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MOSFETs
NMOS Field Effect Transistor



MOSFET = Metal Oxide Semiconductor Field Effect Transistor
Four terminal device (gate, source, drain, substrate)
Unipolar transistor – one type of charge carrier

FET is a current control mechanism based on an electric field established by the voltage
applied to the control terminal
Source (S)
Gate (G)
Drain (D)
Metal
n+
Oxide (SiO2)
Channel
region
n+
L
p-type substrate (Body)
Body (B)
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Page MOSFET 5.1-1
MOSFETs
NMOS Cross-section
S
Metal
G
Metal or Polysilicon
W
D
Oxide (SiO2)
n+
Source
Region
L
n+
P-type substrate
(body)
Channel
Region
B
Drain
Region
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Page MOSFET 5.1-2
MOSFETs
Creating a Channel for Current Flow
+
vGS
-
S
Induced
n-type
channel
D
G

n+
n+
A positive voltage is applied to the
gate which forms an inversion layer,
or an n-type channel
L
p-type substrate (Body)
B
Depletion region
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Page MOSFET 5.1-3
MOSFETs
Operation with Small vDS
+
vGS
-
S
iS=iD
iG=0
+
vDS (small)
-
iD
D
G
iD(mA)
n+
iD
n+
0.4
vGS=Vt+4V
0.3
vGS=Vt+3V
0.2
vGS=Vt+2V
Induced n-type channel
p-type substrate (Body)
B
vGS=Vt+1V
0.1
vGS Vt
50
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100
150
200 v (mV)
DS
Page MOSFET 5.1-4
MOSFETs
Exercise 5.1

Note that in the small signal linear range, iD is proportional to (vGS-Vt)vDS Find the
constant of proportionality for the device below and the range of drain-to-source
resistance for vGS=2V to 5V
Constant of proportionality
iD(mA)
point1 : 0.4mA  4V(.2V)  0.5mA/V 2
point 1
0.4
vGS=Vt+4V
point2 : 0.1mA  1V(.2V)  0.5mA/V 2
0.3
vGS=Vt+3V
0.2
vGS=Vt+2V
Drain-to-source resistance
v
200mV
v GS  Vt  4V : RDS  DS 
 2 K
iD
.1mA
point 1
v
200mV
v GS  Vt  4V : RDS  DS 
 500
iD
.4mA
0.1
vGS=Vt+1V
vGS Vt
50
100
150
200 v (mV)
DS
Vt=1V
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Page MOSFET 5.1-5
MOSFETs
Uniform Channel Approximation
W=width
VS

VG>>VD and VS E
y

Ex
inversion channel
y
VD > VS
x=L
x=0
n+


Ey
p-type
n+
x
depletion
region edge
Assumptions
 Uniform behavior in the z (channel width)
direction
 The mobility is a constant
 The x directed electric field is
approximately a constant within the
channel thickness (ych) at a given x. This
is known as the gradual channel
n
approximation.
At a given x,
The current is constant, independent the
location of the chosen cross-section (i.e.
In x
independent of x)
Channel charge

J  x, y, z   q n x, y, zn Ex  x, y, z 


Area
y  ych
z W
   y 0 z 0
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
qn  x, y , z n E x  x dydz
Page MOSFET 5.1-6
MOSFETs
Charge per unit length in the channel (VGS >> VDS)
VGS
Gate Electrode (metal)
dx
v(x)
channel
source
Gate Oxide
velocity 
dx
dt
drain
v(x) = the voltage at point x
Charge dq
dv(x)=the incremental voltage about point x

dv  x 
Ex  
dx
dqx   CoxW vGS  vx   Vt dx
Cox 
 ox
tox

 rSiO  0
2
tox
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
3.9 8.85x1014 F cm 
tox
The voltage drop between the
gate and the channel, in excess
of the threshold voltage Vt
determines the amount of charge
Page MOSFET 5.1-7
MOSFETs
Derivation of the MOSFET current equation in the
Linear Region of Operation
dx
dv  x 
  n E  x   n
dt
dx

 W   ych
I n  n   dz    qn x , y dy  E x x 

 w 0   y  0
dx
dv  x 
  n E  x   n
dt
dx
dq dq x  dx
dx
dv  x 
i

 velocity  n
dt
dx dt
dt
dx
dq x 
 CoxW vGS  v  x   Vt 
dx
iDdx  nCoxW vGS  vx   Vt dvx 
L
i
D
0
dx 
vDS
  C W v
n
ox
GS
 v  x   Vt dv x 
0
2

W
vDS
iD  nCox vGS  Vt vDS 
L
2 
Process
Layout Geometries
Technology
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Terminal
potentials
W
iDS
Q 'inversion  x
vGS=Vt + 3V
vGS=Vt + 2V
vGS=Vt + 1V
vGS  Vt
200mV
Page MOSFET 5.1-8
vDS
MOSFETs
Exercise 5.2

Find the expression for rDS=vDS/iD when vDS is small. Find the value of rDS for an NMOS
transistor having k’n=20mA/V2,Vt=1V, and W/L=100m/10m when operated at vGS=5V
2

vDS
W
iD  k 'n vGS  Vt vDS 

L
2 
W
for small vDS : iD  k 'n vGS  Vt vDS
L
v
 W

 rDS  DS  1 k 'n vGS  Vt 
L
iD


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

2 100 m


20

A
V
5
V

1
V


10m


 1 800 106  1.25k
rDS 
rDS
vDS
1
iD
Page MOSFET 5.1-9
MOSFETs
Operation as vDS is Increased
+
vGS
-
S
+
vDS
D
G
n+
n+
p-type substrate
B
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Page MOSFET 5.1-10
MOSFETs
Channel pinch off



Increasing vDS causes the channel to acquire a tapered shape
Eventually, as vDS reaches vGS-Vt, the channel is pinched off at the drain end
Increasing vDS above vGS-Vt has little effect (theoretically no effect) on the channel’s
shape
vDS  vGS-Vt
Source
Channel
vDS
Drain
vDS 0
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Page MOSFET 5.1-11
MOSFETs
Operation as vDS is Increased (cont.)
iD
Triode
Curve bends
because the
channel resistance
increases with vDS
Almost a straight line
with slope proportional
to (vGS-Vt)
vDS < vGS Vt
Saturation
vDS  vGS Vt
Current saturates because the channel is
pinched off at the drain end, and vDS no
longer affects the channel
vGS >Vt
vDSsat  vGS Vt
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vDS
Page MOSFET 5.1-12
MOSFETs
Higher Drain Voltages (pinch-off)
For Example VG=3V Vt=1V
VS=0
VGS=VG-VS
VGS=3V

Ey

Ey
VDS=VGS-Vt=2V
x=L
x=0
n+
VGD=VG-VD=1V just Vt
inversion channel
y

Ex
p-type
n+
VD > VS
x
depletion
region edge
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Page MOSFET 5.1-13
MOSFETs
The Saturation Region of Operation
2

VDS 
W
I D   C 'ox  n VGS  Vt VDS 

L
2


and substitute in
VDS  VGS  Vt
2


VGS  Vt  
W
I D  C 'ox  n VGS  Vt VGS  Vt  

L
2


ID 
1W
2
C 'ox  n VGS  Vt 
2 L
Square Law -
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i.e The current is proportional to the
voltage in excess of the threshold squared
Page MOSFET 5.1-14
MOSFETs
MOSFET Transistor Operating Regions Summary
Linear (triode)
Region
Pinchoff - onset of
Saturation Region
Saturation Region
Figure taken from Semiconductor Devices,
Physics and Technology, S. M. Sze,1985, John
Wiley & Sons
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Page MOSFET 5.1-15
MOSFETs
MOSFET Operation Summary
Let: k 'n   nCox
Triode Region: vGS >Vt, vDS < vGS-Vt
v 
W
iD  k 'n vGS  Vt vDS 

