Question

2. (14 marks total) This question deals with the series RLC circuit discussed in the classroom and in the labs. Assume that the voltage source is arbitrary and there is a non-zero charge, g(0), on the capacitor at time t 0 when a switch is closed to start current flow. For this question assume variable R, L and C values. (a) Write down the differential equation that describes the charge on the capacitor as a function of time. (2 marks) (b) Derive the Laplace Transform of the current as a function of time. (3 marks) (c) Derive the Unit Impulse Response for this linear system if the input is an arbitrary voltage source and the output is the charge on the capacitor as a function of time? (d) Derive the Transfer Function for the voltage drop across the inductor as a function of (e) Derive the Laplace Transform of the charge on the capacitor if the input voltage source Assume the system is not critically damped. (3 marks) time. (3 marks) is a cosine curve of amplitude A, angular frequency w and zero phase ofiset. (3 marks)

Answer 1

Let q the instantaneous charge on capacitor and i is the current in the circuit. L di/dt + q/c + Ri = v since i = dq/dt we can write, L d^2 q/dt^2 + q/c + R dq/dt = v - (1) let me write the eqn (1) in terms of i L di/dt + iR + 1/c integral^t_0 i(t) dt = v using laplace transformation, L[L di/dt] + L[iR] + L[1/c integral^t_0 i(t) dt] = L v(t) + /v - (2) Now, L[L di(t)/dt] = L[S I(S) - I_0] here, I(s) = L[i(t)], L v(t) = v(s) i_0 = 0 as initially switch was open L[1/c integral^t_0 i(t) dt] = I(s)/sc so eq(2) can be written as, L[S I(s) - I_0] + I(s) R + I(s)/SL + q_0/sc = v_(s) I(s) = S v(s) + v_s I_0 - q(0)/L/S^2 + R/L S + 1/L c \r\n From eqn (1) L d^2 q/dt^2 + R dq/dt + q/c = v d^2 q/dt^2 + 2b dq/dt +