This image describes an RLC circuit with a sinusoidal voltage source $v_s(t)$, and the task is to find the steady-state current $i(t)$ for three different cases of $v_s(t)$.

Step-by-Step Solution

The circuit consists of a resistor $R$, an inductor $L$, and a capacitor $C$ in series. The input voltage is given as $v_s(t)$, and we are tasked with finding the steady-state current $i(t)$ for different forms of $v_s(t)$.

General Approach for RLC Circuit:

1. Impedance Calculation: The total impedance $Z$ of the series RLC circuit is given by: $Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$ where:

   • $R$ is the resistance,
   • $L$ is the inductance,
   • $C$ is the capacitance,
   • $\omega$ is the angular frequency of the source.

   The magnitude of the impedance is: $|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$

2. Steady-State Current: The steady-state current $i(t)$ is related to the voltage by Ohm’s law in phasor form: $I = \frac{V_s}{Z}$ Once the phasor current is found, it can be converted back to the time domain.

Case 1: $v_s(t) = 1 \text{ V (DC)}$

For a DC voltage, $\omega = 0$. The impedance becomes: $Z = R + j\left(0 - \frac{1}{0}\right) = \infty$ This means that the capacitor acts as an open circuit in the steady state, and the current is zero: $i(t) = 0$

Case 2: $v_s(t) = \cos(\omega t)$

For an AC voltage with $v_s(t) = \cos(\omega t)$, the phasor form of the voltage is: $V_s = 1 \angle 0^\circ$

The impedance is: $Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$

Thus, the phasor current is: $I = \frac{1 \angle 0^\circ}{Z}$ The time-domain current is: $i(t) = |I| \cos(\omega t + \angle I)$

Case 3: $v_s(t) = \cos(\omega t) + 2 \cos(2\omega t + \frac{\pi}{3})$

This is a sum of two sinusoidal sources, so we solve for the response to each source separately and then sum the results.

For $v_{s1}(t) = \cos(\omega t)$, follow the same process as in Case 2.

For $v_{s2}(t) = 2 \cos(2\omega t + \frac{\pi}{3})$, the phasor voltage is: $V_{s2} = 2 \angle \frac{\pi}{3}$

The impedance for this frequency is: $Z_2 = R + j\left(2\omega L - \frac{1}{2\omega C}\right)$

The phasor current for this term is: $I_2 = \frac{2 \angle \frac{\pi}{3}}{Z_2}$

The total current is the sum of the currents from both components: $i(t) = i_1(t) + i_2(t)$

Let me know if you want me to calculate further with numerical values or need additional details.

Follow-up Questions:

1. What is the phasor representation for a sinusoidal voltage source?
2. How does the capacitor behave in a DC steady-state condition?
3. What is the significance of resonance in an RLC circuit?
4. How does the inductor’s impedance change with frequency?
5. Can you describe how to sum multiple sinusoidal signals in an RLC circuit?

Tip:

When solving AC circuits with sinusoidal sources, always convert time-domain voltages to phasor form first before applying Ohm’s law!