# Circuit Theory/2Source Excitement/Example46

Find the time domain expression for i_{o} given that I_{s} = 1μ(t) amp.

## Contents

## Steady State Particular Solution[edit]

After a long time, the cap opens and inductor shorts putting the two resistors in parallel and splitting the current making i_{o} 1/2 amp.

## Transient Particular Solution[edit]

Start writing a node equation:

Substitute voltage terminal relationships:

Find V in terms of i_{o} through the L R branch:

Substitute to get I_{s} in terms of i_{o}:

Substituting numbers from the problem:

## Time constant[edit]

Guess: Substituting:

Does this equal zero?

No. Rats. Need to evaulate the above quadratic in order to guess another solution.

So the next guess is:

## Finding the Constants[edit]

After a very long time, the capacitor is going to open and the inductor is going to short. This leaves two equal resistors in parallel that are going to split the current in half.

So now the expression for i_{o} is:

Initially the current through the conductor is 0, so i_{o}(0_{+}) = 0:

Which means that:

The other initial condition affecting i_{o} is the voltage across the inductor .. which is zero. We can find an expression for V_{L}:

Setting all this equal to 0 at t=0 yields:

So:

Thus i_{o} is:

This solution is used to find i_{o} for a complicated source using the convolution integral.