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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

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

Circuit Structure

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**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.

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**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

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**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

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**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

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**Hybrid-π Model Structure Hybrid-π Model**

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

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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

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**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

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**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

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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

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**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| =

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**Butterworth Unity-Feedback Response(2)**

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**Circuit Implementation**

Schematic of a three-stage NMC amplifier

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**Structure of NMC with Null Resistor (NMCNR)**

Hybrid-π Model

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

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**Structure of Nested Gm-C Compensation (NGCC)**

Hybrid-π Model

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

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