Download Small-signal circuit Analysis

EE 330
Lecture 28
Small-Signal Model Extension
Applications of the Small-signal Model
Review from Last Lecture
Consider 4-terminal network
I1
I2
4-Terminal
Device
I3
V1
V2
V3
I1  f1 V1,V 2 , V3 
I2  f2 V1,V 2 , V3 
I3  f3 V1,V 2 , V3 





Nonlinear network characterized by 3 functions each
functions of 3 variables
y 11 v1  y12 v2  y13 v3
Review from Last Lecture
i1
Consider now 3 functions, each a function of 3 variables
Extending this approach to the two nonlinear functions I2 and I3
i1  y11v1  y12 v2  y13v3
i2  y 21v1  y22 v2  y23v3
i3  y 31v1  y32 v2  y33v3
where
yij 
fi V1,V 2 , V3 
Vj


V  VQ
This is a small-signal model of a 4-terminal network and it is linear
9 small-signal parameters characterize the linear 4-terminal network
Small-signal model parameters dependent upon Q-point !
Review from Last Lecture
A small-signal equivalent circuit of a 4-terminal nonlinear network
i1
V1
y11
y12V2
y13V3
y21V1
y23V3
i2
V2
y22
yij 
i3
V3
y33
y31V1
y32V2
Equivalent circuit is not unique
Equivalent circuit is a three-port network
fi V1,V 2 , V3 
Vj


V  VQ
Review from Last Lecture
Consider 3-terminal network
Small-Signal Model
3
4-Terminal
Device

I1
I2
I3
V1
V2
V3
i1  y11V1  y12V2  y13V3
i2  y 21V1  y22V2  y23V3
i3  y 31V1  y32V2  y33V3

i1  g1 V 1,V 2,V 3 

i2  g2 V 1,V 2,V 3 

i3  g3 V 1,V 2,V 3 




yij 


fi V1,V 2 , V3 
Vj


V  VQ
Review from Last Lecture
Consider 3-terminal network
Small-Signal Model
I1
I2
3-Terminal
Device
V2
i2 iy g21VV,V1    y22V2
1
1
1
2
i2  g 2 V1,V 2 
fi V1 ,V 2 
Vj
yij 
i1  y11V1  y12V2
V1
 
V  VQ
  V1Q 

V  

V
 2Q 


A Small Signal Equivalent Circuit
i2
i1
V1
y11
y12V2
y21V1
y22
V2
4 small-signal parameters characterize this 3-terminal (two-port) linear network
Small signal parameters dependent upon Q-point
Review from Last Lecture
Consider 2-terminal network
Small-Signal Model
2
4-Terminal
Device

I1
I2
I3
V1
V2
V3
i1  y11V1  y12V2  y13V3
i2  y 21V1  y22V2  y23V3
i3  y 31V1  y32V2  y33V3

i1  g1 V 1,V 2,V 3 

i2  g2 V 1,V 2,V 3 

i3  g3 V 1,V 2,V 3 




yij 


fi V1,V 2 , V3 
Vj


V  VQ
Review from Last Lecture
Consider 2-terminal network
Small-Signal Model
I1
2-Terminal
Device
i y V
1
11
y 
11
1
V1
f  V 
V
1
1
A Small Signal Equivalent Circuit
i1
V1
y11
VV
1
1Q
V  VQ
Review from Last Lecture
Small Signal Model of MOSFET
Large Signal Model
I 0
G
3-terminal device

0

V 
W

I  μC
V

V


V
L
2




W
 V  V  1   V
μC

2L
V V
GS
V V
DS
D
ID
Region
OX
VGS5
Saturation
Region
VGS4
GS
T
DS
2
VGS6
Triode
OX
Cutoff
GS
T
DS
T
GS

T
V  V V
DS
GS
V V
GS
T
T
V  V V
DS
Region
VGS3
VGS2
VGS1
VDS
MOSFET is usually operated in saturation region in linear applications
where a small-signal model is needed so will develop the small-signal
model in the saturation region
GS
T
Review from Last Lecture
Small Signal Model of MOSFET
g  μC
OX
m
W
V
L
GSQ
V
T

g  I
DQ
O
D
G
VGS
gO
gmVGS
S
Alternate equivalent expressions:
I =μC
DQ
OX
W
V
2L
GSQ
V
T
 1   V   μC
2
DSQ
OX
W
V  V
L
W
g  2μC
 I
L
2I
g 
V V
g  μC
OX
m
GSQ
OX
m
DQ
DQ
m
GSQ
T
T

