The BJT
BJT Device Equations
Figure 1 shows the circuit symbols for the npn and pnp BJTs. In the active mode, the collectorbase
junction is reverse biased and the baseemitter junction is forward biased. Because of recombina
tions of the minority and majority carriers, the equations for the currents can be divided into three
regions: low, mid, and high. For the npn device, the currents are given by
Low Level:
i
B
=
I
SE
·
exp
μ
v
BE
nV
T
¶
−
1
¸
(1)
i
C
=
I
S
·
exp
μ
v
BE
V
T
¶
−
1
¸
(2)
Mid Level:
i
B
=
I
S
β
F
·
exp
μ
v
BE
V
T
¶
−
1
¸
(3)
i
C
=
I
S
·
exp
μ
v
BE
V
T
¶
−
1
¸
(4)
High Level:
i
B
=
I
S
β
F
·
exp
μ
v
BE
V
T
¶
−
1
¸
(5)
i
C
=
I
S
r
I
K
I
SO
·
exp
μ
v
BE
2
V
T
¶
−
1
¸
(6)
where all leakage currents that are a function of
v
CB
have been neglected. In the current equations,
I
S
is the saturation current and
β
F
is the midlevel basetocollector current gain.
These are
functions of the collectorbase voltage and are given by
I
S
=
I
SO
μ
1 +
v
CB
V
A
¶
=
I
SO
μ
1 +
v
CE
−
v
BE
V
A
¶
(7)
β
F
=
β
FO
μ
1 +
v
CB
V
A
¶
=
β
FO
μ
1 +
v
CE
−
v
BE
V
A
¶
(8)
Figure 1: BJT circuit symbols.
In the equations for
i
B
and
i
C
,
V
A
is the Early voltage and
I
SO
and
β
FO
, respectively, are the
zero bias values of
I
S
and
β
F
. The constant
n
is the emission coe
ﬃ
cient or ideality factor of the
baseemitter junction.
It accounts for recombinations of holes and electrons in the baseemitter
1
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junction at low levels. Its value, typically in the range
1
≤
n
≤
4
, is determined by the slope of
the plot of
ln (
i
C
)
versus
v
BE
at low levels. The default value in SPICE is
n
= 1
.
5
. The constant
I
SE
is determined by the value of
i
B
where transition from the lowlevel to the midlevel region
occurs. The constant
I
K
is determined by the value of
i
C
where transition from the midlevel to
the highlevel region occurs. Note that
I
S
/
β
F
=
I
S
0
/
β
F
0
so that
i
B
is not a function of
v
CB
in the
midlevel region. The equations apply to the pnp device if the subscripts
BE
and
CB
are reversed.
Figure 2 shows a typical plot of
i
C
versus
v
BE
for
v
CE
constant. The plot is called the transfer
characteristics. There is a threshold voltage above which the current appears to increase rapidly.
This voltage is typically
0
.
5
to
0
.
6
V. In the forward active region, the basetoemitter voltage is
typically
0
.
6
to
0
.
7
V. Figure 3 shows typical plots of
i
C
versus
v
CE
for
i
B
constant. The plots are
called the output characteristics. Note that the slope approaches a constant as
v
CE
is increased. If
the straight line portions of the curves are extended back so that they intersect the
v
CE
axis, they
would intersect at the voltage
v
CE
=
−
V
A
+
v
BE
'
−
V
A
. For
v
CE
small,
v
BE
> v
CE
and the BJT
is in the saturation region.
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 Summer '08
 HOLLIS
 Direct Current, Bipolar junction transistor, Norton's theorem, RTE

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