AC circuits

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Alternating Current (AC) Circuits Containing Inductance

Courtesy of Getty Images, Inc.

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Why You Need to Know

I

nductance is one the three major types of loads found in alternating current circuits. Electricians need to understand the impact on a pure inductive circuit and how current lags voltage when the effect is applied in an AC circuit. This unit ■ ■

explains how properties other than resistance can limit the flow of current. introduces another measurement called impedance. Impedance is the total current-limiting effect in an AC circuit and can be comprised of more than one element, such as resistance and inductance. Without an understanding of inductance, you will never be able to understand many of the concepts to follow in later units.

OUTLINE 17–1 Inductance 17–2 Inductive Reactance 17–3 Schematic Symbols 17–4 Inductors Connected in Series 17–5 Inductors Connected in Parallel 17–6 Voltage and Current Relationships in an Inductive Circuit 17–7 Power in an Inductive Circuit 17–8 Reactive Power 17–9 Q of an Inductor

KEY TERMS Current lags voltage Induced voltage Inductance (L) Inductive reactance, (XL) Quality (Q) Reactance Reactive power (VARs)

Courtesy of Niagara Mohawk Power Corporation.

UNIT

17

Inductance in AC Circuits

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OBJECTIVES

After studying this unit, you should be able to ■

discuss the properties of inductance in an AC circuit.

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discuss inductive reactance.

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compute values of inductive reactance and inductance.

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discuss the relationship of voltage and current in a pure inductive circuit.

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be able to compute values for inductors connected in series or parallel.

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discuss reactive power (VARs).

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determine the Q of a coil.

PREVIEW

T

his unit discusses the effects of inductance on AC circuits. The unit explains how current is limited in an inductive circuit as well as the effect inductance has on the relationship of voltage and current. ■

17–1

Inductance Inductance (L) is one of the primary types of loads in AC circuits. Some amount of inductance is present in all AC circuits because of the continually changing magnetic field (Figure 17–1). The amount of inductance of a single

Expansion of magnetic field

Magnetic field

Collapse of magnetic field

FIGURE 17–1 A continually changing magnetic field induces a voltage into any conductor.

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UNIT 17

FIGURE 17–2 As current flows through a coil, a magnetic field is created around the coil.

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FIGURE 17–3 As current flow decreases, the magnetic field collapses.

conductor is extremely small, and, in most instances, it is not considered in circuit calculations. Circuits are generally considered to contain inductance when any type of load that contains a coil is used. For circuits that contain a coil, inductance is considered in circuit calculations. Loads such as motors, transformers, lighting ballast, and chokes all contain coils of wire. In Unit 14, it was discussed that whenever current flows through a coil of wire, a magnetic field is created around the wire (Figure 17–2). If the amount of current decreases, the magnetic field collapses (Figure 17–3). Recall from Unit 14 several facts concerning inductance: 1. When magnetic lines of flux cut through a coil, a voltage is induced in the coil. 2. An induced voltage is always opposite in polarity to the applied voltage. This is often referred to as counter-electromotive force (CEMF). 3. The amount of induced voltage is proportional to the rate of change of current. 4. An inductor opposes a change of current. The inductors in Figure 17–2 and Figure 17–3 are connected to an alternating voltage. Therefore the magnetic field continually increases, decreases, and reverses polarity. Since the magnetic field continually changes magnitude and direction, a voltage is continually being induced in the coil. This induced voltage is 180⬚ out of phase with the applied voltage and is always in opposition to the applied voltage (Figure 17–4). Since the induced voltage is always in opposition to the applied voltage, the effective applied voltage is reduced by the induced voltage. For example, assume an inductor is connected to a 120-volts AC line. Now assume that the inductor has an induced voltage of 116 volts. Since the induced voltage subtracts from the applied voltage, there are only 4 volts to push current through the wire resistance of the coil (120 V ⫺ 116 V ⫽ 4).

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Applied voltage

Induced voltage

Applied voltage Induced voltage

FIGURE 17–4 The applied voltage and induced voltage are 180 degrees out of phase with each other.

6 Ohmmeter

FIGURE 17–5 Measuring the resistance of a coil.

