An alternating voltage
applied to a resistor R drives a current
in the resistor,
The current is in phase with the applied voltage.
For an alternating current
passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is
. To express it in the same form as the dc power (P = I 2R), a special value of current is used. It is called root mean square (rms) current and is denoted by I:
Similarly, the rms voltage is defined by
We have P = IV = I2R
An ac voltage applied to a pure inductor L, drives a current in the inducto , where
is called inductive reactance. The current in the inductor lags the voltage by The average power supplied to an inductor over one complete cycle is zero.
An ac voltage
applied to a capacitor drives a current in the capacitor:
is called capacitive reactance.
The current through the capacitor is ahead of the applied voltage. As in the case of inductor, the average power supplied to a capacity or over one complete cycle is zero.
For a series RLC circuit driven by voltage
, the current isgiven by
is called the impedance of the circuit.
The average power loss over a complete cycle is given by
The term is called the power factor.
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,
The quality factor Q defined by
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:
and therefore, the frequency ω of free oscillation is
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 turns and a secondary coil of turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by and the currents are related by
If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up 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.
A sinusoidal voltage of peak value 283 V and frequency 50 Hz is applied to a series LCR circuit in which R = 3 Ω, L = 25.48 mH, and C = 796 μF. Find (a) the impedance of the circuit; (b) the phase difference between the voltage across the source and the current; (c) the power dissipated in the circuit; and (d) the power factor.
(a) To find the impedance of the circuit, we first calculate XL and XC.
Since is negative, the current in the circuit lags the voltage across the source.
(c) The power dissipated in the circuit is
(d) Power factor =