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

3 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).

4 Reference books Signals and Systems by S. Haykin and B. Van Veen, John Wiley &Sons, ISBN Feedback Control of Dynamic Systems, 4th edition, by F.G. Franklin, J.D. Powell, and A. Emami-Naeini, Prentice Hall, ISBN

5 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.

6 Oscillation Conditions A negative feedback system may oscillate at ω 1 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

7 Time-domain Response vs. Close-loop Pole Positions

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

9 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°

10 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

11 Phase Margin To ensure stability, |βH| must drop to unity beforethe phase crosses -180° Phase margin (PM):, where w 1 is the unity gain frequency PM<0, unstable PM>0, stable Usually require PM > 45°, prefer 60°

12 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

13 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

14 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

15 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

16 Illustration of the Two Methods

17 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

18 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

19 Translating the Dominant Pole toward origin

20 Outline Stability and Compensation Operational Amplifier-Compensation

21 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)

22 Miller Compensation CcCc CcCc

23 Pole Splitting as a Result of Miller Compensation R L =r o9 || r o11 C E : capacitance from node E to gnd CS stage