Signal Circuit and Transistor Small-Signal Model
Lecture notes: Sec. 5
Sedra & Smith (6th Ed): Sec. 5.5 & 6.7 Sedra & Smith (5th Ed): Sec. 4.6 & 5.6
F. Najmabadi, ECE65, Winter 2012
Transistor Amplifier Development
F. Najmabadi, ECE65, Winter 2012
Bias & Signal
.....
:
,...)( ,,, : MOS
rRR
rRRD
gsGSGS
DDSGS
iIivVvR
vVvivv
+=+=
+=
....., :
,,, : MOS
RRD
DDSGS
IVR
IVV
....., :
,,, : MOS
rrD
ddsgs
ivR
ivv
+
Bias Signal only = (Bias + Signal) - Bias
?
Finding signal circuit elements -- Resistor
F. Najmabadi, ECE65, Winter 2012
)( RRRRRRr IiRRIRiVvv −=−=−=
Resistor Voltage Current iv Equation
Bias + Signal: vR iR vR = R iR
Bias: VR IR VR = R IR
Signal: vr = vR − VR ir = iR − IR ??
rr Riv =
A resistor remains as a resistor in the signal circuit.
Finding signal circuit elements -- Capacitor
F. Najmabadi, ECE65, Winter 2012
dtVvdC
dtdVC
dtdvCIii CCCC
CCc)( −
=−=−=
Capacitor Voltage Current iv Equation
Bias + Signal: vC iC iC = C dvC /dt
Bias: VC IC IC = C dVC /dt
Signal: vc = vC − VC ic = iC − IC ??
dtdvCi c
c =
A capacitor remains as a capacitor in the signal circuit. o Since VC = const., IC = 0 , i.e., A capacitor acts as an open circuit for bias circuit.
Finding signal circuit elements – IVS & ICS
F. Najmabadi, ECE65, Winter 2012
0=−=−= SSIVSIVSivs VVVvv
Independent voltage source
Voltage Current iv Equation
Bias + Signal: vIVS iIVS vIVS = VS = const
Bias: VIVS IIVS VIVS = VS = const
Signal: vivs = vIVS − VIVS iivs = iIVS − IIVS ??
0 ,0 ≠= ivsivs iv
An independent voltage source becomes a short circuit!
An independent current source becomes an open circuit!
Similarly:
Exercise: Show that dependent sources remain as dependent sources
Summary of signal circuit elements
Resistors& capacitors: The Same o Capacitor act as open circuit in the bias circuit.
Independent voltage source (e.g., VDD) : Effectively grounded
Independent current source: Effectively open circuit o Careful about current mirrors as they are NOT “ideal” current sources (early
effect and/or channel width modulation was ignored!)
Dependent sources: The Same
Non-linear Elements: Different! o Diodes & transistors ?
F. Najmabadi, ECE65, Winter 2012
Diode Signal Response
F. Najmabadi, ECE65, Winter 2012
−
×
=
−
+=−=
1expexp
expexp :Signal
T
d
T
Dsd
T
Ds
T
dDsDDd
nVv
nVVIi
nVVI
nVvVIIii
VD
ID
vd
id ?
=+
T
DsD nV
vIi exp :Signal Bias
=
T
DsD nV
VII exp :Bias
vD
iD
A different iv equation! iv equation is non-linear! Related to bias value, ID!
1exp
−
×=
T
dDd nV
vIi
Diode small-signal model:
F. Najmabadi, ECE65, Winter 2012
1exp
−
×=
T
dDd nV
vIi
vd
id ?
