Question 5.12: Characterize the composite waveform obtained by multiplying ......

In this case the composite waveform is expressed as

v(t) = \sin ω_{0}t[V_{A}e^{-t/T_{C}}]u(t)  V

= V_{A}[e^{-t/T_{C}} \sin ω_{0}t]u(t)  V

Figure 5–30 shows a graph of this waveform for T_{0} = 2T_{C}. For t < 0 the step function forces the waveform to be zero. At t = 0, and periodically thereafter, the waveform passes through zero because sin (nπ) = 0. The waveform is not periodic, however, because the decaying exponential gradually reduces the amplitude of the oscillation. for all practical purposes the oscillations become negligibly small for t>5T_{C}. The waveform obtained by multiplying a sinusoid by a decaying exponential is called a damped sine.