408
Compact Models for Integrated Circuit Design
the bulk-ohmic resistances, we get from Equation 11.69 in the normal active
region, we have
I
I
I
q
V
v
I
V V
V
C
C T
SS
c
BE
kT
SS
BC
AF
B
≡
= +
(
)
−
= + (
)
1
1
1
exp
exp
E E
kT
v
−
1
(11.93)
From Equation 11.93, we can show
g
dI
dV
I
V
o
C
CE V
C
AF
BE
=
≅
=constant
( )
0
(11.94)
where:
I C (0) is the collector current at V CE = 0
Evaluation of q f : The parameter q f can be considered as the normalized excess
carrier density in the base with EB-junction voltage V BE only and models the
high-level injection. From the charge neutrality condition, total excess majority
carriers = total excess minority carriers. Therefore, for an npn-BJT in the normal
active mode of operation with |V BE | > 0 and V BC = 0
Q
qA p x N x dx
qA n x
n
N x
dx
F
F
a
x
x
F
i
a
x
x
E
C
E
C
=
−
=
−
∫
∫
( )
( )
( )
( )
2
(11.95)
We have shown in Equation 11.42 that the minority carrier forward base transit time τ B of a BJT and the base charge Q B are given by Q B = τ B I CC . Therefore,
from Equation 11.95, the forward base transit time can be expressed as
Q
I
F
B F CC
= τ
(11.96)
where:
I CC = I CT given by Equation 11.69 at V BC = 0
τ BF is the forward base transit time
Therefore, the normalized forward injection charge is given by
q
Q
Q
I
Q
Q
I
q
V
v
f
F
B
BF CC
B
BF
B
SS
b
BE
kT
=
=
=
−
0
0
0
1
τ
τ
exp
(11.97)
Evaluation of q r : Similar to q f , q r can be considered as the normalized excess
carrier density in the base due to CB-junction voltage V BC only and models
the high-level injection. Again, from the charge neutrality condition, total
Compact Models for Integrated Circuit Design
the bulk-ohmic resistances, we get from Equation 11.69 in the normal active
region, we have
I
I
I
q
V
v
I
V V
V
C
C T
SS
c
BE
kT
SS
BC
AF
B
≡
= +
(
)
−
= + (
)
1
1
1
exp
exp
E E
kT
v
−
1
(11.93)
From Equation 11.93, we can show
g
dI
dV
I
V
o
C
CE V
C
AF
BE
=
≅
=constant
( )
0
(11.94)
where:
I C (0) is the collector current at V CE = 0
Evaluation of q f : The parameter q f can be considered as the normalized excess
carrier density in the base with EB-junction voltage V BE only and models the
high-level injection. From the charge neutrality condition, total excess majority
carriers = total excess minority carriers. Therefore, for an npn-BJT in the normal
active mode of operation with |V BE | > 0 and V BC = 0
Q
qA p x N x dx
qA n x
n
N x
dx
F
F
a
x
x
F
i
a
x
x
E
C
E
C
=
−
=
−
∫
∫
( )
( )
( )
( )
2
(11.95)
We have shown in Equation 11.42 that the minority carrier forward base transit time τ B of a BJT and the base charge Q B are given by Q B = τ B I CC . Therefore,
from Equation 11.95, the forward base transit time can be expressed as
Q
I
F
B F CC
= τ
(11.96)
where:
I CC = I CT given by Equation 11.69 at V BC = 0
τ BF is the forward base transit time
Therefore, the normalized forward injection charge is given by
q
Q
Q
I
Q
Q
I
q
V
v
f
F
B
BF CC
B
BF
B
SS
b
BE
kT
=
=
=
−
0
0
0
1
τ
τ
exp
(11.97)
Evaluation of q r : Similar to q f , q r can be considered as the normalized excess
carrier density in the base due to CB-junction voltage V BC only and models
the high-level injection. Again, from the charge neutrality condition, total