Transistors.
Small-Signal Models
 Small-signal operation
 Small- signal parameters
 Small- signal models
1/13
Small-signal model is necessary to deduce
vo as a function of vi
2/13
Small-signal operation
The transistor for the small-signal regime:
regime
 differential parameters (small-signal parameters)
 small-signal equivalent circuit of the transistor.
 the values of the small-signal parameters depend on the OP
• transistor model for low frequency:
 input resistance
 output resistance
 controlled source showing the input-output transfer
• the model for high frequency will be enhanced
with parasitic capacitances between its terminals
3/13
Small-signal MOSFET
CS topology
The full circuit of the amplifier with
one MOST (biasing + small-signal)
The equivalent small signal
circuit results by setting to zero
the dc sources
4/13
Small-signal parameters
• Transconductance
id
i D
gm 
vDS cst 
vDS cst
vGS
v gs
 (  (vGS  VTh ) 2
gm 
vGS
Q
 2  (vGS  VTh ) Q
2I D
gm 
 2  ID
VGS  VTh
integrated transistors:
id  g m v gs
W
g m  2K I D
L
MOSFET: voltage-controlled current
source for small signal
5/13
• Input resistance
the gate is electrically insulated from the rest of
structure: the input resistance is infinite
• Output resistance
the output characteristics are not
perfect horizontally, the drain current
increases slightly with the drain-source
voltage at vGS=cst.
v DS
1
ro 

go
i D
vGS
vds
 cst 
id
VA – Early voltage
iD   (vGS
 vDS
 VTh ) 1 
 VA
2
vGS  cst



VA
ro 
ID
6/13
dc regime
MOST:
small-signal regime
id  g m v gs
g m  2  (VGS  VTh )
I D   (VGS  VTh ) 2
VDS
RO 
ID
id  2  (VGS  VTh )v gs
VA
ro 
ID
7/13
Small-signal model of the MOSFET
• low frequency:
2I D
gm 
 2  ID
vGS  VTh
VA
ro 
ID
• high frequency:
the parasitic capacitances
appear between terminals;
typically pF or fractions of pF
8/13
Small-signal parameters of the BJT
 Transconductance
iC
gm 
v BE
ic
vCE cst 
vbe
vCE cst
iC  I S e vBE / VT
VT  25mV @ 20 o C
IC
gm 
 40 I C @ 20o C
VT
VT 
KT
q
temp.  g m 
• Current gain
iC

i B
ic
vCE cst 
ib
vCE cst
Even if some differences can appear
between the dc current gain and
differential current gain we will use the
same notation and the same value (e.g.
β =100) for the first order analysis.
9/13
Small-signal parameters of the BJT
• Output resistance
vCE
ro 
iC
vBE cst
iC  I S e
vBE
VT
vce

ic
 vCE
1 
VA

vBE cst



• Input resistance
v BE
rbe 
i B
rbe 
vbe
vCE cst 
ib
vCE cst

gm
vA
ro 
IC
10/13
Small-signal model of the BJT (low frequency)
g m  40 I C
rbe 

gm
vA
ro 
IC
simplified
hybrid-π
model
11/13
Small-signal model of the BJT
(high frequency)
hybrid-π
model
 parasitic capacitances between the terminals
 the effect of the capacitors: decreasing the gain at high
frequency
 one can also use the model with the CCCS
12/13
Numerical example for MOSFET
MOSFET : K=100μA/V2 , W/L=1, VA=100V ; biased at ID=100μA.
What are the values of the small signal parameters at low frequency?
g m  2K
W
L
I D  2 100  1  100  0.14mS
VA 100

 1MΩ
ro 
I D 0.1
Numerical example for BJT
BJT biased in OP at IC=100μA, VA=100V, β=100.
What are the values of the small signal parameters at low frequency?
gm=40·IC=40·0.1=4mS

100

 25KΩ
rbe 
gm
4
VA 100

 1MΩ
ro 
I C 0.1
13/13