Consider the parallel RLC circuit shown in Fig. 7.16. The second-order differential equation that describes the voltage v(t) is
\frac{d^{2} v}{d t^{2}}+\frac{1}{R C} \frac{d v}{d t}+\frac{v}{L C}=0
A comparison of this equation with Eqs. (7.14) and (7.15) indicates that for the parallel RLC circuit the damping term is 1/2 RC and the undamped natural frequency is 1 / \sqrt{L C}. If the circuit parameters are R = 2 Ω, C = 1/5 F, and L=5 H, the equation becomes
\frac{d^{2} x(t)}{d t^{2}}+2 \zeta \omega_{0} \frac{d x(t)}{d t}+\omega_{0}^{2} x(t)=0 7.14
s^{2}+2 \zeta \omega_{0} s+\omega_{0}^{2}=0 7.15
\frac{d^{2} v}{d t^{2}}+2.5 \frac{d v}{d t}+v=0
Let us assume that the initial conditions on the storage elements are i_{L}(0)=-1 \mathrm{~A} and v_{C}(0)=4 \mathrm{~V}. Let us find the node voltage v(t) and the inductor current.