23.3

Circuits Containing Resistance, Capacitance, and Inductance

Capacitors and inductors can be combined along with resistors in a single circuit. The simplest combination contains a resistor, a capacitor, and an inductor in series, as Figure 23-9 shows. In a series RCL circuit the total opposition to the flow of charge is called the impedance of the circuit and comes partially from (1) the resistance R, (2) the capacitive reactance XC, and (3) the inductive reactance XL. It is tempting to follow the analogy of a series combination of resistors and calculate the impedance by simply adding together R, XC, and XL. However, such a procedure is not correct. Instead, the phasors shown in Figure 23-10 must be used. The lengths of the voltage phasors in this drawing represent the maximum voltages VR, VC, and VL across the resistor, the capacitor, and the inductor, respectively. The current is the same for each device, since the circuit is wired in series. The length of the current phasor represents the maximum current I0 . Notice that the drawing shows the current phasor to be (1) in phase with the voltage phasor for the resistor, (2) ahead of the voltage phasor for the capacitor by 90°, and (3) behind the voltage phasor for the inductor by 90°. These three facts are consistent with our earlier discussion in Sections 23.1 and 23.2.

Figure 23-9 A series RCL circuit contains a resistor, a capacitor, and an inductor.

Figure 23-10 The three voltage phasors (VR, VC, and VL) and the current phasor (I0) for a series RCL circuit.

The basis for dealing with the voltage phasors in Figure 23-10 is Kirchhoff’s loop rule. In an ac circuit this rule applies to the instantaneous voltages across each circuit component and the generator. Therefore, it is necessary to take into account the fact that these voltages do not have the same phase; that is, the phasors VR, VC, and VL point in different directions in the drawing. Kirchhoff’s loop rule indicates that the phasors add together to give the total voltage V0 that is supplied to the circuit by the generator. The addition, however, must be like a vector addition, to take into account the different directions of the phasors. Since VL and VC point in opposite directions, they combine to give a resultant phasor of , as Figure 23-11 shows. In this drawing the resultant is perpendicular to VR and may be combined with it to give the total voltage V0. Using the Pythagorean theorem, we find

In this equation each of the symbols stands for a maximum voltage and when divided by gives the corresponding rms voltage. Therefore, it is possible to divide both sides of the equation by and obtain a result for . This result has exactly the same form as that above, but involves the rms voltages , , and . However, to avoid such awkward symbols, we simply interpret VR, VC, and VL as rms quantities in the following expression:

| (23.5)

The last step in determining the impedance of the circuit is to remember that , , and . With these substitutions Equation 23.5 can be written as

Therefore, for the entire RCL circuit, it follows that

| (23.6)

where the impedance Z of the circuit is defined as

| (23.7)

The impedance of the circuit, like R, XC, and XL, is measured in ohms. In Equation 23.7, and .

Figure 23-11 This simplified version of Figure 23-10 results when the phasors VL and VC, which point in opposite directions, are combined to give a resultant of VL − VC.

The phase angle between the current in and the voltage across a series RCL combination is the angle between the current phasor I0 and the voltage phasor V0 in Figure 23-11. According to the drawing, the tangent of this angle is

| (23.8)

The phase angle is important because it has a major effect on the average power delivered to the circuit. Remember that, on the average, only the resistance consumes power; that is, (Equation 20.15b). According to Figure 23-11, , so that . Therefore,

| (23.9)

where is the rms voltage of the generator, according to Equation 23.6. The term is called the power factor of the circuit. As a check on the validity of Equation 23.9, note that if no resistance is present, , and . Consequently, , a result that is expected since, on the average, neither a capacitor nor an inductor consumes energy. Conversely, if only resistance is present, , and . In this case, , which is the expression for the average power delivered to a resistor. Examples 3 and 4 deal with the current, voltages, and power for a series RCL circuit.