Computing the Induced Voltage The amount of induced voltage in an inductor can be computed if the resistance of the wire in the coil and the amount of circuit current are known. For example, assume that an ohmmeter is used to measure the actual amount of resistance in a coil and the coil is found to contain 6 ohms of wire resistance (Figure 17–5). Now assume that the coil is connected to a 120-volts AC circuit and an ammeter measures a current flow of 0.8 ampere (Figure 17–6). Ohm’s law can now be used to determine the amount of voltage necessary to push

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Ammeter

0.8 A 120 VAC

FIGURE 17–6 Measuring circuit current with an ammeter.

0.8 ampere of current through 6 ohms of resistance: E⫽I⫻R E ⫽ 0.8 A ⫻ 6 E ⫽ 4.8 V Since only 4.8 volts are needed to push the current through the wire resistance of the inductor, the remainder of the 120 volts is used to overcome the ⫺苶4.8 ( 苶苶V) 苶2 ⫽ 119.904 volts). coil’s induced voltage of 119.904 volts (兹苶 (120 苶苶) V苶2苶

17–2

Inductive Reactance Notice that the induced voltage is able to limit the flow of current through the circuit in a manner similar to resistance. This induced voltage is not resistance, but it can limit the flow of current just as resistance does. This current-limiting property of the inductor is called reactance and is symbolized by the letter X. This reactance is caused by inductance, so it is called inductive reactance and is symbolized by XL, pronounced “X sub L.” Inductive reactance is measured in ohms just as resistance is and can be computed when the values of inductance and frequency are known. The following formula can be used to find inductive reactance: XL ⫽ 2 ␲ fL where XL ⫽ inductive reactance 2 ⫽ a constant ␲ ⫽ 3.1416 f ⫽ frequency in hertz (Hz) L ⫽ inductance in henrys (H)

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Coil with more inductance

Coil with less inductance FIGURE 17–7 Coils with turns closer together produce more inductance than coils with turns far apart.

The inductive reactance, in ohms, is caused by the induced voltage and is, therefore, proportional to the three factors that determine induced voltage: 1. the number of turns of wire; 2. the strength of the magnetic field; 3. the speed of the cutting action (relative motion between the inductor and the magnetic lines of flux). The number of turns of wire and the strength of the magnetic field are determined by the physical construction of the inductor. Factors such as the size of wire used, the number of turns, how close the turns are to each other, and the type of core material determine the amount of inductance (in henrys, H) of the coil (Figure 17–7). The speed of the cutting action is proportional to the frequency (hertz). An increase of frequency causes the magnetic lines of flux to cut the conductors at a faster rate and, thus, produces a higher induced voltage or more inductive reactance.

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EXAMPLE 17–1

The inductor shown in Figure 17–8 has an inductance of 0.8 H and is connected to a 120-V, 60-Hz line. How much current will flow in this circuit if the wire resistance of the inductor is negligible?

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120 VAC 60 Hz

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0.8 H

FIGURE 17–8 Circuit current is limited by inductive reactance.

Solution The first step is to determine the amount of inductive reactance of the inductor: XL ⫽ 2 ␲ fL XL ⫽ 2 ⫻ 3.1416 ⫻ 60 Hz ⫻ 0.8 XL ⫽ 301.594 ⍀ Since inductive reactance is the current-limiting property of this circuit, it can be substituted for the value of R in an Ohm’s law formula: I⫽

E XL

I⫽

120 V 301.594 ⍀

I ⫽ 0.398 A If the amount of inductive reactance is known, the inductance of the coil can be determined using the formula L⫽

XL 2␲f ■

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EXAMPLE 17–2

Assume an inductor with a negligible resistance is connected to a 36-V, 400-Hz line. If the circuit has a current flow of 0.2 A, what is the inductance of the inductor?

Solution The first step is to determine the inductive reactance of the circuit: E I 36 V XL ⫽ 0.2 A

XL ⫽

XL ⫽ 180 ⍀ Now that the inductive reactance of the inductor is known, the inductance can be determined: L⫽

XL 2␲f

L⫽

180 ⍀ 2 ⫻ 3.1416 ⫻ 400 Hz

L ⫽ 0.0716 H ■

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EXAMPLE 17–3

An inductor with negligible resistance is connected to a 480-V, 60-Hz line. An ammeter indicates a current flow of 24 A. How much current will flow in this circuit if the frequency is increased to 400 Hz?