W
V
2L
GSQ
V
T

2
Review from Last Lecture
Small Signal Model of BJT
3-terminal device
IC
Forward Active Model:
 V 
I  J A e 1 

V


V
VBE
Vt
Saturation
C
Forward Active
S
CE
E
AF
VCE
IB 
Cutoff
Usually operated in Forward Active Region
when small-signal model is needed
JS A E
e
β
BE
Vt
Review from Last Lecture
Small Signal Model of BJT
i y V yV
i y V y V
B
11
C
21
BE
12
BE
22
CE
CE
i  g V
i g V g V
B
I
g 
βV
CQ
t
I
g 
V
g 
CQ
O
m
C
I
V
BE
m
BE
O
CQ
AF
t
B
C
VBE
E
gπ
gmVBE
gO
CE
Review from Last Lecture
Active Device Model Summary
Element
ss equivalent
dc equivalnet
Diodes
Simplified
MOS
transistors
Simplified
Simplified
Bipolar
Transistors
Simplified
Simplified
What are the simplified dc equivalent models?
Review from Last Lecture
Active Device Model Summary
What are the simplified dc equivalent models?
dc equivalent
0.6V
Simplified
G
D
VGSQ
μCOX W
 VGSQ -VTn 2
2L
S
Simplified
G
D
VGSQ
2
μCOX W
VGSQ -VTp
2L

Simplified
S
B
IBQ
0.6V
βIBQ
E
Simplified
B
IBQ
Simplified
C
0.6V
C
βIBQ
E

Recall:
Alternative Approach to small-signal
analysis of nonlinear networks
1. Linearize nonlinear devices
(have small-signal model for key devices!)
2. Replace all devices with small-signal
equivalent
3. Solve linear small-signal network
Remember that the small-signal model is operating point dependent!
Thus need Q-point to obtain values for small signal parameters
Comparison of Gains for MOSFET and BJT Circuits
MOSFET
BJT
VDD
VCC
R1
R
VOUT
VOUT
M1
Q1
VIN(t)
VIN(t)
VSS
VEE
I R
A 
V
CQ
VB
t
A
2I R

V  V
DQ
VM
SS
`1
T

Verify the gain expression obtained for the BJT using a small signal analysis
VOUT
VCC
R
R
VIN
VOUT
Q1
VIN(t)
Neglect λ effects to be consistent with earlier analysis
VOUT
VEE
Vbe
VIN
VOUT = -gmRVBE
V IN
=
V BE
gmVbe
gπ
AV =
R
VOUT
= -gmR
V IN
gm =
ICQ
Vt
AV = -
ICQR
Vt
Note this is identical to what was obtained with the direct nonlinear analysis
Example:
VDD
M2
VOUT
VIN
M1
VSS
Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
Example:
Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
VDD
M2
M1
VOUT
VIN
M1
M2
VIN
VOUT
VSS
Small-signal circuit
Example:
Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
VDD
M1
M2
VOUT
VIN
VIN
VOUT
M1
VSS
M2
Small-signal circuit
G
D
VGS
gmVGS
S
Small-signal MOSFET model for λ=0
Example:
Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
M1
M2
VIN
VOUT
Small-signal circuit
VIN
VGS1
gm1VGS1
VGS2
gm2VGS2
VOUT
Small-signal circuit
Example:
VIN
VGS1
gm1VGS1
VGS2
gm2VGS2
VOUT
Small-signal circuit
Analysis:
By KCL
g V g V
m1
GS 1
m2
GS 2
but
V V
GS 1
IN
-V  V
GS 2
OUT
thus:
A 
V
V
g

V
g
OUT
m1
IN
m2
Example:
VIN
VGS1
VGS2
gm1VGS1
gm2VGS2
VOUT
Small-signal circuit
Analysis:
A 
V
Recall:
D
OUT
m1
IN
m2
W
L
g   2 I C
m
V
g

V
g
1
OX
1
2I C
D
A 
V
OX
W
L
1
1
W
2 I C
L
2
D
OX
2

W
W
1
2
L
L
2
1
Example:
VIN
VGS1
VGS2
gm1VGS1
gm2VGS2
VOUT
Small-signal circuit
Analysis:
A 
V
Recall:
V
g

