Question 5.14: Characterize the composite waveform obtained as the differen......
Characterize the composite waveform obtained as the difference of two exponentials with the same amplitude.
The equation for this composite waveform is
υ(t) = [V_{A} e^{−t/T_{1}}] u(t) − [V_{A} e^{−t/T_{2}}] u(t)
= V_{A} (e^{−t/T_{1}} – e^{−t/T_{2}}) u(t)
For T1 > T2 the resulting waveform is illustrated in Figure 5–35 (plotted for T1 = 2T2). For t < 0 the waveform is zero. At t = 0 the waveform is still zero, since
υ(0) = V_{A} (e^{−0} − e^{−0})
= V_{A}(1 − 1) = 0
For t \gg T_{1} the waveform returns to zero because both exponentials decay to zero. For 5T1 > t > 5T2 the second exponential is negligible and the waveform essentially reduces to the first exponential. Conversely, for t \ll T_{1} the first exponential is essentially constant, so the second exponential determines the early time variation of the waveform. The waveform is called a double exponential, since both exponential components make important contributions to the waveform.
