# Circuit Theory/2Source Excitement/Example45

## Contents

## Particular/Steady State solution[edit]

Inductor short, cap open, V_{s} = 5 μ(t),find i_{r}

## Homogeneous/Transient Solution[edit]

Loop equation:

Differential equation that needs to be solved:

Guess:

Substitute to check if possible:

So the answer is going to be second order, thus guess was wrong, but can guess more accurately now by computing roots of the above second order equation:

Both roots are negative and equal, so the new guess is:

Checking again by plugging into s^{2} + 2s + 1 = 0:

Yes it equals zero now! So can go on. Have to add a constant to the differential equation solution so V_{cr} is:

## Without Initial Conditions .. Finding the Constants[edit]

Have initial conditions: V_{CR}(0_{+}) = 0 since initially cap is a short and impedance times the derivative of the inductor current i_{t}(0_{+}) = 5. Turning this into an equation:

The final voltage across the parallel RC combination is going to be 5 volts (after a very long time) because the capacitor opens and the inductor shorts.

This is the matlab code that computes the limit:

syms A B C1 t f = A*exp(-t) + B*t*exp(-t) + C1; limit(f,t,inf)

Only B is unknown now:

The initial voltage across the inductor is going to be 5 volts. But this does not lead to the value of B. Another initial condition is that the initial current through the capacitor (even though it is initially a short) is zero because the inductor is initially an open. This leads to B:

Now V_{CR} is:

Which means that i_{r} is:

## Without C_1 constant[edit]

Trying to do this problem without the C_1 constant ends in something like this:

Which has no solution. Or it can lead to 5=5 where the constant disappears from the equation without finding a number for it. Or it can lead to A or B equaling infinity. Any of these non-answers means a mistake was made somewhere.