411
Bipolar Junction Transistor Compact Models
Then substituting for q b in Equation 11.69, we get for high-level injection at
V BC = 0 and V BE >> v kT
I
I
I
V v
I Q
V
v
C high level
CT
SS
BE
kT
f SS
B
BE
kT
(
)
exp
exp
−
=
≅
(
)−
(
1
2
0
τ
) )
≅
Q I
V
v
B SS
f
BE
kT
0
2
τ
exp
(11.109)
Again, for low-level injection, if we assume q e = q c = 0, then from Equation 11.78
q b = 1. Then from Equation 11.69, we get for low-level injection @ V BC = 0 and
V BE >> v kT as
I
I
V
v
C low level
S S
BE
kT
(
)
−
≅
exp
(11.110)
At the transition point from the low-level injection to high-level injection, the
collector current must be continuous and, therefore, equal. Let us assume
that the intersection of high current and low current asymptote is given by
(I KF , V KF ) at I C(high-level) = I C(low-level) . Therefore, from Equations 11.109 and 11.110 at
the transition point (I KF , V KF ), we can show
I
Q I
V
v
KF
B SS
f
KF
kT
=
0
2
τ
exp
(11.111)
and
I
I
V
v
KF
SS
KF
kT
≅
exp
(11.112)
From the above equations, we can show that the collector current at the transition from the low-level to high-level injection, called as the forward kneecurrent, I KF is given by
I
Q
KF
B
f
=
0
τ
(11.113)
Figure 11.26 shows the knee-point (I KF , V KF ) in the ln(I C ) versus V BE plot at
V BC = 0. It is observed from Figure 11.26 that the slope of ln(I C ) − V BE /v kT plot
for high-level injection is, clearly, smaller (theoretically, about 1/2) than that
due to low-level injection.
Similarly, we can show that in the reverse mode of BJT operation, the inverse
knee-current I KR is given by
I
Q
KR
B
r
=
0
τ
(11.114)
Bipolar Junction Transistor Compact Models
Then substituting for q b in Equation 11.69, we get for high-level injection at
V BC = 0 and V BE >> v kT
I
I
I
V v
I Q
V
v
C high level
CT
SS
BE
kT
f SS
B
BE
kT
(
)
exp
exp
−
=
≅
(
)−
(
1
2
0
τ
) )
≅
Q I
V
v
B SS
f
BE
kT
0
2
τ
exp
(11.109)
Again, for low-level injection, if we assume q e = q c = 0, then from Equation 11.78
q b = 1. Then from Equation 11.69, we get for low-level injection @ V BC = 0 and
V BE >> v kT as
I
I
V
v
C low level
S S
BE
kT
(
)
−
≅
exp
(11.110)
At the transition point from the low-level injection to high-level injection, the
collector current must be continuous and, therefore, equal. Let us assume
that the intersection of high current and low current asymptote is given by
(I KF , V KF ) at I C(high-level) = I C(low-level) . Therefore, from Equations 11.109 and 11.110 at
the transition point (I KF , V KF ), we can show
I
Q I
V
v
KF
B SS
f
KF
kT
=
0
2
τ
exp
(11.111)
and
I
I
V
v
KF
SS
KF
kT
≅
exp
(11.112)
From the above equations, we can show that the collector current at the transition from the low-level to high-level injection, called as the forward kneecurrent, I KF is given by
I
Q
KF
B
f
=
0
τ
(11.113)
Figure 11.26 shows the knee-point (I KF , V KF ) in the ln(I C ) versus V BE plot at
V BC = 0. It is observed from Figure 11.26 that the slope of ln(I C ) − V BE /v kT plot
for high-level injection is, clearly, smaller (theoretically, about 1/2) than that
due to low-level injection.
Similarly, we can show that in the reverse mode of BJT operation, the inverse
knee-current I KR is given by
I
Q
KR
B
r
=
0
τ
(11.114)