集成电路设计自动化 讲义 Lect04_DPI_SFG_Method_(Short)

2015-09
Lecture 4. DPI
slide 2
Outline
• Driving Point Impedance (DPI) method • Signal Flow Graph (SFG) method • Examples
Two-stage opamp transfer function
Vo2
( ) sCm2 + Y2 −1
gm3
sCm1
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Lecture 4. DPI
Vout
( ) sCm1 + Y3 −1
slide 22
Transfer Function
sCm1
Vin
gm1
sCm2
Vo1
( ) sCm1 + sCm2 + Y1 −1
-gm2
Loop 1
Vo2
( ) sCm2 + Y2 −1
IM 1 + IM 2 = 0 (small-signal Ibias = 0)
gm1(Va − V1) + gm2(Vb − V1) = 0
gm1Va + gm2Vb =( gm1 + gm2 )V1
= V2
(IM
4
−
IM
2)
g ds 2
1 +
g ds 4
DPI at node “2”
[ ] =
gm1(Va − V1) − gm2 (Vb − V1)
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Lecture 4. DPI
slide 21
DPI Analysis
= Y1 1/ R1 + sC1 = Y2 1/ R2 + sC2 = Y3 1/ R3 + sCL
sCm1
Vin
gm1
sCm2
Vo1
( ) sCm1 + sCm2 + Y1 −1
-gm2
sCm2
Three loops in the SFG
C1 v2 −
(gm2 )v2
R2
C2
2 V2
gm1v1
sCcvo
G1 + s(C1 + Cc )
3 Vo
(gm2 − sCc )v2
3 Vo
sCcv2
gm2v2
G2 + s(C2 + Cc )
G2 + s(C2 + Cc )
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Lecture 4. DPI
slide 18
Two-stage opamp (SFG)
Lecture 4. DPI
slide 8
Deriving TF from SGF
V1
G1
V2
1/ (G1 + G2 + G3 )
G3
V3
1/ (G3 + G4 )
G3
Mason’s Rule
feedforward gain
loop gain
V3 V1
=
G1G3 1− G32
/ (G1 + G2 / (G1 + G2
Yf
Vi H (s)
Vo
yi
gm yo
Vout YL
V1
Y1
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Vi
1 (Y1 + Yf + yi )
−gm + Yf
Yf
1 (Yf + YL + yo )
Vout
Lecture 4. DPI
From Ochoa (1998)
slide 16
Two-Stage Opamp Analysis
Fall 2015
Foreword
• The DPI method introduced in this lecture is based on the following paper
• A. Ochoa, Jr., “A systematic approach to the analysis of general and feedback circuits and systems using signal flow graphs and driving-point impedance,” IEEE Trans. on CAS-II: Analog and Digital Signal Processing, vol. 45, no. 2, Feb. 1998, pp. 187-195.
G3V3
G3V2
V1
G1
ADD currents entering node ‘2’
V2
Z2
G3
3 V3
DP impedance
( ) = Z3 G3 + G4 −1
V3
Z3
∑ ID,m = Gm,nVn n
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G3
Gm,n = admittance driving node m from node n
M3
M4
V2
2
Va
M1 M 2
Vb
1
Va
gm1
Vb
gm2
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gm1 -gm1
1/(gm1+gm2)
gm2
V1
-gm2
Lecture 4. DPI
gm1(Va − V1)
V2
1/(gds2+gds4)
−gm2(Vb − V1)
slide 13
Example 3
1 V1
M 2 go2
Y1
Vi
Circuit Conversion
ID,n denotes the driving point (DP) current seen at node “n”.
1
R1 2
R3
3
2
R3
3
IS1 = G1V1
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R2 0
(total DP current)
2 V2 I= D,2 G1V1 + G3V3
Lecture 4. DPI
slide 5
Circuit Example
1
Vin
R1 2
R2 0
R3 3
+
R4 Vout
-
Consider two driving points at nodes “2” and
“3”
1
R1 2
R3
3
2
R3
3
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R2 0
Lecture 4. DPI
R4 0 slide 6
I D,3 = G3V2
IS 3 = G3V3 Gˆ = G1 + G2 + G3 (lumped DP admittance)
Lecture 4. DPI
R4 0
3 V3 G=ˆ G3 + G4
slide 7
Drawing SFG
G1V1
DP impedance
( ) 2 V2Z2 = G1 + G2 + G3 −1
From Ochoa (1998)
slide 14
Example 4
V2
2
Y2
1
Yx
Vx
Ibias
V1 Y1
Vx
−gm
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V1
1 (Y1 + Yx )
Yx
Yx
Lecture 4. DPI
1 (Y2 + Yx )
V2
From Ochoa (1998)
slide 15
Example 5
Y1 V1
G1(V2-V1) + G2(V2) + G3(V2-V3) = 0
Separating the voltages
IS1 = G1V1
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2 V2
-G1V1 + (G1+G2+ G3) V2 - G3V3 = 0
IS3 = G3V3
viewed as
current srcs
Gˆ = G1 + G2 + G3 (sum of admittances)
Y. –J. Kim and S. –H. Lee, “A 10-b 120-MS/s 45 nm CMOS ADC using a re-configurable threestage switched amplifier,” Analog Integrated Circuits and Signal Processing, vol. 72, 75-87, 2012.