L
2 
2
DS
Saturation Region: vGS >Vt, vDS  vGS-Vt
iD 
1
W
2
k 'n VGS  Vt  1 λVDS 
2
L
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Typical Process-Technology
Parameter Values
Electron mobility: n  580 cm2/Vs
Oxide permittivity: ox = 3.97o
= 3.97 X 8.85 X 10-14
=3.5 X 10-13 F/cm
Oxide Capacitance: Cox = ox/tox
= 1.75 fF/m2 for tox=0.02m
= 0.35 fF/m2 for tox=0.1m
Process
transconductance
parameter: k’n= nCox = ox/tox
= 100A/V2 for tox=0.02m
= 20A/V2 for tox=0.1m
Page MOSFET 5.1-16
MOSFETs
PMOS Field Effect Transistors
Gate (G)
Drain (D)
Source (S)
Metal
p+
Oxide (SiO2)
Channel
region
p+
L
n-type substrate (Body)
Body (B)
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Page MOSFET 5.1-17
MOSFETs
Sub-threshold Region
MOS
behavior
ID
Moderate inversion
Small
current
level < ~1A
weak inversion
VGS sub-threshold
junction leakage
3t
I D   constant  e
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VDS
VGB
n0t
Bipolar-like
behavior
In the sub-threshold regime
Page MOSFET 5.1-18
MOSFETs
CMOS technology
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Page MOSFET 5.1-19
Figure taken from supplemental material for Digital Integrated Circuits, A Design Perspective, Jan M. Rabaey,1996, Prentice Hall
MOSFETs
MOSFET Circuit symbols
NMOS
D
D
B
G
PMOS
S
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S
G
G
S
S
B
D
G
D
Page MOSFET 5.1-20
MOSFETs
The iD-vDS Characteristics (NMOS)
V DS = V
+
iG=0
t
V GS = 5V
Saturation
Triode
VGS = 4V
1
V GS = 3V
vDS
+
vGS
-V
iS=iD
-
-
V GS= 2V
Square Dependence
iD
i D (m A )
2
GS
V GS = 1V
0.0
1.0
2.0
v
3.0
DS
4.0
5.0
(V)
i D as a function of v DS
Figure taken from supplemental material for Digital Integrated Circuits, A Design Perspective, Jan M. Rabaey,1996, Prentice Hall
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-21
MOSFETs
iD vs. vGS Characteristic for an NMOS transistor in saturation
iDS
vDS  vGS-Vt
Vt
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iD 
1
W
2
k 'n VGS  Vt 
2
L
vGS (V)
Page MOSFET 5.1-22
MOSFETs
Large Signal Model of a MOSFET in Saturation
iG=0
G
iD
+
+
iD 
vGS
1
W
2
k 'n VGS  Vt 
2
L
vDS
-
S
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D
vGS  Vt
vDS  vGS-Vt
Page MOSFET 5.1-23
MOSFETs
Finite Output Resistance in Saturation
Source
Channel
Drain
-
vDSsat  vGS-Vt
+
-
+
vDS-vDSsat
L
DL
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Page MOSFET 5.1-24
MOSFETs
Channel Length (Drain Current) Modulation
due to changes in VDS
VDS  VGS  Vt
triode saturation
VA = -(1/l)
iDS 
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1 'W
2
kn VGS  Vt  1  lvDS 
2 L
Page MOSFET 5.1-25
MOSFETs
Large Signal Model of the MOSFET Incorporating the Output
Resistance
iG=0
G
iD
+
vGS
iD 
1
W
2
k 'n VGS  Vt 
2
L
rO
vDS
-
S
rO 
© REP 5/23/2017 ENGR224
D
+
vGS  Vt
vDS  vGS-Vt
VA
1
 lI D 
ID
Page MOSFET 5.1-26
MOSFETs
Exercises

5.3 An enhancement mode transistor with Vt=2V has its source terminal grounded and a
3V DC source connected to the gate. In which region of operation does the device
operate for:
+
a) VD=0.5V VDS = 0.5V < VGS-Vt,  in triode region

b) VD=1V
VDS = 1V = VGS-Vt,  in saturation region (pinch-off)
c) VD=5V
VDS = 5V > VGS-Vt,  in saturation region
3V
-
5.4 If the transistor above has k’n=20A/V2, W=100m and L=10m, find the value of iD in
the above cases - ignore the dependence of iD on vDS in saturation.
2

vDS
W
0.52 
6 100 
a) iD  k 'n vGS  Vt vDS 
  20 10
3  20.5 
  75μA
L
2 
10 
2 
1
W
20 10 6 100
2
3  22  100A
b) iD  k 'n vGS  Vt  
2
L
2
10
c) iD  100A
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vD
+
Page MOSFET 5.1-27
-
MOSFETs
Exercises

5.6 An enhancement MOSFET with k’n(W/L)=0.2mA/V2, Vt=1.5V, and l=0.02V-1 is operated
with vGS=3.5V. Find iD at vDS=2V and vDS=10V. Determine rO at this value of vGS.
iDS
1 'W
0.2 103
2
3.5  1.52 1  0.02  2  0.416mA
 kn VGS  Vt  1  lvDS  
2 L
2
iDS
1 'W
0.2 103
2
3.5  1.52 1  0.02 10  0.480mA
 kn VGS  Vt  1  lvDS  
2 L
2
rO 
DvDS
10V  2V

 125k
DiD
0.480mA  0.416mA
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Page MOSFET 5.1-28
MOSFETs
The iD-vDS Characteristics (PMOS)
i D (m A )
V GS = -1V
iD
V GS = -2V
VGS = -3V
-1
+
iG=0
vDS
+
vGS
V GS = -4V
iS=iD
-
-
Saturation
-2
Triode
V GS = -5V
V DS = V
-1.0
-2.0
GS
-3.0
-4.0
v DS (V)
-V
t
-5.0
0.0
Figure taken from supplemental material for Digital Integrated Circuits, A Design Perspective, Jan M. Rabaey,1996, Prentice Hall
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-29
MOSFETs
n-channel Enhancement Mode MOS Transistors

Enhancement Mode Transistors. A normally open switch. At zero volts on the gate no
current flows ( a positive Voltage must be applied to the gate to enhance a channel of
electrons)
Source
Gate
Drain
Drain
Gate
n
n
p substrate
IDS
Substrate
Source
n-channel
Source - where electrons come from (-)
Drain - where electrons flow to (+)
Channel is enhanced (resistive)
-VGS
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Channel is off
+VGS
0
Page MOSFET 5.1-30
MOSFETs
n-channel Depletion Mode MOS Transistors

Depletion Mode Transistors. A normally closed switch. At zero volts on the gate a
current flows ( a negative Voltage must be applied to the gate to deplete the channel of
electrons)
Source
Gate
Drain
Drain
Gate
n
Substrate
n
p substrate
n-channel
Source
Source - where electrons come from (-)
Drain - where electrons flow to (+)
IDS
Channel is made stronger
Channel is depleted (Off)
-VGS
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+VGS
0
Page MOSFET 5.1-31
MOSFETs
p-channel Enhancement Mode MOS Transistors

Enhancement Mode Transistors. A normally open switch. At zero volts on the gate no
current flows ( a negative Voltage must be applied to the gate to enhance a channel of
holes)
Source
Gate
Source
Drain
Gate
p
Substrate
p
n substrate
p-channel
Drain
Source - where holes come from (+)
Drain - where holes flow to (-)
0
-VGS
+VGS
Channel is enhanced
IDS
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Page MOSFET 5.1-32
MOSFETs
p-channel Depletion Mode MOS Transistors

Depletion Mode Transistors. A normally closed switch. At zero volts on the gate a
current flows ( a positive Voltage must be applied to the gate to deplete the channel of
electrons)
Source
Gate
Source
Drain
Gate
p
Drain
p
n substrate
-VGS
Substrate
0
p-channel
Source - where holes come from (+)
Drain - where holes flow to (-)
+VGS
Channel is depleted
IDS
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Page MOSFET 5.1-33
MOSFETs
The Body or Back Gating Effect on Threshold Voltage
Source (S)
Vbody-Source
Slightly
(<0.4V)
Forward
Biased

Gate (G)
n+
Channel
region
Drain (D)
n+
p-type substrate (Body)
Gate (G)
Source (S)
Vbody-Source=0
n+
Channel
region
Drain (D)
n+
p-type substrate (Body)
A slight body to source forward bias raises
the potential of the electrons in the substrate
reducing the gate voltage necessary to invert
the surface
A reverse body to
source bias lowers
the potential of the
electrons in the
substrate
increasing the gate
voltage necessary
to invert
the surface
Body (B)
Source (S)
Vbody-Source
Reverse
Biased

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n+
Gate (G)
Channel
region
Vt0
Drain (D)
Vt  Vt 0  
n+
p-type substrate (Body)
Vt
Gamma Body Effect
Parameter

 2
f
 VSB  2 f
2qN A Si
Cox
Page MOSFET 5.1-34

MOSFETs
Example 5.1

Design the circuit shown below so that the transistor operates at ID = 0.4 mA and VD= 1V.
The NMOS transistor has Vt = 2 V, nCox = 20 uA/V2, L = 10m, and W = 400 m. Assume l
= 0.
V
I
DD
D
 5V
R
Since VD = 1V, we are operating in
the saturation region.
D
V
I
V
D
SS
R
1
W
 n C ox VGS  Vt 2
2
L
1
400
VGS  22
0.4   20  10 3 
2
10
2
2
0  VGS  2   VGS  4VGS  3
ID 
D
S
 5V
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0  VGS  3VGS  1
VGS  3 or VGS  1
Choose VGS = 3V. Since gate is
grounded, potential at the source
must be -3V.
RS 
VS  VSS  3  (5)

 5k
ID
0.4
To establish +1V at the drain,
RD 
VDD  VD 5  1

 10k
ID
0.4
Page MOSFET 5.1-35
MOSFETs
Example 5.2

Design the circuit shown below to obtain a current ID = 0.4 mA. Find the value required
for R and find the dc voltage VD. Let the NMOS transistor have Vt = 2 V, nCox = 20 uA/V2,
L = 10m, and W = 100 m. Assume l = 0.
V
I
DD
D
 10V
Since VDG = 0V, we are operating in
the saturation region.
1
W
 n C ox VGS  Vt 2
2
L
1
100
VGS  22
0.4   20  10 3 
2
10
2
2
4  VGS  2   VGS  4VGS
Choose VGS = 4V since 0V < Vt and
VD = 4V.
ID 
R
V
D
Thus,
R 
VDD  VD 10  4

 15k
ID
0.4
0  VGS  4VGS  0
VGS  4 or VGS  0
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-36
MOSFETs
Example 5.3

Design the circuit shown below to establish a drain voltage of 0.1V. What is the effective
resistance between drain and source at this operating point? Let Vt = 1 V, and kn’(W/L) =
1 mA/V2.
The MOSFET is operating in the triode region, since the
drain voltage is lower than the gate voltage, and Vt is 1V.
V
I
DD
 5V
Triode, if
D
R
D
V
D
 0.1V
VDS  VGS  Vt
1





