Instructor ： Po-Yu Kuo

教師：郭柏佑

Lecture1: Frequency Compensation and Multistage Amplifiers I

EL 6033類比濾波器 ( 一 )

Analog Filter (I)

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Outline

Stability and Compensation Operational Amplifier-Compensation

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Stability

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).

)(1)(

)(1

)(

)(

)()(

sHsA

sH

sH

sX

sYsT

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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

If , X(s) = 0, then gain goes to infinity.

The circuit can amplify its own noise until it eventually

begins to oscillates.

)(1)(

)(1

)(

)(

)()(

sHsA

sH

sH

sX

sYsT

1)( sH

1)( 1 jwH

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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, is more positive than -180°H

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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): , where w1 is

the unity gain frequency PM<0, unstable PM>0, stable Usually require PM > 45°, prefer 60°

)(180 1wwHPM

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One-pole System In order to analyze the stability of the system, we plot

Single pole cannot contribute phase shift greater than 90° and the system is unconditionally stable

)(

)(

jwsH

jwsH

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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 CE: large R (cascode), large C (Miller effect)A: normal R, large C (load)

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Miller Compensation

CcCc

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Pole Splitting as a Result of Miller Compensation

RL=ro9 || ro11

CE: capacitance from node E to gnd CS stage