# Circuit Theory/Convolution Integral

## Impulse Response edit

So far circuits have been driven by a DC source, an AC source and an exponential source. If we can find the current of a circuit generated by a Dirac delta function or impulse voltage source δ, then the convolution integral can be used to find the current to any given voltage source!

## Example Impulse Response edit

The current is found by taking the derivative of the current found due to a DC voltage source! Say the goal is to find the δ current of a series LR circuit .. so that in the future the convolution integral can be used to find the current given any arbitrary source.

Choose a DC source of 1 volt (the real Vs then can scale off this).

The particular homogeneous solution (steady state) is 0. The homogeneous solution to the non-homogeneous equation has the form:

Assume the current initially in the inductor is zero. The initial voltage is going to be 1 and is going to be across the inductor (since no current is flowing):

If the current in the inductor is initially zero, then:

Which implies that:

So the response to a DC voltage source turning on at t=0 to one volt (called the unit response μ) is:

Taking the derivative of this, get the impulse (δ) current is:

Now the current due to any arbitrary V_{S}(t) can be found using the convolution integral:

Don't think i_{δ} as current. It is really . V_{S}(τ) turns into a multiplier.

## LRC Example edit

Find the time domain expression for i_{o} given that I_{s} = cos(t + π/2)μ(t) amp.

Earlier the step response for this problem was found:

The impulse response is going to be the derivative of this:

The Mupad code to solve the integral (substituting x for τ) is:

f := exp(-(t-x)) *sin(t-x) *(1 + cos(x)); S := int(f,x = 0..t)

## Finding the integration constant edit

This implies:

## TO DO edit

This was created with matlab, turned into a gif with ImageMagick, cropped with a photo editor and then released into the public domain.

Several others have created an alternative animation.

- The blue symbol represents .
- The red symbol represents the arbitrary .
- The current due to the V
_{S}black (on top of the yellow). - The turn on event occurs at t = 5 seconds, not 0.
- The voltage of the source is not on indefinitely. It turns on at zero and off at 5 time constants.