Simulation Lab (EC-406) - Important Questions

Derive the expression for the capacitor voltage $v_C(t)$ and the circuit current $i(t)$ for a series RC circuit when a step voltage $V_s$ is applied at $t=0$ with the capacitor initially uncharged. State the time constant $\tau$ and show that $$v_C(t)=V_s\left(1- e^{-\frac{t}{RC}}\right)\quad\text{and}\quad i(t)=\frac{V_s}{R}e^{-\frac{t}{RC}}.$$

A capacitor of capacitance $C$ is charged to voltage $V_0$ and at $t=0$ it is connected across a resistor $R$. Derive the expression for the capacitor voltage $v_C(t)$ for $t>0$ and determine the expression for energy dissipated in the resistor.

In the series RC circuit shown, the switch is closed at $t=0$. The initial capacitor voltage is $V_0$. Determine $v_C(t)$ and $i(t)$ for $t>0$ and compute the time $t$ at which $v_C(t)=0.63\,V_s$. (Assume a step source $V_s$ applied at $t=0$.)

For a series RL circuit with resistance $R$ and inductance $L$, a step voltage $V_s$ is applied at $t=0$ with zero initial current. Derive the expression for the inductor current $i_L(t)$ and the voltage across the inductor $v_L(t)$. Define the time constant $\tau$ and show that $$i_L(t)=\frac{V_s}{R}\left(1-e^{-\frac{R}{L}t}\right)\quad\text{and}\quad v_L(t)=V_s e^{-\frac{R}{L}t}.$$

An inductor carrying current $I_0$ is suddenly disconnected from its source at $t=0$ and connected across a resistor $R$. Find the expression for $i_L(t)$ for $t>0$ and the energy initially stored in the inductor. Also find the time required for the energy to fall to one‑fourth of its initial value.

Solve for the current $i(t)$ in the series RL circuit when a voltage $V_s$ is applied at $t=0$ and the inductor has an initial current $I_0$. Provide the complete solution showing both steady‑state and transient parts.

For a series RLC circuit, derive the homogeneous solution for the circuit current $i(t)$ and obtain the characteristic equation. Define the damping factor $\alpha$ and resonant frequency $\omega_0$, and derive expressions for the three cases: overdamped, critically damped and underdamped. Provide the general form of $i(t)$ for the underdamped case.

Using Laplace transform techniques, determine the complete response of the series RLC circuit to a unit step voltage applied at $t=0$ (zero initial conditions). Show the algebraic solution in $s$‑domain and perform inverse transform to obtain $i(t)$. Express your result in terms of $\alpha$ and $\omega_0$.

Analyze the natural response of a parallel RLC circuit with initial capacitor voltage $V_0$ and zero initial inductor current. Derive the voltage $v(t)$ across the parallel combination and discuss conditions for oscillation. Also compute the total energy stored initially and show how it evolves with time.