electrical engineering

3. AC Circuit Analysis

3.1. Sinusoidal Waveforms
- Represented by ext{v(t)} = ext{V}m in(\omega t + \phi) where ext{V}m is amplitude, \omega is angular frequency (\text{2\pi f}), and \phi is phase angle.
- RMS Value: For a sinusoidal voltage/current, \text{X}{rms} = \frac{\text{X}m}{\sqrt{2}}. Used for power calculations.
3.2. Phasors
- Complex number representation of sinusoidal signals, simplifying AC circuit analysis. \text{V} = \text{V}m \angle \phi or \text{V} = \text{V}m e^{j\phi}.
3.3. Impedance (Z)
- Generalization of resistance for AC circuits. Measured in Ohms (\Omega). Complex quantity. \text{Z} = \text{R} + \text{jX}.
- Resistor: \text{Z}R = \text{R}. - Inductor: ZL = j\omega L (Inductive Reactance \text{X}L = \omega L). - Capacitor: ZC = \frac{1}{j\omega C} = -j \frac{1}{\omega C} (Capacitive Reactance \text{X}C = \frac{1}{\omega C}). 3.4. Admittance (Y) - The reciprocal of impedance. \text{Y} = \frac{1}{\text{Z}} = \text{G} + \text{jB} (Conductance + Susceptance). 3.5. Power in AC Circuits - Apparent Power (S): \text{S} = \text{VI}^* (Complex Power). Measured in Volt-Amperes (VA). - Real Power (P): Average power consumed by resistive elements. \text{P} = \text{V}{rms} \text{I}{rms} \cos(\thetaV - \thetaI). Measured in Watts (W). - Reactive Power (Q): Power exchanged between source and reactive elements. $$\text{Q} = \text{V}{rms} \text