P 8.7-3 Find v_c(t) for t > 0 for the circuit shown in Figure P 8.7-3 when i_s = [2 cos 2t] u(t) mA.

Figure P 8.7-3

Let’s consider a current junction and write a sum of currents

Do a voltage walk around the right loop and we can find the voltage across the 5 kΩ resistor.

The current in the 10 kΩ resistor is the current going into the capacitor so

so

This is our forcing equation so we “guess” a solution of

Taking derivative

Putting these into diff - eq

So comparing coefficients of terms gives the two equations

From term

From term

This equation tells us

Use that in the first relation

So

And our equation for

Now we need to add in the natural part

To find R_t deactivate current source

So R_t is the two resistors in series, so

We find K by using the initial condition that since

So if in the steady state before t = 0, there is no current, there could not be any charge on the capacitor, so

therefore

The final solution is

An alternative way of solving this can be done by finding a Thevenin equivalent circuit.

First do a source transformation to simplify circuit and we get

We can now identify the Thevenin circuit as

Now we can do a voltage walk around the circuit and we get

Again

Knowing

simplify

This is the same differential equation we got above so we would from here follow the same solution and again get

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This page last updated on July 23, 2019