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
Bias & Signal
Bias
Signal only
= (Bias + Signal) - Bias
+
?
MOS : VGS ,VDS , I D ,
MOS : vgs , vds , id ,
vR = VR + vr
RD :
RD :
iR = I R + ir
.....
MOS : vGS , vDS , iD ,
(vGS = VGS + vgs ,...)
RD :
.....
F. Najmabadi, ECE65, Winter 2012
VR , I R
.....
vr , ir
Finding signal circuit elements -- Resistor
Resistor
Voltage
Current
iv Equation
Bias + Signal:
vR
iR
vR = R iR
Bias:
VR
IR
VR = R IR
vr = vR − VR
ir = iR − IR
??
Signal:
vr = vR − VR = RiR − RI R = R (iR − I R )
vr = Rir
 A resistor remains as a resistor in the signal circuit.
F. Najmabadi, ECE65, Winter 2012
Finding signal circuit elements -- Capacitor
Capacitor
Voltage
Current
iv Equation
Bias + Signal:
vC
iC
iC = C dvC /dt
Bias:
VC
IC
IC = C dVC /dt
vc = vC − VC
i c = i C − IC
??
Signal:
ic = iC − I C = C
dvC
dV
d (v − V )
−C C = C C C
dt
dt
dt
ic = C
dvc
dt
 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.
F. Najmabadi, ECE65, Winter 2012
Finding signal circuit elements – IVS & ICS
Independent
voltage source
Voltage
Current
iv Equation
Bias + Signal:
vIVS
iIVS
vIVS = VS = const
Bias:
VIVS
IIVS
VIVS = VS = const
vivs = vIVS − VIVS
iivs = iIVS − IIVS
??
Signal:
vivs = vIVS − VIVS = VS − VS = 0
vivs = 0, iivs ≠ 0
 An independent voltage source becomes a short circuit!
Similarly:
 An independent current source becomes an open circuit!
F. Najmabadi, ECE65, Winter 2012
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
 v
Bias + Signal : iD = I s exp D
 nVT
Bias :
Signal :
 VD
I D = I s exp
 nVT
vD



iD
VD



ID
V
V + v 
id = iD − I D = I s exp D d  − I s exp D
 nVT
 nVT 
 VD    vd  
 − 1
 × exp
id = I s exp
nV
nV
 T   T 
  vd
id = I D × exp
  nVT
F. Najmabadi, ECE65, Winter 2012
 
 − 1
 



id
vd
?
 A different iv equation!
 iv equation is non-linear!
 Related to bias value, ID!
Diode small-signal model:
vd
  vd
id = I D × exp
  nVT
 
 − 1
 
id
 v  1 v
 v 
Taylor Series Exapnsion : exp d  = 1 +  d  +  d
 nVT  2!  nVT
 nVT 
 v 
 v 
vd
<< 1 :
If
exp d  ≈ 1 +  d 
nVT
 nVT 
 nVT 
  vd
id ≈ I D × 1 + 
  nVT
vd =
F. Najmabadi, ECE65, Winter 2012
   ID
 − 1 = 
   nVT
nVT
id = rd id
ID
2

 + ....


vD

?
Formal derivation of small signal model
 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)
i A = f (v A )
f ( 2 ) (VA )
2
= f (VA ) + f (VA ) ⋅ (v A − VA ) +
⋅ (v A − VA ) + ...
2!
f ( 2 ) (VA ) 2
(1)
= f (VA ) + f (VA ) ⋅ va +
⋅ va + ...
2!
≈ f (VA ) + f (1) (VA ) ⋅ va
(1)
i A = ia + I A = I A + f (1) (VA ) ⋅ va
ia = g (va ) = f
F. Najmabadi, ECE65, Winter 2012
(1)
(VA ) ⋅ va
(Taylor Series
Expansion)
Small signal means:
f
(1)
f ( 2 ) (VA ) 2
⋅ va
(VA ) ⋅ va >>
2!
f (1) (VA )
va << 2 ⋅ ( 2 )
f (VA )
Derivation of diode small signal model
 nVvD

T

i D = I S ⋅ e − 1 = f ( v D )





 nVv
T

f (v) = I S e − 1




v
1
× I S e nVT
f (1) (v) =
nVT
VD
VD
 nV

nV
I D = f (VD ) = I S ⋅  e T − 1 = I S e T − I S




v

nVT
I ⋅e
id = f (1) (VD ) × vd =  S
nVT


VD


nV
 IS ⋅ e T

× vd = 

nVT


 v =VD

 I 
I + I 
id =  D S  × vd ≈  D  × vd
 nVT 
 nVT 
id =
vd
rd
rd ≈
F. Najmabadi, ECE65, Winter 2012
nVT
ID

