23.4

Resonance in Electric Circuits

CONCEPTS AT A GLANCE The behavior of the current and voltage in a series RCL circuit can give rise to a condition of resonance. Resonance occurs when the frequency of a vibrating force exactly matches a natural (resonant) frequency of the object to which the force is applied. When resonance occurs, the force can transmit a large amount of energy to the object, leading to a large-amplitude vibratory motion. As the Concepts-at-a-Glance chart in Figure 23-15 illustrates, we have already encountered several examples of resonance. First, resonance can occur when a vibrating force is applied to an object of mass m that is attached to a spring whose spring constant is k (Section 10.6). In this case there is one natural frequency f0, whose value is . Second, resonance occurs when standing waves are set up on a string (Section 17.5) or in a tube of air (Section 17.6). As the right-hand side of the concept chart shows, the string and tube of air have many natural frequencies, one for each allowed standing wave. As we will now see, a condition of resonance can also be established in a series RCL circuit. In this case there is only one natural frequency, and the vibrating force is provided by the oscillating electric field that is related to the voltage of the generator.

Figure 23-15 CONCEPTS AT A GLANCE The phenomenon of resonance can occur when an object vibrates on a spring or when standing waves are established on a string or in a tube of air. Resonance can also occur in a series RCL circuit. The antennas attached to the telephone communications tower in this photograph broadcast electromagnetic waves at specific frequencies, each of which corresponds to a frequency of a resonant circuit. (Peter Timmermans/Stone/Getty Images)

Figure 23-16 helps us to understand why there is a resonant frequency for an ac circuit. This drawing presents an analogy between the electrical case (ignoring resistance) and the mechanical case of an object attached to a horizontal spring (ignoring friction). Part a shows a fully stretched spring that has just been released, with the initial speed v of the object being zero. All the energy is stored in the form of elastic potential energy. When the object begins to move, it gradually loses potential energy and picks up kinetic energy. In part b, the object moves with maximum kinetic energy through the position where the spring is unstretched (zero potential energy). Because of its inertia, the moving object coasts through this position and eventually comes to a halt in part c when the spring is fully compressed and all kinetic energy has been converted back into elastic potential energy. Part d of the picture is like part b, except the direction of motion is reversed. The resonant frequency f0 of the object on the spring is the natural frequency at which the object vibrates and is given as according to Equations 10.6 and 10.11. In this expression, m is the mass of the object, and k is the spring constant.

Figure 23-16 The oscillation of an object on a spring is analogous to the oscillation of the electric and magnetic fields that occur, respectively, in a capacitor and in an inductor.

In the electrical case, Figure 23-16a begins with a fully charged capacitor that has just been connected to an inductor. At this instant the energy is stored in the electric field between the capacitor plates. As the capacitor discharges, the electric field between the plates decreases, while a magnetic field builds up around the inductor because of the increasing current in the circuit. The maximum current and the maximum magnetic field exist at the instant when the capacitor is completely discharged, as in part b of the figure. Energy is now stored entirely in the magnetic field of the inductor. The voltage induced in the inductor keeps the charges flowing until the capacitor again becomes fully charged, but now with reversed polarity, as in part c. Once again, the energy is stored in the electric field between the plates, and no energy resides in the magnetic field of the inductor. Part d of the cycle repeats part b, but with reversed directions of current and magnetic field. Thus, an ac circuit can have a resonant frequency because there is a natural tendency for energy to shuttle back and forth between the electric field of the capacitor and the magnetic field of the inductor.
To determine the resonant frequency at which energy shuttles back and forth between the capacitor and the inductor, we note that the current in a series RCL circuit is (Equation 23.6). In this expression Z is the impedance of the circuit and is given by (Equation 23.7). As Figure 23-17 illustrates, the rms current is a maximum when the impedance is a minimum, assuming a given generator voltage. The minimum impedance of occurs when the frequency is f0, such that or . This result can be solved for f0 , which is the resonant frequency:

| (23.10)

The resonant frequency is determined by the inductance and the capacitance, but not the resistance.

Figure 23-17 In a series RCL circuit the impedance is a minimum, and the current is a maximum, when the frequency f equals the resonant frequency f0 of the circuit.

