning of “behaving badly”
and determine the sources of instability.

12.8.1 Review of Bode’s Rules (1/2)
Example 12.34: Phase Response
➢ Note: Since the phase begins to change at 1/10 of a pole or zero
frequency, even poles or zeros that seem far may affect it
significantly—a point of contrast to the behavior of the magnitude.
Assumption:
10 ωp1 < 0.1 ωz
10 ωz < 0.1 ωp2
➢ The phase of a transfer
function:
– begins to decrease (increase)
at 1/10 of the pole (zero)
frequency,
– incurs a change of −45◦ (+45◦)
at the pole (zero) frequency,
– and experiences a total change
of nearly −90◦ (+90◦) at 10
times the pole (zero)
frequency.
0.1
ω
p1
10
ω
p1
0.1
ω
z

12.8.1 Review of Bode’s Rules (2/2)
Example 12.35: Three-Pole System
For a three-pole system, a
finite frequency w1
produces a phase of -180o,
which means an input
signal that operates at this
frequency will have its
output inverted.
-135°
-225°
-45°
𝑵𝒐𝒕𝒆: 𝑨𝒆−𝒋𝟏𝟖𝟎°
= 𝑨 𝒄𝒐𝒔 −𝟏𝟖𝟎 + 𝒔𝒊𝒏 −𝟏𝟖𝟎 = −𝑨

12.8.2 Problem of Instability (1/5)
Instability of a Negative Feedback Loop
➢ Substitute jω for s. If for a certain ω1, KH(jω1) reaches -1, the closed
loop gain becomes infinite. This implies for a very small input signal
at ω1, the output can be very large. Thus the system becomes
unstable.
➢ It is also possible to understand the above oscillation phenomenon
intuitively
)
(
1
)
(
)
(
s
KH
s
H
s
X
Y
+
=

Instability of a Negative Feedback Loop
)
(
1
)
(
)
(
s
KH
s
H
s
X
Y
+
=
Intuitively
➢ From example 12.35: A three-pole system introducing a phase
shift of −180° reverses the sign of the input signal at ω1.
➢ Now, if H(s) in the figure above contains three poles such that
∠H = −180◦ at ω1, then the feedback becomes positive at this
frequency, thereby producing a feedback signal that enhances
the input.
➢ Circulating around the loop, the signal may thus continue to
grow in amplitude.
➢ In practice, the final amplitude remains bounded by the supply
voltage or other “saturation” mechanisms in the circuit.

“Barkhausen’s Criteria” for Oscillation
∠𝑲𝑯 𝒋𝝎𝟏 = −𝟏𝟖𝟎
|𝑲𝑯(𝒋𝝎𝟏)| = 𝟏
𝐾𝐻(𝑗𝜔1) = −1 ⇒
Guarantees positive feedback (sufficient delay)
Ensures sufficient loop gain for the circulating signal to grow

Time Evolution of Instability

Example 12.37: Oscillation Example
➢ This system oscillates,
since there’s a finite
frequency at which the
phase is -180o and the gain
is greater than unity. In
fact, this system exceeds
the minimum oscillation
requirement.

12.8.3 Stability Condition (1/5)
Condition for Oscillation
➢ Although for both systems above, the frequencies at which |KH|=1
and KH= -180o are different, the system on the left is still unstable
because at KH= -180o, |KH|>1. Whereas the system on the right is
stable because at KH= -180o, |KH|<1.
➢ Thus, to avoid instability, we must ensure that these two conditions
do not occur at the same frequency.

