Instructor ： Po-Yu Kuo
教師：郭柏佑



Introduction
Structure and Hybridπ Model
 Stability Criteria
 Circuit Structure
2

 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

 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
4

 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

 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
6

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

Transfer Function
 Assuming g m3
>> g m2 and C
L
, C m1
, C m2
>> C
1
, C
2
A v
( s )

V o
( s )
V in
( s )


1
 g m 1 g m 2 g mL r
1 r
2 r
L

 1
 sC m 1 g m 2 g mL r
1 r
2 r
L s
C m 2


 1
 g mL s

C m 2 g m 2 s
2

C m 1
C m 2 g m 2 g mL s
2


C
L
C m 2 g m 2 g mL




NMC has 3 poles and 2 zeros
UGF = DC gain p
-3dB
= g m1
/C m1
8

Review on Quadratic Polynomial (1)
 When the denominator of the transfer function has a quadratic polynomial as
D ( s )

1
 s
Qw
0 s
2
 w
0
2


The amplifier has either 2 separate poles (real roots of D(s)) or 1 complex pole pair (complex roots)
Complex pole pair exists if
 


1
Qw
0


2

4 w
0
2

0

Q

1
2
9

Review on Quadratic Polynomial (2)
 The complex pole can be expressed using the s-plane:
 The position of poles: p
2 , 3
 w
0
 2 poles are located at p
2 , 3
 w
0
2 Q
 j

 w
0
2 Q
4 Q
2 
1


 p
2 , 3
 w
0
2
 j w
0
2
10

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
11

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
1 / 2
 PM of the amplifier
PM

180
  tan

1



UGF p

3 dB



 tan

1






Q 1




UGF

UGF
/ p
2 , 3
/ p
2 , 3
2











60
 where |p
2,3
| =
( g m 2 g m 3
/ C
L
C m 2
)
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
Hybridπ Model
15

 Assume g mL
>> g m2
, C
L
, C m1
, C m2
>> C
1
, C
2

A v
( s )

V o
( s )
V in
( s )
 g m 1 g m 2 g mL r
1 r
2 r
L

 1
 s

C m 1
R m

C m 2
( R m

1 / g mL
)

 s
2

1
 sC m 1 g m 2 g mL r
1 r
2 r
L



1
 s
C m 2 g m 2
 s
2
C m 1
C m 2
( g mL
R m
C
L
C m 2 g m 2 g mL

 g m 2 g mL

1
 g m 1 g m 2 g mL r
1 r
2 r
L


1
 s sC m 1 g m 2 g mL r
1 r
2 r
L



1
 s
C m 2 g m 2
C m 1 g mL
 s
2


C
L
C m 2 g m 2 g mL

 if R m

1 g mL

1 )


16

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

 Assume C
L
, C m1
, C m2
>> C
1
, C
2
A v
( s )

V o
( s )
V in
( s )

 g m 1 g m 2 g mL r
1 r
2 r
L
1



1
 s sC m 1 g m 2 g mL r
1 r
2 r
L
C m 2
( g mf


 1
 s
2
 g m 2 g mL g m 2
)
C m 2
( g mf 2

 s
2 g m 2 g m 2 g mL
C m 1
C m 2
( g mf 1 g m 1 g m 2 g
 g m 1
) mL


 g mL
)
 s
2
C g
L m 2
C g m 2 mL




1
 g m 1 g m 2 g mL r
1 r
2 r
L sC m 1 g m 2 g mL r
1 r
2 r
L


 1
 s
C m 2 g m 2
 s
2
C
L
C m 2 g m 2 g mL

 if g mf 1
 g m 1
& g mf 2
 g m 2
18