V
g
OUT
m1
IN
m2
A 
V
W
W
1
2
L
L
2
1
If L1=L2, obtain
A 
V
W
W
1
2
L
W

L
W
2
1
1
2
The width and length ratios can be accurately set when designed in a standard
CMOS process
Graphical Analysis and Interpretation
Consider Again
VDD
R1
VOUT
M1
VIN(t)
VSS
IDQ 
V =V -I R
OUT
DD
D
μC W
I 
V  V
2L
OX
DQ
SS
T

2
μ COX W
VSS  VT 2
2L
μC W
I 
 V -V -V
2L
OX
D
IN
SS
T

2
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
VDD
 1   V 
2
R1
DS
VOUT
ID
VGS6
M1
VIN(t)
VGS5
VSS
VGS4
VGS3
VGS2
VGS1
VDS
Load Line
V =V -I R
Device Model at Operating Point
μC W
I 
V  V
2L
OUT
DD
D
OX
DQ
SS
T

2
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
VDD
 1   V 
2
R1
DS
VOUT
ID
VGS6
M1
VIN(t)
VGS5
VGS4
VGS3
VSS
VGSQ=-VSS
VGS2
VGS1
VDS
Load Line
Q-Point
IDQ
V =V -I R
OUT
I 
DQ
DD
D
μC W
V  V
2L
OX
SS
T

2
μ COX W
VSS  VT 2

2L
I 
D
μC W
 V -V -V
2L
OX
IN
SS
T

2
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
 1   V 
2
DS
VDD
R1
VOUT
ID
VGS6
M1
VGS5
VIN(t)
VGS4
VGS3
VGSQ=-VSS
VGS2
VGS1
VDS
Q-Point
Load Line
Saturation region
VSS
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
 1   V 
2
DS
VDD
R1
VOUT
ID
VGS6
M1
VGS5
VIN(t)
VGS4
VGS3
VGSQ=-VSS
VSS
VGS2
VGS1
VDS
Q-Point
Saturation region
Load Line
• Linear signal swing region smaller than saturation region
• Modest nonlinear distortion provided saturation region operation maintained
• Symmetric swing about Q-point
• Signal swing can be maximized by judicious location of Q-point
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
 1   V 
2
DS
VDD
R1
VOUT
ID
VGS6
M1
VGS5
VGS4
VIN(t)
VGSQ=-VSS
VSS
VGS3
VGS2
VGS1
VDS
Q-Point
Saturation region
Load Line
Very limited signal swing with non-optimal Q-point location
Graphical Analysis and Interpretation
Device Model (family of curves)
I 
DQ
μC W
 V -V
2L
OX
GS
T
 1   V 
2
DS
VDD
R1
VOUT
ID
VGS6
VGS5
M1
VIN(t)
VGS4
VGS3
VGS2
VGS1
VDS
Q-Point
VSS
VGSQ=-VSS
Saturation region
Load Line
• Signal swing can be maximized by judicious location of Q-point
• Often selected to be at middle of load line in saturation region
Small-Signal MOSFET Model Extension
Existing model does not depend upon the bulk voltage !
Observe that changing the bulk voltage will change the electric field in the
V
channel region !
DS
ID
VGS
VBS
IG
IB
(VBS small)
E
Further Model Extensions
Existing model does not depend upon the bulk voltage !
Observe that changing the bulk voltage will change the electric field in the
channel region !
VDS
ID
VGS
VBS
IG
IB
(VBS small)
E
Changing the bulk voltage will change the thickness of the inversion layer
Changing the bulk voltage will change the threshold voltage of the device

VT  VT0     VBS  

Typical Effects of Bulk on Threshold Voltage for n-channel Device

VT  VT0     VBS  

0.4V
-
1
2


0.6V
VT
VT0
VBS
~ -5V
Bulk-Diffusion Generally Reverse Biased (VBS< 0 or at least less than 0.3V) for nchannel
Shift in threshold voltage with bulk voltage can be substantial
Often VBS=0
Typical Effects of Bulk on Threshold Voltage for p-channel Device
V  V 
T