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 3 | | | Current in a Series RCL Circuit

A series RCL circuit contains a resistor, a capacitor, and a 35.7-mH inductor. The generator has a frequency of 512 Hz and an rms voltage of 35.0 V. Determine the rms current in the circuit. Reasoning The rms current in the circuit is equal to the rms voltage of the generator divided by the impedance of the circuit, according to Equation 23.6. The impedance of the circuit can be found from the resistance of the resistor and the reactances of the capacitor and the inductor via Equation 23.7.
Knowns and Unknowns The following table summarizes the data that we have:

Description | Symbol | Value | Comment
Resistance of resistor | R |  |
Capacitance of capacitor | C |  |
Inductance of inductor | L | 35.7 mH |
Frequency of generator | f | 512 Hz |
Rms voltage of generator | Vrms | 35.0 V |
Unknown Variable |  |  |
Rms current in circuit | Irms | ? |

Modeling the Problem

| Step 1 Current
| According to Equation 23.6, the rms voltage Vrms of the generator, the rms current Irms, and the impedance Z of the circuit are related according to

Solving for the current gives Equation 1 at the right, in which Vrms is known. The impedance is unknown, but it can be dealt with as in Step 2.

| (1)

| Step 2 Impedance
| For a series RCL circuit, the impedance Z is related to the resistance R, the inductive reactance XL, and the capacitive reactance XC, according to

| (23.7)

As indicated at the right, this expression can be substituted into Equation 1. The resistance is given, and we turn to Step 3 to deal with the reactances.

| Step 3 Inductive and Capacitive Reactances
| The inductive reactance XL and the capacitive reactance XC are given by Equation 23.4 and Equation 23.2 as

where L is the inductance, C is the capacitance, and f is the frequency. Using these two expressions, we find that

This result can now be substituted into Equation 23.7, as shown at the right.

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

The rms current in the circuit is

Related Homework: Problems 19, 39

| Example 4 | | | Voltages and Power in a Series RCL Circuit

For the series RCL circuit discussed in Example 3, the resistance, capacitance, and inductance are , , and , respectively. The generator has a frequency of 512 Hz and an rms voltage of 35.0 V. In Example 3, it is found that the rms current in the circuit is . Find (a) the rms voltage across each circuit element and (b) the average electric power delivered by the generator.
Reasoning The rms voltages across each circuit element can be determined from , , and . In these expressions, the rms current is known, and the resistance R is given. The capacitive reactance XC and inductive reactance XL can be determined from Equations 23.2 and 23.4. The average power delivered to the circuit by the generator is specified by Equation 23.9.
Solution

a. | The individual reactances are

| (23.2)

| (23.4)

The rms voltages across each circuit element are

| (20.14)

| (23.1)

| (23.3)

Observe that these three voltages do not add up to give the generator’s rms voltage of 35.0 V. Instead, the rms voltages satisfy Equation 23.5. It is the sum of the instantaneous voltages across R, C, and L that equals the generator’s instantaneous voltage, according to Kirchhoff’s loop rule. The rms voltages do not satisfy the loop rule.

Problem solving insight In a series RCL circuit, the rms voltages across the resistor, capacitor, and inductor do not add up to equal the rms voltage across the generator.

b. | The average power delivered by the generator is (Equation 23.9). Therefore, a value for the phase angle is needed and can be obtained from Equation 23.8 as follows:

The phase angle is negative since the circuit is more capacitive than inductive (XC is greater than XL), and the current leads the voltage. The average power delivered by the generator is

This amount of power is delivered only to the resistor, since neither the capacitor nor the inductor uses power, on average.

In addition to the series RCL circuit, there are many different ways to connect resistors, capacitors, and inductors. In analyzing these additional possibilities, it helps to keep in mind the behavior of capacitors and inductors at the extreme limits of the frequency. When the frequency approaches zero (i.e., dc conditions), the reactance of a capacitor becomes so large that no charge can flow through the capacitor. It is as if the capacitor were cut out of the circuit, leaving an open gap in the connecting wire. In the limit of zero frequency the reactance of an inductor is vanishingly small. The inductor offers no opposition to a dc current. It is as if the inductor were replaced with a wire of zero resistance. In the limit of very large frequency, the behaviors of a capacitor and an inductor are reversed. The capacitor has a very small reactance and offers little opposition to the current, as if it were replaced by a wire with zero resistance. In contrast, the inductor has a very large reactance when the frequency is very large. The inductor offers so much opposition to the current that it might as well be cut out of the circuit, leaving an open gap in the connecting wire. Conceptual Example 5 illustrates how to gain insight into more complicated circuits using the limiting behaviors of capacitors and inductors.

| Conceptual Example 5 | | | The Limiting Behavior of Capacitors and Inductors

Figure 23-12a shows two circuits. The rms voltage of the generator is the same in each case. The values of the resistance R, the capacitance C, and the inductance L in these circuits are the same. The frequency of the ac generator is very near zero. In which circuit does the generator supply more rms current?
Reasoning and Solution According to Equation 23.6, the rms current is given by . However, the impedance Z is not given by Equation 23.7, since the circuits in Figure 23-12a are not series RCL circuits. Since Vrms is the same in each case, the greater current is delivered to the circuit with the smaller impedance Z. In the limit of very small frequency, the capacitors behave as if they were cut out of the circuit, leaving gaps in the connecting wires. In this limit, however, the inductors behave as if they were replaced by wires with zero resistance. Figure 23-12b shows the circuits as they would appear according to these changes. Circuit I behaves as if it contained only two identical resistors in series, with a total impedance of . In contrast, circuit II behaves as if it contained two identical resistors in parallel, in which case the total impedance is given by , or . At a frequency very near zero, then, circuit II has the smaller impedance and its generator supplies the greater rms current.

Figure 23-12 (a) These circuits are discussed in the limit of very small or low frequency in Conceptual Example 5. (b) For a frequency very near zero, the circuits in part a behave as if they were as shown here.

Related Homework: Conceptual Questions 6 and 7; Problems 20, 24

The physics of body-fat scales. The impedance of an ac circuit contains important information about the resistance, capacitance, and inductance in the circuit. As an example of a very complex circuit, consider the human body. It contains muscle, which is a relatively good conductor of electricity due to its high water content, and also fat, which is a relatively poor conductor due to its low water content. The impedance that the body offers to ac electricity is referred to as bioelectrical impedance and is largely determined by resistance and capacitive reactance. Capacitance enters the picture because cell membranes can act like tiny capacitors. Bioelectric impedance analysis provides the basis for the determination of body-fat percentage by the body-fat scales (see Figure 23-13) that are widely available for home use. When you stand barefoot on such a scale, electrodes beneath your feet send a weak ac current (approximately ) through your lower body in order to measure your body’s impedance. The scale also measures your weight. A built-in computer combines the impedance and weight with information you provide about height, age, and sex to determine the percentage of fat in your body to an accuracy of about 5% . For men (age 20 to 39) a percentage of 8 to 19% is considered average, whereas the corresponding range of values for women of similar ages is 21 to 33%.

Figure 23-13 Bathroom scales are now widely available that can provide estimates of your body-fat percentage. When you stand barefoot on the scale, electrodes beneath your feet send a small ac current through your lower body that allows the body’s electrical impedance to be measured. This impedance is correlated with the percentage of fat in the body.

Weak ac electricity with a much lower frequency than that used to measure bioelectrical impedance is used in a technique called transcutaneous electrical nerve stimulation (TENS). TENS is the most commonly used form of electroanalgesia in pain-management situations and, in its conventional form, uses an ac frequency between 40 and 150 Hz. Ac current is passed between two electrodes attached to the body and inhibits the transmission of pain-related nerve impulses. The technique is thought to work by affecting the “gates” in a nerve cell that control the passage of sodium ions into and out of the cell (see Section 19.6). Figure 23-14 shows TENS being applied during assessment of pain control following suspected damage to the radial nerve in the forearm.

Figure 23-14 Transcutaneous electrical nerve stimulation (TENS) is shown here being applied to the forearm, in an assessment of pain control following suspected damage to the radial nerve. (Martin Dohrn/SPL/Photo Researchers, Inc.)

The physics of transcutaneous electrical nerve stimulation (TENS).

Check Your Understanding 3

A long wire of finite resistance is connected to an ac generator. This wire is then wound into a coil of many loops and reconnected to the generator. Is the current in the circuit with the coil greater than, less than, or the same as the current in the circuit with the uncoiled wire? (The answer is given at the end of the book.)
Background: The current in a circuit depends on its impedance. The impedance depends not only on the resistance that is present, but also on the capacitance and inductance.
For similar questions (including calculational counterparts), consult Self-Assessment Test 23.2, which is described at the end of Section 23.4.

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