7.4 AC Voltage Applied to an Inductor
An ac source connected to an inductor.
Figure 7.5 shows an ac source connected to an inductor. Usually, inductors have appreciable resistance in their windings, but we shall assume that this inductor has negligible resistance. Thus, the circuit is a purely inductive ac circuit. Let the voltage across the source be
where the second term is the self-induced Faraday emf in the inductor; and L is the self-inductance of the inductor. The negative sign follows from Lenz’s law (Chapter 6). Combining Eqs. (7.1) and (7.10), we have
Equation (7.11) implies that the equation for i(t), the current as a function of time, must be such that its slope di/dt is a sinusoidally varying quantity, with the same phase as the source voltage and an amplitude given by
and get,
The integration constant has the dimension of current and is time- independent. Since the source has an emf which oscillates symmetrically about zero, the current it sustains also oscillates symmetrically about zero, so that no constant or time-independent component of the current exists. Therefore, the integration constant is zero.
Using
we have
where
The amplitude of the current is, then
The dimension of inductive reactance is the same as that of resistance and its SI unit is ohm (
(a) A Phasor diagram for the circuit in Fig. 7.5. (b) Graph of v and i versus ω t.
A comparison of Eqs. (7.1) and (7.12) for the source voltage and the current in an inductor shows that the current lags the voltage by
We see that the current reaches its maximum value later than the voltage by one-fourth of a period
The instantaneous power supplied to the inductor is
So, the average power over a complete cycle is
since the average of
Thus, the average power supplied to an inductor over one complete cycle is zero.