Summary

1. 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.
2. 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 $(1/2)i_m^2\overline{R}$. To express it in the same form as the dc power $(P=I^{2}R)$, a special value of current is used. It is called root mean square (rms) current and is denoted by I: $$I=\frac{i_m}{\sqrt{2}}=0.707i_m$$ Similarly, the rms voltage is defined by $$V=\frac{v_m}{\sqrt{2}}=0.707v_m$$ We have $P=IV=I^{2}R$
3. 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.
4. 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.
5. For a series RLC circuit driven by voltage $v=v_m\sin \omega t$, the current is given by $i=i_m\sin (\omega t+\varphi )$ $$where\text{(}{i}_m=\frac{v_m}{\sqrt{R^2+{(X_C-X_L)}^2}}\text{)}$$ $$\text{ and }\varphi ={\tan }^{-1}\frac{X_C-X_L}{R}$$ $Z=\sqrt{R^2+{(X_C-X_L)}^2}$ is called the impedance of the circuit. The average power loss over a complete cycle is given by $P=VI\cos \varphi$. The term cos φ is called the power factor.
6. 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.
7. 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.
8. 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, ${\omega }_0=\frac{1}{\sqrt{LC}}$. The quality factor Q defined by $Q=\frac{{\omega }_{0}L}{R}=\frac{1}{{\omega }_{0}CR}$ is an indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.
9. 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: $$\frac{{\mathrm{d}}^{2}q}{d{t}^2}+\frac{1}{LC}q=0$$ and therefore, the frequency ω of free oscillation is ${\omega }_0=\frac{1}{\sqrt{LC}}$. The energy in the system oscillates between the capacitor and the inductor but their sum or the total energy is constant in time.
10. 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.