gen-define-the-behavior-of-an-rlc-circuit-and-give-one-short-wor-153b

Define the behavior of an RLC circuit and give one short worked example.

# RLC Circuit Behavior An **RLC circuit** contains a resistor (R), inductor (L), and capacitor (C). Its behavior is governed by a second-order differential equation: $$L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}i = V(t)$$ ## Key Characteristics: The circuit's response depends on the **damping ratio**, determined by comparing R to the critical resistance: - **Underdamped** (low R): Oscillates before settling - **Critically damped** (optimal R): Fastest settling without oscillation - **Overdamped** (high R): Slow return to equilibrium without oscillation --- ## Worked Example: Series RLC Circuit **Given:** R = 100 Ω, L = 0.1 H, C = 10 μF, with initial capacitor voltage V₀ = 10 V **Find:** The damping type and natural frequency **Solution:** 1. **Natural frequency:** $$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{0.1 \times 10 \times 10^{-6}}} = \frac{1}{\sqrt{10^{-6}}} = 1000 \text{ rad/s}$$ 2. **Damping coefficient:** $$\alpha = \frac{R}{2L} = \frac{100}{2(0.1)} = 500 \text{ s}^{-1}$$ 3. **Since α < ω₀**, the circuit is **underdamped** and will oscillate at: $$\omega_d = \sqrt{\omega_0^2 - \alpha^2} = \sqrt{1000^2 - 500^2} = 866 \text{ rad/s}$$ The voltage will oscillate at **866 rad/s** while decaying exponentially.