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EL 6033 類比濾波器 ( 一 ). Analog Filter (I). Instructor : Po-Yu Kuo 教師 : 郭柏佑. Lecture1: Frequency Compensation and Multistage Amplifiers I. Outline. Stability and Compensation Operational Amplifier-Compensation. Stability. The stability of a feedback system, like any other LTI system, is - PowerPoint PPT Presentation
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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)
2
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
4
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