Solution The first step in solving this problem is to determine the amount of inductance of the coil. Since the resistance of the wire used to make the inductor is negligible, the current is limited by inductive reactance. The inductive reactance can be found by substituting XL for R in an Ohm’s law formula: E I 480 V XL ⫽ 24 A

XL ⫽

XL ⫽ 20 ⍀

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Now that the inductive reactance is known, the inductance of the coil can be found using the formula L⫽

XL 2␲f

NOTE: When using a frequency of 60 Hertz, 2 ⫻ ␲ ⫻ 60 ⫽ 376.992. To simplify calculations, this value is generally rounded to 377. Since 60 hertz is the major frequency used throughout the United States, 377 should be memorized because it is used in many calculations: L⫽

20 ⍀ 377 Hz

L ⫽ 0.053 H Since the inductance of the coil is determined by its physical construction, it does not change when connected to a different frequency. Now that the inductance of the coil is known, the inductive reactance at 400 Hertz can be computed: XL ⫽ 2␲fL XL ⫽ 2 ⫻ 3.1416 ⫻ 400 Hz ⫻ 0.053 XL ⫽ 133.204 ⍀ The amount of current flow can now be found by substituting the value of inductive reactance for resistance in an Ohm’s law formula: I⫽

E XL

I⫽

480 V 133.204 ⍀

I ⫽ 3.603 A ■

17–3

Schematic Symbols The schematic symbol used to represent an inductor depicts a coil of wire. Several symbols for inductors are shown in Figure 17–9. The symbols shown with the two parallel lines represent iron-core inductors, and the symbols without the parallel lines represent air-core inductors.

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Air-core inductors

Iron-core inductors

FIGURE 17–9 Schematic symbols for inductors.

17–4

Inductors Connected in Series When inductors are connected in series (Figure 17–10), the total inductance of the circuit (LT) equals the sum of the inductances of all the inductors: LT ⫽ L1 ⫹ L2 ⫹ L3 The total inductive reactance (XLT) of inductors connected in series equals the sum of the inductive reactances for all the inductors: XLT ⫽ XL1 ⫹ XL2 ⫹ XL3

FIGURE 17–10 Inductors connected in series.

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EXAMPLE 17– 4

Three inductors are connected in series. Inductor 1 has an inductance of 0.6 H, Inductor 2 has an inductance of 0.4 H, and Inductor 3 has an inductance of 0.5 H. What is the total inductance of the circuit?

Solution LT ⫽ 0.6 H ⫹ 0.4 H ⫹ 0.5 H LT ⫽ 1.5 H ■

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EXAMPLE 17–5

Three inductors are connected in series. Inductor 1 has an inductive reactance of 180 ⍀, Inductor 2 has an inductive reactance of 240 ⍀, and Inductor 3 has an inductive reactance of 320 ⍀. What is the total inductive reactance of the circuit?

Solution XLT ⫽ 180 ⍀ ⫹ 240 ⍀ ⫹ 320 ⍀ XLT ⫽ 740 ⍀ ■

17–5

Inductors Connected in Parallel When inductors are connected in parallel (Figure 17–11), the total inductance can be found in a manner similar to finding the total resistance of a parallel circuit. The reciprocal of the total inductance is equal to the sum of the reciprocals of all the inductors: 1 1 1 1 ⫽ ⫹ ⫹ LT L1 L2 L3 or LT ⫽

1 1 1 1 ⫹ ⫹ L1 L2 L3

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FIGURE 17–11 Inductors connected in parallel.

Another formula that can be used to find the total inductance of parallel inductors is the product-over-sum formula: LT ⫽

L 1 ⫻ L2 L 1 ⫹ L2

If the values of all the inductors are the same, total inductance can be found by dividing the inductance of one inductor by the total number of inductors: LT ⫽

L N

Similar formulas can be used to find the total inductive reactance of inductors connected in parallel: 1 1 1 1 ⫽ ⫹ ⫹ XLT XL 1 XL 2 XL 3 or XLT ⫽

1 1 1 1 ⫹ ⫹ XL 1 XL 2 XL 3

or XLT ⫽

XL 1 ⫻ XL 2 XL 1 ⫹ XL 2

or XLT ⫽

XL N

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EXAMPLE 17–6

Three inductors are connected in parallel. Inductor 1 has an inductance of 2.5 H, Inductor 2 has an inductance of 1.8 H, and Inductor 3 has an inductance of 1.2 H. What is the total inductance of this circuit?

