P 8.7-4 Many have witnessed the use of an electrical megaphone for amplification of speech to a crowd. A model of a microphone and speaker is shown in Figure P 8.7-4a, and the circuit model is shown in Figure P 8.7-4b. Find v(t) for v_s = 10(sin 100t)u(t) which could represent a person whistling or singing a pure tone.

Figure P 8.7-4

v(t) will be found from the current passing through the 8 Ω resistor. The circuit is a simple series circuit. At steady state the inductor acts like a short circuit so, the steady state current is just

So for t < 0

Deactivating power source yields the Thevenin resistance as the two resistors in series, so

And

So the Thevenin circuit is

So a voltage walks provides

Since

Assume a forced solution of

Collecting terms

This provides two equations to solve for A and B

And

Using the first equation we get

Plug this into second equation and then solve for A

So

And

So the forced solution is

The natural solution is

So full solution is

Now use the value at t = 0

So

Thus

Now

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