Let's solve **both problems step by step** and **develop expressions for the damping factor (α)** and the **undamped natural frequency (ω₀)** for both the **series** and **parallel RLC circuits**. --- # 1. **Series RLC Circuit** **Circuit:** A resistor (R), inductor (L), and capacitor (C) are connected in series. ## **Step 1: Write the Differential Equation** By KVL (Kirchhoff's Voltage Law): \[ V(t) = V_R + V_L + V_C = Ri(t) + L \frac{di(t)}{dt} + \frac{1}{C}\int i(t)dt \] Taking the derivative with respect to time: \[ L \frac{d^2i(t)}{dt^2} + R \frac{di(t)}{dt} + \frac{1}{C}i(t) = \frac{dV(t)}{dt} \] For the homogeneous case (\( V(t) = \)): \[ L \frac{d^2i(t)}{dt^2} + R \frac{di(t)}{dt} + \frac{1}{C}i(t) = \] Divide by \( L \): \[ \frac{d^2i(t)}{dt^2} + \frac{R}{L} \frac{di(t)}{dt} + \frac{1}{LC}i(t) = \] --- ## **Step 2: Standard Second-Order Form** This matches the standard form: \[ \frac{d^2x}{dt^2} + 2\alpha \frac{dx}{dt} + \omega_^2 x = \] Comparing coefficients: - \( 2\alpha = \frac{R}{L} \implies \boxed{\alpha = \frac{R}{2L}} \) - \( \omega_^2 = \frac{1}{LC} \implies \boxed{\omega_ = \frac{1}{\sqrt{LC}}} \) --- ## **Summary: Series RLC** \[ \boxed{ \begin{align*} \alpha &= \frac{R}{2L} \\ \omega_ &= \frac{1}{\sqrt{LC}} \end{align*} } \] --- # 2. **Parallel RLC Circuit** **Circuit:** A resistor (R), inductor (L), and capacitor (C) are connected in parallel. ## **Step 1: Write the Differential Equation** By KCL (Kirchhoff's Current Law): \[ I(t) = I_R + I_L + I_C = \frac{v(t)}{R} + \frac{1}{L}\int v(t)dt + C\frac{dv(t)}{dt} \] Taking the derivative with respect to time: \[ \frac{dI(t)}{dt} = \frac{1}{R}\frac{dv(t)}{dt} + \frac{v(t)}{L} + C\frac{d^2v(t)}{dt^2} \] For the homogeneous case (\( I(t) = \)): \[ C\frac{d^2v(t)}{dt^2} + \frac{1}{R}\frac{dv(t)}{dt} + \frac{1}{L}v(t) = \] Divide by \( C \): \[ \frac{d^2v(t)}{dt^2} + \frac{1}{RC}\frac{dv(t)}{dt} + \frac{1}{LC}v(t) = \] --- ## **Step 2: Standard Second-Order Form** This matches the standard form: \[ \frac{d^2x}{dt^2} + 2\alpha \frac{dx}{dt} + \omega_^2 x = \] Comparing coefficients: - \( 2\alpha = \frac{1}{RC} \implies \boxed{\alpha = \frac{1}{2RC}} \) - \( \omega_^2 = \frac{1}{LC} \implies \boxed{\omega_ = \frac{1}{\sqrt{LC}}} \) --- ## **Summary: Parallel RLC** \[ \boxed{ \begin{align*} \alpha &= \frac{1}{2RC} \\ \omega_ &= \frac{1}{\sqrt{LC}} \end{align*} } \] --- # **Final Answers Table** | Circuit Type | Damping Factor (\(\alpha\)) | Natural Frequency (\(\omega_\)) | |---------------------|----------------------------|-----------------------------------| | Series RLC | \(\frac{R}{2L}\) | \(\frac{1}{\sqrt{LC}}\) | | Parallel RLC | \(\frac{1}{2RC}\) | \(\frac{1}{\sqrt{LC}}\) |

# Expressions for Damping Factor (α) and Natural Frequency (ω₀) This document outlines the derivation of expressions for the damping factor (α) and the undamped natural frequency (ω₀) for both series and parallel RLC circuits. ## 1. Series RLC Circuit ### Step 1: Write the Differential Equation Using Kirchhoff's Voltage Law (KVL): \[ V(t) = Ri(t) + L \frac{di(t)}{dt} + \frac{1}{C}\int i(t) dt \] Taking the derivative with respect to time leads to: \[ L \frac{d^2i(t)}{dt^2} + R \frac{di(t)}{dt} + \frac{1}{C}i(t) = 0 \] Dividing by \( L \): \[ \frac{d^2i(t)}{dt^2} + \frac{R}{L} \frac{di(t)}{dt} + \frac{1}{LC}i(t) = 0 \] ### Step 2: Standard Second-Order Form This resembles the standard form: \[ \frac{d^2x}{dt^2} + 2\alpha \frac{dx}{dt} + \omega_0^2 x = 0 \] Comparing coefficients gives: - \( 2\alpha = \frac{R}{L} \) → \( \alpha = \frac{R}{2L} \) - \( \omega_0^2 = \frac{1}{LC} \) → \( \omega_0 = \frac{1}{\sqrt{LC}} \) ### Summary: Series RLC \[ \begin{align*} \alpha &= \frac{R}{2L} \\ \omega_0 &= \frac{1}{\sqrt{LC}} \end{align*} \] --- ## 2. Parallel RLC Circuit ### Step 1: Write the Differential Equation Using Kirchhoff's Current Law (KCL): \[ I(t) = \frac{v(t)}{R} + \frac{1}{L}\int v(t) dt + C\frac{dv(t)}{dt} \] Taking the derivative leads to: \[ \frac{dI(t)}{dt} = \frac{1}{R}\frac{dv(t)}{dt} + \frac{v(t)}{L} + C\frac{d^2v(t)}{dt^2} \] For the homogeneous case: \[ C\frac{d^2v(t)}{dt^2} + \frac{1}{R}\frac{dv(t)}{dt} + \frac{1}{L}v(t) = 0 \] Dividing by \( C \): \[ \frac{d^2v(t)}{dt^2} + \frac{1}{RC}\frac{dv(t)}{dt} + \frac{1}{LC}v(t) = 0 \] ### Step 2: Standard Second-Order Form This matches the standard form: \[ \frac{d^2x}{dt^2} + 2\alpha \frac{dx}{dt} + \omega_0^2 x = 0 \] Comparing coefficients gives: - \( 2\alpha = \frac{1}{RC} \) → \( \alpha = \frac{1}{2RC} \) - \( \omega_0^2 = \frac{1}{LC} \) → \( \omega_0 = \frac{1}{\sqrt{LC}} \) ### Summary: Parallel RLC \[ \begin{align*} \alpha &= \frac{1}{2RC} \\ \omega_0 &= \frac{1}{\sqrt{LC}} \end{align*} \] --- ## Final Answers Table | Circuit Type | Damping Factor (α) | Natural Frequency (ω₀) | |---------------------|--------------------------|---------------------------------| | **Series RLC** | \( \frac{R}{2L} \) | \( \frac{1}{\sqrt{LC}} \) | | **Parallel RLC** | \( \frac{1}{2RC} \) | \( \frac{1}{\sqrt{LC}} \) |