1

5

1

0
.
1

 0.01  0.395 mA
ID

2

The required resistor value,
RD 
VDD  VD 5  0.1

 12.4 k
ID
0.395
Effective drain-to-source resistance,
r DS 
© REP 5/23/2017 ENGR224
VDS
0.1

 253 
ID
0.395
Page MOSFET 5.1-37
MOSFETs
Example 5.4 NMOS

Analyze the following circuit to determine all the node voltages and branch currents, given
that Vt=1V and k’n(W/L) is 1mA/V2. Neglect the channel length modulation effect (i.e.
assume l=0)
VDD = +10 V



Since the gate current is zero (why?),
the voltage at the gate is simply
determined by voltage division between
RG1 And RG2, and since they are equal
VG is VDD/2 or 5 Volts.
Since the gate voltage is significantly
higher than ground it is likely that the
transistor is on, but we don not know if
it is in the triode region of operation or
in saturation.
We will assume that it is saturated and
solve the problem and then check the
validity of our assumptions (often the
hard part for beginners). The saturation
equations are easier to work with and
that makes a good choice for starting
out. If our assumptions do not check
out we have to go back and use the
triode region equations
© REP 5/23/2017 ENGR224
RG1 = 10 M
ID =IS
RD = 6 k
D
+5 V
RG2 = 10 M



G
IS =ID
assumed
S
RS = 6 k
The drain current has to be equal to the
source current since IG is zero
VGS = 5 - 6,000(ID)
And in saturation
ID 
1 W
2
k 'n VGS  Vt 
2
L
Page MOSFET 5.1-38
+
vS
-
MOSFETs
Example 5.4 continued

Again, in saturation
ID 
1 W
2
k 'n VGS  Vt 
2
L
VDD = +10 V
1
A
2
I D   0.001 2 5  6,000I D   1
2
V 

Which is a quadratic Eq. in ID
18I D2  25I D  8  0



This yields two values for ID, 0.89 mA
and 0.5 mA
Which is valid for our assumption of
saturation?
For ID of 0.89 mA the source voltage
would be 6,000(0.00089) or 5.34 Volts
which is higher than the gate voltage
and since the gate to source voltage
to turn the device on (i.e. threshold
voltage) is +1 Volt the device would
be off not saturated this answer is
not valid.
© REP 5/23/2017 ENGR224
RD = 6 k
RG1 = 10 M
D
+5 V
RG2 = 10 M




G
assumed
S
RS = 6 k
+
vS
-
For ID of 0.5 mA the source voltage would be
6,000(0.0005) or 3 Volts which means the gate
to source voltage is 5-3 or 2 Volts which is
greater than the threshold voltage (+1 Volt)
and the device is on. But is it saturated?
The drain is at 10-(6,000)(0.0005) or 7 Volts
VDS=7-3 or 4 Volts
Since VDS (4V) is greater than VGS-Vt (2V-1V)
the device is by definition in saturation so
our initial assumption was correct
Page MOSFET 5.1-39
MOSFETs
Example 5.5 (PMOS)


Design the following circuit so that the transistor
operates in saturation with ID=0.5mA and VD=+3V.
Let the enhancement type PMOS transistor have
Vt=-1V and k’p=1mA/V2. Assume l = 0. What is the
largest value that RD can have while still maintaining
saturation-region operation?
We were given the conditions to be met so lets start
there, ID=0.5mA
VDD = +5 V
RG1
S
assumed
1
A
2
D
 0.001 2 VGS   1
+
2
V 
RG2
RD
vD
20.0005
 VGS  1
0.001
VGS  1  1
VGS  0,2V For PMOS VGS < Vt  VGS is negative
0.0005 



VG to ground is VDD -VSG = 5-(2)=+3V
The gate voltage can be set by picking appropriate
values of RG1 and RG2 in a voltage divider, for
example, RG1 = 2M and RG2 = 3M
The drain resistor value can be found from
V
3
RD  D 
 6k 
I D 0.0005
© REP 5/23/2017 ENGR224
G
=3V
The border for saturation to linear
occurs at VDS=VGS-Vt so
VDSmax=-2-(-1) = -1V, therefore
VD to ground max is VDD-VSD=5-(1)=4V
At VD=4V and ID=0.5mA, RD=8k
Page MOSFET 5.1-40
MOSFETs
Example 5.6 (depletion NMOS)


The depletion MOSFET in the circuit is required to
supply the variable resistor RD with a constant
current of 100A. If k’n = 20 A/V2 and Vt = -1 V, find
the W/L ratio required. Also find the range that RD
can have while the current through it remains
constant at 100A. Assume l = 0.
RD
The MOSFET in this circuit is conducting (VGS = 0). It
must be operated in the saturation mode in order to
conduct a constant current ID while RD (and VD) is
varying.
1 W
2
I D  k n VGS  Vt 
2 L
1
W
2
100   20  0   1
2
L
W
 10
L

VDD = +5 V
The saturated mode of operation will be maintained
for
VDS  VGS  Vt  0   1  1,
VD  1V
© REP 5/23/2017 ENGR224
+
D
G
assumed
S
vD
-
VD  VDD  RD I D
RD max 
VDD  VD min 5  1

 40k
ID
0.1
Thus, RD can vary in range from 0 to 40 k
Page MOSFET 5.1-41
MOSFETs
Example 5.7 (depletion mode FETs)


Design the circuit shown below to establish a dc
voltage of +9.9V at the source. At this operating
point, what is the effective resistance between the
source and the drain of the transistor? Let Vt = -1V
and K’n(W/L) be 1 mA/V2.
In this case the gate and the source are just slightly
below the drain (0.1V) and VGS=0. VDS is not greater
than VGS-Vt (0.1 is not greater than 0-(-1)) so the
transistor is in the triode region and the current is
1

2
I D  0.001 0   10.1  0.1 
2


I D  0.1mA


The effective source to drain resistance is
© REP 5/23/2017 ENGR224
assumed
S
RS
VS
9.9

 99k  100k
I D 0.0001
rDS 
rDS  1k
D
G
We can now find RS by
RS 
VDD = +10 V
+
VS=9.9V
-
VDS
0.1

 1k
I D 0.0001
Page MOSFET 5.1-42
MOSFETs
Exercise 5.13

Consider the circuit below where the voltage VD1 is applied to the gate of another
transistor, Q2. Assume that Q1 and Q2 are identical. Find the drain current and voltage of
Q2. Assume l = 0. (see example 5.3)
From example 5.3, VD=4V and ID1=0.4mA.
V
R
2
 10 k
Q
DD
V
 10V
Q1 and Q2 are identical and have equal VGS.
R  12.4k
D1
2
V
Q
1
Assuming that Q2 is also operating in the
saturation mode, it’s drain current will be
identical to that of Q1, 0.4mA
D1
Thus,
VD2  10  0.4 10  6 V
Since VD2 > VG2 (4V), we are
indeed operating in the
saturation region.
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-43
MOSFETs
MOSFET as an Amplifier
VDS  VGS  Vt
DIGS
Linear
Output
Change
saturation
IDS-VDS Characteristic
DVGS=> Linear Input Change
DVDS does not change output much

Since changes in the drain to source voltage does not change the output much we will
focus on the IDS-VGS characteristic
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-44
MOSFETs
MOSFET Amplifier Configuration




We can obtain amplification of a small analog signal by use of an enhancement mode
MOSFET.
A dc voltage (bias) VGS, is applied along with the input signal to be amplified, vgs
superimposed on it. The output voltage is taken at the drain and consists of a dc and ac
response.
A circuit for amplification is shown below
This circuit is not practical because
 The dc voltage source at the input is difficult to implement
 Integrated circuit resistors take up too much room
 MOSFETs are used for loads
VDD


To be used as an amplifier the MOSFET must
be biased in the saturation region
To find the dc bias we set the ac component
of the input to zero and determine the dc
drain current in saturation (we will neglect
channel length modulation in this case, that ac
is we will assume l=0)
1 W
2
I D  k 'n VGS  Vt 
2
L
vgs(t)
VGS +
DC
© REP 5/23/2017 ENGR224
iDS(t)
(DC+ac)
RD
D
G
S
+
vDS(t)
(DC+ac)
Page MOSFET 5.1-45
MOSFETs
The Signal Current at the Drain


The dc voltage at the drain, VD will be equal to VDD - RDID
To ensure saturation we must have
VDS > VGS  Vt

Now we go back to the situation where we have both the dc bias and the ac signal
vGS  VGS  vgs

The resulting total instantaneous drain current to be
1 'W
2
k n VGS  v gs  Vt 
2 L
1 W
VGS  Vt 2  kn' W VGS  Vt vgs  1 kn' W vgs 2
 k n'
2 L
L
2 L
iDS 
iDS
dc bias

ac response
non-linear ac response
We can focus on the ac response if we keep the input signal small, such that
1 'W 2
W
VGS  Vt vgs
k n vgs < k n'
2 L
L
© REP 5/23/2017 ENGR224
or
vgs << 2VGS  Vt 
Page MOSFET 5.1-46
MOSFETs
Transconductance

If the small-signal condition specified on the previous page is satisfied we can neglect
the last non-linear term in the current equation and express iD as
iDS  I DS  ids

Where
W
VGS  Vt vgs
id  k
L
'
n

And we know that the ratio of id to vgs is the transconductance gm
gm 

id
W
VGS  Vt 
 kn'
vgs
L
In general
gm 
iD
vGS
© REP 5/23/2017 ENGR224
vGS VGS
Page MOSFET 5.1-47
MOSFETs
Voltage Gain

