Circuit Theory/Convolution Integral

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Examples of arbitrary sources that could drive a circuit

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.

Series LR circuit with impulse δ function as voltage source

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

Series LR circuit with unit μ step function voltage source

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 VS(t) can be found using the convolution integral:

Don't think iδ as current. It is really . VS(τ) turns into a multiplier.

LRC Example[edit]

RLC problem for circuit theory wikibook

Find the time domain expression for io given that Is = 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]

Convolution of spiky function with box2.gif

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 VS 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.