Post on 07-May-2022
EE 330 Lecture 27
Small-Signal Modelsss
models of BJT
Quiz 20 Obtain the small signal model of the following circuit. Assume MOSFET is operating in the saturation region
And the number is ….
6
31
2
45
7
8
9
And the number is ….
6
31
2
4
5
7
8
9
Quiz 20 Obtain the small signal model of the following circuit. Assume MOSFET is operating in the saturation region
Solution:
( )0mV g g I+ =
0
1 1EQ
m mR
g g g=
+
Small
Signal Model of BJT
3-terminal device
Usually operated in Forward Active Region when small-signal model is needed
1BE
t
VV CE
C S E
AF
VI J A eV
⎛ ⎞= +⎜ ⎟
⎝ ⎠t
BEV
VES
B eβAJI =
Forward Active Model:
Review from Last Time
Small
Signal Model of BJT
CQ
t
IβV
gπ= CQ
t
IVm
g = CQ
AF
IVO
g =
21 22C BE CEy y= +i V V
11 12B BE CEy y= +i V V
C m BE O CEg g= +i V V
B BEg
π=i V
Review from Last Time
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 Time
Active Device Model SummaryWhat are the simplified dc equivalent models?
dc equivalent
( )2OXGSQ Tn
μC W V -V2L
G D
S
VGSQ ( )2OXGSQ Tp
μC W V -V2L
B C
E
BQβI0.6V
IBQ
B C
E
BQβI0.6V
IBQ
Review from Last Time
Example: Determine the small signal voltage gain AV
=VOUT
/VIN
.
Assume M1 and M2are operating in the saturation region and that λ=0
Small-signal circuit
Small-signal circuit
Review from Last Time
Example:
Small-signal circuit Analysis:
1
2
OUT mV
IN m
gAg
= = −VV
1 2
2 1
V
W LAW L
= −Recall:
If L1 =L2 , obtain
1 2 1
2 1 2
V
W L WAW L W
= − = −
The width and length ratios can be accurately set when designed in a standard CMOS processBut care must be taken in the layout to obtain this accuracy!
Review from Last Time
1
VIN
OUT
DD
SS
1
OUT DD DV =V -I R
( )2OX
DQ SS T
μ C WI V V2L
= + ( )2OX
D IN SS T
μC WI V -V -V2L
=
( )2TSSOX
DQ VV2L
WC μI +=
Graphical Analysis and Interpretation Consider Again
OUT DD DV =V -I R
( )2OX
DQ SS T
μ C WI V V2L
= +
Graphical Analysis and Interpretation Device Model (family of curves) ( ) ( )2 1 λ= +OX
DQ GS T DS
μ C WI V -V V2L
Load Line
Device Model at Operating Point
OUT DD DV =V -I R
( )2OX
DQ SS T
μ C WI V V2L
= +( )2
OXD IN SS T
μC WI V -V -V2L
=
( )2TSSOX
DQ VV2L
WC μI +=
Graphical Analysis and Interpretation
VGSQ
=-VSS
Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS
μ C WI V -V V2L
Graphical Analysis and Interpretation
VGSQ
=-VSS
Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS
μ C WI V -V V2L
Saturation region
Graphical Analysis and Interpretation
VGSQ
=-VSS
Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS
μ C WI V -V V2L
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
Graphical Analysis and Interpretation
VGSQ
=-VSS
Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS
μ C WI V -V V2L
Saturation region
Very limited signal swing with non-optimal Q-point location
Graphical Analysis and Interpretation
VGSQ
=-VSS
Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS
μ C WI V -V V2L
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 ExtensionExisting 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)
EChanging the bulk voltage will change the thickness of the inversion layer
Changing the bulk voltage will change the threshold voltage of the device
( )φφγ −−+= BST0T VVV
T
T0
BS
( )φφγ −−+= BST0T VVV
Typical Effects of Bulk on Threshold Voltage for n-channel Device
1-20.4Vγ 0.6Vφ
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 Vγ φ φ= − + −
Typical Effects of Bulk on Threshold Voltage for p-channel Device
1-20.4Vγ 0.6Vφ
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
( ) ( )1
GS T
DSD OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V VVWI μC V V V V V V V V
L 2WμC V V V V V V V V2L
T
λ
⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟
⎝ ⎠⎪⎪
− • + ≥ ≥ −⎪⎩
( )φφγ −−+= BST0T VVV