The total instantaneous drain voltage is
vD  VDD  RDiD

Using the small signal condition
vD  VDD  RD I D  id 

or
vD  VD  RDid
The small-signal component of the drain voltage is
vd  id RD   g m RD vgs

The voltage gain is then given by
vout vd

  g m RD
vin vgs
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-48
MOSFETs
DC Bias with an ac small signal
iDS
The DC bias level determines the
ac parameters
By restricting the input signal
swing to small values we can
“linearize” the characteristic like
we did for amplifier transfer
characterisitcs
tangent at Q
slope  g m
Bias Point - Q
IDS
t
ids (t)
VDD
iDS(t)
(DC+ac)
ac
vgs(t)
VGS +
DC
© REP 5/23/2017 ENGR224
D
G
VGS
RD
S
0
0
vGS (V)
Vt
+
vgs (t)
vDS(t)
(DC+ac)
-
t
input
Page MOSFET 5.1-49
MOSFETs
The Input and Output Signals

vGS
vgs
vGS
VGS
2
<< 2VGS  Vt 
t
vD max  VDD
vD
vgs

to prevent cutoff

gmRD
vD min  vG max  Vt
VD

t
0
© REP 5/23/2017 ENGR224
When the gate to source
signal is at its maximum the
drain current is at its
maximum. Maximum drain
current means that the drop
across the drain resistor is at
its maximum or that the drain
to source voltage of the
MOSFET is at its minimum
The output is therefore 180
degrees out of phase from the
input. We have an inverting
small signal amplifier in this
configuration
In order for the transistor to
operate in the saturation
region at all times there is a
minimum drain voltage that
must be maintained.
The input signal must be
small enough to keep the
output above the minimum
vout vd

  g m RD
vin vgs
Page MOSFET 5.1-50
MOSFETs
Small-Signal Saturation Equivalent Circuit Models

G
+
gmvgs
vgs
-

D

S

ro 
VA
ID

G
D
+
gmvgs
vgs
S
© REP 5/23/2017 ENGR224
ro

From a small-signal point of view the
FET behaves like a voltage controlled
current source as shown on the top
figure to the left.
Th input resistance is high (ideally
infinite)
After the DC analysis is done to
determine the ac parameters and then
the ac equivalent circuit is drawn.
A DC voltage source is ideally immune
to changes in current so small ac
voltage changes do not cause change
in current (dI/dV is zero or R is zero) it
is replaced by a SHORT circuit.
Current source are replaced by OPEN
circuits
In the first model is was assumed that
the drain current did not change with
increasing VDS in saturation but we
know that it does. The dependence
can be modeled by a finite resistance
ro, between source and drain, whose
value is approximated by the equation
shown at the left.
Page MOSFET 5.1-51
MOSFETs
Applying the small-signal model

On the previous page VA = 1/l . Which we can determine from the Id-Vd characteristic,
typically in the 10k to 1,000k range

Remember, the ac parameters, gm and ro depend on the DC bias point

The gain for the following circuit is given below (note that ro reduces the gain)
vd
  g m RD || ro 
vgs
VDD
iDS(t)
(DC+ac)
ac
vgs(t)
VGS +
RD
DC
© REP 5/23/2017 ENGR224
D
G
D
G
ac small-signal circuit
S
+
vDS(t)
vgs
+
vgs(t)
-
gmvgs
ro
RD
(DC+ac)
Page MOSFET 5.1-52
+
vds
-
MOSFETs
The Transconductance - gm

The transconductance as we have seen
already is the incremental change in drain
current due to an incremental change in
gate voltage or
W
g m  k n' 
L


VGS  Vt 


2I D
W 
kn'   
2
 L  VGS  Vt 
and by substitution
g m  2kn'


The transconductance of a bipolar
transistor is proportional to the bias
current (not the square root of it) and does
not depend on the physical size or
geometry
We can write gm in another useful way, as
shown
If we solve the saturation current equation
for VGS-Vt we get
VGS  Vt   2 I D
W 
k n'  
L


W
L
1 ID
2 Veff
gm 
IC
ID
So gm is proportional to the square root of
the dc bias current
At any bias current gm is proportional to
the square root of W/L
© REP 5/23/2017 ENGR224
gm 


2I D
ID

VGS  Vt 1 V  V 
GS
t
2
for Veff  VGS  Vt

gm 
which we can compare to
for BJT transisto rs
1
Vt is ~25mV but
VGS  Vt  ~ 100mV
2
VT
MOSFETs are smaller and use less power
Page MOSFET 5.1-53
MOSFETs
Example 5.8 - Complete Amplifier Analysis

In the following circuit, a discrete MOSFET amplifier is shown in which the input signal vi is
coupled to the gate via a large capacitor, and the output signal at the drain is coupled to
the load resistance RL via another large capacitor. We will assume that the coupling
capacitors are large enough so that they act as short circuits for the ac signal frequencies
of interest. We wish to analyze the amplifier circuit to determine its small-signal voltage
gain and its input resistance. The transistor has Vt=1.5V, kn’(W/L) = 0.25mA/V2, and VA = 50V.

Start by doing the dc analysis
VDD=15V
1
0.00025VGS  1.52
2
 VD since I G  0
ID 
VGS
RG=10M
RD=10k
D
1
2
I D  0.00025VGS  1.5
2
2
I D  0.000125VD  1.5
ac
Rin
+
vi
-

G
S
+

vDS(t)
RL=10k
VD  15  RD I D  15 10,000I D 
VD  4.4V
I D  1.06mA
© REP 5/23/2017 ENGR224
The other solution is not valid (contradictory)
Page MOSFET 5.1-54
MOSFETs
ac equivalent circuit for Example 5.8
VDD=15V
RG=10M
RD=10k
D
ac
Rin
+
vi

G
S
+

vDS(t)
-
VDD is shorted to ground
in the ac circuit
RL=10k
-
RG
ac small-signal equivalent circuit
D
G
+
vgs
-
vi
+
vgs
-
gmvgs
ro
RD
RL
+
vo
-
Rin
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-55
MOSFETs
ac analysis for Example 5.8

The value of gm is given by

W 
g m  k n'  VGS  Vt 
L
g m  0.000254.4  1.5  0.725


Av 
mA
V
The MOSFET output resistance ro is
ro 
The value of the voltage gain Av is given
by
VA
50

 47k
I D 0.00106
vo
  g m RD || RL || ro 
vi
Av  0.00072510,000 || 10,000 || 47 
Av  3.3
V
V
We can now us the ac equivalent circuit
to determine the expression for the
output voltage in terms of the gate to
source small-signal voltage
vo   g m vgs RD || RL || ro 
vgs  vi
Since RG is so large
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-56
MOSFETs
Evaluation of the Input Resistance for Example 5.8

To find the input resistance we note that the input current is given by
ii 
vi  vo  
Rin 
RG
vi
RG
 vo  vi
1-  
1- 3.3  4.3vi
RG
 vi  RG
vi RG 10,000,000


 2.33M
ii 4.3
4.3
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-57
MOSFETs
The Source Absorption Theorem (Appendix E)
Used to derive the “T” Model for a MOSFET
I1  Y V1  V2 
I1
Node 1
I1  YV1 1  V2 V1 
I1  YV1 1  K 
I2
Y

V1
Node 2

V2  KV1


Y1V1  I1
Y1  Y 1  K 
I 2  Y V2  V1 
I 2  YV 2 1  V1 V2 
I 2  Y 1  1 K V2
Y2V2  I 2
Y2  Y 1  1 K 
© REP 5/23/2017 ENGR224
Node 1

V1

I1
I2
Y1
Y2
Node 2

V2  KV1

Page MOSFET 5.1-58
MOSFETs
The “T” Equivalent Circuit Model
ig = 0
G
G
+
gmvgs
vgs
-
id
ig = 0
G
ig = 0
gmvgs
+
1
gm
vgs
S
gmvgs
gmvgs
S
S
G
X
+
vgs
-
D
D
ro
id
D
is
X
+
id
gmvgs
gmvgs
vgs
S
D
is
is
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-59
MOSFETs
Modeling the Body Effect

The body effect (or back gate effect) occurs when the source is NOT at the same
potential as the substrate (body). The substrate acts as a second gate and therefore we
include a second dependent current source in our model, as shown below
g mb 
iD
vBS
the dependence is linked to the threshold voltage
D
vGS  constant,v DS  constant
g mb  g m
Vt



VSB 2 2 f  VSB
G
G
0.1 <  < 0.3
S
D
+
gmvgs
vgs
-
ro
gmbvbs
vbs +
B
S
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-60
MOSFETs
Exercise 5.17






For the following amplifier, let VDD = 5V, RD = 10k, Vt = 1 V, kn’ = 20 A/V2, W/L = 20, VGS
= 2V and l = 0.
Find the dc current ID and the dc voltage VD
Find gm
find the voltage gain
If vgs = 0.2sinwt volts, find vd assuming that the small-signal approximation holds. What
are the maximum and minimum values of vD.
Use the following equation to find the various components of the drain current
iDS 
1 'W
W
2
VGS  Vt vgs  1 kn' W vgs 2
k n VGS  Vt   k n'
2 L
L
2 L
VDD
iDS(t)
(DC+ac)
ac
vgs(t)
VGS +
DC
© REP 5/23/2017 ENGR224
RD
D
G
S
+
vDS(t)
(DC+ac)
Page MOSFET 5.1-61
MOSFETs
Exercise 5.17 continued