Condition for Stability
➢ ωPX, (“phase crossover”), is the frequency at which KH= -180o.
➢ ωGX, (“gain crossover”), is the frequency at which |KH|= 1.
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛: 𝜔𝐺𝑋 < 𝜔𝑃𝑋
i. e. we must ensure that
|KH| falls below 1 at or
even before the phase
crossover frequency
𝝎𝑮𝑿, frequency at which KH = 1
𝝎𝑷𝑿, frequency at which ∠KH= −180°

Example 12.38: Stability Example I (1/2)
➢ The circuit exhibits a low-frequency gain
of (gmRD)3 and three coincident poles given
by (RDC1)−1.
➢ Thus, as depicted in (b), |H| begins to fall
at a rate of 60 dB/dec at ωp = (RDC1)−1.
➢ The phase begins to change at 1/10 of ωp
(actually even below), reaches −135° at ωp,
and approaches −270° at 10ωp.
➢ To guarantee that a unity-feedback system
incorporating this amplifier remains stable,
we must ensure that |KH|( = |H|) falls
below unity at the phase crossover
frequency.
➢ In (c), the procedure entails identifying
ωPX on the phase response, finding the
corresponding point, P, on the gain
response, and requiring that |HP| < 1.

Example 12.38: Stability Example I (2/2)

Example 12.40: Stability Example II
5
.
0
1
|
|
5
.
0
=

K
H p
the weaker feedback
permits a greater
open-loop gain.
6 dB

12.8.4 Phase Margin (1/2)
Marginally Stable vs. Stable
Marginally Stable Stable
By how much wGX < wPX?

12.8.4 Phase Margin (2/2)
➢ A measure commonly used to
quantify the stability of
feedback systems is the “phase
margin” (PM)
➢ Phase Margin = H(ωGX)+180
➢ The larger the phase margin,
the more stable the negative
feedback becomes

45
=
PM
Example 12.41
-180 ⁰
𝑷𝑴 −
+
𝝎𝑮𝑿, frequency at which KH = 1
➢ For a well-behaved response,
we typically require a phase
margin of 60◦.
➢ Thus, the above example is not
considered an acceptable
design.
➢ In other words, the gain
crossover must fall below the
second pole.

12.8.5 Frequency Compensation
➢ Phase margin can be improved by moving ωGX closer to
origin while maintaining ωPX unchanged.
𝑆ℎ𝑖𝑓𝑡𝑖𝑛𝑔 𝝎𝑮𝑿 𝑐𝑙𝑜𝑠𝑒𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛
⋘⋘
𝑃ℎ𝑎𝑠𝑒 𝑚𝑜𝑣𝑒𝑑 𝑓𝑟𝑜𝑚
" < −180°" 𝑡𝑜" > −180°"
𝝎𝑷𝑿

Example 12.42 (1/2)
➢ Ccomp is added to lower the dominant
pole so that ωGX occurs at a lower
frequency than before, which means
phase margin increases.

Example 12.42 (2/2)
➢ Ccomp is added to lower the dominant pole so that ωGX
occurs at a lower frequency than before, which means
phase margin increases.
𝒃𝒆𝒇𝒐𝒓𝒆 𝒂𝒅𝒅𝒊𝒏𝒈 𝑪𝒄𝒐𝒎𝒑
𝒂𝒇𝒕𝒆𝒓 𝒂𝒅𝒅𝒊𝒏𝒈 𝑪𝒄𝒐𝒎𝒑

12.8.5 Frequency Compensation Procedure
➢ 1) We identify a PM, then -180o+PM gives us the new ωGX, or ωPM.
➢ 2) On the magnitude plot at ωPM, we extrapolate up with a slope of
+20dB/dec until we hit the low frequency gain then we look “down”
and the frequency we see is our new dominant pole, ωP’.

12.8.5 Frequency Compensation Example
Example: 45o Phase Margin Compensation
2
p
PM w
w =
new ωGX, or ωPM after compensation
ωGX before
compensation

Miller Compensation
➢ To save chip area, Miller multiplication of a smaller capacitance
creates an equivalent effect.
c
O
O
m
eq C
r
r
g
C )]
||
(
1
[ 6
5
5
+
=

12.8 Problems
Problem 12.65 (1/2)
Phase Margin = H(ωGX)+180°

12.8 Problems
Problem 12.68

12.8 Problems
Problem 12.68 (2/2)

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