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

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

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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.

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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 [g m r o ]~20-40dB Gain of two-stage amplifier [(g m r o ) 2 ]~40-80dB Gain of three-stage amplifier [(g m r o ) 3 ]~80-120dB, which is sufficient for most applications

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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

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6 Structure of NMC DC gain=(-A 1 )x(A 2 )x(-A 3 )=(-g m1 r 1 ) x(g m2 r 2 ) x(-g mL r L ) Pole splitting is realized by both Both C m1 and C m2 realize negative local feedback loops for stability

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7 Hybrid-π Model Structure Hybrid- π Model Hybrid- model is used to derive small-signal transfer function (V o /V in )

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8 Transfer Function Assuming g m3 >> g m2 and C L, C m1, C m2 >> C 1, C 2 NMC has 3 poles and 2 zeros UGF = DC gain p -3dB = g m1 /C m1

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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

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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

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11 Stability Criteria Stability criteria are for designing C m1, C m2, g m1, g m2, g mL 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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12 Butterworth Unity-Feedback Response(1) Assume zeros are negligible 1 dominant pole (p -3dB ) located within the passband, and 2 nondominant poles (p 2,3 ) are complex and |p 2,3 | is beyond the UGF of the amplifier Butterworth unity-feedback response ensures the Q value of p 2,3 is PM of the amplifier where |p 2,3 | =

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13 Butterworth Unity-Feedback Response(2)

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14 Circuit Implementation Schematic of a three-stage NMC amplifier

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

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16 Transfer function Assume g mL >> g m2, C L, C m1, C m2 >> C 1, C 2

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

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18 Transfer function Assume C L, C m1, C m2 >> C 1, C 2

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