Using the following identity show that there is a slight shift in ID (by how much?) and
that there is a second harmonic component (2nd harmonic has a frequency of 2w)
Express the amplitude of the second harmonic component as a percentage of the
amplitude of the fundamental (this is known as the second-harmonic distortion)
sin 2 wt 
1 1
 cos 2wt
2 2
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-62
MOSFETs
Biasing MOSFET Amplifier Circuits


Biasing is the establishment of an appropriate dc operating point for the transistor.
An appropriate dc bias point has a stable and predictable dc drain current ID, and a dc
drain to source voltage that ensures operation in the saturation mode for all expected
input signal levels.
VDS  VGS  Vt
DIGS
Linear
Output
Change
saturation
IDS-VDS Characteristic
DVGS=> Linear Input Change
DVDS does not change output much

Discrete component MOSFET circuits are not common but we will use them to introduce
various biasing techniques.
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-63
MOSFETs
MOSFET Biasing - Single Supply





Plus side of MOSFET bias circuit design - the gate current is
zero (easier to design)
Negative side of MOSFET bias circuit design - Vt (VGS response)
varies more than vBE does in BJT circuits.
The circuit shown at the right is commonly used when a single
power supply is available. A voltage divider is used to establish
a fixed dc voltage at the gate.
 Since the gate current is zero the two gate bias resistors
RG1 and RG2 can be selected to be very large (M range)
 This will provide a large amplifier input resistance
A resistor, called a self bias resistor is connected to the source.
If the device is turned on more (more source current) the drop
across source resistor will increase and reduce the current. A
balance (negative feedback) situation will be created.
RD is selected to be as high as possible to obtain high gain but
small enough to allow for a large enough output signal swing at
the drain while still keeping the MOSFET in saturation at all
times.
© REP 5/23/2017 ENGR224
V
R
G1
DD
R
I
D
D
R
G2
R
Page MOSFET 5.1-64
S
MOSFETs
MOSFET Biasing - Dual Supplies




For symmetric supplies (positive and negative of the same
magnitude) a simpler bias arrangement can be used.
The resistor RG establishes a dc ground at the gate of the
transistor.
Directly grounding the gate will also establish a dc bias but RG is
used to increase the input resistance seen by a signal source
that may be capacitively coupled to the gate
RD is selected to be as high as possible to obtain high gain but
small enough to allow for a large enough output signal swing at
the drain while still keeping the MOSFET in saturation at all
times.
V
DD
R
I
D
R
R
G
V SS
© REP 5/23/2017 ENGR224
D
Page MOSFET 5.1-65
S
MOSFETs
MOSFET Biasing with a Constant Current Source





Having a constant current source establishes the source and
drain current level.
We will look at how to construct a constant current source
shortly
As before RD is selected to be as high as possible to obtain
high gain but small enough to allow for a large enough output
signal swing at the drain while still keeping the MOSFET in
saturation at all times.
As before, The resistor RG establishes a dc ground at the gate
of the transistor.
Directly grounding the gate will also establish a dc bias but RG
is used to increase the input resistance seen by a signal source
that may be capacitively coupled to the gate
R
I
D
R
G
I
V SS
© REP 5/23/2017 ENGR224
D
Page MOSFET 5.1-66
MOSFETs
Common-Source Circuit with Resistive Gate Feedback




The feedback resistor RG forces the dc voltage at the gate to be the
same as that of the drain (since IG = 0).
The input can be capacitively coupled to the gate and the output
can be taken at the drain to form a simple common-source
amplifier.
The output signal swing is limited on the low side since the drain
(which is tied to the gate) must be high enough to satisfy the
threshold voltage gate to source. If the output goes too low the
transistor slips out of saturation into the triode region of operation
and the output will be distorted.
As before RD is selected to be as high as possible to obtain high
gain but small enough to allow for a large enough output signal
swing at the drain while still keeping the MOSFET in saturation at
all times.
© REP 5/23/2017 ENGR224
V
0
R
DD
R
D
G
Page MOSFET 5.1-67
I
D
MOSFETs
Biasing of Discrete MOSFET Amplifiers

Four circuits for biasing the MOSFET in discrete-circuit design.
V
V
R
G1
DD
R
I
R
I
D
D
R
G2
R
D
0
R
R
G
DD
D
D
D
R
R
I
DD
V
R
R
G
I
G
D
I
D
S
S
V SS
classical arrangement
2 power supplies
© REP 5/23/2017 ENGR224
V SS
constant current source
common-source circuit
Page MOSFET 5.1-68
MOSFETs
Biasing of Discrete MOSFET Amplifiers

Exercise 5.22 Design the circuit below for a MOSFET having k’n(W/L) = 0.5 mA/V2 and Vt = 2V,
and utilizing two power supplies, ± 10V. Design for ID = 1mA and allow for a signal swing at
the drain of ±2V. The amplifier is required to present a 1M input resistance to a signal
source that is capacitively coupled to the gate. Assume l=0.
 10V
W
k 'n
 0.5mA/V 2 and Vt  2V
L
RG  1M, I D  1mA
Thus,
1
2
1   0.5VGS  2
2
VGS  4V
R
I
D
D

1M
R
R
G
S
VG  0, VS  4V
RS 
 4   10
 6k
.001
Where we have neglected the signal
component of VG
VD min  VG  Vt  0  2  2V
© REP 5/23/2017 ENGR224
 10V
To allow for a ±2V signal swing at the drain,
VD  0V, thus RD 
10  0
 10k
.001
Page MOSFET 5.1-69
MOSFETs
Design Philosophy




The circuits that we have just looked at are not suitable for integrated circuit design.
They use too many resistors which take up room and are therefore expensive.
It turns out that since we are making small MOSFETs anyways, that if we can use
transistors that act sort of like resistors we can make much smaller (more dense)
circuits.
The coupling capacitors also take up way to much room so coupling and bypass
capacitors are not used in the design of MOSFET amplifier circuits.
 You can learn more about integrated circuit fabrication in EGRE435 during the fall
of 2000.
 You can learn more about MOS analog circuit design in EGRE307 during the fall of
2000.
 You can learn more about MOS digital circuit design in VLSI Design EGRE429.
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-70
MOSFETs
Basic MOSFET Constant Current Source

Use a reference current through one transistor (Generated through Q1) to set the
voltage across the gate to source of another transistor (Q2) and hence replicate the
reference current through the drain of the second transistor.
Q1 is saturated since VGS = VDS

V
I
R
REF
I
I D1 
O
0
I
Q
1
V
0
0
D1
V
For Q1,
DD

V
Q
O
1
W 
2
k ' n   VGS  Vt 
2  L 1
I D1  I REF 
I
VDD  VGS
R
2
GS
-
I O  I D2
constant current source
IO
I REF
© REP 5/23/2017 ENGR224
REF
I
I
Q

1
W 
2
 k ' n   VGS  Vt 
2  L 2
W L 2
W L 1

V
GS
-
current mirror
Page MOSFET 5.1-71
Q
O

D1
1
For Q2,
DD
2
V
O
_
MOSFETs
Effect of VO on IO

In the previous current source description we assumed that the transistor Q2 is operting
in saturation
IO
Slope 
VO  VGS  Vt
1
rO
DVO
V A2
RO 
 rO 2 
DI O
IO
I REF
Region of
“constant” current operation
VGS  Vt VGS
© REP 5/23/2017 ENGR224
VO
Page MOSFET 5.1-72
MOSFETs
Example 5.9

Given VDD=5V and using IREF=100A, it is required to design the circuit shown below to
obtain an output current whose nominal value is 100A. Find R if Q1 and Q2 are matched,
have channel lengths of 10m and channel widths of 100m, Vt=1V and k’n=20A/V2. What
is the lowest possible value of VO? Assuming the fabrication technology results in an Early
voltage that can be expressed as VA=10L, where L is in microns and VA in volts, find the
output resistance of the current source. Also find the change in output current resulting
from a 3V change in VO.
VGS  2V
V
I
R
REF
52
 30k
100A
VO min  VGS  Vt  2  1  1V
R
DD
I
O
Since L=10m,
0
I
Q
1
V
0
0
D1
Q

V
O
2
GS
-
I D1  I REF  100 
© REP 5/23/2017 ENGR224
1
100
VGS  12
 20 
2
10
V A  10  10  100V
rO 
100V
 1M
100 A
The output current will be 100A at
VO=VGS=2V. If VO changes by +3V, the
corresponding change in IO will be 3%,
DI O 
DVO
3V

 3A
rO
1M
Page MOSFET 5.1-73
MOSFETs
Current-Steering Circuits
V
I 2  I REF
DD

V
I
V
DD
Q
REF
I
R
I
2
I
I
D1
Q
1

V
Q
2
Q
3
Q
threshold voltage for n-channel devices
5
I5  I4
3
W L5
,
W L4
where I 4  I 3
VD 5 < VDD  VGS 5  Vtp
GS1
-
© REP 5/23/2017 ENGR224
 V SS
W L3
W L1
5
I
4
and I 3  I REF
VD 2 , VD3 > VSS  VGS1  Vtn
GS 5
4
W L2
W L1
threshold voltage for p-channel devices
Page MOSFET 5.1-74
MOSFETs
Basic Configurations of Single-Stage IC MOS Amplifiers
V
V
DD
v
I
I
v
v
V
DD
v
O
DD
I
v
O
I
v
© REP 5/23/2017 ENGR224
I
V SS
Page MOSFET 5.1-75
O
MOSFETs
Resistive Load Common Source Amplifier

An ideal current source has an infinite resistance, the current does not depend on the
voltage at the node to which it is connected, but what happens if we use a resistive load

