402
Compact Models for Integrated Circuit Design
In Equation 11.69, q b is the normalized base charge and is defined as
q
Q
Q
b
B
B
=
0
(11.70)
Equation 11.69 is the generalized expression for current source at all injection levels, where I SS is a fundamental constant @ V BE = V BC = 0 and is
given by Equation 11.66. The normalized majority carrier charge q b in the
neutral base region accurately models the base-width modulation. In the
next section, we will express q b in terms of bias-dependent measurable
model parameters.
11.5.4.1 Components of Injected Base Charge
In order to evaluate q b , we first determine the components of Q B . For the simplicity of Q B analysis, we assume that
• npn-BJT is in saturation, that is, V BE > 0 and V BC > 0, then
• Minority carriers are injected into the base both from the emitter
and from the collector
• From the charge neutrality, the total increase in the majority carriers
in the base = total increase in the minority carrier concentration
• Superposition of carriers in different regions holds, that is
• Total excess majority carrier density = sum of the excess majority
carrier density due to each junction separately
• Excess majority carrier concentration in the base = excess carriers due to the forward voltage (V BE + V BC )
• Depletion approximation holds
With the above assumptions, we define
• p F (x) as the majority carrier concentration at any point x in the base
at V BC = 0
• p R (x) as the majority carrier concentration at any point x in the base
at V BE = 0
• N a (x) as the base doping concentration at any point x
Then the excess majority carrier concentration in the base region is shown in
Figure 11.25 and is given by
′ =
−
+
−
p x
p x N x
p x N x
F
a
R
a
( )
( )
( )
( )
( )
(11.71)
Compact Models for Integrated Circuit Design
In Equation 11.69, q b is the normalized base charge and is defined as
q
Q
Q
b
B
B
=
0
(11.70)
Equation 11.69 is the generalized expression for current source at all injection levels, where I SS is a fundamental constant @ V BE = V BC = 0 and is
given by Equation 11.66. The normalized majority carrier charge q b in the
neutral base region accurately models the base-width modulation. In the
next section, we will express q b in terms of bias-dependent measurable
model parameters.
11.5.4.1 Components of Injected Base Charge
In order to evaluate q b , we first determine the components of Q B . For the simplicity of Q B analysis, we assume that
• npn-BJT is in saturation, that is, V BE > 0 and V BC > 0, then
• Minority carriers are injected into the base both from the emitter
and from the collector
• From the charge neutrality, the total increase in the majority carriers
in the base = total increase in the minority carrier concentration
• Superposition of carriers in different regions holds, that is
• Total excess majority carrier density = sum of the excess majority
carrier density due to each junction separately
• Excess majority carrier concentration in the base = excess carriers due to the forward voltage (V BE + V BC )
• Depletion approximation holds
With the above assumptions, we define
• p F (x) as the majority carrier concentration at any point x in the base
at V BC = 0
• p R (x) as the majority carrier concentration at any point x in the base
at V BE = 0
• N a (x) as the base doping concentration at any point x
Then the excess majority carrier concentration in the base region is shown in
Figure 11.25 and is given by
′ =
−
+
−
p x
p x N x
p x N x
F
a
R
a
( )
( )
( )
( )
( )
(11.71)