Solution LT ⫽

1 1 1 1 ⫹ ⫹ 2.5 H 1.8 H 1.2 H

1 (1.788) L T ⫽ 0.559 H LT ⫽

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17–6

Voltage and Current Relationships in an Inductive Circuit In Unit 16, it was discussed that when current flows through a pure resistive circuit, the current and voltage are in phase with each other. In a pure inductive circuit, the current lags the voltage by 90⬚. At first this may seem to be an impossible condition until the relationship of applied voltage and induced voltage is considered. How the current and applied voltage can become 90⬚ out of phase with each other can best be explained by comparing the relationship of the current and induced voltage (Figure 17–12). Recall that the induced voltage is proportional to the rate of change of the current (speed of cutting action). At the beginning of the waveform, the current is shown at its maximum value in the negative direction. At this time, the current is not changing, so induced voltage is zero. As the current begins to decrease in value, the magnetic field produced by the flow of current decreases or collapses and begins to induce a voltage into the coil as it cuts through the conductors (Figure 17–3). Circuit current

Induced voltage

FIGURE 17–12 Induced voltage is proportional to the rate of change of current.

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The greatest rate of current change occurs when the current passes from negative, through zero and begins to increase in the positive direction (Figure 17–13). Since the current is changing at the greatest rate, the induced voltage is maximum. As current approaches its peak value in the positive direction, the rate of change decreases, causing a decrease in the induced voltage. The induced voltage will again be zero when the current reaches its peak value and the magnetic field stops expanding. It can be seen that the current flowing through the inductor is leading the induced voltage by 90⬚. Since the induced voltage is 180⬚ out of phase with the applied voltage, the current lags the applied voltage by 90⬚ (Figure 17–14).

Point of no current change

Induced voltage

Circuit current

Greatest rate of current change FIGURE 17–13 No voltage is induced when the current does not change.

Applied voltage Current flow

FIGURE 17–14 The current lags the applied voltage by 90⬚.

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17–7

Power in an Inductive Circuit In a pure resistive circuit, the true power, or watts, is equal to the product of the voltage and current. In a pure inductive circuit, however, no true power, or watts, is produced. Recall that voltage and current must both be either positive or negative before true power can be produced. Since the voltage and current are 90⬚ out of phase with each other in a pure inductive circuit, the current and voltage will be at different polarities 50% of the time and at the same polarity 50% of the time. During the period of time that the current and voltage have the same polarity, power is being given to the circuit in the form of creating a magnetic field. When the current and voltage are opposite in polarity, power is being given back to the circuit as the magnetic field collapses and induces a voltage back into the circuit. Since power is stored in the form of a magnetic field and then given back, no power is used by the inductor. Any power used in an inductor is caused by losses such as the resistance of the wire used to construct the inductor, generally referred to as I2R losses, eddy current losses, and hysteresis losses. The current and voltage waveform in Figure 17–15 has been divided into four sections: A, B, C, and D. During the first time period, indicated by A, the current is negative and the voltage is positive. During this period, energy is being given to the circuit as the magnetic field collapses. During the second time period, B, both the voltage and current are positive. Power is being used to produce the magnetic field. In the third time period, C, the current is positive and the voltage is negative. Power is again being given back to the circuit as the field

A

B

C

D

FIGURE 17–15 Voltage and current relationships during different parts of a cycle.

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collapses. During the fourth time period, D, both the voltage and current are negative. Power is again being used to produce the magnetic field. If the amount of power used to produce the magnetic field is subtracted from the power given back, the result will be zero.