The resistor constrains the current flow through the transistor and vice-versa.
For the solid resistor load curve plotted on the transistor characteristic shown
below the resistance is less than that of the dashed curve.
On the solid curve the valid operating points (curve intersections) are labeled
with a letter and on the dashed curve the letters have a prime
On the next page we plot the voltage transfer characteristic of this amplifier


V
DD
R
Vtn  0.8V
v
G
D
iD1
+
VR
-
v
Q1 in
triode
Load curve
Q1 in
saturation
vGS1  4V  VID
vGS1  3V  VIC
D
O
I REF
I
S
C
vGS1  2V  VIB
B
A
D’
vGS1  1V  VIA
Vtn  0.8V
A’
0
common-source
VR=VDD-VDS
© REP 5/23/2017 ENGR224
1
2
Vout
3
4
5V
VDD
Vin
vDS1  vO
Page MOSFET 5.1-76
MOSFETs
Resistive Load Common Source Amplifier continued



The plot of the transfer characteristic of the common-source amplifier with a resistive load
is shown below
Since the resistor is not an ideal current source the gain varies with the load resistance
The higher the load resistance the higher the gain but the smaller the allowed input signal
swing (an still have the transistor saturated)
VDD 5
saturation
A
4
R1>R2
3
Vout
2
gain is proportional to the slope
Gain1>Gain2
B
A’
1
B’
C
C’
0
0
1
Vtn
© REP 5/23/2017 ENGR224
2
3
D
triode
D’
4
5
Vin
Page MOSFET 5.1-77
MOSFETs
The CMOS Common-Source Amplifier
(PMOS current source load)

If the PMOS active load device is made with a long channel then lambda is small (the
magnitude of VA is large or the transistor output resistance ro is large). In other words a
long channel load acts more like an ideal current source.
rO 2 
V
V
I
3
I REF
DD

Q
VA 2
i
Q
2
v
v
REF
v
1
rO 2
one curve
not a family of curves
O
Q
1
CMOS common-source amplifier circuit
© REP 5/23/2017 ENGR224
Slope 
v SG  VSG
-
i
+
I
-
Q2 in
saturation
I REF

SG
-
Q2 in
triode
0
V
SG
 Vtp

VSG
v
i-v characteristic of the active-load Q2
Page MOSFET 5.1-78
MOSFETs
Graphical construction to determine transfer characteristic



We expect to have a transfer characteristic that has a high gain since output resistance
of the load transistor can be made high (long channel)
The load curve is like a high value resistor that has been translated upwards
We are most interested in the area of intersection between A and B for amplification
iD1
Q1 in
saturation
Q1 in
triode
Load curve
vGS1  V1B
B
A
I REF
0
VOB
VDD  VSG

vGS1  V1 A
VDD
VDD  VSG  Vtp
© REP 5/23/2017 ENGR224

vDS1  vO
Page MOSFET 5.1-79
MOSFETs
Transfer Characteristic of the
active load common source amplifier


When the input is low the load transistor is in the triode region, as the input voltage is
increased the active load becomes saturated.
The gain is relatively high and depends on the output resistance of the transistor used in
the current mirror
VDD
I

VDD  VSG  Vtp
II
III

IV
I
A’
II
III
IV
Q1
Q1
Cutoff
Saturation
Q2
Triode
Q1
Saturation
Q2
Saturation
Q1
Triode
Q2
Saturation



B’
VOB
Vtn
V1 A
© REP 5/23/2017 ENGR224
V1B
Page MOSFET 5.1-80
MOSFETs
Small-signal equivalent circuit of the common-source config.
D1,D2

vi
+
-
v
gs1

g m1vgs1
ro1
ro 2
v
o
-
vo
AV    g m1 ro1 ro 2 
vi
S1,S2
Since ro1 and ro2 are usually large the gain can be large
without taking up a lot of room on the chip with an
integrated resistor.
W 
g m1  2k 'n   I REF
 L 1
The gain then becomes
If we use physically based parameters for the transistors
we get a design equation in terms of the transistor’s length,
width, transconductance, output resistance and reference
current source value
W 
2k 'n  
 L 1
Av  
1
1

VA1 VA 2
Voltage gains on the order of 20-100 are obtained using
CMOS common-source configuration
If we assume that VA1  VA 2
The source of each device is connected to the body
so the body effect on the threshold voltage is not a
factor
AV  
© REP 5/23/2017 ENGR224
1
I REF
1  W  VA
k 'n  
2  L 1 I REF
Page MOSFET 5.1-81
ro1 
VA1
I REF
MOSFETs
Effect of RS on AC gain

In section 5.6 we discussed various DC biasing schemes and in those schemes we saw
that the source resistor, RS (like its counterpart RE in the BJT case) provides negative
feedback to the gate to source voltage and helps to stabilize the DC value of ID.
If the DC bias point of the gate is increased the source current will increase but not by
as much as when there is no source resistance
This source resistance DOES affect the AC gain!
Consider the case of a MOSFET biased to have a gm of 30mA/V, rO=16K, and RD=470
AC model WITHOUT RS:




V
+
vgs(t)
-
vi
+
VRD
D -
D
G
gmvgs
ro
RD
DD
R
+
vo
-
v
G
I
D
v
O
S
RS
vo
vi
  g m RD rO   30 X 103 (470 16 K)  13.7 V
© REP 5/23/2017 ENGR224
V
common-source with a
source resistance
Page MOSFET 5.1-82
MOSFETs
Effect of RS on AC gain (cont.)

G
vi
Now consider AC gain WITH RS, let
RS=10:

More easily analyzed with the T model:
D
D
+
vgs(t)
-
gmvgs ro
RD
+
vo
-
+
G
gmvgs
+
1


gm
vgs  vi 
 1
 g  RS
 m


  v  33.33   0.7692v
i
 i  33.33  10 


vi
vgs
S
1
gm
RS
vO   gmvgs RD rO   30 X 103 0.7692vi (470 16K)  10.54vi
vo
vi
 10.54V
© REP 5/23/2017 ENGR224
V
Page MOSFET 5.1-83
ro
RD
vo
-
MOSFETs
The use of a Source By-Pass Capacitor


In the circuit shown on this page, at low frequencies, the
source capacitor CS is an open circuit and the source
resistance has a voltage drop across it which reduces the gate
to source voltage across the transistor.
On the next page the ac equivalent circuit is shown
V
DD
RD
v
G
I
D
+
VRD
-
v
O
S
ZS
RS
CS
common-source with a
source resistance
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-84
MOSFETs
The use of a Source By-Pass Capacitor (ac circuit)

vi
+
-
v

g m v gs
gs
-
v
o
S1
ZS
RD
RS
© REP 5/23/2017 ENGR224
CS
Z Cs 
1
jwC
as w   Z Cs  0
CS short circuit (high frequencies)
CS open circuit (low frequencies)
Note that Vgs does not equal vi now
Page MOSFET 5.1-85
MOSFETs
The CMOS Common-Gate Amplifier

V
In this case a constant dc level is applied to the gate of the transistor and the
input signal is applied to the source
The signal source at the gate will be zero (hence the name common gate)
There will be a potential difference between the source and the bulk (body) so
we need to use the model which includes that effect.


DD

V
Q
I
SG
-
3
Q
G1
2

i
v
REF
v
D1,D2
v
O
g m1vgs1
ro1
gs1
g mb1vbs1
-
Q
1
BIAS

v
I
+
S1
-
vi +-
v
bs1
B1
-
common-gate amplifier
vi=VSG vo=VDG
Body effect
Ri
Vgs1=-vi
vbs1=-vi
small-signal equivalent circuit
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-86

v
ro 2
o
-
MOSFETs
Simplified circuit
vi  vo  ro1
ro1
ii
vi
D1,D2
S1
+
-
g m1  g mb1 vi

ro 2
v
o
-
Ri
output node
equation
vi  vo
v
 g m1  g mb1 vi  o
ro1
ro 2
vo 
1
  g m1  g mb1  ro1 // ro 2 
vi 
ro1 
Av  g m1  g mb1 ro1 // ro 2 
Av 
if 1/ro1<<gm1




Input resistance
(input node equation)
ii  g m1  g mb1 vi 
Ri 
vi  vo
ro1
 ro 2 
vi
1
1  

ii g m1  g mb1   ro 2 
The body effect adds to the gain but reduces the input resistance
The active load (ro2) reduces the gain but slightly increases the input resistance
By comparison the Common-Gate configuration has a gain similar to that of the common-source
amplifier but the input resistance is much lower
The common-gate configuration is used in a combination circuit called a cascode amplifier that we will
study later in EGRE307.
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-87
MOSFETs
The Common-Drain or Source-Follower Configuration

V
V
I


DD
v
REF
DD
i
Q
5


v
Q
Q
3
The common-drain or source follower configuration is used as buffer
amplifier. Although its voltage gain is less than unity it has a low output
resistance and is therefore capable of driving low impedance loads with
little loss of gain.
Typically found in the output stage of a multi-stage amplifier.
The fact that its impedance is buffered can be used to extend the highfrequency response range of amplifiers and speed up digital circuits.
Vdd is at signal ground hence the name common drain
The input impedance is very high since it is the gate of a MOSFET (THIS
IS A VERY BIG ADVANTAGE OVER BJTs)
G1
o

