EL 6033
類比濾波器 (一)
Analog Filter (I)
Lecture1: Frequency Compensation and
Multistage Amplifiers I
Instructor：Po-Yu Kuo
教師：郭柏佑
Outline


Stability and Compensation
Operational Amplifier-Compensation
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Stability
Y (s)
H ( s)
T ( s) 

X ( s) 1  H ( s )
A( s)  1  H ( s)
The stability of a feedback system, like any other LTI system, is
completely determined by the location of its poles in the S-plane. The
poles (natural frequencies)of a linear feedback system with closed-loop
Transfer function T(s) are defined as the roots of the characteristic
equation A(s)=0, where A(s) is the denominator polynomial of
T(s).
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Reference books


Signals and Systems by S. Haykin and B. Van Veen,
John Wiley &Sons, 1999. ISBN 0-471-13820-7
Feedback Control of Dynamic Systems, 4th edition, by
F.G. Franklin, J.D. Powell, and A. Emami-Naeini,
Prentice Hall, 2002. ISBN 0-13-032393-4
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Bode Diagram Method
Y (s)
H ( s)
T ( s) 

X ( s) 1  H ( s )
A( s)  1  H ( s)
If  H (s)  1 , X(s) = 0, then gain goes to infinity.
The circuit can amplify its own noise until it eventually
begins to oscillates.
 H ( jw1 )  1
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Oscillation Conditions

A negative feedback system may oscillate at ω1 if
 The phase shift around the loop at this frequency is
so much that the feedback becomes positive
 And the loop gain is still enough to allow signal
buildup
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Time-domain Response vs. Close-loop Pole Positions
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Bode Plot of Open-loop Gain for Unstable and Stable Systems
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Unstable Condition


The situation can be viewed as
 Excessive loop gain at the frequency for which the
phase shift reaches -180°
 Or equivalently, excessive phase at the frequency for
which the loop gain drops to unity
To avoid instability, we must minimize the total phase
shift so that for |βH|=1,  H is more positive than -180°
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Gain Crossover point and Phase Crossover Point



Gain crossover point
 The frequencies at which the magnitude of the loop
gain are equal to unity
Phase crossover point
 The frequencies at which the phase of the loop gain
are equal to -180°
A stable system, the gain crossover point must occur
before the phase crossover
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Phase Margin





To ensure stability, |βH| must drop to unity beforethe
phase crosses -180°
Phase margin (PM): PM  180  H ( w  w1 ) , where w1 is
the unity gain frequency
PM<0, unstable
PM>0, stable
Usually require PM > 45°, prefer 60°
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One-pole System

In order to analyze the stability of the system, we plot
H ( s  jw)
H ( s  jw)
Single pole cannot contribute phase shift greater
than 90° and the system is unconditionally stable
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Tow-pole System
System is stable since the
open loop gain drops to
below unity at a frequency for
which the phase is smaller
than -180°
Unity gain frequency move
closer to the original


Same phase, improved
stability, gain crossover point
is moved towards original,
resulting more stable system
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Frequency Compensation


Typical opamp circuits contain many poles
Opamp must usually be “compensated” - open-loop
transfer function must be modified such that
 The closed loop circuit is stable
 And the time response is well-behaved
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Compensation Method


The need for compensation arises because the
magnitude does not drop to unity before the phase
reaches -180°
Two methods for compensation:
 Minimize the overall phase shift
 Drop the gain
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Illustration of the Two Methods
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Trade-offs

Minimizing phase shift
 Minimize the number of poles in the signal path
 The number of stages must be minimized
 Low voltage gain, limited output swing

Dropping the gain
 Retains the low-frequency gain and output swing
 Reduces the bandwidth by forcing the gain to fall at
lower frequencies
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General Approach

First try to design an opamp so as to minimize the
number of poles while meeting other requirements

The resulting circuit may still suffer from insufficient
phase margin, we then compensate the opamp
 i.e. modify the design so as to move the gain
crossover point toward the origin
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Translating the Dominant Pole toward origin
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Outline


Stability and Compensation
Operational Amplifier-Compensation
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Compensation of Two-stage Opamp
Input: small R, reduced miller effect due to
cascode – small C, ignored
X: small R, normal C
E: large R (cascode), large C (Miller effect)
A: normal R, large C (load)
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Miller Compensation
Cc
Cc
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Pole Splitting as a Result of Miller Compensation


RL=ro9 || ro11
CE: capacitance from
node E to gnd CS stage
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