2015-09
Lecture 4. DPI
slide 4
DPI Intuition
• DPI is closely related to the modified nodal analysis (MNA) method.
V1
1
Vin
V2
R1 2
R2 0
V3
R3 3
+
R4 Vout
-
The sum of currents at node “2” (all leaving):
1 gds2 + gds4
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Lecture 4. DPI
M3 Va
M1
M4 V2
2
Vb M2
1
Ibias
Both IM4 and (- IM2) entering node “2”
slide 12
SFG for the Current Mirror
From Ochoa (1998)
common src point
slide 19
Two-stage opamp -- I/O TF
+
v1
R1
−
(gm1 )v1
Cc
+ C1 v2
−
R2 (gm2 )v2
vo
C2
One zero in RHP
Vo =
gm1(gm2 − sCc )
V1 [G1 + s(C1 + Cc )][G2 + s(C2 + Cc )] − sCc (sCc − gm2 )
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D(s) ≡ G1G2 + s[G1C2 + G2C1 + Cc (G1 + G2 + gm2 )] + s2[C1C2 + (C1 + C2 )Cc ]
Lecture 4. DPI
Two poles in LHP
slide 20
RNMC opamp
Reversed nested Miller compensation (RNMC) amplifier
gm3
Vout
( ) sCm1 + Y3 −1
sCm2
2 V2
gm1v1
sCcvo
G1 + s(C1 + Cc )
(gm2 − sCc )v2
3 Vo
G2 + s(C2 + Cc )
V1
− g m1
V2
1/ (G1 + s(C1 + Cc ))
sCc − gm2
Vo
1/ (G2 + s(C2 + Cc ))
sCc
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Lecture 4. DPI
Cc
vo
+
+
v1 −
gm1v1
R1
C1 v2 −
g m 2v 2
R2 C2
output impedance of 1st stage
output impedance of 2nd stage
2015-09
Lecture 4. DPI
slide 17
DPI Analysis
2
Cc
3
vo
+
+
v1 −
R1 (g m1 )v1
Mixed-Signal Design and Automation Methods 混合信号电路设计与自动化方法
Lecture 4
Driving Point Impedance (DPI) Method
Prof. Guoyong Shi shiguoyong@sjtu.edu.cn School of Micro/Nano-electronics Shanghai Jiao Tong University
+ G3 )(G3 + G3 )(G3
+ G4 ) + G4 )
1
=
(G1
+
G2
+
G1G3 G3 )(G3
+
G4 )
−
G32
Vin
R1 2 R2
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0 Lecture 4. DPI
R3
3
+
R4 Vout
-
slide 9
Example 1
Y2
2
Vo
Vin
V1
Y1 1
gmV1 Yo
No need to substitute smallsignal mode.
• Appendix 1: Mason’s rule
2015-09
Lecture 4. DPI
slide 3
Basic Steps for DPI
1. Introduce internal voltage variables 2. Derive current-driven impedances in Norton form 3. Draw Signal Flow G百度文库aph (SFG) 4. Derive I/O functions
Vin
Y1
I D,1
1/ (Y1 + Y2 ) V1
Y2 −gm
currents entering node (2)
I D,2
Vo
1/ (Y2 + Yo )
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Y2
From Ochoa (1998)
Lecture 4. DPI
slide 10
Highlights of Steps
• Mark the nodes (as the Driving Points) • Capture the Driving Point Impedance (DPI) @ each
M1 go1
2 V2
go1 & go2 are output admittance of M1 & M2
current entering node “1”
VSG,2 = -V1
Vi
- gm1
V1
1/(Y1+go1)
- gm2 + Y1
V2
1/(Y1+go2)
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Y1
Lecture 4. DPI
node. • Collect the currents entering each DPI node. • Complete the SFG
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Lecture 4. DPI
slide 11
Example 2: Current Mirror
= IM1 gm1(Va − V1); = IM 2 gm2 (Vb − V1); I= M 4 I= M 3 IM 1