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Circuit Theory/Convolution Integral/Examples/2R1LExample/2 Resistor, 1 inductor example
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thumb|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
-- or --
Homogeneous Solution
First order so τ is:
Particular Solution
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
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
The impulse solution is the derivative of the above:
Convolution Integral
f := (exp((t-y)/x)/(C*R2))*2*exp(2*y/z);
S :=int(f,y=0..t)
Evaluate Integration Constant
Know that VRC=0 at t=0 so:
So:
And finally: