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Lecture II
Lecture II:
• Linear circuit theory review
• Amplifier basics
• MOS small signal model
A. Rivetti – INFN Sezione di Torino
Nodal analysis
R2
Is
R1
R4
R3
Vs
Nodal analysis provides a systematic and reliable method to calculate
all voltages and currents in a linear circuit
A. Rivetti – INFN Sezione di Torino
Nodal analysis
Writing nodal equations
v1
Is
R1
R2
v2
R3
R4
Vs


v
v
1
1 v2

0
I s 

R
R
1
2

 v2  v1  v2  v2 V s  0

R3 R 4
 R2
A. Rivetti – INFN Sezione di Torino
Nodal analysis
Writing the circuit matrix
R2
v1
Is
1
 1


 R1 R2

1
 
R2

R1
v2
R3
R4
Vs

 I s 


 v1 
R2      
1
1
1    V s 


 v2  
 R4 
R 2 R3 R 4 

1
A. Rivetti – INFN Sezione di Torino
Nodal analysis
Solving the circuit matrix
v1 


v2 
1
Is
V
R
1
s
4

1



R
1
  R 1R
1

R

2
2 
2
1
4
1

1
R
2
1

1
R R R
2
2
1

3
1

1
R R R
2
2
1

1
R R
1

1
R
2
3
4
Is
2
V
R
s
4
A. Rivetti – INFN Sezione di Torino
Nodal analysis
Another example
Vs
R2
Is
R1
R3
R4
A. Rivetti – INFN Sezione di Torino
Nodal analysis
Lecture II
Lecture II:
• Linear circuit theory review
• Amplifier basics
• MOS small signal model
A. Rivetti – INFN Sezione di Torino
Amplifier characteristic
 The input-output characteristic of an amplifier is usually a non-linear
function
 Over some interval of the input signal, this function can be
approximated by a polynomial:
2
n
y(t )  a  a x(t )  a x(t )  ...  a x(t )
0
1
2
n
 For narrow range of the input signal, we may write:
y(t )  a  a x(t )
0
1
The above expression does not obey the superposition principle
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Small signal model
 If a0 does not depend on the signal, we can write:
y(t )  a x(t )
1
 This is an expression that obeys the superposition principle
 The small signal model takes into account only variations of signals
within a circuit
 The small signal equivalent circuit can be studied with the methods
of linear circuit analysis
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Voltage amplifier
Rs
Vs(t)
Vi(t)
Ri
Vout
 AV = Vout/Vi
 Input impedance high (ideally infinite)
 Output impedance small (ideally zero)
A. Rivetti – INFN Sezione di Torino
Amplifier basics
VA small signal model
RS
Vs(t)
RO
Vi(t)
RI
AVVi
RL Vout
Note: impedances may also be complex
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Current amplifier
Is(t)
Rs
Ii(t)
Ri
Iout
 AV = Iout/Ii
 Input impedance small (ideally zero)
 Output impedance high (ideally infinite)
A. Rivetti – INFN Sezione di Torino
Amplifier basics
CA small signal model
Is(t)
Rs
Ii(t)
Ri
Is(t)
Ro
Iout(t)
RL
Note: impedances may also be complex
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Transconductance amplifier
Rs
Vs(t)
Vi(t)
Ri
Iout
 AV = Iout/Vi
 Input impedance high (ideally infinite)
 Output impedance high (ideally infinite)
 Important: the gain is not a number
A. Rivetti – INFN Sezione di Torino
Amplifier basics
TCA small signal model
RS
Vs(t)
Vi(t)
RI
Is(t)
Ro
Iout(t)
RL
Note: impedances may also be complex
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Transimpedance amplifier
Is(t)
Rs
Ii(t)
Ri
Vout
 AV = Vout/Ii
 Input impedance small (ideally zero)
 Output impedance small (ideally zero)
 Note: Gain is not a number
A. Rivetti – INFN Sezione di Torino
Amplifier basics
TA small signal model
RO
Is(t)
Rs
Ii(t)
Ri
AVVi
RL Vout
Note: impedances may also be complex
A. Rivetti – INFN Sezione di Torino
Amplifier basics
Lecture II
Lecture II:
• Linear circuit theory review
• Amplifier basics
• MOS small signal model
A. Rivetti – INFN Sezione di Torino
Simplified small signal DC model
The MOS transistor in saturation can be seen as a voltage controlled
current source
RS
Vs(t)
Vs(t)
gm =
IDS
VGS
gmVs
W
m
C
n
(VGS – VTH) =
OX
=
L
2 mn COX W IDS
L
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Practical example
What is the equivalent small signal model of this?
Vdrain
Vgate
W=100 mm
L=10 mm
mnCOX=190 mA/V2
VTH=0.6 V
Vdrain=2.5 V
Vgate=1.25 V
Vs
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
current (mA)
Gm simulation(1)
356.7
355.7
0
1
2
time (mS)
Vs=1mV pk-pk
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
current (mA)
Gm simulation (2)
660
355
0
1
2
time (mS)
Vs=250mV pk-pk
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Output impedance
Vdrain
r0
Vgate
Vs
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Including the output impedance
The MOS transistor in saturation can be seen as a voltage controlled
current source with finite output impedance
RS
Vs(t)
Vs(t)
gm =
ro =
IDS
VGS
1
gmVs
ro
W
m
C
n
(VGS – VTH) =
OX
=
L
2 mn COX W IDS
L
lIDS
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Bulk transconductance
For a more accurate model, the bulk effect must also be taken into
account
RS
Vs(t)
Vs(t)
gmb =
IDS
VBS
gmVs
ro
gmbvbs
VTH
W
m
(VGS – VTH)
= n COX
=
L
VSB
gm g
2fF + VSB
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Small signal DC model
The saturated MOS transistor is a voltage controlled current
source with finite output impedance
RS
Vs(t)
Vs(t)
gmVs
ro
gmbvbs
gm models the gate transconductance
gmb models the bulk transconductance (the bulk effect)
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model
Some numbers…
gm =
IDS
VGS
=
2 mn COX W IDS
L
ro =
1
lIDS
IDS = 100mA, W/L=50, mnCOX=190mA/V2
l=0.01V-1
gm = 1mS
ro = 1MW
A. Rivetti – INFN Sezione di Torino
MOS small signal DC model