The effect of resistance on electrical resonance is to make the “sharpness” of the circuit response less pronounced, as Figure 23-18 indicates. When the resistance is small, the current-versus-frequency graph falls off suddenly on either side of the maximum current. When the resistance is large, the falloff is more gradual, and there is less current at the maximum.

Figure 23-18 The effect of resistance on the current in a series RCL circuit.

The following example deals with one application of resonance in electric circuits. In this example the focus is on the oscillation of energy between a capacitor and an inductor. Once a capacitor/inductor combination is energized, the energy will oscillate indefinitely as in Figure 23-16, provided there is some provision to replace any dissipative losses that occur because of resistance. Circuits that include this type of provision are called oscillator circuits.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 6 | | | A Heterodyne Metal Detector

Figure 23-19 shows a heterodyne metal detector being used. As Figure 23-20 illustrates, this device utilizes two capacitor/inductor oscillator circuits, A and B. Each produces its own resonant frequency, and . Any difference between these frequencies is detected through earphones as a beat frequency . In the absence of any nearby metal object, the inductances LA and LB are the same and f0A and f0B are identical. There is no beat frequency. When inductor B (the search coil) comes near a piece of metal, the inductance LB decreases, the corresponding oscillator frequency f0B increases, and a beat frequency is heard. Suppose that initially each inductor is adjusted so that , and each oscillator has a resonant frequency of 855.5 kHz. Assuming that the inductance of search coil B decreases by 1.000% due to a nearby piece of metal, determine the beat frequency heard through the earphones.

Figure 23-19 A heterodyne metal detector can be used to locate buried metal objects. Researchers (like the one in this photograph) also use metal detectors to locate meteorite fragments, because many of the fragments contain large amounts of metals such as iron or nickel. (David Parker/Photo Researchers, Inc.)

Figure 23-20 A heterodyne metal detector uses two electrical oscillators, A and B, in its operation. When the resonant frequency of oscillator B is changed due to the proximity of a metallic object, a beat frequency, whose value is , is heard in the headphones.

The physics of a heterodyne metal detector.

Reasoning The beat frequency is , and to find it we need to determine the effect that the 1.000% decrease in the inductance LB has on the resonant frequency f0B. We will do this by a direct application of (Equation 23.10).
Knowns and Unknowns The following table summarizes the available data:

Description | Symbol | Value | Comment
Resonant frequency of circuit A | f0A | 855.5 kHz | Does not change.
Amount by which metal object causes inductance LB to decrease | – | 1.000% |
Unknown Variable |  |  |
Beat frequency |  | ? |

Modeling the Problem

| Step 1 Beat Frequency
| The beat frequency is the magnitude of the difference between the two resonant frequencies, as indicated by Equation 1 at the right. The vertical bars, as usual, indicate that the magnitude or absolute value of the enclosed quantity is needed. The frequency f0A is given, but the frequency f0B is determined by the presence of the metal object and is unknown. We deal with it in Step 2.

| (1)

| Step 2 Effect of Metal Object on Frequency f0B
| According to Equation 23.10, the resonant frequency of circuit B is

| (23.10)

Initially, each inductor is adjusted so that . Since the metal object causes the inductance of coil B to decrease by 1.000%, the new value of LB becomes . Substituting this expression into Equation 23.10 gives

Rearranging this result, we find that

Finally, we recognize that and can then express the resonant frequency of circuit B in the presence of the metal object as follows:

This result can now be substituted into Equation 1, as shown at the right.

Solution Combining the results of each step algebraically, we find that

The beat frequency, then, is

Related Homework: Problem 32

Check Your Understanding 4

The resistance in a series RCL circuit is doubled. (a) Does the resonant frequency increase, decrease, or remain the same? (b) Does the maximum current in the circuit increase, decrease, or remain the same? (The answers are given at the end of the book.)
Background: Answering this question requires a knowledge of which circuit elements (resistor, capacitor, inductor) determine the resonant frequency and which control the maximum current in the circuit.
For similar questions (including calculational counterparts), consult Self-Assessment Test 23.2, which is described on page 740.

Test your understanding of the material in Sections 23.3 and 23.4:

· | Circuits Containing Resistance, Capacitance, and Inductance
· | Resonance in Electric Circuits

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.