ee 330 lecture 28 small-signal model extension applications of the small-signal model
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EE 330Lecture 28
Small-Signal Model Extension
Applications of the Small-signal Model
32133
32122
32111
V,,VVfI
V,,VVfI
V,,VVfI
Nonlinear network characterized by 3 functions each functions of 3 variables
Consider 4-terminal networkI1
I2
I3 V1
V2
V3
4-TerminalDevice
Review from Last Lecture
Consider now 3 functions, each a function of 3 variables
1 11 1 12 2 13 3y y y i v v v
Extending this approach to the two nonlinear functions I2 and I3
QVVj
321iij V
V,,VVfy
3232221212 vyyy vvi
3332321313 vyyy vvi
3132121111 vyyy vvi
This is a small-signal model of a 4-terminal network and it is linear9 small-signal parameters characterize the linear 4-terminal networkSmall-signal model parameters dependent upon Q-point !
where
Review from Last Lecture
y11y12V2
V1
i1
y13V3
y22y21V1
V2
i2
y23V3
y33y31V1
V3
i3
y32V2
A small-signal equivalent circuit of a 4-terminal nonlinear network
QVVj
321iij V
V,,VVfy
Equivalent circuit is not uniqueEquivalent circuit is a three-port network
Review from Last Lecture
Small-Signal Model
1 1 1 2 3
2 2 1 2 3
3 3 1 2 3
, ,
, ,
, ,
g
g
g
i V V V
i V V V
i V V V
I1
I2
I3 V1
V2
V3
4-TerminalDevice
3232221212 Vyyy VVi
3332321313 Vyyy VVi
3132121111 Vyyy VVi
QVVj
321iij V
V,,VVfy
3
Consider 3-terminal networkReview from Last Lecture
Small-Signal Model
2122
2111
,
,
VVi
VVi
g
g
I1
I2
V1V2
3-TerminalDevice
2221212 VVi yy 2121111 VVi yy
QVVj
21iij V
,VVfy
y11 y22
y12V2
y21V1
V1 V2
i1 i2A Small Signal Equivalent Circuit
2Q
1Q
V
VV
Consider 3-terminal network
4 small-signal parameters characterize this 3-terminal (two-port) linear networkSmall signal parameters dependent upon Q-point
Review from Last Lecture
Small-Signal Model
1 1 1 2 3
2 2 1 2 3
3 3 1 2 3
, ,
, ,
, ,
g
g
g
i V V V
i V V V
i V V V
I1
I2
I3 V1
V2
V3
4-TerminalDevice
3232221212 Vyyy VVi
3332321313 Vyyy VVi
3132121111 Vyyy VVi
QVVj
321iij V
V,,VVfy
2
Consider 2-terminal networkReview from Last Lecture
Small-Signal Model
1 11 1yi V
Q
1 1
11
1 V V
f Vy
V
y11V1
i1
A Small Signal Equivalent Circuit
1QV V
Consider 2-terminal network
I1
V1
2-TerminalDevice
Review from Last Lecture
Small Signal Model of MOSFET
1
GS T
DS
D OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V V
VWI μC V V V V V V V V
L 2
WμC V V V V V V V V
2L
T
GI 0
3-terminal device
Large Signal Model
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
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
SaturationRegion
CutoffRegion
TriodeRegion
Review from Last Lecture
Small Signal Model of MOSFET
DQI
Og
OX GSQ T
WμC V V
Lmg
gOgmVGS
VGS
G D
S
Alternate equivalent expressions:
12 2
DQ OX GSQ T DSQ OX GSQ T
W WI =μC V V V μC V V
2L 2L
OX GSQ T
WμC V V
Lmg
OX DQ
W2μC I
Lmg
2DQ
m
GSQ T
Ig
V V
Review from Last Lecture
Small Signal Model of BJT
3-terminal device
VCE
IC
Forward Active
Cutoff
Saturation
Usually operated in Forward Active Region when small-signal model is needed
1BE
t
V
V CE
C S E
AF
VI J A e
V
t
BE
V
V
ESB e
β
AJI
Forward Active Model:
Review from Last Lecture
Small Signal Model of BJT
CQ
t
I
βVg
CQ
t
I
Vmg CQ
AF
I
VOg
21 22C BE CEy y i V V11 12B BE CEy y i V V
C m BE O CEg g i V V
B BEg
i V
gOgmVBE
VBEgπ
B
E
C
Review from Last Lecture
Active Device Model Summary
Simplified
Simplified
MOStransistors
Diodes
Simplified
Simplified
Simplified
Bipolar Transistors
Element ss equivalent dc equivalnet
What are the simplified dc equivalent models?
