Circuit Theory/Convolution Integral/Examples/2R1LExample/2 Resistor, 1 inductor example

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Given Vs = 2e2t/to find the drop across the C R1 parallel combination

Find drop across CR1 pair given that Vs 2e2t/to.

Outline of solution:

Transfer Function[edit]

-- or --

Homogeneous Solution[edit]

First order so τ is:

Particular Solution[edit]

After a long time, due to Vs = 1, the capacitor opens. So VRC is part of a voltage divider consisting of just two resistors:

Evaluate Initial Conditions[edit]

Combining the homogeneous and particular:

At t=0, the voltage across the capacitor is zero so:

Initially the cap is a short, so the current through the cap is limited by R2 so:

Since Vs = 1 (doing this for the unit step function because using convolution integral):

and

In summary:

Find impulse solution[edit]

The impulse solution is the derivative of the above:

Convolution Integral[edit]

f := (exp((t-y)/x)/(C*R2))*2*exp(2*y/z);
S :=int(f,y=0..t)

Evaluate Integration Constant[edit]

Know that VRC=0 at t=0 so:

So:

And finally: