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 1 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 1. (a) DC voltage and current are constant in time, once the current is established. (b) A graph of voltage and current versus time for 60-Hz AC power. The voltage and current are sinusoidal and are in phase for a simple resistance circuit. The frequencies and peak voltages of AC sources differ greatly.
Figure 2. The potential difference V between the terminals of an AC voltage source fluctuates as shown. The mathematical expression for V
is given by
Figure 2 shows a schematic of a simple circuit with an AC voltage source. The voltage between the terminals fluctuates as shown, with the AC voltage
is the voltage at time t, V0
is the peak voltage, and f
is the frequency in hertz. For this simple resistance circuit, I=V/R
, and so the AC current
is the current at time t
, and I0 = V0/R
is the peak current. For this example, the voltage and current are said to be in phase, as seen in Figure 1(b).
Current in the resistor alternates back and forth just like the driving voltage, since I = 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 = IV
. Using the expressions for I
above, we see that the time dependence of power is
, as shown in Figure 3.
Making Connections: Take-Home Experiment—AC/DC Lights
Wave your hand back and forth between your face and a fluorescent light bulb. Do you observe the same thing with the headlights on your car? Explain what you observe. Warning: Do not look directly at very bright light.
Figure 3. AC power as a function of time. Since the voltage and current are in phase here, their product is non-negative and fluctuates between zero and I0V0
. Average power is (1/2)I0V0
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 3, the average power Pave
This is evident from the graph, since the areas above and below the (1/2)I0V0
line are equal, but it can also be proven using trigonometric identities. Similarly, we define an average or rms current Irms
and average or rms voltage Vrms
to be, respectively,
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,
Pave = IrmsVrms,
as stated above. It is standard practice to quote Irms
, and Pave
rather than the peak values. For example, most household electricity is 120 V AC, which means that Vrms
is 120 V. The common 10-A circuit breaker will interrupt a sustained Irms
greater than 10 A. Your 1.0-kW microwave oven consumes Pave
= 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
The various expressions for AC power Pave
Pave = Irms Vrms,
Example 1. Peak Voltage and Power for AC
(a) What is the value of the peak voltage for 120-V AC power? (b) What is the peak power consumption rate of a 60.0-W AC light bulb?
We are told that Vrms
is 120 V and Pave
is 60.0 W. We can use
to find the peak voltage, and we can manipulate the definition of power to find the peak power from the given average power.
Solution for (a)
Solving the equation
for the peak voltage V0
and substituting the known value for Vrms
Discussion for (a)
This means that the AC voltage swings from 170 V to –170 V and back 60 times every second. An equivalent DC voltage is a constant 120 V.
Solution for (b)
Peak power is peak current times peak voltage. Thus,
We know the average power is 60.0 W, and so
P0 = 2(60.0 W) = 120 W.
So the power swings from zero to 120 W one hundred twenty times per second (twice each cycle), and the power averages 60 W.