SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

324 CHAPTER 7. TEMPORAL RESPONSE OF DIODES AND BIPOLAR TRANSISTORS

so that we have

iB(t)−

IC(EOS)

βF

=

QS

τS

+

dQS

dt

(7.4.22)

We will see later that in the switching characteristics of the BJT, the overdrive charge and the
time constantτSappearing in the equation above play a critical role.

7.4.1 Junction Voltages at Saturation ......................

Having discussed the various operating modes of the BJT, we will now obtain expressions for
the junction voltages as the device goes into the saturation mode. These voltages are useful in
studying the behavior of BJTs for logic elements. In figure 7.9 we show a simple model of the
BJT in the saturation mode. Let us apply Kirchhoff’s voltage law (KVL) to the voltage values:

VCE = VCB+VBE

= −VBC+VBE (7.4.23)

Thus
VCE(sat)=VBE(sat)−VBC (7.4.24)

To obtainVBE(sat)we multiply the second of Eqns. 6.68 byαRand subtract the resulting
equation from the first of Eqns. 5.68. This gives

IE−αRIC=IES(1−αFαR)eeVBE/kBT (7.4.25)

UsingIE=IB+IC, we find

IB+IC(1−αR)=IES(1−αFαR)eeVBE/kBT (7.4.26)

This gives forVBE(sat)

VBE(sat)=

kBT

e

ln

[

IB+IC(1−αR)

IEO

]

(7.4.27)

where

IEO=IES(1−αFαR) (7.4.28)

In a similar manner, the value ofVBC(sat)is

VBC(sat)=

kBT

e

ln

[

αFIB−IC(1−αF)

ICO

]

(7.4.29)

with
ICO=ICS(1−αFαR) (7.4.30)

From these values ofVBE(sat)andVBC(sat)we have

VCE(sat)=

kBT

e

ln

[

IB+IC(1−αR)

αFIB−IC(1−αF)

·

ICO

IEO

]

(7.4.31)