Summary

An alternating voltage \(v=v_{\mathrm{m}} \sin \omega t\) applied to a resistor R drives a current \(i=i_{m} \sin \omega t\) in the resistor, \(i_{\mathrm{m}}=\frac{v_{m}}{R}\). The current is in phase with the applied voltage.
For an alternating current \(i=i_{m} \sin \omega t\) passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is \((1 / 2) i_{m}^{2} \bar{R}\). To express it in the same form as the dc power \(\left(P=I^{2} R\right)\), a special value of current is used. It is called root mean square (rms) current and is denoted by I: \[ I=\frac{i_{m}}{\sqrt{2}}=0.707 i_{m} \] Similarly, the rms voltage is defined by \[ V=\frac{v_{m}}{\sqrt{2}}=0.707 v_{m} \] We have \(P=I V=I^{2} R\)
An ac voltage \(v=v_{m} \sin \omega t\) applied to a pure inductor L, drives a current in the inductor \(i=i_{m} \sin (\omega t-\pi / 2)\), where \(i_{m}=v_{m} / X_{L} \cdot X_{L}=\omega L\) is called inductive reactance. The current in the inductor lags the voltage by π/2. The average power supplied to an inductor over one complete cycle is zero.
An ac voltage \(v=v_{m} \sin \omega t\) applied to a capacitor drives a current in the capacitor: \(i=i_{\mathrm{m}} \sin (\omega t+\pi / 2)\) Here, \[ i_{m}=\frac{v_{m}}{X_{C}}, X_{C}=\frac{1}{\omega C} \] is called capacitive reactance. The current through the capacitor is π/2 ahead of the applied voltage. As in the case of inductor, the average power supplied to a capacitor over one complete cycle is zero.
For a series RLC circuit driven by voltage \(v=v_{m} \sin \omega t\), the current is given by \(i=i_{m} \sin (\omega t+\phi)\) \[ where \(i_{m}=\frac{v_{m}}{\sqrt{R^{2}+\left(X_{C}-X_{L}\right)^{2}}}\) \] \[ \text { and } \phi=\tan ^{-1} \frac{X_{C}-X_{L}}{R} \] \(Z=\sqrt{R^{2}+\left(X_{C}-X_{L}\right)^{2}}\) is called the impedance of the circuit. The average power loss over a complete cycle is given by \(P=V I \cos \phi\). The term cos φ is called the power factor.
In a purely inductive or capacitive circuit, cos φ = 0 and no power is dissipated even though a current is flowing in the circuit. In such cases, current is referred to as a wattless current.
The phase relationship between current and voltage in an ac circuit can be shown conveniently by representing voltage and current by rotating vectors called phasors. A phasor is a vector which rotates about the origin with angular speed ω . The magnitude of a phasor represents the amplitude or peak value of the quantity (voltage or current) represented by the phasor.

The analysis of an ac circuit is facilitated by the use of a phasor diagram.
An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e., the amplitude of the current is maximum at the resonant frequency, \(\omega_{0}=\frac{1}{\sqrt{L C}}\). The quality factor Q defined by \(Q=\frac{\omega_{0} L}{R}=\frac{1}{\omega_{0} C R}\) is an indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.
A circuit containing an inductor L and a capacitor C (initially charged) with no ac source and no resistors exhibits free oscillations. The charge q of the capacitor satisfies the equation of simple harmonic motion: \[ \frac{\mathrm{d}^{2} q}{d t^{2}}+\frac{1}{L C} q=0 \] and therefore, the frequency ω of free oscillation is \(\omega_{0}=\frac{1}{\sqrt{L C}}\). The energy in the system oscillates between the capacitor and the inductor but their sum or the total energy is constant in time.
A transformer consists of an iron core on which are bound a primary coil of \(N_{p}\) turns and a secondary coil of \(N_{s}\) turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by \[ V_{s}=\left(\frac{N_{s}}{N_{p}}\right) V_{p} \] and the currents are related by \[ I_{s}=\left(\frac{N_{p}}{N_{s}}\right) I_{p} \] If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up (\(V_{s}\) > \(V_{p}\) ). This type of arrangement is called a step- up transformer. If the secondary coil has turns less than the primary, we have a step-down transformer.