Consider a single loop AC circuit with a function generator, a capacitor and an ideal inductor in series (i.e. there is no resistance in the circuit). The function generator supplies a time varying voltage ℰ().
I was asked to find particular and homogeneous solutions to V_c_(t). I was able to solve this.
I am struggling with finding the transfer function H(s)
Here is the question: a.) Write the differential equation describing the circuit in the linear operator form () = () with () as the input (voltage supplied by the function generator) and () as the output (the voltage across the capacitor). b.) Take the Laplace transform of the linear operator equation. c.) Solve the Laplace transformed equation for (). In doing so, identify the Φ() and Ψ() parts of the solution. d.) From Ψ(), determine the transfer function () for this LC circuit. e.) In the case where both the initial charge and initial current are both zero, take the inverse Laplace transform of () to find () (which is () again). As before, assume that the function generator is generating a voltage: () = ℰ() = cos() my calculated Y(s)
I know I need to decompose Y(s) into partial fractions in order to be able to take the inverse Laplace transform. However, whenever I do this math I get an answer which the inverse Laplace transform does not work on.