Model Parameters : {μ,COX
,VT0
,φ,γ,λ}
Design Parameters : {W,L} but only one degree of freedom W/L
0I0I
B
G
==
Small-Signal Model Extension
( ) ( )1
GS T
DSD OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V VVWI μC V V V V V V V V
L 2WμC V V V V V V V V2L
T
λ
⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟
⎝ ⎠⎪⎪
− • + ≥ ≥ −⎪⎩
( )φφγ −−+= BST0T VVV
000 =∂∂
==∂∂
==∂∂
==== QQQ VVGS
G13
VVDS
G12
VVGS
G11 V
IyVIy
VIy
000 =∂∂
==∂∂
==∂∂
==== QQQ VVGS
B33
VVDS
B32
VVGS
B31 V
IyVIy
VIy
Q Q Q
D D D21 12 13
GS DS GSV V V V V V
I I Iy y yV V Vm o mb
g g g= = =
∂ ∂ ∂= = = = = =
∂ ∂ ∂
GI =0
BI =0
( ) ( )DS2
TGSOXD VVV2LWμCI λ+•−= 1
( )φφγ −−+= BST0T VVV
( ) ( )11
OX GS T DS OX EBQ
W WμC 2 V V V μC V2L L
Dm
V VGS V V
IgV
λ==
∂= = − • + ≅∂
( )2
OX GS T DQ
WμC 2 V V λ λI2L
Do
V VDS V V
IgV ==
∂= = − • ≅∂
( ) ( )1
OX GS T DS
WμC 2 V V 1+λV2L
Q Q
D Tmb
BS BSV V V V
I VgV V
= =
⎛ ⎞∂ ∂= = − • − •⎜ ⎟∂ ∂⎝ ⎠
( ) ( ) ( )12
11 12 Q
Q Q
D Tmb BS
V VBS BSV V V V
I Vg VV V
γ φ −
== =
∂ ∂ ⎛ ⎞= ≅ • = − − −⎜ ⎟∂ ∂ ⎝ ⎠OX EBQ OX EBQ
W WμC V μC VL L
2m
BSQ
g-Vmb
g γφ
≅
Small Signal Model Summary
OXm EBQ
μC Wg VL
=
DQo λIg =
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
BSQmmb V
ggφγ
2
G
S
gs
bs
ds
d
g
bB
D
dsobsmbgsmd
b
g
vgvgvgii
i
++==
=
0
0
Relative Magnitude of Small Signal MOS Parameters
OXm EBQ
μC Wg VL
=
o DQg λI 5E-7= =
2mb m m
BSQ
g g .26gV
γφ
⎛ ⎞= =⎜ ⎟⎜ ⎟−⎝ ⎠
d m gs mb bs o dsi g v g v g v= + +
DQOX
m IL
WC2μg =DQ
m
EBQ
2IgV
=
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
22OXDQ EBQ
μC W 10 WI V 1V =5E-52L 2L
≅ =
OXm EBQ
μC Wg V 1E-4L
= =0 m mb
g <<g ,g
mb mg < g
In this example
This relationship is commonIn many circuits, VBS
=0 as well
Small Signal Model Summary
dsobsmbgsmd
b
g
vgvgvgii
i
++==
=
0
0
( ) ( )1
GS T
DSD OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V VVWI μC V V V V V V V V
L 2WμC V V V V V V V V2L
T
λ
⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟
⎝ ⎠⎪⎪
− • + ≥ ≥ −⎪⎩
( )φφγ −−+= BST0T VVV
Large Signal Model Small Signal Model
OXm EBQ
μC Wg VL
=
DQo λIg =
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
BSQmmb V
ggφγ
2
where
How does gm
vary with IDQ
?
DQOX
m IL
WC2μg =
( )TGSQOX
m VVL
WμCg −=
EBQ
DQ
TGSQ
DQm V
2IVV
2Ig =−
=
Varies with the square root of IDQ
Varies linearly with IDQ
Doesn’t vary with IDQ
How does gm
vary with IDQ
?
All of the above are true –
but with qualification
gm
is a function of more than one variable (IDQ
) and how it varies depends upon how the remaining variables are constrained
Small Signal Model Summary
( )TGSQOX
m VVL
WμCg −=
DQo λIg =
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
BSQmmb V
ggφγ
2
An
equivalent circuit
This contains absolutely no more information than the previous model
Small Signal Model Summary
More convenient representation
Small Signal Model Summary
Simplification that is often adequate
Small Signal Model Summary
Even further simplification that is often adequate
Small Signal Model Summary
Alternate equivalent representations for gm
DQOX
m IL
WC2μg =
( )TGSQOX
m VVL
WμCg −=
EBQ
DQ
TGSQ
DQm V
2IVV
2Ig =−
=
( )2TGSOXD VV2LWμCI −≅from
Alternate Equivalent Small Signal Model of the BJT
Small Signal BJT Model
t
CQm V
Ig =t
CQ
βVIg =π
AF
CQ
VIg ≅o
bbe iv =πg
π
mm g
gg bbe iv =
β
βVIVI
gg
t
Q
t
Q
π
m =
⎥⎦
⎤⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡
=
bbe iv βgm =
Observe :
Small Signal BJT Model
t
CQm V
Ig =t
CQ
βVIg =π
AF
CQ
VIg ≅o
t
CQ
βVIg =π
AF
CQ
VIg ≅o
B
E
C
Vbe
ic
o Vce
ib
π biβ
Alternate equivalent small signal model
Relative Magnitude of Small Signal BJT Parameters
t
CQm V
Ig =t
CQ
βVIg =π
AF
CQ
VIg ≅o
β
βVIVI
gg
t
Q
t
Q
π
m =
⎥⎦
⎤⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡
=
7726mV100
200VVβ
