gen-define-impedance-and-reactance-and-give-one-short-worked-exa-3753

Define impedance and reactance and give one short worked example.

# Impedance and Reactance ## Definitions **Impedance (Z)**: The total opposition to current flow in an AC circuit, measured in ohms (Ω). It combines resistance and reactance and is a complex quantity: $$Z = R + jX$$ **Reactance (X)**: The opposition to current flow caused by capacitors and inductors in an AC circuit, measured in ohms (Ω). Unlike resistance, it doesn't dissipate energy. - **Inductive reactance**: $X_L = 2\pi fL = \omega L$ - **Capacitive reactance**: $X_C = \frac{1}{2\pi fC} = \frac{1}{\omega C}$ ## Worked Example **Problem**: Find the impedance of a series circuit containing a 50 Ω resistor and a 0.1 H inductor at f = 60 Hz. **Solution**: 1. Calculate inductive reactance: $$X_L = 2\pi fL = 2\pi(60)(0.1) = 37.7 \text{ Ω}$$ 2. Find impedance magnitude: $$|Z| = \sqrt{R^2 + X_L^2} = \sqrt{50^2 + 37.7^2} = \sqrt{2500 + 1421} = 62.6 \text{ Ω}$$ 3. Find phase angle: $$\theta = \tan^{-1}\left(\frac{X_L}{R}\right) = \tan^{-1}\left(\frac{37.7}{50}\right) = 37.0°$$ **Answer**: $Z = 62.6\angle37.0°$ Ω or $Z = 50 + j37.7$ Ω