Instructor ： Po-Yu Kuo 教師：郭柏佑

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 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).

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

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.

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

Time-domain Response vs. Close-loop Pole Positions

Bode Plot of Open-loop Gain for Unstable and Stable Systems

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°

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

Phase Margin • To ensure stability, |βH| must drop to unity beforethe phase crosses -180° • Phase margin (PM): , where w1isthe unity gain frequency • PM<0, unstable • PM>0, stable • Usually require PM > 45°, prefer 60°

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

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

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

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

Illustration of the Two Methods

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

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

Translating the Dominant Pole toward origin

Outline • Stability and Compensation • Operational Amplifier-Compensation

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)

Miller Compensation Cc Cc

Pole Splitting as a Result of Miller Compensation • RL=ro9 || ro11 • CE: capacitance from node E to gnd CS stage

Instructor ： Po-Yu Kuo 教師：郭柏佑