V
VIVβ
I
gg
t
AF
AF
Q
t
Q
o
π =•
≈=
⎥⎦
⎤⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡
=
oπm ggg >>>>
Often the go term can be neglected in the small signal model because it is so small
Relative Magnitude of Small Signal Parameters
t
CQm V
Ig =t
CQ
βVIg =π
AF
CQ
VIg ≅o
β
βVIVI
gg
t
Q
t
Q
π
m =
⎥⎦
⎤⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡
=
7726mV100
200VVβ
V
VIVβ
I
gg
t
AF
AF
Q
t
Q
o
π =•
≈=
⎥⎦
⎤⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡
=
oπm ggg >>>>
Often the go term can be neglected in the small signal model because it is so small
Simplified small signal model
Vbe
ic
mVbe Vce
ib
π
Comparison of BJT and MOSFET
Comparison of MOSFET and BJT
t
CQm V
Ig =
EBVL
WμCg OXm =
DQOX
m IL
WC2μg =
EBQ
DQm V
2Ig =
BJT MOSFET
⎪⎪⎩
⎪⎪⎨
⎧
==>
==>===
2VVif50mV
2V50mV
V
100mVV if250mV
100mV50mV
V
2VV
V2IVI
gg
EBEB
EBEB
t
EB
EB
DQ
t
CQ
mMOS
mBJT
40
The transconductance
of the BJT is typically much larger than that of the MOSFET (and larger is better!) This is due to the exponential rather than quadratic output/input relationship
Comparison of MOSFET and BJT
t
CQm V
Ig =
EBVL
WμCg OXm =
DQOX
m IL
WC2μg =
EBQ
DQm V
2Ig =
BJT MOSFET
⎪⎪⎩
⎪⎪⎨
⎧
==>
==>===
2VVif50mV
2V50mV
V
100mVV if250mV
100mV50mV
V
2VV
V2IVI
gg
EBEB
EBEB
t
EB
EB
DQ
t
CQ
mMOS
mBJT
40
The transconductance
of the BJT is typically much larger than that of the MOSFET (and larger is better)This is due to the exponential rather than quadratic output/input relationship
Comparison of MOSFET and BJT
AF
CQ
VIg ≅oDQIλ=og
BJT MOSFET
5.020001.
11 =≈== − VVAFDQ
AF
CQ
oMOS
oBJT
V1
IVI
gg
λλ
The output conductances
are comparable but that of the BJT is usually modestly smaller (and smaller is better!)
Comparison of MOSFET and BJT
t
CQ
βVIg =π
BJT MOSFET
0=πg
gπ
of a MOSFET is much smaller than that of a BJT (and smaller is better!)
gπ
is the reciprocal of the input impedance
Standard Approach to small-signal analysis of nonlinear networks
NonlinearNetwork
dc EquivalentNetwork
Q-point
Values for small-signal parameters
Small-signalequivalent network
Small-signal output
Total output(good approximation)
Systematic Approach to Small-Signal Circuit Analysis
•
Obtain dc equivalent circuit by replacing all elements with large-signal (dc) equivalent circuits
•
Obtain dc operating points (Q-point)
•
Obtain ac equivalent circuit by replacing all elements with small-signal equivalent circuits
•
Analyze linear small-signal equivalent circuit
Dc and small-signal equivalent elementsRecall
Square-Law Model
( ) ( )1
GS T
DSD OX GS T DS GS DS GS T
2
OX GS T DS GS T DS GS T
0 V VVWI μC V V V V V V V V
L 2WμC V V V V V V V V2L
T
λ
⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟
⎝ ⎠⎪⎪
− • + ≥ ≥ −⎪⎩
( )φφγ −−+= BST0T VVV
0I0I
B
G
==
Simplified MOS Model for Q-point Analysis
( )2
D OX GS T
WI μC V V2L
= −
0I0I
B
G
==
( )2OX
GS T
μC W V -V2L
⎛ ⎞⎜ ⎟⎝ ⎠
Simplified dc equivalent circuit
dc BJT model
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
AF
CEBC V
V1βII
t
BE
VV
ESB e
βAJI =
qkTVt =
VBE
=0.7VVCE
=0.2V
IC
=IB
=0
Forward Active
Saturation
Cutoff
VBE
>0.4V
VBC
<0
IC
<βIB
VBE
<0
VBC
<0
A small portion of the operating region is missed with this model but seldom operate in the missing region
Simplified dc BJT model for Q- point Analysis
C BI βI=
t
BE
VV
ESB e
βAJI =
C BI βI=
BEV 0.6V=
Simplified dc equivalent circuit
Examples
Not convenient to have multiple dc power suppliesVOUTQ
very sensitive to VEE
Examples
Not convenient to have multiple dc power suppliesVOUTQ
very sensitive to VEE
B
E
C
CC
Vin
1
Vout
1
B
1
ExamplesDetermine VOUTQ
, AV
, RIN
B
E
C
CC
Vin
1
Vout
1
B
1
ExamplesDetermine VOUT
and VOUT
(t) if VIN
=.002sin(400t)
B
E
C
CC
Vin
1
Vout
1
B
1
End of Lecture 27