Key Points

An alternating voltage is represented by $v = v_m \sin(\omega t)$ and the current by $i = i_m \sin(\omega t + \phi)$. Here, $v_m$ and $i_m$ are the peak values, $\omega$ is the angular frequency, and $\phi$ is the phase difference.

The root mean square (rms) value is the effective value of AC. The rms voltage is $V_{rms} = \frac{v_m}{\sqrt{2}} \approx 0.707 v_m$, and the rms current is $I_{rms} = \frac{i_m}{\sqrt{2}} \approx 0.707 i_m$.

In a purely resistive AC circuit, the voltage and current are in the same phase ($\phi = 0$). The relationship between peak values is given by Ohm's law, $v_m = i_m R$.

In a purely inductive circuit, the current lags behind the voltage by a phase angle of $\frac{\pi}{2}$. The opposition offered is the inductive reactance, $X_L = \omega L$, where $\omega$ is the angular frequency.

In a purely capacitive circuit, the current leads the voltage by a phase angle of $\frac{\pi}{2}$. The opposition offered is the capacitive reactance, $X_C = \frac{1}{\omega C}$.

Phasors are rotating vectors used to represent AC voltage and current. The length of the phasor represents the amplitude of the quantity, and the angle between phasors represents the phase difference.

For a series LCR circuit, the total effective opposition to the current is called impedance, Z. It is calculated as $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

The phase angle $\phi$ between the source voltage and the circuit current in a series LCR circuit is given by $\tan \phi = \frac{X_L - X_C}{R}$.

Resonance occurs when inductive reactance equals capacitive reactance ($X_L = X_C$). At this point, impedance is minimum ($Z=R$), the current is maximum, and the resonant angular frequency is $\omega_0 = \frac{1}{\sqrt{LC}}$.

The average power dissipated in an AC circuit is given by $P = V_{rms} I_{rms} \cos \phi$. The term $\cos \phi$ is known as the power factor.

The power factor $\cos \phi$ ranges from 0 to 1. For a purely inductive or capacitive circuit, $\phi = \pm \frac{\pi}{2}$, so $\cos \phi = 0$, and the power dissipated is zero. The current in such a circuit is called wattless current.

A transformer is a device used to change AC voltage levels based on mutual induction. For an ideal transformer, the ratio of secondary to primary voltages equals the ratio of turns in their coils: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$.

In an ideal transformer with 100% efficiency, the input power equals the output power ($P_p = P_s$). This leads to the current relationship $\frac{I_s}{I_p} = \frac{N_p}{N_s}$.

A step-up transformer increases voltage ($N_s > N_p$) but decreases current. A step-down transformer decreases voltage ($N_s < N_p$) but increases current.