Review from Last Lecture
Active Device Model SummaryWhat are the simplified dc equivalent models?
Simplified
Simplified
Simplified
Simplified
Simplified
0.6V
dc equivalent
G D
S
VGSQ 2OXGSQ Tn
μC WV -V
2L
G D
S
VGSQ 2OXGSQ Tp
μC WV -V
2L
B C
E
BQβI0.6V
IBQ
B C
E
BQβI0.6V
IBQ
Review from Last Lecture
Recall:
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
Alternative Approach to small-signal analysis of nonlinear networks
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
R
Q1
VIN(t)
VOUT
VCC
VEE
BJT MOSFET
R1
VIN(t)
VOUT
VDD
VSS
M1
DQ `1
VM
SS T
2I RA
V V CQ
VB
t
I RA
V
Verify the gain expression obtained for the BJT using a small signal analysis
R
Q1
VIN(t)
VOUT
VCC
VEE
R
VIN
VOUT
RVIN
VOUT
Vbe gmVbegπ
OUT m BE
IN BE
= -g R
=
V V
V V
OUTV m
IN
A = -g RV
=V
CQIm
t
g =V
ICQV
t
RA -
V=
Note this is identical to what was obtained with the direct nonlinear analysis
Neglect λ effects to be consistent with earlier analysis
M1
M2
VIN
VOUT
VDD
VSS
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
VDD
VSS
Example: Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
VIN
VOUT
M1 M2
Small-signal circuit
M1
M2
VIN
VOUT
VDD
VSS
Example: Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2
are operating in the saturation region and that λ=0
VIN
VOUT
M1 M2
Small-signal circuit
gmVGSVGS
G
S
D
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
VIN
VOUT
M1 M2
Small-signal circuit
Small-signal circuit
VGS1 gm1VGS1VIN
VOUT
VGS2 gm2VGS2
Example:
Small-signal circuit
VGS1 gm1VGS1VIN
VOUT
VGS2 gm2VGS2
Analysis:
By KCL
1 1 2 2m GS m GSg gV V
but
1GS INV V
2GS OUT-V V
thus:
1
2
OUT m
V
IN m
gA
g VV
Example:
Small-signal circuit
VGS1 gm1VGS1VIN
VOUT
VGS2 gm2VGS2
Analysis:
1
2
OUT m
V
IN m
gA
g VV
1
1 1 2
2 12
2
2
2
D OX
V
D OX
WI C
L W LA
W LWI C
L
Recall: 1
1
2m D OX
Wg I C
L
Example:
Small-signal circuit
VGS1 gm1VGS1VIN
VOUT
VGS2 gm2VGS2
Analysis:1
2
OUT m
V
IN m
gA
g VV
1 2
2 1
V
W LA
W LRecall:
If L1=L2, obtain
1 2 1
2 1 2
V
W L WA
W L W
The width and length ratios can be accurately set when designed in a standard CMOS process
R1
VIN(t)
VOUT
VDD
VSS
M1
OUT DD DV =V -I R
2OX
DQ SS T
μ C WI V V
2L 2
OX
D IN SS T
μC WI V -V -V
2L
2TSSOX
DQ VV2L
WC μI
Graphical Analysis and Interpretation Consider Again
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
R1
VIN(t)
VOUT
VDD
VSS
M1
OUT DD DV =V -I R
2OX
DQ SS T
μ C WI V V
2L
Graphical Analysis and Interpretation Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
Load Line
Device Model at Operating Point
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
Load LineQ-Point
R1
VIN(t)
VOUT
VDD
VSS
M1
OUT DD DV =V -I R
2OX
DQ SS T
μ C WI V V
2L
2OX
D IN SS T
μC WI V -V -V
2L
2TSSOX
DQ VV2L
WC μI
Graphical Analysis and Interpretation
VGSQ=-VSS
Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
Load LineQ-Point
R1
VIN(t)
VOUT
VDD
VSS
M1
Graphical Analysis and Interpretation
VGSQ=-VSS
Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
Saturation region
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
Load LineQ-Point
R1
VIN(t)
VOUT
VDD
VSS
M1
Graphical Analysis and Interpretation
VGSQ=-VSS
Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
Saturation region
•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
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
Load LineQ-Point
R1
VIN(t)
VOUT
VDD
VSS
M1
Graphical Analysis and Interpretation
VGSQ=-VSS
Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
Saturation region
Very limited signal swing with non-optimal Q-point location
ID
VDS
VGS1
VGS6
VGS5
VGS4
VGS3
VGS2
Load LineQ-Point
R1
VIN(t)
VOUT
VDD
VSS
M1
Graphical Analysis and Interpretation
VGSQ=-VSS
Device Model (family of curves) 2
1 OX
DQ GS T DS
μ C WI V -V V
2L
Saturation region
•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 channel region !
