# Instructor：Po-Yu Kuo 教師：郭柏佑

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

Outline Introduction Structure and Hybrid-π Model Stability Criteria
Circuit Structure

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.

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

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

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

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

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

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

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

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

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| =

Butterworth Unity-Feedback Response(2)

Circuit Implementation
Schematic of a three-stage NMC amplifier

Structure of NMC with Null Resistor (NMCNR)
Hybrid-π Model

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

Structure of Nested Gm-C Compensation (NGCC)
Hybrid-π Model

Transfer function Assume CL, Cm1, Cm2 >> C1, C2

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