Transistors.
Small-Signal Models
 Small-signal operation
 Small-signal parameters
 Small-signal models
hibrid  
Necessity for dc transistor biasing
 transistor utilization as amplifier (CS, CE)
 in active region (aF), the transistor operates around the dc
operating point (OP)
VPS – dc supply
VI – sets the OP: (VO, IO)
vi – input voltage
(to be amplified)
vo – output voltage
(amplified voltage)
• superposition of the
variable signal over the
dc voltage
Small-signal model (linear model) is necessary to deduce
vo as a function of vi
Small-signal operation
The transistor for the small-signal regime:
 small-signal parameters (differential parameters)
 small-signal equivalent circuit of the transistor.
 the values of the small-signal parameters depend on the OP
(they are calculated in the OP)
• transistor model for low and medium 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
T – small-signal model
• two-port network
 input resistance
 transfer: a controlled current source (by a voltage) - VCCS
 output resistance
Small-signal MOSFET
CS topology
- linear model The full circuit of the amplifier with one
MOST (dc biasing + small signal)
The small-signal equivalent circuit
results by setting to zero all dc
voltage and/or current sources
Small-signal parameters
• Transconductance
(it shows the transfer from the variable input
voltage to the variable output current)
i D
gm 
vGS
vDS cst
id

v gs
 (  (vGS  VTh ) 2
gm 
vGS
Q
iD   vGS  VTh 
vDS cst
 2  (VGS  VTh )
2I D
g m  2  (VGS  VTh ) 
 2 I D
VGS  VTh
integrated transistors:
id  g m v gs
W
g m  2K I D
L
MOSFET: voltage-controlled current
source for small signal
2
• Input resistance
the gate is electrically insulated from the rest of structure:
the input resistance is infinite (open-circuit)
• Output resistance
the output characteristics are not
perfectly horizontal, the drain current
slightly increases with the drain to
source voltage at vGS=cst.
iD   (vGS
 vDS
 VTh ) 1 
 VA
2



VA – Early voltage
1 v DS
ro 

go
iD
vGS
vds
 cst 
id
vGS  cst
VA
ro 
ID
dc regime
MOST:
small-signal regime
id  g m v gs
g m  2 VGS  VTh  
2I D

 2 I D
VGS  VTh
I D   (VGS  VTh ) 2
VDS
RO 
ID
id  2 (VGS  VTh )vgs
VA
ro 
ID
Small-signal model of the MOSFET
• low and medium frequency:
g m  2 VGS  VTh  
2I D

 2 I D
VGS  VTh
VA
ro 
ID
• high frequency:
the parasitic capacitances
appear between terminals;
typically pF or fractions of pF
linear models (valid around OP)
hibrid  
Small-signal parameters of the BJT
 Transconductance
iC
gm 
v BE
ic
vCE cst 
vbe
vCE cst
iC  I S evBE /VT
• Current gain
VT  25mV @ 20o C
IC
gm 
 40 I C @ 20o C
VT
g m [mS]
I C [mA]
KT
VT 
q
temp.  g m 
iC

i B
ic
vCE cst 
ib
vCE cst
Even if some differences can appear
between in the values of dc current gain
and small-signal current gain, for the first
order analysis, we will use the same
notation and the same value
(e.g. β =100)
Small-signal parameters of the BJT – cont.
• Output resistance
vCE
ro 
iC
vBE cst
iC  I S e
vBE
VT
vce

ic
 vCE
1 
VA

VA
ro 
IC
vBE cst



• Input resistance
v BE
rbe 
iB
rbe 
vbe
vCE cst 
ib

gm
vCE cst
Small-signal model of the BJT
(low and medium
frequency)
g m  40I C
rbe 

gm
VA
ro 
IC
hybrid-π
models
simplified
hybrid-π
models
linear models
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
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
ro 

 1MΩ
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
rbe 
gm

4
 25KΩ
VA 100
ro 

 1MΩ
I C 0.1