17–8

Reactive Power Although essentially no true power is being used, except by previously mentioned losses, an electrical measurement called volt-amperes-reactive (VARs) is used to measure the reactive power in a pure inductive circuit. VARs can be computed in the same way as watts except that inductive values are substituted for resistive values in the formulas. VARs is equal to the amount of current flowing through an inductive circuit times the voltage applied to the inductive part of the circuit. Several formulas for computing VARs are VARs ⫽ EL ⫻ IL EL2 XL VARs ⫽ IL2 ⫻ xL VARs ⫽

where EL ⫽ voltage applied to an inductor IL ⫽ current flow through an inductor XL ⫽ inductive reactance

17–9

Q of an Inductor So far in this unit, it has been generally assumed that an inductor has no resistance and that inductive reactance is the only current-limiting factor. In reality, that is not true. Since inductors are actually coils of wire, they all contain some amount of internal resistance. Inductors actually appear to be a coil connected in series with some amount of resistance (Figure 17–16). The amount of resistance compared with the inductive reactance determines the quality (Q) of the coil. Inductors that have a higher ratio of inductive reactance to resistance are considered to be inductors of higher quality. An inductor constructed with a large wire will have a low wire resistance and, therefore, a higher Q (Figure 17–17). Inductors constructed with many turns of small wire have a much higher resistance

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Pure inductance Internal resistance

FIGURE 17–16 Inductors contain internal resistance.

Large wire produces an inductor with a high Q.

Small wire produces an inductor with a low Q. FIGURE 17–17 The Q of an inductor is a ratio of inductive reactance as compared to resistance. The letter Q stands for quality.

and, therefore, a lower Q. To determine the Q of an inductor, divide the inductive reactance by the resistance: Q⫽

XL R

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Although inductors have some amount of resistance, inductors that have a Q of 10 or greater are generally considered to be pure inductors. Once the ratio of inductive reactance becomes 10 times as great as resistance, the amount of resistance is considered negligible. For example, assume an inductor has an inductive reactance of 100 ohms and a wire resistance of 10 ohms. The inductive reactive component in the circuit is 90⬚ out of phase with the resistive component. This relationship produces a right triangle (Figure 17–18). The total current-limiting effect of the inductor is a combination of the inductive reactance and resistance. This total current-limiting effect is called impedance and is symbolized by the letter Z. The impedance of the circuit is represented by the hypotenuse of the right triangle formed by the inductive reactance and the resistance. To compute the value of impedance for the coil, the inductive reactance and the resistance must be added. Since these two components form the legs of a right triangle and the impedance forms the hypotenuse, the Pythagorean theorem discussed in Unit 15 can be used to compute the value of impedance: Z ⫽ R2 ⫹ XL2 Z ⫽ 102 ⫹ 1002 Z ⫽ 10, 100 Z ⫽ 100.499 ⍀

Total coil impedance (Z)

Inductive reactance (XL)

Resistance (R) FIGURE 17–18 Coil impedance is a combination of wire resistance and inductive reactance.

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Notice that the value of total impedance for the inductor is only 0.5 ohm greater than the value of inductive reactance. If it should become necessary to determine the true inductance of an inductor, the resistance of the wire must be taken into consideration. Assume that an inductor is connected to a 480-volt 60-Hz power source and that an ammeter indicates a current flow of 0.6 ampere. Now assume that an ohmmeter measures 150 ⍀ of wire resistance in the inductor. What is the inductance of the inductor? To determine the inductance, it will be necessary to first determine the amount of inductive reactance as compared to the wire resistance. The total current-limiting value (impedance) can be found with Ohm’s law: E I 480 Z⫽ 0.6 Z ⫽ 800 ⍀ Z⫽

The total current-limiting effect of the inductor is 800 ⍀. This value is a combination of both the inductive reactance of the inductor and the wire resistance. Since the resistive part and the inductive reactance part of the inductor are 90⬚ out of phase with each other, they form the legs of a right triangle with the impedance forming the hypotenuse of the triangle (Figure 17-19). To determine the amount of inductive reactance, use the following formula: X L ⫽ Z 2 ⫺ R2 XL ⫽ 8002 ⫺ 1502 XL ⫽ 785.812 ⍀ Now that the amount of inductive reactance has been determined, the inductance can be computed using the formula L⫽

XL 2␲f

L⫽

785.812 2 ⫻ 3.1416 ⫻ 60

L ⫽ 2.084 henrys

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R (Resistance) 150 W

XL (Inductive reactance)

Z (Impedance) 800 W

FIGURE 17–19 The inductive reactance forms one leg of a right triangle.

Summary ■

Induced voltage is proportional to the rate of change of current.

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Induced voltage is always opposite in polarity to the applied voltage.

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Inductive reactance is a countervoltage that limits the flow of current, as does resistance.

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Inductive reactance is measured in ohms.

