Circuit Theory/Convolution Integral/Examples/Example43

thumb|2R 1L 1C circuit to solve using the convolution integral .. for wikibook circuit analysis Given that is = 1 + cos(t), find io using the convolution integral. Outline: Transfer Function simplify(1/((s+1)*(s + 1 + 1/(s+1)))) Homogeneous Solution Setting the denominator to zero and find the values of s: solve(s^2 + 2*s + 2) This means the solution has the form: Particular Solution After a long time the cap opens and the inductor shorts. The current is split between the two resistors. Each will get 1/2 of the unit step function source which would be 1/2 amp: Initial Conditions The particular solution still has to apply so at t= ∞: Initially the current has to be zero in this leg so: The initial voltage across the cap is zero, and across the leg is zero and across the inductor is zero. So: f := 1/2 + exp(-t)*((-1/2)*cos(t) + B*sin(t)); g = diff(f,t) So now: Impulse Response Taking the derivative of the above f := 1/2*(1-exp(-t)*(cos(t) + sin(t))); g = diff(f,t) get: Convolution Integral f := exp(x-t)*sin(t-x)*(1 + cos(x)); S =int(f,x=0..t);