2
v
vi
 V SS
common-drain
or
source-follower
D1
+
-
g m1vgs1
ro1
gs1
g mb1vbs1
S1

ro 2
v
o
+
-
v
bs1
B1 Signal Ground
-
small-signal equivalent circuit
(again the body effect is included)
Use the source absorption theorem to transform the dependent source into a resistance 1/gmb1
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-88
MOSFETs
Simplified circuit
G1
vo  vs1  g m1 RS v gs1

vi  v gs1  vs1  v gs1  g m1 RS v gs1
v
vi
+
-
vi  1  g m1 RS v gs1
g m1vgs1
gs1
S1
RS

v
o
-
Av 
RS 
1
g mb1
// ro1 // ro 2
The body effect reduces the gain by ~10 to 30 percent
Av 
vo
g m1 RS

vi 1  g m1 RS
g m1
1
1
g m1  g mb 2  
ro1 ro 2
Av 
g m1
,
g m1  g mb1
Av 
1
1 
g mb1  g m1
typically 0.1<<0.3
© REP 5/23/2017 ENGR224
0
Page MOSFET 5.1-89
MOSFETs
NMOS Load Devices - Saturated Enhancement Mode
VDD
Saturated Enhancement Mode Load
VGS=VDS therefore always saturated
i
locus of points on
many curves
i
+
D2
VGS=VDS
i
D
S2
v
S
0
i
G1
VGS1=Vin
V
t
1 W
2
k 'n v  Vt  , v  Vt
2
L
VDS2
-
G2
G
+
D1
+
VDS1=Vout
S1 -
v
VGS2=VDS2
i
x
x
x
0
© REP 5/23/2017 ENGR224
VDD  Vt 2 VDD
Page MOSFET 5.1-90
MOSFETs
NMOS Amplifier with Enhancement Load
V
DD
Q
iD1
A
2
I
V3 V1
0
-
I Q1 cutoff
VDD
A’
VDD  Vt 2
II
Q1 in
saturation
L 1
L 2
© REP 5/23/2017 ENGR224
B’
V3
0
Vt1
L 1 1
L 2 1   2
Due to the body effect on Vt
VSB for Q2 is not equal to zero
(reduces the gain)
III
Q1 in
triode region
V1
W
W
W
W
W L 1  W L 1
V 
v
W L 2 t  W L 2 I
vGS1  Vt1
vDS1
Vt 2 VDD
vO

vO  VDD  Vt 


Av  
vGS1  ...

Q vO

Av  
vGS1  V
B
2
iD 2  iD1
v
Load curve
V
vI
Page MOSFET 5.1-91
MOSFETs
NMOS Amplifier with Enhancement Load
Positives
Uses the same type of device for the load as the driver (enhancement)
Negatives
Lower Gain than the Depletion Mode Load
Smaller output signal swing (The output does not go all the way up to VDD)
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-92
MOSFETs
NMOS Load Devices - Saturated Depletion Mode
DepletionMode Load,
VGS=0
Always on
IDS
i
Depletion
Mode Load
VGS=0
Always on
+
D
v
G
S
VGS
triode
-
In saturation
i

v 
i  I DSS 1  
 VA 
1 W
I DSS  k 'n V 2 tD
2
L
I DSS 
© REP 5/23/2017 ENGR224
V
tD
Slope 
saturation
i
0
1 W 2 
v 
k 'n V tD 1  
2
L
 VA 
1 W 2
k 'n V tD
2
L
v
Page MOSFET 5.1-93
1
rO 2
MOSFETs
NMOS Amplifier with Depletion Load
V
Q
DD
G1
2
D1,G2.S2

vi
iD 

v
Q
1
v
v
gs1
g m1vgs1
-
O
ro1
g mb2vbs 2
ro 2
D2,B2
S1,B1
-
I
+
-

-
Av 
 1 

vO
 // ro1 // ro 2 
  g m1 
vI
 g mb1 

Av  
Av  
© REP 5/23/2017 ENGR224
g m1
g
  m1
g mb 2
g m 2
W L 1
W L 2
1

Page MOSFET 5.1-94
v
o
-
MOSFETs
NMOS Amplifier with Depletion Load
vO
Load curve
iD1
vGS1  ...
C
B
vGS1  ...
VDD
I
II
III
IV
A’
VDD  VtD
B’
A
V
0
DD
Positives
 VtD 
VDD
vO
C’
VOB
Higher Gain than the Saturated Enhancement Mode Load
Larger output signal swing (all the way up to VDD)
VtE
I
Negatives
II
Requires a different type of device (depletion) for the load
and is therefore more complicated
© REP 5/23/2017 ENGR224
Off
Q1
Q2 Triode
Saturation
Q1
Q2 Triode
vI
Q1 Saturation
III Q2 Saturation

Triode
IV Q1
Q2 Saturation
Page MOSFET 5.1-95
MOSFETs
The CMOS Digital Logic Inverter


For any IC technology used in digital circuit design, the basic circuit element is the logic
inverter.
The inverter uses two matched enhancement-type MOSFETS; an n-channel and a pchannel. The body of each device is connected to its source which eliminates any body
effect.
V
V
DD
Q
v
Q
P
v
I
Q
N
CMOS inverter
© REP 5/23/2017 ENGR224
DD
O
v
P
i
i v
Q
DP
I
O
DN
N
simplified inverter circuit
Page MOSFET 5.1-96
MOSFETs
Circuit Operation


We assume that the n-channel device is the driver, and the p-channel device is the load.
When the input is high, vO=VOL=0 Volts, and the power dissipation in the inverter is 0.

v
V
SGP
V
0
-
DD
DD
i
Q
v
SGN
DD
P
i
v
Q

V
vGSN  VOH  VDD
Operating
point
O
0
N
VOL  0
Load curve
vSGP  0
VDD
 VDD
r
v
O
0
DSN
vO
 W 

rDS  1 k 'n   VDD  Vtn 
  L n

-
Operation of the CMOS inverter when vI is high
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-97
MOSFETs
Circuit Operation, cont’d


We assume that the n-channel device is the driver, and the p-channel device is the load.
When the input is low, vO=VDD, and the power dissipation in the inverter is 0.
V

DD
Load curve
vSGP  VDD
-
i
Q
v
SGN
v
Q
DD
r
P
i

V
vGSP  VDD 
vGSN  VOL  0
O
0
N
VOL  0
DSP
v
Operating point
O
VOH  VDD
VDD
0
 VDL
vO
 W 
rDSP  1 k ' p   VDD  Vtp
  L  p

-
Operation of the CMOS inverter when vI is low
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-98


MOSFETs
Circuit Operation, cont’d

The basic CMOS logic inverter behaves as an ideal inverter
 The output voltage levels are 0 and VDD, and the signal swing is the maximum
possible. This results in wide noise margins.
 The static power dissipation in the inverter is 0
 A low-resistance path exists between the output terminal and ground (in the lowoutput state) or VDD (in the high-output state). The low output resistance makes the
inverter less sensitive to the effects of noise and other disturbances.
 The active pull-up and pull-down devices provide the inverter with high outputdriving capability in both directions.
 The input resistance of the inverter is infinite (because IG=0). Thus the inverter can
drive an arbitrarily large number of similar inverters with no loss in signal level.
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-99
MOSFETs
Voltage Transfer Characteristic
PMOS Load Lines
V in = V DD -V GSp
IDn = - I Dp
IDn
V out = V DD -V DSp
V out
IDp
IDn
IDn
Vin=0
Vin=0
Vin=3
Vin=3
V DSp
V DSp
Vout
VGSp=-2
VGSp=-5
Vin = V DD-V GSp
IDn = - IDp
© REP 5/23/2017 ENGR224
Vout = V DD-V DSp
Page MOSFET 5.1-100
Figure taken from supplemental material for Digital Integrated Circuits, A Design Perspective, Jan M. Rabaey,1996, Prentice Hall
MOSFETs
CMOS Inverter Load Characteristics
In,p
V in = 0
Vin = 5
NMOS
PMOS
Vin = 4
Vin = 1
Vin = 4
Vin = 3
Vin = 2
Vin = 3
Vin = 4
V in = 3
Vin = 2
Vin = 2
Vin = 1
Vin = 0
Vin = 5
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-101
Figure taken from supplemental material for Digital Integrated Circuits, A Design Perspective, Jan M. Rabaey,1996, Prentice Hall
MOSFETs
The Voltage Transfer Characteristic
For QN,
1
W  

iDN  k 'n   vI  Vtn vO  v 2 O  for vO  vI  Vtn
2
 L n 

and
iDN 
1 W 
2
k 'n   vI  Vtn  for vO  vI  Vtn
2  L n
For QP,


1
W  
2
iDP  k ' p    VDD  vI  Vtp VDD  vO   VDD  vO   for vO  vI  Vtp
2
 L p 

and
iDP 
Wp
Wn

1
W 
k ' p   VDD  vI  Vtp
2
 L p

n
p
© REP 5/23/2017 ENGR224

2
for vO  vI  Vtp
W 
W 
k 'n    k ' p  
 L n
 L p
Page MOSFET 5.1-102
MOSFETs
The Voltage Transfer Characteristic, cont’d
QN in saturation
QP in triode
vO
QN off
VOH  VDD
A
 VDD

 Vt 

 2

© REP 5/23/2017 ENGR224
NM H  VOH  VIH
NM L  VIL  VOL
B
QN in saturation
QP in saturation
 VDD

 Vt 

 2

VOL  0
Slope = -1
C
Slope = -1
QP in saturation
QN in triode
QP off
D
0
Vt
VIL VIH V  V  V
DD
t
DD
V
Vth  DD
2
vI
Page MOSFET 5.1-103
MOSFETs
The Complementary MOS (CMOS) Inverter


Complementary means
+5 V
both nMOS and pMOS
transistors are used
source of holes
A polite “tug O’War”! Only
gate
p channel (n well)
one device pulls at a time
drain for holes
 A High Voltage on Vin
Vout
turns On the NMOS Vin
drain for electrons
device and turns Off
n channel (p wafer)
the PMOS device
gate
source of electrons
 A Low Voltage on Vin
turns off the NMOS
and turns On the
The source is where charge
PMOS
carriers come from and the
 Power is dissipated
drain is where they flow to,
only when the output
holes come from the higher
is switching from low
voltage and flow towards a
to high or high to low
more negative terminal,
electrons come from the more
negative terminal and flow
towards the positive
© REP 5/23/2017 ENGR224
Vgsp= Vgp-Vsp
= 5-5 = 0
g=5
+5 V
s=5
open
Vin = + 5V
Vout = 0 V
closed
g=5
s=0
Vgsn= Vgn-Vsn
= 5-0 = 5
+5 V
Vgsp= Vgp-Vsp
= 0-5 = -5
s=5
g=0
closed
Vin = 0
Vgsn= Vgn-Vsn
= 0-0 = 0
Vout = + 5V
open
g=0
s=0
Page MOSFET 5.1-104
MOSFETs
How are Noise Margins Determined?


