Electromagnetic Induction and Alternating Currents - Alternating Current (LCR Circuits, Resonance)

Alternating Current (AC) is the current whose magnitude changes continuously with time and direction reverses periodically, represented as $I = I_0 \sin \omega t$.

RMS (Root Mean Square) value, also known as the effective or virtual value, is given by $I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$.

Phasors are rotating vectors used to represent sinusoidally varying voltages and currents in AC circuits to show phase relationships.

Inductive Reactance ($X_L$) is the opposition offered by an inductor to AC, where $X_L = \omega L$. It increases with frequency.

Capacitive Reactance ($X_C$) is the opposition offered by a capacitor to AC, where $X_C = \frac{1}{\omega C}$. It decreases as frequency increases.

In a series $LCR$ circuit, the total opposition to current is called Impedance ($Z$), which is the phasor sum of $R$, $X_L$, and $X_C$.

Resonance occurs in a series $LCR$ circuit when $X_L = X_C$. At this condition, the impedance is minimum ($Z = R$) and the current is maximum.

The Quality Factor ($Q$-factor) measures the sharpness of resonance. A higher $Q$ indicates a narrower and sharper resonance peak.

Power Factor ($\cos \phi$) is the ratio of real power to apparent power. In a purely resistive circuit $\cos \phi = 1$, while in purely inductive or capacitive circuits, $\cos \phi = 0$.

Wattless Current is the component of AC that consumes no power in the circuit, occurring when the phase difference between voltage and current is $\frac{\pi}{2}$.