 ID + IS 

v
×
=
 d  nV  × vd
T




vD
iD
Diode can be replaced with a
resistor in the signal circuit!
vd
id
rd = nVT/ID
Small signal model vs iv characteristics
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
id = f (1) (VD ) × vd
rd =
1
nVT
≈
f (1) (VD ) I D
F. Najmabadi, ECE65, Winter 2012
Derivation of MOS small signal model (1)
MOS iv equations: iD = f (vGS, vDS)
iG = 0
 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),
(Taylor Series Expansion in 2 variables)
I D + id = iD = f (vGS , vDS )
= f (VGS , VDS ) +
≈
ID
+
∂f
∂vGS
∂f
∂vGS
⋅ (vGS − VGS ) +
VGS ,VDS
× v gs +
VGS ,VDS
∂f
∂vDS
id ≈
F. Najmabadi, ECE65, Winter 2012
ig = 0
∂f
∂vDS
⋅ (vDS − VDS ) + ...
VGS ,VDS
× vds
VGS ,VDS
∂f
∂vGS
× v gs +
VGS ,VDS
∂f
∂vDS
× vds
VGS ,VDS
Derivation of MOS small signal model (2)
iD = 0.5µ nCox
id =
∂f
∂vGS
∂f
∂vGS
W
(vGS − Vt ) 2 (1 + λvDS ) = f (vGS , vDS )
L
⋅ v gs +
VGS ,VDS
VGS ,VDS
0.5µ nCox
=λ×
0.5µ nCox
VGS ,VDS
VGS ,VDS
W
(VGS − Vt ) 2 (1 + λVDS )
2I
L
= D ≡ gm
(VGS − Vt )
VOV
= λ × 0.5µ nCox
VGS ,VDS
⋅ vds
W
(vGS − Vt )(1 + λvDS )
L
= 2 × 0.5µ nCox
= 2×
∂f
∂vDS
∂f
∂vDS
W
(vGS − Vt ) 2
L
VGS ,VDS
W
(VGS − Vt ) 2 (1 + λVDS )
1
λI D
L
=
≈ λI D ≡
(1 + λVDS )
(1 + λVDS )
ro
id = g m ⋅ v gs +
F. Najmabadi, ECE65, Winter 2012
vds
ro
ig = 0
MOS small signal “circuit” model
ig = 0 and id = g m ⋅ v gs +
vds
ro
Statement of KCL
Two elements in parallel
Input open circuit
gm =
2⋅ ID
VOV
F. Najmabadi, ECE65, Winter 2012
ro ≈
1
λ ⋅ ID
g m ro =
2
2V
= A >> 1
λVOV VOV
PMOS “circuit” small signal model is identical to NMOS
PMOS*
NMOS
=
 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!
F. Najmabadi, ECE65, Winter 2012
Exercise: Derive PMOS small signal model (follow
derivation of NMOS small-signal model)
Derivation of BJT small signal model (1)
iB = ( I s / β ) e
BJT iv equations: iB = f1 (vBE)
iC = f2 (vBE, vCE)
iC = I s e
v BE
VT
v BE
VT
 vCE
1 +
 VA



 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)
We need to perform Taylor Series Expansion in 2 variables for both iB and iC.
iB ≈
df1
dvBE
× vbe
VBE ,VCE
F. Najmabadi, ECE65, Winter 2012
iC ≈
∂f 2
∂vBE
× vbe +
VBE ,VCE
∂f 2
∂vDCE
× vce
VBE ,VCE
Derivation of BJT small signal model (2)
iB = ( I s / β ) e
v BE
VT
I B = (I s / β ) e
df1
dvBE
iC = I s e
= f1 (vBE )
VBE
VT
IC = I s e
v BE
VBE ,VCE
df
iB ≈ 1
dvBE
v BE
VT
1
( I s / β ) e VT
=
VT
× vbe
VBE ,VCE
VBE
1
I
= B ≡
VT rπ
df 2
dvCE
1
= × vbe
rπ
ib =
F. Najmabadi, ECE65, Winter 2012
df 2
dvBE
vbe
rπ
VBE
VT



 vCE
1 +
 VA
 VCE
1 +
VA

BE

V +V
VT
 = I s e × A CE
VA

v BE
VBE ,VCE
I
= s e VT
VT
VBE ,VCE
I
= s e VT
VA
V
 vCE
1 +
 VA
CE
v BE
ic = g m vbe +

I

= C ≡ gm
VT
 VBE ,V
vce
ro
=
VBE ,VCE
IC
1
≡
VA + VCE ro
BJT small signal “circuit” model
vbe
ib =
rπ
A resistor, rπ ,
between B & E
vce
ic = g m vbe +
ro
Statement of KCL
Two elements in parallel
IC
gm =
VT
VT
rπ =
IB
We follow S&S:
vbe is denoted as vπ
ro =
Similar to NMOS/PMOS, the small circuit model
for a PNP BJT is the same as that of a NPN.
F. Najmabadi, ECE65, Winter 2012
VA + VCE VA
≈
IC
IC
Alternative BJT small signal “circuit”
model
gm Model
gm =
IC IC I B β
=
×
=
VT I B VT rπ
g m vπ =
β ib Model
F. Najmabadi, ECE65, Winter 2012
β
rπ
× vπ = β iπ
Summary of transistor small signal models
NMOS/PMOS
gm =
2⋅ ID
VOV
ro ≈
NPN/PNP BJT
1
λ ⋅ ID
rπ =
VT
IB
gm =
IC β
=
VT rπ
ro ≈
VA
IC
 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.
F. Najmabadi, ECE65, Winter 2012