Summary

• Root Mean Square (RMS) Values: For a sinusoidal AC, $V_{\text{rms}} = \frac{V_0}{\sqrt{2}}$ and $I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$, where $V_0$ and $I_0$ are peak values. RMS value gives the equivalent DC heating effect.

• AC through a Resistor: Voltage and current are in phase. Instantaneous power $p = i^2 R$ is always positive; average power $P_{\text{avg}} = I_{\text{rms}}^2 R$.

• AC through an Inductor: Current lags voltage by $90^\circ$ ($\frac{\pi}{2}$). Inductive reactance $X_L = \omega L$ (in ohms). Average power over a cycle is zero (no dissipation).

• AC through a Capacitor: Current leads voltage by $90^\circ$ ($\frac{\pi}{2}$). Capacitive reactance $X_C = \frac{1}{\omega C}$ (in ohms). Average power over a cycle is zero.

• Series LCR Circuit: Impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$. Phase angle $\phi$ given by $\tan \phi = \frac{X_L - X_C}{R}$.

• Resonance: Occurs when $X_L = X_C$, i.e., $\omega_0 = \frac{1}{\sqrt{LC}}$. At resonance, $Z = R$ (minimum), current is maximum, and circuit is purely resistive ($\phi = 0$).

• Quality Factor (Q-factor): $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R}$. It measures sharpness of resonance — higher $Q$ means narrower bandwidth.

• Power in AC Circuits: Average power $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi$, where $\cos \phi$ is the power factor. For purely resistive load, $\cos \phi = 1$; for purely reactive, $\cos \phi = 0$.

• Transformer: Works on mutual induction. For an ideal transformer, $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$. Step-up increases voltage, step-down decreases it. Power is conserved (no losses assumed).

Physical quantities, symbols, dimensions and units used in this chapter.