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!
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.