T0
0.4V
-
1
2

V  
BS


0.6V
VBS
VT0
VT
Bulk-Diffusion Generally Reverse Biased (VBS > 0 or at least greater than -0.3V)
for n-channel
Same functional form as for n-channel devices but VT0 is now negative and
the magnitude of VT still increases with the magnitude of the reverse bias
Model Extension Summary
IG  0
IB  0

0

V 
W

I  μC
V

V


V
L 
2 


W
μC
 V  V   1   V


2L
V V
GS
V V
DS
D
OX
GS
T
DS
2
OX
GS
VT  VT0  
T
DS
  V
BS
 
T
T
V  V V
V V
V  V V
GS

GS
T
DS
DS
GS
GS
T
T

Model Parameters : {μ,COX,VT0,φ,γ,λ}
Design Parameters : {W,L} but only one degree of freedom W/L
Small-Signal Model Extension
I =0
G
I =0

0

V 
W

I  μC
V

V


V
L 
2 


W
 V  V   1   V
μC

2L
B
V V
GS
V V
DS
D
OX
GS
T
DS
GS
VT  VT0  
y11 
y31 
y 
21
IG
VGS
 
V  VQ
IB
VGS
I
V
0
 
V  VQ
D
GS V  VQ
m
DS
  V
BS
y12 
0
g
T
y32 
y 
12
IG
VDS
 
 
V  VQ
IB
VDS
I
V
D
DS V  VQ
V  V V

V V
V  V V
GS
DS
T
DS

0
 
V  VQ
T
GS
2
OX
T
0
g
IG
VGS
y13 
y33 
o
IB
VGS
y 
13
 
V  VQ
0
 
V  VQ
I
V
0
D
GS V  VQ
g
mb
GS
GS
T
T
ID  μC OX
W
VGS  VT 2  1  VDS 
2L
VT  VT0  
g 
m
I
V
 μC
D
OX
o
I
V
 μC
D
OX
DS V VQ
I
g 
V
D
BS V VQ
I
g 
V
D
BS V VQ
W
2V  V
2L

T
GS
T
EBQ
BS V VQ
m
BSQ

 1   V
DS

 μC
V VQ
T
GS
W
V
 μC
V 
L
V

g g
2  -V
mb

1
2
λ
OX
W
V
L
EBQ
 λI
DQ
V VQ
W
 μC
2 V  V
2L
OX
mb
W
2 V  V
2L
OX
mb
 
BS
GS
GS V VQ
g 
 V
T

1
 V 

 1+λV

 V 
T
DS

BS
V VQ
W
1


  μC
V   1     V
L
2



OX
EBQ
BS

 1 
1
2
V VQ
Small Signal Model Summary
ig
G
Vgs
B
Vbs
S
ib
i
D d
Vds
Small
Signal
Network
ib  0
μC W
g 
V
L
go  λIDQ
id  g mvgs  g mbvbs  govds
gmb
OX
ig  0
m
EBQ



 gm
2  V
BSQ





Relative Magnitude of Small Signal MOS
Parameters
Consider:
i g v g v gv
d
m
gs
mb
bs
o
ds
3 alternate equivalent expressions for gm
μC W
g 
V
L
2μ COX W
gm 
L
OX
m
EBQ
IDQ
2I
g 
V
DQ
m
EBQ
If μCOX=100μA/V2 , λ=.01V-1, γ = 0.4V0.5, VEBQ=1V, W/L=1, VBSQ=0V
μC W
I 
V
2L
OX
DQ
-4
2
EBQ
10 W

1V  =5E-5
2L
2
μC W
g 
V
L
EBQ
g  λI  5E-7


g g 
2  V

o
mb
g <<g ,g
g <g
0
OX
m
In this example
 1E-4
DQ
m
BSQ
mb

  .26g

m
m
mb
m
This relationship is
common
In many circuits,
VBS=0 as well
Large and Small Signal Model Summary
Large Signal Model

0


I  μC


μC

D
Small Signal Model
ig  0
V V
GS
V 
W
V

V


V
L
2 
W
 V  V   1   V
2L
V V V  V V
DS
OX
GS
T
DS
2
OX
GS
T
VT  VT0  
DS
  V
BS
ib  0
T
GS

T
DS
GS
T
V V V  V V
GS
 

T
DS
GS
id  g mvgs  g mbvbs  govds
where
T
g 
m
μC W
V
L
OX


gmb  gm 
2  V
BSQ

go  λIDQ
EBQ




End of Lecture 28