The slope of the voltage
transfer characteristic of an
inverter is the gain.
There are three key points on
the gain plot
 The point at which the
magnitude of the gain is
first equal to unity
(45 degrees)
 The point at which the
magnitude is maximum
 The point second point at
which the gain is again
equal to unity
VOH
5
|slope| = 1
|slope| = Maximum = 5
4
VOUT
(Volts)
VOL
3
2
|slope| = 1
1
0
|gain|
slope
1 2
VIL
3
VIH
4
5
VIN (Volts)
max
5
1
© REP 5/23/2017 ENGR224
VIN (Volts)
Page MOSFET 5.1-105
MOSFETs
What Noise Margins really mean


On the previous page VIL was equal to 1.2V and VIH was equal to 3V
VOL was equal to 0.7V and VOH was equal to 4.9V

The Noise Margins are defined as follows
NML = VIL - VOL
in our case = 1.2 - 0.7 = 0.5 Volts
NMH = VOL - VIL
= 4.9 - 3.0 = 1.9 Volts
VOH 5
VOUT
4
3
(Volts) 2
VOL 1
0
|slope| = 1
|slope| = Maximum = 5
|slope| = 1
1 2 3 4
VIL
VIH
5
VIN (Volts)
What the output
What the input
produces
accepts
solid high
+5V
High VOH =
(5- 4.9) V
Marginal High
VIH = 3V
NMH = (5-3)-(5-4.9)
and up
“solid”
=1.9V
signals
Marginal Low
VIL = 1.2V
Low VOL =
NML = 1.2-9.7=0.5V
(0.7- 0) V
solid low
0V
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-106
MOSFETs
How does an Inverter (with gain) restore
a “poor” signal level?





Assume that we have two identical inverters in series and that they both have the same
voltage transfer characteristic given below.
Lets say that the input to the first inverter is 3.1 Volts, which is about as marginal a high
signal as will be recognized as a high by the inverter.
The output of the first
inverter will be _____??
If we take that as the input
to the second inverter the
output of the second
inverter will be ___ ??
Volts.
Is it a solid high?
The inverter supplies (+5
and ground) and the gain
drives unknown and
marginal signals towards
solid levels
© REP 5/23/2017 ENGR224
VOH
5
??
3.1
4
VOUT
(Volts)
VOL
3
2
|slope| = 1
1
0
1 2
VIL
3
VIH
4
VIN (Volts)
5
Input to
inverter # 1
= 3.1V
Page MOSFET 5.1-107
MOSFETs
Solution to previous page
VOH
5
0.65
3.1
5
4
VOUT
(Volts)
VOL
3
2
0
Output of
1 2
inverter # 1
VIL
= 0.65V
Input to
INV #2
© REP 5/23/2017 ENGR224
|slope| = 1
1
3
VIH
4
VIN (Volts)
5
Input to
inverter # 1
= 3.1V
Page MOSFET 5.1-108
MOSFETs
Propagation Delay and Rise and Fall Times of a signal
vI
VOH
90% of (VOH-VOL)
Input Signal
50% 1/2(VOL+VOH)
10% of (VOH-VOL)
VOL
tr
vO
tf
tPHL
Time
tPLH
VOH
Output Signal
VOL
tTHL
© REP 5/23/2017 ENGR224
tTLH
Time
Page MOSFET 5.1-109
MOSFETs
The Analog Switch


The MOSFET is often used as a voltage-controlled switch.
The voltage applied to the gate of each QN and QP turns them on and off.
 In the off position, the MOSFET behaves as an open circuit between drain and source
 In the on position, the MOSFET presents a resistance rDS between drain and source
rDS 


1
W
k n' 
L

VGS  Vt 

for small vDS
More stringent requirements are placed on an analog switch, relative to a digital switch.
 When the switch is open, we want it to operate as an open circuit--the off-resistance
switch should be very high (ideally infinite)
 A high on-resistance would result in signal attenuation
The switch should be bidirectional (able to conduct in both directions)

vA
+
-
RL
CL
v
O
Analog Switch
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-110
MOSFETs
Circuit Operation


Notice that terminals are not labeled; they are interchangeable because the MOSFET is a
symmetrical device.
In general, the drain is the terminal which is at the higher voltage.
v
A
 4V
RL
CL
a
b
v
c
b
 7V +
-
RL
v
CL
© REP 5/23/2017 ENGR224
O
O
a
+
-
v
vC high; vA positive
v
A
 4V
vC high; vA negative
Page MOSFET 5.1-111
MOSFETs
Equivalent Circuits for Transmission Gate
vA (positive)
RL
a’
5V
+
-
b’
RL
CL
a
b
v
v
O
+
b’
b
a’
a
O
5V
CL
5V
+
-
+
vA (negative)
© REP 5/23/2017 ENGR224
Page MOSFET 5.1-112
5V
MOSFETs
CMOS Transmission Gate and Circuit Symbol

Compared to the single NMOS switch, the transmission gate provides better performance
at the expense of greater circuit complexity and chip area.
C
V
DD
C
Output
Input
P
Output
Input
N
C
(clock)
© REP 5/23/2017 ENGR224
V
C
C
Characteristics
0
1
Bidirectional open circuit
Bidirectional short circuit
SS
Page MOSFET 5.1-113
MOSFETs
MOSFET Internal Capacitances and High-Frequency Model





The MOSFET has internal capacitances, however, they are neglected in the small-signal
model.
The gain of every MOSFET amplifier falls off at some high frequency.
The MOSFET model must be augmented by including internal capacitances.
There are basically two types of internal capacitances in the MOSFET:
 The gate capacitive effect: the gate electrode (polysilicon) forms a parallel-plate
capacitor with the channel, with the oxide layer serving as the capacitor dielectric.
 The source-body and drain-body depletion-layer capacitances: these are the
capacitances of the reversed-biased pn junctions formed by the n+ source region
(source diffusion) and the p-type substrate, and by the n+ drain region (drain diffusion)
and the substrate.
There will be five capacitances in total:
 Cgs, Cgd, Cgb, Csb, and Cdb, where the substrates indicate the location of the
C gd
capacitances in the model.
G 
D
v
gs
g m v gs
C gs
-
g mbvbs
ro

C gb
S
v
C sb
bs

Cdb
B
© REP 5/23/2017 ENGR224
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MOSFETs
The Gate Capacitive Effect

The gate capacitive effect can be modeled by three capacitances Cgs, Cgd, and Cgb.
 In the triode region at small vDS
1
C gs  C gd  WLC ox
2
 In saturation
2
C gs  WLCox
3
C gd  0
 In the cutoff region
Cgs  Cgd  0
Cgb  WLCox
May be zero depending on the bulk potential
 Overlap capacitance
Cov  WLovCox
© REP 5/23/2017 ENGR224
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MOSFETs
The Junction Capacitances
For source diffusion, the source-body capacitance, Csb
Csb 
Csb0
V
1  SB
V0
Csb0 is the value of Csb at zero body bias
VSB is the magnitude of the reverse-bias voltage
V0 is the junction built-in voltage
For drain diffusion, the drain-body capacitance, Cdb
Cdb 
Cdb0
V
1  DB
V0
Cdb0 is the capacitance value at zero reverse-bias voltage
VDB is the magnitude of the reverse-bias voltage
V0 is the junction built-in voltage
The above formulas assume small-signal operation.
© REP 5/23/2017 ENGR224
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MOSFETs
The High-Frequency MOSFET Model
C gd
G
D

v
gs
g m v gs
C gs
-
ro
g mbvbs

C gb
S
v
Csb
bs
Cdb

B
C gd
G
C gd
D

v
gs
C gs
-
g m v gs
ro
Cdb
G
D

v
gs
C gs
g m v gs
S
© REP 5/23/2017 ENGR224
S
Page MOSFET 5.1-117
ro
MOSFETs
Unity-Gain Frequency (fT)

The unity-gain frequency (fT) is defined as the frequency at which the short-circuit
current-gain of the common-source configuration becomes unity.
C gd
Ii
Io

v
gs
C gs
g m v gs
ro
-
I o  g mVgs  sC gdVgs
I o  g mVgs
Vgs 
Ii
sC gs  C gd 
Io
gm

I i sCgs  Cgd 
© REP 5/23/2017 ENGR224
The magnitude of the current gain becomes unity at the
frequency
wT 
fT 
gm
Cgs  Cgd 
fT 
wT
2
gm
2Cgs  Cgd 
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