VBS
VGS
VDS
IDIG
IB
(VBS small)
E
Further Model ExtensionsExisting model does not depend upon the bulk voltage !
Observe that changing the bulk voltage will change the electric field in the channel region !
VBS
VGS
VDS
IDIG
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
BST0T VVV
VT
VT0
VBS
~ -5V
BST0T VVV
Typical Effects of Bulk on Threshold Voltage for n-channel Device
1-20.4V 0.6V
Bulk-Diffusion Generally Reverse Biased (VBS< 0 or at least less than 0.3V) for n-channel Shift in threshold voltage with bulk voltage can be substantialOften VBS=0
T T0 BSV V V
Typical Effects of Bulk on Threshold Voltage for p-channel Device
1-20.4V 0.6V
Bulk-Diffusion Generally Reverse Biased (VBS > 0 or at least greater than -0.3V) for n-channel
VT
VT0
VBS
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
1
GS T
DS
D OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V V
VWI μC V V V V V V V V
L 2
WμC V V V V V V V V
2L
T
BST0T VVV
Model Parameters : {μ,COX,VT0,φ,γ,λ}
Design Parameters : {W,L} but only one degree of freedom W/L
0I
0I
B
G
Small-Signal Model Extension
1
GS T
DS
D OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V V
VWI μC V V V V V V V V
L 2
WμC V V V V V V V V
2L
T
BST0T VVV
000
QQQ VVGS
G13
VVDS
G12
VVGS
G11 V
Iy
V
Iy
V
Iy
000
QQQ VVGS
B33
VVDS
B32
VVGS
B31 V
Iy
V
Iy
V
Iy
Q Q Q
D D D
21 12 13
GS DS GSV V V V V V
I I Iy y y
V V Vm o mbg g g
GI =0
BI =0
DS2
TGSOXD VVV2L
WμCI 1
BST0T VVV
11
OX GS T DS OX EBQ
W WμC 2 V V V μC V
2L LQQ
D
m
V VGS V V
Ig
V
2
OX GS T DQ
WμC 2 V V λ λI
2LQQ
D
o
V VDS V V
Ig
V
1
OX GS T DS
WμC 2 V V 1+λV
2LQ Q
D T
mb
BS BSV V V V
I Vg
V V
1
21
1 12 Q
Q Q
D T
mb BSV V
BS BSV V V V
I Vg V
V V
OX EBQ OX EBQ
W WμC V μC V
L L
2m
BSQ
g-V
mbg
Small Signal Model Summary
OX
m EBQ
μC Wg V
L
DQo λIg
BSQ
mmbV
gg
2
G
S
Vgs
Vbs
Vds
id
ig
ibB
D
Small Signal
Network
dsobsmbgsmd
b
g
vgvgvgi
i
i
0
0
Relative Magnitude of Small Signal MOS Parameters
OX
m EBQ
μC Wg V
L
o DQg λI 5E-7
2mb m m
BSQ
g g .26gV
d m gs mb bs o dsi g v g v g v
DQOX
m IL
WC2μg
DQ
m
EBQ
2Ig
V
Consider:
3 alternate equivalent expressions for gm
If μCOX=100μA/V2 , λ=.01V-1, γ = 0.4V0.5, VEBQ=1V, W/L=1, VBSQ=0V
-4
22OX
DQ EBQ
μC W 10 WI V 1V =5E-5
2L 2L
OX
m EBQ
μC Wg V 1E-4
L
0 m mbg <<g ,g
mb mg < g
In this example
This relationship is common
In many circuits, VBS=0 as well
Large and Small Signal Model Summary
dsobsmbgsmd
b
g
vgvgvgi
i
i
0
0
1
GS T
DS
D OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V V
VWI μC V V V V V V V V
L 2
WμC V V V V V V V V
2L
T
BST0T VVV
Large Signal Model Small Signal Model
OX
m EBQ
μC Wg V
L
DQo λIg
BSQ
mmbV
gg
2
where
End of Lecture 28