Instructor ： Po-Yu Kuo 教師 ： 郭柏佑

1. EL 6033類比濾波器 (一) Analog Filter (I) Instructor：Po-Yu Kuo 教師：郭柏佑 Lecture3: Design Technique for Three-Stage Amplifiers

2. Outline • Introduction • Structure and Hybrid-πModel • Stability Criteria • Circuit Structure

3. Why We Need Three-Stage Amplifier? • Continuous device scaling in CMOS technologies lead to decrease in supply voltage • High dc gain of the amplifier is required for controlling different power management integrated circuits such as low-dropout regulators and switched-capacitor dc/dc regulators to maintain the constant of the output voltage irrespective to the change of the supply voltage and load current.

4. High DC Gain in Low-Voltage Condition • Cascode approach: enhance dc gain by stacking up transistors vertically by increasing effective output resistance (X) • Cascade approach: enhance dc gain by increasing the number of gain stages horizontally (Multistage Amplifier) • Gain of single-stage amplifier [gmro]~20-40dB • Gain of two-stage amplifier [(gmro)2]~40-80dB • Gain of three-stage amplifier [(gmro)3]~80-120dB, which is sufficient for most applications

5. Challenge and Soultion • Three-stage amplifier has at least 3 low-frequency poles (each gain stage contributes 1 low-frequency pole) • Inherent stability problem • General approach: Sacrifice UGF for achieving stability • Nested-Miller compensation (NMC) is a classical approach for stabilizing the three-stage amplifier

6. Structure of NMC • DC gain=(-A1)x(A2)x(-A3)=(-gm1r1) x(gm2r2) x(-gmLrL) • Pole splitting is realized by both • Both Cm1 and Cm2 realize negative local feedback loops for stability

7. Hybrid-π Model Structure Hybrid-πModel Hybrid- model is used to derive small-signal transfer function (Vo/Vin)

8. Transfer Function • Assuming gm3 >> gm2 and CL, Cm1, Cm2 >> C1, C2 • NMC has 3 poles and 2 zeros • UGF = DC gain p-3dB = gm1/Cm1

9. Review on Quadratic Polynomial (1) • When the denominator of the transfer function has a quadratic polynomial as • The amplifier has either 2 separate poles (real roots of D(s)) or 1 complex pole pair (complex roots) • Complex pole pair exists if

10. Review on Quadratic Polynomial (2) • The complex pole can be expressed using the s-plane: • The position of poles: • 2 poles are located at • If , then

11. Stability Criteria • Stability criteria are for designing Cm1, Cm2, gm1, gm2, gmL to optimize unity-gain frequency (UGF) and phase margin (PM) • Stability criteria: • Butterworth unity-feedback response for placing the second and third non-dominant pole • Butterworth unity-feedback response is a systematic approach that greatly reduces the design time of the NMC amplifier

12. Butterworth Unity-Feedback Response(1) • Assume zeros are negligible • 1 dominant pole (p-3dB) located within the passband, and 2 nondominant poles (p2,3) are complex and |p2,3| is beyond the UGF of the amplifier • Butterworth unity-feedback response ensures the Q value of p2,3 is • PM of the amplifier • where |p2,3| =

13. Butterworth Unity-Feedback Response(2)

14. Circuit Implementation Schematic of a three-stage NMC amplifier

15. Structure of NMC with Null Resistor (NMCNR) Structure Hybrid-πModel

16. Transfer function • Assume gmL >> gm2, CL, Cm1, Cm2 >> C1, C2

17. Structure of Nested Gm-C Compensation (NGCC) Structure Hybrid-πModel

18. Transfer function • Assume CL, Cm1, Cm2 >> C1, C2