403
Bipolar Junction Transistor Compact Models
From Equation 11.68, the total majority carrier charge in the base consists of
equilibrium charge due to N a and excess components due to p′(x) and is given by
Q qA
p x dx
q A N x dx
q A
B
E
x
x V
E a
x
x V
E
E VBE
C BC
E VBE
C BC
=
=
+
′
∫
∫
( )
( )
(
)
(
)
(
)
(
)
p p x x dx
x
x V
E VBE
C BC
( )( )
(
)
(
)
∫
(11.72)
The equilibrium base charge includes three components: Q E due to the
decrease in the depletion region by the forward-biased EB-junction space
charge region, Q B0 due to the neutral base charge, and Q C due to the decrease
in the depletion charge by the forward-biased BC-junction depletion region
as shown in Figure 11.25. Therefore, the total equilibrium component of the
base charge under the saturation condition is given by
qA N x dx Q Q
Q
qA N x dx
E a
x
x V
E
B
C
E a
x
x
E VBE
C BC
E VBE
E
( )
( )
(
)
(
)
(
)
∫
∫
=
+
+
=
+
0
0
q qA N x dx
qA N x dx
E a
x
x
E a
x
x V
E
C
C
C BC
( )
( )
(
)
0
0
0
∫
∫
+
(11.73)
Therefore, from Equations 11.72 and 11.73, the total base charge due to EB
and CB-junction forward biases is given by
Q
q A N x dx
qA N x dx
qA N x dx
B
E a
x
x
E a
x
x
E a
x
x
E VBE
E
E
C
C
C
=
+
+
∫
∫
( )
( )
( )
(
)
0
0
0
0
( (
)
(
)
( )
(
)
V
E
x
x V
BC
E VBE
C BC
qA p x dx
∫
∫
+
′
(11.74)
E
B
C
X jE
X jC
x E0
x C0
Q F
Q B0
Q R
Q C
Q E
p(x)
p F (x)
p R (x)
N a (x)
x E (V BE )
x C (V BC )
x
FIGURE 11.25
The components of base charge in an npn-BJT in saturation: Q B0 is the base charge in the neutral
base region without applied biases; Q E and Q C are the increase in the base charge due to the
EB and CB forward biases, respectively; Q F and Q R are the excess charge due to the high-level
injection from the emitter and collector, respectively, to the base; p F (x) and p R (x) are the excess
majority carrier concentration in the base region from emitter and collector, respectively