Question 5.11: Characterize the composite waveform obtained by multiplying ......
Characterize the composite waveform obtained by multiplying the ramp r(t)/T_{C} times an exponential.
The equation for this composite waveform is
v(t) = \frac{r(t)}{T_{C}}[V_{A}e^{-t/T_{C}} ]u(t) V
= V_{A}[(t/T_{C})e^{-t/T_{C}} ]u(t) V
For t < 0 the waveform is zero because of the step function. At t = 0 the waveform is zero because r(0) = 0. For t > 0 there is a competition between two effects—the ramp increases linearly with time while the exponential decays to zero. Since the composite waveform is the product of these terms, it is important to determine which effect dominates. In the limit, as t → ∞, the product of the ramp and exponential takes on the indeterminate form of infinity times zero. A single application of l’Hôpital’s rule, then, shows that the exponential dominates, forcing the v(t) to zero as t becomes large. That is, the exponential decay overpowers the linearly increasing ramp, as shown by the graph in Figure 5–29. The waveform obtained by multiplying a ramp by a decaying exponential is called a damped ramp.
