SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

340 CHAPTER 7. TEMPORAL RESPONSE OF DIODES AND BIPOLAR TRANSISTORS

= rπ Cπ CBE

CBC

Iin ( jω)

vBE

Iin ( jω) Io ( jω)

vBE gmVBE

Io ( jω)

Figure 7.17: Bipolar equivalent circuit for calculatingfτ.

Now that we have derived all of the small-signal currents in the device and expressed them in
terms of conductive and capacitative components, it is relatively straightforward to construct a
small-signal equivalent model. This model is shown in figure 7.16.

7.5.3 Small Signal Figures of Merit ......................

Current gain cutoff frequencyfτ

As stated earlier, the current gain cutoff frequencyfτis defined as the frequency at which the
short circuit current gain becomes 1. We assumed earlier thatfτcould be found by summing all
the delays in the device (see equation 7.5.1 and equation 7.5.2). We will now show why this is
the case.
The value offτis obtained by applying nodal analysis to the bipolar equivalent circuit for
the termination shown in figure 7.17. The input capacitanceCin=Cπ+CBE. The frequency
dependent current gain of the deviceβ(jω)is given by

β(jω)=

Io(jω)

Iin(jω)

(7.5.72)

We define the input impedancezinas

zin=rπ

∣∣

∣∣

∣∣

∣∣^1

jωCin

=

rπ

1+jωrπCin

(7.5.73)

We can then writeIinandIoas

Iin(jω)=

[

vBE

zin

+jωvBECBC

]

(7.5.74)

Io(jω)=vBE[gm−jωCBC] (7.5.75)

Usinggm=r−e^1
Io(jω)
Iin(jω)

=

zin

re

[

1 −jωCBCre

1+jωCBCzin

]

(7.5.76)