Topic: Network Theory
The network shown in below figure consist of two coupled coils and a capacitor. At $t=0$, the switch is closed connecting a voltage generator, $V_{in}=V~\sin(\frac{t}{\sqrt{MC}})$. What will be the value of $\frac{dv_{a}}{dt}(0^{+})$?
- A. $(\frac{V}{L})\sqrt{\frac{M}{C}}$
- B. $(\frac{V}{L})$
- C. 0
- D. $(\frac{V}{L})(\frac{M}{C})$
Answer: A. $(\frac{V}{L})\sqrt{\frac{M}{C}}$
Explanation: By taking the Laplace transform of the circuit and solving for $V_a(s)$, and then taking the inverse Laplace transform, we get $V_a(t) = V\sqrt{\frac{M}{L}} \sin\left(\frac{t}{\sqrt{LC}}\right)$. Differentiating with respect to $t$, $\frac{dV_a(t)}{dt} = \frac{V}{L}\sqrt{\frac{M}{C}} \cos\left(\frac{t}{\sqrt{LC}}\right)$. Substituting $t=0$, we obtain $\frac{dV_a(0^+)}{dt} = \frac{V}{L}\sqrt{\frac{M}{C}}$.