DT
D
T
dDd
T
d
T
d
T
d
T
d
T
d
T
d
vnVI
nVvIi
nVv
nVv
nVv
nVv
nVv
nVv
=
−
+×≈
+≈
<<
+
+
+=
11
1exp :1 If
.... !2
11exp :Exapnsion SeriesTaylor 2
dddD
Td iri
InVv ==
Formal derivation of small signal model
F. Najmabadi, ECE65, Winter 2012
2)2(
)1(
!2)()( a
AaA vVfvVf ⋅>>⋅
)()(2 )2(
)1(
A
Aa Vf
Vfv ⋅<<
Small signal means:
aAaa vVfvgi ⋅== )()( )1(
Signal + Bias for element A (iA, vA) : iA = f (vA) Bias for element A (IA, VA) : IA = f (VA) Signal for element A (ia, va) : ia = g (va)
( ) ( )
aAA
aA
aAA
AAA
AAAA
AA
vVfVf
vVfvVfVf
VvVfVvVfVf
vfi
⋅+≈
+⋅+⋅+=
+−⋅+−⋅+=
=
)()(
...!2
)()()(
...!2
)()()(
)(
)1(
2)2(
)1(
2)2(
)1( (Taylor Series Expansion)
aAAAaA vVfIIii ⋅+=+= )()1(
Derivation of diode small signal model
F. Najmabadi, ECE65, Winter 2012
)(1 DnVv
SD vfeIi T
D
=
−⋅= TT nV
v
ST
nVv
S eInV
vfeIvf 1)( 1)( )1( ×=
−=
1)( SnVV
SnVV
SDD IeIeIVfI T
D
T
D
−=
−⋅==
dT
SDd
T
nVV
Sd
VvT
nVv
SdDd v
nVIIv
nVeIv
nVeIvVfi
T
D
D
T
×
+=×
⋅
=×
⋅
=×=
=
)()1(
dT
Dd
T
SDd v
nVIv
nVIIi ×
≈×
+=
d
dd r
vi =D
Td I
nVr ≈ vd
id rd = nVT/ID
vD
iD
Diode can be replaced with a resistor in the signal circuit!
Small signal model vs iv characteristics
F. Najmabadi, ECE65, Winter 2012
Small signal model is equivalent to approximating the non-liner iv characteristics curve by a line tangent to the iv curve at the bias point
D
T
Dd
dDd
InV
Vfr
vVfi
≈=
×=
)(1
)(
)1(
)1(
Derivation of MOS small signal model (1)
F. Najmabadi, ECE65, Winter 2012
Signal + Bias for MOS (iD, vGS , vDS) : iD = f (vGS, vDS), iG = 0 Bias for MOS (ID, VGS , VDS) : ID = f (VGS, VDS), IG = 0 Signal for MOS (id, vgs , vds) : id = g (vgs , vds), ig = 0
MOS iv equations: iD = f (vGS, vDS) iG = 0
dsVVDS
gsVVGS
D
DSDSVVDS
GSGSVVGS
DSGS
DSGSDdD
vvfv
vfI
VvvfVv
vfVVf
vvfiiI
DSGSDSGS
DSGSDSGS
×∂∂
+×∂∂
+≈
+−⋅∂∂
+−⋅∂∂
+=
==+
,,
,,
...)()(),(
),(
,,
dsVVDS
gsVVGS
d vvfv
vfi
DSGSDSGS
×∂∂
+×∂∂
≈
(Taylor Series Expansion in 2 variables)
Derivation of MOS small signal model (2)
F. Najmabadi, ECE65, Winter 2012
dsVVDS
gsVVGS
d vvfv
vfi
DSGSDSGS
⋅∂∂
+⋅∂∂
=,,
),()1()( 5.0 2DSGSDStGSoxnD vvfvVv
LWCi =+−= λµ
mOV
D
tGS
DStGSoxn
VVDStGSoxnVVGS
gV
IVV
VVVL
WC
vVvL
WCvf
DSGS
DSGS
≡=−
+−×=
+−×=∂∂
2)(
)1()( 5.02
)1)(( 5.02
2
,,
λµ
λµ
oD
DS
D
DS
DStGSoxn
VVtGSoxn
VVDS
rI
VI
V
VVVL
WC
VvL
WCvf
DSGSDSGS
1)1()1(
)1()( 5.0
)( 5.0
2
,
2
,
≡≈+
=+
+−×=
−×=∂∂
λλ
λλ
λµλ
µλ
0 =+⋅= go
dsgsmd i
rvvgi
MOS small signal “circuit” model
F. Najmabadi, ECE65, Winter 2012
and 0 o
dsgsmdg r
vvgii +⋅==
Do I
r⋅
≈λ
1
OV
Dm V
Ig ⋅=
2 122>>==
OV
A
OVom V
VV
rgλ
Statement of KCL Two elements in parallel Input open circuit
PMOS “circuit” small signal model is identical to NMOS
F. Najmabadi, ECE65, Winter 2012
=
PMOS small-signal circuit model is identical to NMOS o We will use NMOS circuit model for both! o For both NMOS and PMOS, while iD ≥ 0 and ID ≥ 0, signal quantities: id,
vgs, and vds , can be negative!
