162
Compact Models for Integrated Circuit Design
Q y
C
V
V y
V
C
V
b
o x
B
sb
B
s b
ox
B
s b
( )
( )
= −
+
+
+
−
≅ −
+
+
γ
φ
φ
γ
φ
2
1
2 2
2
1
2
2 2
2
φ
γ
φ
δ
B
s b
ox
B
s b
V
V y
C
V
V y
+
≅ −
+
+ ⋅
( )
( )
(4.91)
Equation 4.91 is called the square root approximation of Q b (y), where δ accounts
for the bulk-charge effect in MOSFETs and is given by
δ
φ
≡
+
1
2 2 B
s b
V
(4.92)
It is found that the value of δ obtained by Equation 4.92 is too large for accurate calculation of I ds at low V sb and high V ds . In order to obtain accurate
value of δ, several semi-empirical expressions for δ have been proposed as
discussed by Arora [13]. It is found that the more appropriate expressions of
δ for circuit CAD are the following semi-empirical relations
δ
φ
≡
+
+
1
2 1 2 B
s b
V
(4.93)
and
δ
φ
φ
≡
+
− +
+
(
)
1
2 2
1
1
2
1
2
B
s b
B
s b
V
a a
V
(4.94)
where a 1 and a 2 are obtained to minimize the error between the exact function 2φ B
s b
V V y
+
+ ( ) and its approximation
2φ
δ
B
s b
V
V
+
+ ⋅
(
) within the
operating range of V sb and V ds .
With the square root approximation of Q b (y) from Equation 4.91 in Equation
4.66, we get
Q y
C V V
Vy
Vy
V
C V
V
i
o x
g s
f b
B
B
s b
ox
gs
( )
( )
( )
= −
−
−
−
−
⋅
+
+
{
}
= −
−
2
2
φ
γ δ
φ
f fb
B
B
sb
ox
gs
th
V
Vy
C V V
V y
+
+
+
(
) − + ⋅
= −
−
−
2
2
1
φ γ φ
δγ
α
(
) ( )
( )
(4.95)
where we have used Equation 4.12 for V th and α is defined as
α
δ γ
= + ⋅
1
(4.96)
α is called the bulk-charge coefficient.
Compact Models for Integrated Circuit Design
Q y
C
V
V y
V
C
V
b
o x
B
sb
B
s b
ox
B
s b
( )
( )
= −
+
+
+
−
≅ −
+
+
γ
φ
φ
γ
φ
2
1
2 2
2
1
2
2 2
2
φ
γ
φ
δ
B
s b
ox
B
s b
V
V y
C
V
V y
+
≅ −
+
+ ⋅
( )
( )
(4.91)
Equation 4.91 is called the square root approximation of Q b (y), where δ accounts
for the bulk-charge effect in MOSFETs and is given by
δ
φ
≡
+
1
2 2 B
s b
V
(4.92)
It is found that the value of δ obtained by Equation 4.92 is too large for accurate calculation of I ds at low V sb and high V ds . In order to obtain accurate
value of δ, several semi-empirical expressions for δ have been proposed as
discussed by Arora [13]. It is found that the more appropriate expressions of
δ for circuit CAD are the following semi-empirical relations
δ
φ
≡
+
+
1
2 1 2 B
s b
V
(4.93)
and
δ
φ
φ
≡
+
− +
+
(
)
1
2 2
1
1
2
1
2
B
s b
B
s b
V
a a
V
(4.94)
where a 1 and a 2 are obtained to minimize the error between the exact function 2φ B
s b
V V y
+
+ ( ) and its approximation
2φ
δ
B
s b
V
V
+
+ ⋅
(
) within the
operating range of V sb and V ds .
With the square root approximation of Q b (y) from Equation 4.91 in Equation
4.66, we get
Q y
C V V
Vy
Vy
V
C V
V
i
o x
g s
f b
B
B
s b
ox
gs
( )
( )
( )
= −
−
−
−
−
⋅
+
+
{
}
= −
−
2
2
φ
γ δ
φ
f fb
B
B
sb
ox
gs
th
V
Vy
C V V
V y
+
+
+
(
) − + ⋅
= −
−
−
2
2
1
φ γ φ
δγ
α
(
) ( )
( )
(4.95)
where we have used Equation 4.12 for V th and α is defined as
α
δ γ
= + ⋅
1
(4.96)
α is called the bulk-charge coefficient.