245
MOSFET Capacitance Models
Integrating Equation 6.62 from (y = 0, Q i = Q is ) to any point (y, Q i ) along the
channel and after simplification we can show
y
W
I
v Q Q
s
ds
kT
i
i s
=
−
(
)
µ
(6.63)
where:
v kT is the thermal voltage
Q is is the mobile charge density at the source end
At the drain end of the channel Q i = Q id .
Let us first calculate the source and drain charge Q S and Q D , respectively.
Substituting for dy and y from Equations 6.62 and 6.63, respectively, to the
expression for Q D in Equation 6.45, we get
Q
W
L
W
I
v
Q Q Q dQ
D
s
ds
kT
i
i
is
i
Q
Q
is
id
=
−
(
)
∫
µ
2
2
(6.64)
which on integration and after simplification using Equation 6.62 for I ds can
be shown as
Q
WL Q Q
D
i d
i s
=
−
(
)
1
6
2
(6.65)
Now, substituting for the charge densities Q is and Q id from Equation 4.113, we
get the expression for the drain charge Q D as
Q
W LC v
V V
nv
V
v
D
d kT
gs
th
kT
ds
kT
= −
−
⋅
−
+
1
6
2
1
exp
e xp
(6.66)
where we have used Equation 4.117 to eliminate f B from the expressions for
Q is and Q id in Equation 4.113. Equation 6.66 can also be expressed by using
Equation 4.121 relating the depletion capacitance C d and the ideality factor
n
C C
d
o x
= +(
)
1
as
Q
W LC n
v
V V
nv
V
v
D
o x
k T
gs
th
kT
ds
kT
= −
−
(
)
−
⋅
−
+
1
6
1
2
exp
e xp
1 1
(6.67)
Using similar procedures we can show that the expression for the source
charge Q S in the weak inversion region is given by
Q
WLC n
v
V V
nv
V
v
S
o x
k T
gs
th
kT
ds
kT
= −
−
(
)
−
⋅
−
+
1
6
1
2
exp
e xp
(6.68)
MOSFET Capacitance Models
Integrating Equation 6.62 from (y = 0, Q i = Q is ) to any point (y, Q i ) along the
channel and after simplification we can show
y
W
I
v Q Q
s
ds
kT
i
i s
=
−
(
)
µ
(6.63)
where:
v kT is the thermal voltage
Q is is the mobile charge density at the source end
At the drain end of the channel Q i = Q id .
Let us first calculate the source and drain charge Q S and Q D , respectively.
Substituting for dy and y from Equations 6.62 and 6.63, respectively, to the
expression for Q D in Equation 6.45, we get
Q
W
L
W
I
v
Q Q Q dQ
D
s
ds
kT
i
i
is
i
Q
Q
is
id
=
−
(
)
∫
µ
2
2
(6.64)
which on integration and after simplification using Equation 6.62 for I ds can
be shown as
Q
WL Q Q
D
i d
i s
=
−
(
)
1
6
2
(6.65)
Now, substituting for the charge densities Q is and Q id from Equation 4.113, we
get the expression for the drain charge Q D as
Q
W LC v
V V
nv
V
v
D
d kT
gs
th
kT
ds
kT
= −
−
⋅
−
+
1
6
2
1
exp
e xp
(6.66)
where we have used Equation 4.117 to eliminate f B from the expressions for
Q is and Q id in Equation 4.113. Equation 6.66 can also be expressed by using
Equation 4.121 relating the depletion capacitance C d and the ideality factor
n
C C
d
o x
= +(
)
1
as
Q
W LC n
v
V V
nv
V
v
D
o x
k T
gs
th
kT
ds
kT
= −
−
(
)
−
⋅
−
+
1
6
1
2
exp
e xp
1 1
(6.67)
Using similar procedures we can show that the expression for the source
charge Q S in the weak inversion region is given by
Q
WLC n
v
V V
nv
V
v
S
o x
k T
gs
th
kT
ds
kT
= −
−
(
)
−
⋅
−
+
1
6
1
2
exp
e xp
(6.68)