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Inductive reactance is proportional to the inductance of the coil and the frequency of the line.

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Inductive reactance is symbolized by XL.

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Inductance is measured in henrys (H) and is symbolized by the letter L.

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When inductors are connected in series, the total inductance is equal to the sum of all the inductors.

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When inductors are connected in parallel, the reciprocal of the total inductance is equal to the sum of the reciprocals of all the inductors.

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The current lags the applied voltage by 90⬚ in a pure inductive circuit.

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All inductors contain some amount of resistance.

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The Q of an inductor is the ratio of the inductive reactance to the resistance.

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Inductors with a Q of 10 are generally considered to be “pure” inductors.

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Pure inductive circuits contain no true power or watts.

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Reactive power is measured in VARs.

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VARs is an abbreviation for volt-amperes-reactive.

Review Questions 1. How many degrees are the current and voltage out of phase with each other in a pure resistive circuit? 2. How many degrees are the current and voltage out of phase with each other in a pure inductive circuit? 3. To what is inductive reactance proportional? 4. Four inductors, each having an inductance of 0.6 H, are connected in series. What is the total inductance of the circuit? 5. Three inductors are connected in parallel. Inductor 1 has an inductance of 0.06 H; Inductor 2 has an inductance of 0.05 H; and Inductor 3 has an inductance of 0.1 H. What is the total inductance of this circuit? 6. If the three inductors in Question 5 were connected in series, what would be the inductive reactance of the circuit? Assume the inductors are connected to a 60-Hz line. 7. An inductor is connected to a 240-V, 1000-Hz line. The circuit current is 0.6 A. What is the inductance of the inductor?

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8. An inductor with an inductance of 3.6 H is connected to a 480-V, 60-Hz line. How much current will flow in this circuit? 9. If the frequency in Question 8 is reduced to 50 Hz, how much current will flow in the circuit? 10. An inductor has an inductive reactance of 250 ⍀ when connected to a 60-Hz line. What will be the inductive reactance if the inductor is connected to a 400-Hz line?

Practical Applications

Y

ou are working as an electrician installing fluorescent lights. You notice that the lights were made in Europe and that the ballasts are rated for operation on a 50-Hz system. Will these ballasts be harmed by overcurrent if they are connected to 60 Hz? If there is a problem with these lights, what will be the most likely cause of the trouble? ■

Practical Applications

Y

ou have the task of ordering a replacement inductor for one that has become defective. The information on the nameplate has been painted over and cannot be read. The machine that contains the inductor operates on 480 V at a frequency of 60 Hz. Another machine has an identical inductor in it, but its nameplate has been painted over also. A clamp-on ammeter indicates a current of 18 A, and a voltmeter indicates a voltage drop across the inductor of 324 V in the machine that is still in operation. After turning off the power and locking out the panel, you disconnect the inductor in the operating machine and measure a wire resistance of 1.2 ⍀ with an ohmmeter. Using the identical inductor in the operating machine as an example, what inductance value should you order and what would be the minimum VAR rating of the inductor? Should you be concerned with the amount of wire resistance in the inductor when ordering? Explain your answers. ■

Practice Problems Inductive Circuits

1. Fill in all the missing values. Refer to the following formulas: XL ⫽ 2 ␲ f L XL L⫽ 2␲ f XL f⫽ 2␲ L

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Inductance (H) 1.2 0.085 0.65 3.6 0.5 0.85 0.45 4.8

Frequency (Hz) 60 1000 600 25 60 20 400 1000

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Inductive Reactance (⍀) 213.628 4712.389 678.584 411.459 6408.849 201.062 2412.743 40.841

2. What frequency must be applied to a 33-mH inductor to produce an inductive reactance of 99.526 ⍀? 3. An inductor is connected to a 120-volt, 60-Hz line and has a current flow of 4 amperes. An ohmmeter indicates that the inductor has a wire resistance of 12 ⍀. What is the inductance of the inductor? 4. A 0.75-henry inductor has a wire resistance of 90 ⍀. When connected to a 60-Hz power line, what is the total current-limiting effect of the inductor? 5. An inductor has a current flow of 3 amperes when connected to a 240-volt, 60-Hz power line. The inductor has a wire resistance of 15 ⍀. What is the Q of the inductor?

489

AC circuits

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