PMOS* NMOS
Exercise: Derive PMOS small signal model (follow derivation of NMOS small-signal model)
Derivation of BJT small signal model (1)
F. Najmabadi, ECE65, Winter 2012
Signal + Bias for BJT (iB, iC, vBE , vCE) : iB = f1 (vBE), iC = f2 (vBE, vCE)
Bias for BJT (IB, IC, VBE , VCE) : IB = f1 (VBE), IC = f2 (VBE, VCE)
Signal for BJT (ib, ic, vbe , vce) : ib = g1 (vbe), ic = g2 (vbe, vce)
BJT iv equations: iB = f1 (vBE) iC = f2 (vBE, vCE)
+=
=
A
CEVv
sC
Vv
sB
VveIi
eIi
T
BE
T
BE
1
)/( β
We need to perform Taylor Series Expansion in 2 variables for both iB and iC.
,
1be
VVBEB v
dvdfi
CEBE
×≈ ,
2
,
2ce
VVDCEbe
VVBEC v
vfv
vfi
CEBECEBE
×∂
∂+×
∂∂
≈
Derivation of BJT small signal model (2)
F. Najmabadi, ECE65, Winter 2012
o
cebemc
beb r
vvgirvi +==
π
)( )/( 1 BEVv
sB vfeIi T
BE
== β
bebeVVBE
B vr
vdvdfi
CEBE
×=×≈π
1 ,
1
π
βrV
IeIVdv
df
T
B
V
Vv
sTVVBE
BE
T
BE
CEBE
1 )/(1
,
1 ≡==
T
BEVV
sB eII )/( β=
+=
A
CEVv
sC VveIi T
BE
1
A
CEAVV
sA
CEVV
sC VVVeI
VVeII T
BE
T
BE +×=
+= 1
mT
C
VA
CEVv
T
s
VVBE
gVI
Vve
VI
dvdf
CEVBE
T
BE
CEBE
≡=
+= 1
,,
2
oCEA
C
V
Vv
A
s
VVCE rVVIe
VI
dvdf
CEVBE
T
BE
CEBE
1
,
,
2 ≡+
==
BJT small signal “circuit” model
F. Najmabadi, ECE65, Winter 2012
C
A
C
CEAo I
VI
VVr ≈+
=
T
Cm V
Ig =
Statement of KCL Two elements in parallel A resistor, rπ ,
between B & E
o
cebemc
beb r
vvgirvi +==
π
B
T
IVr =π
We follow S&S: vbe is denoted as vπ
Similar to NMOS/PMOS, the small circuit model for a PNP BJT is the same as that of a NPN.
Alternative BJT small signal “circuit” model
F. Najmabadi, ECE65, Winter 2012
πππ
π
π
ββ
β
ivr
vg
rVI
II
VIg
m
T
B
B
C
T
Cm
=×=
=×==
gm Model
β ib Model
Summary of transistor small signal models
F. Najmabadi, ECE65, Winter 2012
Do
OV
Dm I
rV
Ig⋅
≈⋅
=λ
1 2
Comparison of MOS and BJT small-signal circuit models: 1. MOS has an infinite resistor in the input (vgs) while BJT has a finite resistor, rπ
(typically several kΩ). 2. BJT gm is substantially larger than that of a MOS (BJT has a much higher gain). 3. ro values are typically similar (10s of kΩ). gm ro >> 1 for both.
NMOS/PMOS
C
Ao
T
Cm
B
T
IVr
rVIg
IVr ≈===
ππ
β
NPN/PNP BJT