SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

338 CHAPTER 7. TEMPORAL RESPONSE OF DIODES AND BIPOLAR TRANSISTORS

If this is inserted into equation 7.5.56, and the integration executed, one finds readily

J=iω(wB)

exp (−jωt)− 1

−jωt

(7.5.60)

where we have introduced the electron transit time through the capacitor,

τ=w/v (7.5.61)

The expression equation 7.5.60 is easily transformed into the product of an amplitude factor and
a phase factor:

J=iω(wB)·

[

sin(ωτ/2)

ωτ/ 2

]

exp (−jωτ/2) (7.5.62)

The two factors followingiω(wB)indicate the attenuation and the phase shift of the current
leaving the capacitor by the finite transit time through the capacitor.
We note first of all that the signal delay is only one-half the transit time of the electrons
themselves. We also note that there is an attenuation, due to the destructive interference between
different portions of the traveling wave. Forω=2π/τthe amplitude factor is zero, and no
current is collected at all. This is the case when the wavelengthλ=2πv/ωof the traveling
wave is equal to the capacitor plate separation w.
The two terms followingiω(wB)indicate that that the signal passing through the base-collector
depletion capacitor has been both attenuated and phase shifted as a result of the finite transit time
through this region.
Substituting foriω(wB)we can express the output currentiCin terms of the input signalvω.

iC=−

vω

re

(

1 −jω

τB

3

)

·

sin (ωτC)

ωτC

·exp (−jωτC) (7.5.63)

If the frequencyωis sufficiently small, this may be written as

iC=−

vω

re

·

sin (ωτC)

ωτC

·exp

[

−jω

(τB

3

+τC

)]

(7.5.64)

The device transconductance,gm, is defined as

gm=

∂iC

∂vω

(7.5.65)

Inserting equation 7.5.65 into equation 7.5.63, we get for the bipolar transistor transconductance

gm=gm 0 ·

sin (ωτC)

ωτC

·exp

[

−jω

(τB

3

+τC

)]

(7.5.66)

wheregm 0 =r−e^1 is the device transconductance at dc.
ThebasecurrentiBin figure 7.14 is simply the difference iniEandiC,or

iB=iE−iC=iω(0)−iC (7.5.67)