249
MOSFET Capacitance Models
Q
WLC
V V
V AB
Q
WLC
V V
D
o x
g s
t h
d s
S
o x
g s
t h
= −
−
(
) −
+ ′ ′
= −
−
(
)
1
2
1
3
1
2
α
− −
+ ′ − ′
(
)
1
6
1
αV A
B
ds
(6.81)
where
′ = ⋅ +
′ = ⋅ +
A A
V
LE
B B
V
LE
ds
c
ds
c
1
1
(6.82)
and, A and B are defined in Equation 6.55.
Comparing the expressions for Q D and Q S in Equation 6.81 for short channel devices with the corresponding expressions for long channel devices in
Equation 6.54, we notice that the two equations have the same form differing
only in parameters A′ and B′. As can be seen from Equation 6.82, A′ and B′
include the velocity saturation factor. Thus, in the case for a long channel
device, the product LE c is very large, then A′ = A, B = B′, and Equation 6.81
converges to Equation 6.54 as is expected.
Again, using the procedure for deriving Q G for long channel devices, we
can show for short channel devices
Q
W
I
Q Q dV
W
E
Q dV
G
s
ds
g
i
V
c
g
Vd
d
=
⋅
−
∫
∫
µ
2
0
0
(6.83)
Substituting for Q i and Q g from Equations 6.46 and 6.49, respectively, and carrying out the integration, we get after simplification
Q WLC V V
V
A
G
o x
g s
f b
B
ds
=
−
−
−
+
′
2
1
2
φ
α
(6.84)
Here again, for long channel devices Equation 6.84 converges to Equation 6.57.
Similarly, we can show the bulk charge expression for short channel devices as
Q
WLC
V
V D
B
o x
B
sb
ds
= −
+
+ −
′
γ φ
α
2
1
(
)
(6.85)
where
′ = −
−
−
⋅
D D
V V
V
V
LE
gs
th
ds
ds
c
1
12
1 2
( )α
α
(6.86)
and, D is given by Equation 6.60.
MOSFET Capacitance Models
Q
WLC
V V
V AB
Q
WLC
V V
D
o x
g s
t h
d s
S
o x
g s
t h
= −
−
(
) −
+ ′ ′
= −
−
(
)
1
2
1
3
1
2
α
− −
+ ′ − ′
(
)
1
6
1
αV A
B
ds
(6.81)
where
′ = ⋅ +
′ = ⋅ +
A A
V
LE
B B
V
LE
ds
c
ds
c
1
1
(6.82)
and, A and B are defined in Equation 6.55.
Comparing the expressions for Q D and Q S in Equation 6.81 for short channel devices with the corresponding expressions for long channel devices in
Equation 6.54, we notice that the two equations have the same form differing
only in parameters A′ and B′. As can be seen from Equation 6.82, A′ and B′
include the velocity saturation factor. Thus, in the case for a long channel
device, the product LE c is very large, then A′ = A, B = B′, and Equation 6.81
converges to Equation 6.54 as is expected.
Again, using the procedure for deriving Q G for long channel devices, we
can show for short channel devices
Q
W
I
Q Q dV
W
E
Q dV
G
s
ds
g
i
V
c
g
Vd
d
=
⋅
−
∫
∫
µ
2
0
0
(6.83)
Substituting for Q i and Q g from Equations 6.46 and 6.49, respectively, and carrying out the integration, we get after simplification
Q WLC V V
V
A
G
o x
g s
f b
B
ds
=
−
−
−
+
′
2
1
2
φ
α
(6.84)
Here again, for long channel devices Equation 6.84 converges to Equation 6.57.
Similarly, we can show the bulk charge expression for short channel devices as
Q
WLC
V
V D
B
o x
B
sb
ds
= −
+
+ −
′
γ φ
α
2
1
(
)
(6.85)
where
′ = −
−
−
⋅
D D
V V
V
V
LE
gs
th
ds
ds
c
1
12
1 2
( )α
α
(6.86)
and, D is given by Equation 6.60.