20.5

Alternating Current

Many electric circuits use batteries and involve direct current (dc). However, there are considerably more circuits that operate with alternating current (ac), in which the charge flow reverses direction periodically. The common generators that create ac electricity depend on magnetic forces for their operation and are discussed in Chapter 22. In an ac circuit, these generators serve the same purpose as a battery serves in a dc circuit; that is, they give energy to the moving electric charges. This section deals with ac circuits that contain only resistance.
Since the electrical outlets in a house provide alternating current, we all use ac circuits routinely. Figure 20-10 shows the ac circuit that is formed when a toaster is plugged into a wall socket. The heating element of a toaster is essentially a thin wire of resistance R and becomes red hot when electrical energy is dissipated in it. The circuit schematic in the picture introduces the symbol that is used to represent the generator. In this case, the generator is located at the electric power company.

Figure 20-10 This circuit consists of a toaster (resistance = R) and an ac generator at the electric power company.

Figure 20-11 shows a graph that records the voltage V produced between the terminals of the ac generator in Figure 20-10 at each moment of time t. This is the most common type of ac voltage. It fluctuates sinusoidally between positive and negative values as a function of time:

| (20.7)

where V0 is the maximum or peak value of the voltage, and f is the frequency (in cycles/s or Hz) at which the voltage oscillates. The angle in Equation 20.7 is expressed in radians, so a calculator must be set to its radian mode before the sine of this angle is evaluated. In the United States, the voltage at most home wall outlets has a peak value of approximately V0 = 170 volts and oscillates with a frequency of f = 60 Hz. Thus, the period of each cycle is , and the polarity of the generator terminals reverses twice during each cycle, as Figure 20-11 indicates.

Figure 20-11 In the most common case, the ac voltage is a sinusoidal function of time. The relative polarity of the generator terminals during the positive and negative parts of the sine wave is indicated.

The current in an ac circuit also oscillates. In circuits that contain only resistance, the current reverses direction each time the polarity of the generator terminals reverses. Thus, the current in a circuit like that in Figure 20-10 would have a frequency of 60 Hz and would change direction twice during each cycle. Substituting into shows that the current can be represented as

| (20.8)

The peak current is given by , so it can be determined if the peak voltage and the resistance are known.
The power delivered to an ac circuit by the generator is given by P = IV, just as it is in a dc circuit. However, since both I and V depend on time, the power fluctuates as time passes. Substituting Equations 20.7 and 20.8 for V and I into P = IV gives

| (20.9)

This expression is plotted in Figure 20-12.

Figure 20-12 In an ac circuit, the power P delivered to a resistor oscillates between zero and a peak value of I0V0, where I0 and V0 are the peak current and voltage, respectively.

Since the power fluctuates in an ac circuit, it is customary to consider the average power , which is one-half the peak power, as Figure 20-12 indicates:

| (20.10)

On the basis of this expression, a kind of average current and average voltage can be introduced that are very useful when discussing ac circuits. A rearrangement of Equation 20.10 reveals that

| (20.11)

Irms and Vrms are called the root mean square (rms) current and voltage, respectively, and may be calculated from the peak values by dividing them by *:

| (20.12)

| (20.13)

Since the maximum ac voltage at a home wall socket in the United States is typically V0 = 170 volts, the corresponding rms voltage is . Thus, when the instructions for an electrical appliance specify 120 V, it is an rms voltage that is meant. Similarly, when we specify an ac voltage or current in this text, it is an rms value, unless indicated otherwise. Likewise, when we specify ac power, it is an average power, unless stated otherwise.
Except for dealing with average quantities, the relation has the same form as Equation 20.6a (P = IV). Moreover, Ohm’s law can be written conveniently in terms of rms quantities:

| (20.14)

Substituting Equation 20.14 into shows that the average power can be expressed in the following ways:

| (20.15a)

| (20.15b)

| (20.15c)

These expressions are completely analogous to for dc circuits. Example 6 deals with the average power in one familiar ac circuit.

| Example 6 | | | Electric Power Sent to a Loudspeaker

A stereo receiver applies a peak ac voltage of 34 V to a speaker. The speaker behaves approximately* as if it had a resistance of , as the circuit in Figure 20-13 indicates. Determine (a) the rms voltage, (b) the rms current, and (c) the average power for this circuit.

Figure 20-13 A receiver applies an ac voltage (peak value = 34 V) to an speaker.

Reasoning The rms voltage is, by definition, equal to the peak voltage divided by . Furthermore, we are assuming that the circuit contains only a resistor. Therefore, we can use Ohm’s law (Equation 20.14) to calculate the rms current as the rms voltage divided by the resistance, and we can then determine the average power as the rms current times the rms voltage (Equation 20.15a).
Solution

a. | The peak value of the voltage is , so the corresponding rms value is

| (20.13)

Problem solving insight The rms values of the ac voltage and the ac current, Vrms and Irms, respectively, are not the same as the peak values V0 and I0. The rms values are always smaller than the peak values by a factor of .

b. | The rms current can be obtained from Ohm’s law:

| (20.14)

c. | The average power is

| (20.15a)

The electric power dissipated in a resistor causes it to heat up. Excessive power can lead to a potential fire hazard, as Conceptual Example 7 discusses.

| Conceptual Example 7 | | | Extension Cords and a Potential Fire Hazard

During the winter, many people use portable electric space heaters to keep warm. Sometimes, however, the heater must be located far from a 120-V wall receptacle, so an extension cord must be used (see Figure 20-14). However, manufacturers often warn against using an extension cord. If one must be used, they recommend a certain wire gauge, or smaller. Why the warning, and why are smaller-gauge wires better than larger-gauge wires?

Figure 20-14 When an extension cord is used with a space heater, the cord must have a resistance that is sufficiently small to prevent overheating of the cord.

Reasoning and Solution An electric space heater contains a heater element that is a piece of wire of resistance R, which is heated to a high temperature. The heating occurs because of the power dissipated in the heater element. A typical heater uses a relatively large current Irms of about 12 A. On its way to the heater, this current passes through the wires of the extension cord. Since these additional wires offer resistance to the current, the extension cord can also heat up, just as the heater element does. The extent of the heating depends on the resistance of the extension cord. If the heating becomes excessive, the insulation around the wire can melt, and a fire can result. Thus, manufacturers of space heaters warn about extension cords in order to prevent afire hazard. To keep the heating of the extension cord to a safe level, the resistance of the wire must be kept small. Recall from Section 20.3 that the resistance of a wire depends inversely on its cross-sectional area, so a large cross-sectional area gives rise to a small resistance. A wire with a large cross-sectional area is one that has a small gauge number, as discussed in Example 3. Smaller-gauge wires are better because they offer less resistance than larger-gauge wires.
Related Homework: Problem 34

| Check your Understanding 2
| An ac circuit contains only a generator and a resistor. Which one of the following changes leads to the greatest average power being delivered to the circuit? (a) Double the peak voltage of the generator and double the resistance. (b) Double the resistance. (c) Double the peak voltage of the generator and reduce the resistance by a factor of two. (d) Double the peak voltage of the generator. (The answer is given at the end of the book.)
Background: The average power delivered to an ac circuit is related to the rms voltage applied to the circuit and the rms current in the circuit. When only a resistor is present in an ac circuit, Ohm’s law applies and can be expressed in terms of rms quantities.
For similar questions (including calculational counterparts), consult Self-Assessment Test 20.1, which is described at the end of Section 20.7.

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