Alternating Current Versus Direct Current

Alternating Current

Most of the examples dealt with so far, and particularly those utilizing batteries, have constant voltage sources. Once the current is established, it is thus also a constant. Direct current (DC) is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit. Most well-known applications, however, use a time-varying voltage source. Alternating current (AC) is the flow of electric charge that periodically reverses direction. If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating current circuit. Examples include the commercial and residential power that serves so many of our needs. This figure shows graphs of voltage and current versus time for typical DC and AC power. The AC voltages and frequencies commonly used in homes and businesses vary around the world.

This figure shows a schematic of a simple circuit with an AC voltage source. The voltage between the terminals fluctuates as shown, with the AC voltage given by

\(V={V}_{0}\phantom{\rule{0.30em}{0ex}}\text{sin}\phantom{\rule{0.30em}{0ex}}\text{2}\pi \text{ft,}\)

where \(V\) is the voltage at time \(t\), \({V}_{0}\) is the peak voltage, and \(f\) is the frequency in hertz. For this simple resistance circuit, \(I=\text{V/R}\), and so the AC current is

\(I={I}_{0}\phantom{\rule{0.30em}{0ex}}\text{sin 2}\pi \text{ft,}\)

where \(I\) is the current at time \(t\), and \({I}_{0}={V}_{0}\text{/R}\) is the peak current. For this example, the voltage and current are said to be in phase, as seen in this figure (b).

Current in the resistor alternates back and forth just like the driving voltage, since \(I=\text{V/R}\). If the resistor is a fluorescent light bulb, for example, it brightens and dims 120 times per second as the current repeatedly goes through zero. A 120-Hz flicker is too rapid for your eyes to detect, but if you wave your hand back and forth between your face and a fluorescent light, you will see a stroboscopic effect evidencing AC. The fact that the light output fluctuates means that the power is fluctuating. The power supplied is \(P=\text{IV}\). Using the expressions for \(I\) and \(V\) above, we see that the time dependence of power is \(P={I}_{0}{V}_{0}\phantom{\rule{0.30em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.30em}{0ex}}\text{2}\pi \text{ft}\), as shown in this figure.

We are most often concerned with average power rather than its fluctuations—that 60-W light bulb in your desk lamp has an average power consumption of 60 W, for example. As illustrated in this figure, the average power \({P}_{\text{ave}}\) is

\({P}_{\text{ave}}=\cfrac{1}{2}{I}_{0}{V}_{0}.\)

This is evident from the graph, since the areas above and below the \((1/2){I}_{0}{V}_{0}\) line are equal, but it can also be proven using trigonometric identities. Similarly, we define an average or rms current \({I}_{\text{rms}}\) and average or rms voltage \({V}_{\text{rms}}\) to be, respectively,

\({I}_{\text{rms}}=\cfrac{{I}_{0}}{\sqrt{2}}\)

and

\({V}_{\text{rms}}=\cfrac{{V}_{0}}{\sqrt{2}}.\)

where rms stands for root mean square, a particular kind of average. In general, to obtain a root mean square, the particular quantity is squared, its mean (or average) is found, and the square root is taken. This is useful for AC, since the average value is zero. Now,

\({P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}},\)

which gives

\({P}_{\text{ave}}=\cfrac{{I}_{0}}{\sqrt{2}}\cdot \cfrac{{V}_{0}}{\sqrt{2}}=\cfrac{1}{2}{I}_{0}{V}_{0},\)

as stated above. It is standard practice to quote \({I}_{\text{rms}}\), \({V}_{\text{rms}}\), and \({P}_{\text{ave}}\) rather than the peak values. For example, most household electricity is 120 V AC, which means that \({V}_{\text{rms}}\) is 120 V. The common 10-A circuit breaker will interrupt a sustained \({I}_{\text{rms}}\) greater than 10 A. Your 1.0-kW microwave oven consumes \({P}_{\text{ave}}=\text{1.0 kW}\), and so on. You can think of these rms and average values as the equivalent DC values for a simple resistive circuit.

To summarize, when dealing with AC, Ohm’s law and the equations for power are completely analogous to those for DC, but rms and average values are used for AC. Thus, for AC, Ohm’s law is written

\({I}_{\text{rms}}=\cfrac{{V}_{\text{rms}}}{R}.\)

The various expressions for AC power \({P}_{\text{ave}}\) are

\({P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}},\)

\({P}_{\text{ave}}=\cfrac{{V}_{\text{rms}}^{2}}{R},\)

and

\({P}_{\text{ave}}={I}_{\text{rms}}^{2}R.\)