Consider the RLC electrical circuit given below, where the input is current \(i_{i}(t)\) and the output is voltage \(v_{o}(t)\).

(a) The input-output differential equation for the circuit is given by \[
C \ddot{v}_{o}+\frac{1}{R} \dot{v}_{o}+\frac{1}{L} v_{o}=\dot{i}(t)
\]
Given that \(R=\frac{1}{4} \Omega, L=\frac{1}{3} H, C=1 F\) and assuming zero initial conditions, find the system transfer function \(H(s)\).

(b) Find the impulse (natural) response of the system.

(c) Find the unit step response of the system.

(d) What is the relationship between the natural and unit step responses?

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vi (t) 252 W 0.5F V C 252 000 19 A 2H An RLC Electrical Circuit