404
Compact Models for Integrated Circuit Design
From Figure  11.25, we find that the excess forward charge is due to the
forward injection carrier profile [p F (x)  −  N a (x)] and reverse injection carrier
density profile [p R (x) − N a (x)]. Therefore, the excess carrier density is given by
qA p x dx
qA p x N x dx
E
x
x V
E
F
a
x
x V
E VBE
C BC
E VBE
C B
′
=
−
 
 
∫
( )
( )
( )
(
)
(
)
(
)
( C C
E VBE
C BC
qA p x N x dx
Q Q
E
R
a
x
x V
F
R
)
(
)
( )
( )
(
)
∫
∫
+
−
 
 
=
+
(11.75)
Therefore, the total base charge due to the applied EB- and CB-junction
biases is given by
Q Q Q
Q Q Q
B
E
B
C
F
R
=
+
+
+
+
0
(11.76)
where:
Q B0 is the charge in the neutral base region at V BE  = 0 = V BC
Q E is the increase in Q B under V BE and is only a mathematical entity
Q C is the increase in Q B under V BC and is only a mathematical entity
Q F is the excess majority charge in the forward biased-device with V BC  = 0. It
is only a mathematical entity and important under high level injection
Q R is the excess majority charge in the forward biased-device with V BE  = 0. It
is only a mathematical entity and important under high-level injection
It is clear from Figure 11.25 that p F (x) > N a (x) and p R (x) > N a (x), that is, Q F and
Q R represent high-level injection. Then from Equation 11.76, we get the normalized components of base charge as
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
B
B
E
B
B
B
C
B
F
B
R
B
0
0
0
0
0
0
0
=
+
+
+
+
(11.77)
After simplification, we can show for the normalized base charge from
Equation 11.77
q
q q q q
b
e
c
f
r
= + + + +
1
(11.78)
where:
q Q Q q Q Q q Q Q
q Q Q
e
E
B
c
C
B
f
F
B
e
R
B
= = = =
0
0
0
0
,
,
, and
are the respective
normalized components of base charge
In order to develop BJT compact model, each component of the normalized
base charge is expressed in terms of measurable device model parameters.
Next, we will evaluate each component of q b given in Equation 11.78.