23.2

Inductors and Inductive Reactance

As Section 22.8 discusses, an inductor is usually a coil of wire, and the basis of its operation is Faraday’s law of electromagnetic induction. According to Faraday’s law, an inductor develops a voltage that opposes a change in the current. This voltage V is given by (see Equation 22.9*), where is the rate at which the current changes and L is the inductance of the inductor. In an ac circuit the current is always changing, and Faraday’s law can be used to show that the rms voltage across an inductor is

| (23.3)

Equation 23.3 is analogous to , with the term XL appearing in place of the resistance R and being called the inductive reactance. The inductive reactance is measured in ohms and determines how much rms current exists in an inductor for a given rms voltage across the inductor. It is found experimentally that the inductive reactance XL is directly proportional to the frequency f and the inductance L, as indicated by the following equation:

| (23.4)

This relation indicates that the larger the inductance, the larger is the inductive reactance. Note that the inductive reactance is directly proportional to the frequency , whereas the capacitive reactance is inversely proportional to the frequency .
Figure 23-6 shows a graph of the inductive reactance versus frequency for a fixed value of the inductance, according to Equation 23.4. As the frequency becomes very large, XL also becomes very large. In such a situation, an inductor provides a large opposition to the alternating current. In the limit of zero frequency (i.e., direct current), XL becomes zero, indicating that an inductor does not oppose direct current at all. The next example demonstrates the effect of inductive reactance on the current in an ac circuit.

Figure 23-6 In an ac circuit the inductive reactance XL is directly proportional to the frequency f, according to .

| Example 2 | | | An Inductor in an AC Circuit

The circuit in Figure 23-6 contains a 3.60-mH inductor. The rms voltage of the generator is 25.0 V. Find the rms current in the circuit when the generator frequency is (a) and (b) . Reasoning The current can be calculated from , provided the inductive reactance is obtained first. The inductor offers more opposition to the changing current when the frequency is larger, and the values for the inductive reactance will reflect this fact.

Problem solving insight The inductive reactance XL is directly proportional to the frequency f of the voltage. If the frequency increases by a factor of 50, for example, the inductive reactance also increases by a factor of 50.

Solution

a. | At a frequency of , we find

| (23.4)

| (23.3)

b. | The calculation is similar when the frequency is :

| (23.4)

| (23.3)

By virtue of its inductive reactance, an inductor affects the amount of current in an ac circuit. The inductor also influences the current in another way, as Figure 23-7 shows. This figure displays graphs of voltage and current versus time for a circuit containing only an inductor. At a maximum or minimum on the current graph, the current does not change much with time, so the voltage generated by the inductor to oppose a change in the current is zero. At the points on the current graph where the current is zero, the graph is at its steepest, and the current has the largest rate of increase or decrease. Correspondingly, the voltage generated by the inductor to oppose a change in the current has the largest positive or negative value. Thus, current and voltage are not in phase but are one-quarter of a wave cycle out of phase. If the voltage varies as , the current fluctuates as . The current reaches its maximum after the voltage does, and it is said that the current in an inductor lags behind the voltage by a phase angle of 90° ( radians). In a purely capacitive circuit, in contrast, the current leads the voltage by 90° (see Figure 23-4).

Figure 23-7 The instantaneous voltage and current in a circuit containing only an inductor are not in phase. The current lags behind the voltage by one-quarter of a cycle or by a phase angle of 90°.

In an inductor the 90° phase difference between current and voltage leads to the same result for average power that it does in a capacitor. An inductor alternately absorbs and releases energy for equal periods of time, so the average power (and, hence, the average energy) used by an inductor in an ac circuit is zero.

Problem solving insight

As an alternative to the graphs in Figure 23-7, Figure 23-8 uses phasors to describe the instantaneous voltage and current in a circuit containing only an inductor. The voltage and current phasors remain perpendicular as they rotate, because there is a 90° phase angle between them. The current phasor lags behind the voltage phasor, relative to the direction of rotation, in contrast to the equivalent picture in Figure 23-5b for a capacitor. Once again, the instantaneous values are given by the vertical components of the phasors.

Figure 23-8 This phasor model represents the voltage and current in a circuit that contains only an inductor.

Check Your Understanding 2

The drawing shows three ac circuits: one contains a resistor, one a capacitor, and one an inductor. The frequency of each ac generator is reduced to one-half its initial value. Which circuit experiences (a) the greatest increase in current and (b) the least change in current? (The answers are given at the end of the book.)

Background: Consider how the inductive reactance XL, the capacitive reactance XC, and the resistance R change with frequency. In addition, think about how changes in XL, XC, and R alter the current in a circuit.
For similar questions (including calculational counterparts), consult Self-Assessment Test 23.1, which is described next.

Test your understanding of the material in Sections 23.1 and 23.2:

· | Capacitors and Capacitive Reactance
· | Inductors and Inductive Reactance

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