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. (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.

(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

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

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

Current in the resistor alternates back and forth just like the driving voltage, since [latex]I=\text{V/R}[/latex]. 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 [latex]P=\text{IV}[/latex]. Using the expressions for [latex]I[/latex] and [latex]V[/latex] above, we see that the time dependence of power is [latex]P={I}_{0}{V}_{0}\phantom{\rule{0.30em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.30em}{0ex}}\text{2}\pi \text{ft}[/latex], as shown in (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 (Figure), the average power [latex]{P}_{\text{ave}}[/latex] is

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

and

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,

which gives

as stated above. It is standard practice to quote [latex]{I}_{\text{rms}}[/latex], [latex]{V}_{\text{rms}}[/latex], and [latex]{P}_{\text{ave}}[/latex] rather than the peak values. For example, most household electricity is 120 V AC, which means that [latex]{V}_{\text{rms}}[/latex] is 120 V. The common 10-A circuit breaker will interrupt a sustained [latex]{I}_{\text{rms}}[/latex] greater than 10 A. Your 1.0-kW microwave oven consumes [latex]{P}_{\text{ave}}=\text{1.0 kW}[/latex], 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

The various expressions for AC power [latex]{P}_{\text{ave}}[/latex] are

and

Why Use AC for Power Distribution?

Most large power-distribution systems are AC. Moreover, the power is transmitted at much higher voltages than the 120-V AC (240 V in most parts of the world) we use in homes and on the job. Economies of scale make it cheaper to build a few very large electric power-generation plants than to build numerous small ones. This necessitates sending power long distances, and it is obviously important that energy losses en route be minimized. High voltages can be transmitted with much smaller power losses than low voltages, as we shall see. (See (Figure).) For safety reasons, the voltage at the user is reduced to familiar values. The crucial factor is that it is much easier to increase and decrease AC voltages than DC, so AC is used in most large power distribution systems.

It is widely recognized that high voltages pose greater hazards than low voltages. But, in fact, some high voltages, such as those associated with common static electricity, can be harmless. So it is not voltage alone that determines a hazard. It is not so widely recognized that AC shocks are often more harmful than similar DC shocks. Thomas Edison thought that AC shocks were more harmful and set up a DC power-distribution system in New York City in the late 1800s. There were bitter fights, in particular between Edison and George Westinghouse and Nikola Tesla, who were advocating the use of AC in early power-distribution systems. AC has prevailed largely due to transformers and lower power losses with high-voltage transmission.