AC Power 10.1

POWER IN THE TIME DOMAIN The instantaneous power entering a two-terminal circuit N (Fig. 10-1) is deﬁned by pðtÞ ¼ vðtÞiðtÞ

ð1Þ

where vðtÞ and iðtÞ are terminal voltage and current, respectively. If p is positive, energy is delivered to the circuit. If p is negative, energy is returned from the circuit to the source.

Fig. 10-1

In this chapter, we consider periodic currents and voltages, with emphasis on the sinusoidal steady state in linear RLC circuits. Since the storage capacity of an inductor or a capacitor is ﬁnite, these passive elements cannot continue receiving energy without returning it. Therefore, in the steady state and during each cycle, all of the energy received by an inductor or capacitor is returned. The energy received by a resistor is, however, dissipated in the form of thermal, mechanical, chemical, and/or electromagnetic energies. The net energy ﬂow to a passive circuit during one cycle is, therefore, positive or zero. EXAMPLE 10.1 Figure 10-2(a) shows the graph of a current in a resistor of 1 k. Find and plot the instantaneous power pðtÞ. From v ¼ Ri, we have pðtÞ ¼ vi ¼ Ri2 ¼ 1000  106 ¼ 103 W ¼ 1 mW. See Fig. 10-2(b). EXAMPLE 10.2 The current in Example 10.1 passes through a 0.5-mF capacitor. Find the power pðtÞ entering the capacitor and the energy wðtÞ stored in it. Assume vC ð0Þ ¼ 0. Plot pðtÞ and wðtÞ. Figure 10-2(a) indicates that the current in the capacitor is a periodic function with a period T ¼ 2 ms. During one period the current is given by  1 mA ð0 < t < 1 msÞ i¼ 1 mA ð1 < t < 2 msÞ