Transcript Hybrid-π Model

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
2
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.
3
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
4
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
5
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
6
Hybrid-π Model
Structure
Hybrid-π Model
Hybrid- model is used to derive small-signal transfer function (Vo/Vin)
7
Transfer Function

Assuming gm3 >> gm2 and CL, Cm1, Cm2 >> C1, C2

C
C C 
g m1 g m2 g mL r1r2 rL 1  s m 2  s 2 m1 m 2 
g mL
g m 2 g mL 
Vo ( s)

Av ( s) 

Vin ( s)

1  sCm1 g m2 g mL r1r2 rL 1  s Cm2  s 2 CLCm2
g m2
g m 2 g mL






NMC has 3 poles and 2 zeros
UGF = DC gain p-3dB = gm1/Cm1
8
Review on Quadratic Polynomial (1)

When the denominator of the transfer function has a quadratic
polynomial as
s
s2
D( s )  1 
 2
Qw0 w0


The amplifier has either 2 separate poles (real roots of D(s)) or 1
complex pole pair (complex roots)
Complex pole pair exists if
2
 1 
4
 
  
0

2
w0
 Qw0 
1
Q
2
9
Review on Quadratic Polynomial (2)

The complex pole can be expressed
using the s-plane:

The position of poles:
p2,3  w0

2 poles are located at
p 2, 3 

w0

2Q
w

j  0 4Q 2  1 
 2Q

If Q  1 / 2 , then
p 2,3 
w0
2
j
w0
2
10
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
11
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
1/ 2

PM of the amplifier


UGF / p2,3

1  UGF 
1 


PM  180  tan 
  tan 
p

 3dB 
 Q1   UGF / p2,3






2



 60
where |p2,3| = ( g g / C C )
m2 m3
L m2
12
Butterworth Unity-Feedback Response(2)
13
Circuit Implementation
Schematic of a three-stage NMC amplifier
14
Structure of NMC with Null Resistor (NMCNR)
Structure
Hybrid-π Model
15
Transfer function

Assume gmL >> gm2, CL, Cm1, Cm2 >> C1, C2

C C ( g R  1) 

g m1 g m 2 g mL r1r2 rL 1  sC m1 Rm  C m 2 ( Rm  1 / g mL )  s 2 m1 m 2 mL m
g
g
V ( s)
m 2 mL


Av ( s )  o

Vin ( s )


1  sCm1 g m 2 g mL r1r2 rL 1  s Cm 2  s 2 C L Cm 2 
g m2
g m 2 g mL 


C 
g m1 g m 2 g mL r1r2 rL 1  s m1 
g mL 



1  sCm1 g m 2 g mL r1r2 rL 1  s Cm 2  s 2 C L Cm 2
g m2
g m 2 g mL




if
Rm 
1
g mL
16
Structure of Nested Gm-C Compensation (NGCC)
Structure
Hybrid-π Model
17
Transfer function

Assume CL, Cm1, Cm2 >> C1, C2
C m 2 ( g mf 2  g m 2 )
C m1C m 2 ( g mf 1  g m1 ) 


g m1 g m 2 g mL r1r2 rL 1  s
 s2

g
g
g
g
g
Vo ( s )
m 2 mL
m1 m 2 mL


Av ( s) 

C (g
g g )
Vin ( s )


1  sCm1 g m 2 g mL r1r2 rL 1  s m 2 mf 2 m 2 mL  s 2 C L Cm 2 
g m 2 g mL
g m 2 g mL 


g m1 g m 2 g mL r1r2 rL

1  sCm1 g m 2 g mL r1r2 rL 1  s Cm 2  s 2 C L Cm 2
g m2
g m 2 g mL




if
g mf 1  g m1 & g mf 2  g m 2
18