Inductance, Reactance, and AC Circuits

12-56 X YAC generator current in the circuit. In review, when current and voltage pass through zero and reach maximum value at the same time, the current and voltage are said to be in phase. [Figure 12-124A] If the current and voltage pass through zero and reach the maximum values at different times, the current and voltage are said to be out of phase. In a circuit containing only inductance, the current reaches a maximum value later than the voltage, lagging the voltage by 90°, or one-fourth cycle. [Figure 12-124B] In a circuit containing only capacitance, the current reaches its maximum value ahead of the voltage and the current leads the voltage by 90°, or one-fourth cycle. [Figure 12-124C] The amount the current lags or leads the voltage in a circuit depends on the relative amounts of resistance, inductance, and capacitance in the circuit.

Inductance

Characteristics of Inductance Michael Faraday discovered that by moving a magnet through a coil of wire, a voltage was induced across the coil. If a complete circuit was provided, then a current was also induced. The amount of induced voltage is directly proportional to the rate of change of the magnetic field with respect to the coil. The simplest of experiments can prove that when a bar magnet is moved through a coil of wire, a voltage is induced and can be measured on a voltmeter. This is commonly known as Faraday’s Law or the Law of Electromagnetic Induction, which states that the induced emf or electromagnetic force in a closed loop of wire is proportional to the rate of change of the magnetic flux through a coil of wire.

Conversely, current flowing through a coil of wire produces a magnetic field. When this wire is formed into a coil, it then becomes a basic inductor. The magnetic lines of force around each loop or turn in the coil effectively add to the lines of force around the adjoining loops. This forms a strong magnetic field within and around the coil. Figure 12-125A shows a coil of wire strengthening a magnetic field. The magnetic lines of force around adjacent loops are deflected into an outer path when the loops are brought close together. This happens because the magnetic lines of force between adjacent loops are in opposition with each other. The total magnetic field for the two loops is shown in Figure 12-125B. As more loops are added close together, the strength of the magnetic field increases. Figure 12-125C illustrates the combined effects of many loops of a coil. The result is a strong electromagnet.

The primary aspect of the operation of a coil is its property to oppose any change in current through it. This property is called inductance. When current flows through any conductor, a magnetic field starts to expand from the center of the wire. As the lines of magnetic force grow outward through the conductor, they induce an emf in the conductor itself. The induced voltage is always in the direction opposite to the direction of the current flow. The effects of this countering emf are to oppose the immediate establishment of the maximum current. This effect is only a temporary condition. Once the current reaches a steady value in the conductor, the lines of magnetic force no longer expand and the countering emf is no longer present.

At the starting instant, the countering emf nearly equals the applied voltage, resulting in a small current flow. However, as the lines of force move outward, the number of lines cutting the conductor per second becomes progressively smaller, resulting in a diminished counter emf. Eventually, the counter emf drops to zero and the only voltage in the circuit is the applied voltage and the current is at its maximum value. The RL Time Constant Because the inductors basic action is to oppose a change in its current, it then follows that the current cannot change instantaneously in the inductor. A certain time is required for the current to make a change from one value to another.

The rate at which the current changes is determined by a time constant represented by the Greek letter τ. The time constant for the RL circuit is: τ = R L Where τ = seconds L = inductance (H) R = resistance (Ω) In a series RL circuit, the current increases to 63 percent of its full value in 1 time constant after the circuit is closed. This buildup is similar to the buildup of voltage in a capacitor when charging an RC circuit. Both follow an exponential curve and reach 99 percent value after the 5th time constant. [Figure 12-126] Physical Parameters Some of the physical factors that affect inductance are: 12-57 Current and voltage in phase Effect of inductance Effect of capacitance A B C 0° 180° 360° emf 0° Lag 180°90° 360°270° emf 0° Lead 180°90° 360°270° emf 1. The number of turns: Doubling the number of turns in a coil produces a field twice as strong if the same current is used. As a general rule, the inductance varies as the square of the number of turns.

2. The cross-sectional area of the coil: The inductance of a coil increases directly as the cross-sectional area of the core increases. Doubling the radius of a coil increases the inductance by a factor of four. 3. The length of a coil: Doubling the length of a coil, while keeping the same number of turns, halves the value of inductance. 4. The core material around which the coil is formed: Coils are wound on either magnetic or nonmagnetic materials. Some nonmagnetic materials include air, copper, plastic, and glass. Magnetic materials include nickel, iron, steel, or cobalt, which have a permeability that provides a better path for the magnetic lines of force and permit a stronger magnetic field.

Self-Inductance The characteristic of self-inductance was summarized by German physicist Heinrich Lenz in 1833, and gives the direction of the induced emf resulting from electromagnetic induction. This is commonly known as Lenz’s Law, which states: The emf induced in an electric circuit always acts in such a direction that the current it drives around a closed circuit produces a magnetic field, which opposes the change in magnetic flux. Self-inductance is the generation of a voltage in an electric circuit by a changing current in the same circuit. Even a straight piece of wire has some degree of inductance because current in a conductor produces a magnetic field.

When the current in a conductor changes direction, there is a corresponding change in the polarity of the magnetic field around the conductor. Therefore, a changing current produces a changing magnetic field around the wire. To further intensify the magnetic field, the wire can be rolled into a coil, which is called an inductor. The changing magnetic field around the inductor induces a voltage across the coil. This induced emf is called self-inductance and tends to oppose any change in current within the circuit. This property is usually called inductance and symbolized with the letter L. Types of Inductors Inductors used in radios can range from a straight wire at UHF to large chokes and transformers used for filtering the ripple from the output of power supplies and in audio amplifiers.

inductors. Values of inductors range from nano-henries to tens of henries. Inductors are classified by the type of core and the method of winding them. The number of turns in the inductor winding and the core material determine the capacity of the inductor. Cores made of dielectric material like ceramics, wood, and paper provide small amounts of stored energy while cores made of ferrite substances have a much higher degree of stored energy. The core material is usually the most important aspect of the inductors construction. The conductors typically used in the construction of an inductor offer little resistance to the flow of current. However, with the introduction of a core, resistance is introduced in the circuit and the current now builds up in the windings until the resistance of the core is overcome. This buildup is stored as magnetic energy in the core. Depending on the core resistance, the buildup soon reaches a point of magnetic saturation, and it can be released when necessary. The most common core materials are: air, solid ferrite, powdered ferrite, steel, toroid, and ferrite toroid.

12-58 63% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 86% 95% 98% Applied voltage

Current

1t Counter emf Current, counter emf, and applied voltage in an inductive circuit. 2t 3t 4t 5t 99% Opposing magnetic fields Combined magnetic fields Coils with some separationA Coils without separationB Strong magnetic field in a coilC Current eee eee

Current

eee eee SouthNorth Units of Inductance The henry is the basic unit of inductance and is symbolized with the letter H. An electric circuit has an inductance of one henry when current changing at the rate of one ampere per second induces a voltage of one volt into the circuit. In many practical applications, millihenries (mH) and microhenries (μH) are more common units. The typical symbol for an inductor is shown in Figure 12-127. Inductors in Series If we connect two inductors in series, the same current flows through both inductors and, therefore, both are subject to the same rate of change of current. [Figure 12-128] When inductors are connected in series, the total inductance LT, is the sum of the individual inductors. The general equation for n number of inductors in series is: LT = L1 + L2 + L3 + … LN Inductors in Parallel When two inductors are connected in parallel, each must have the same potential difference between the terminals.

[Figure 12-129] When inductors are connected in parallel, the total inductance is less than the smallest inductance. The general equation for n number of inductors in parallel is: LT = + +1 L1 1 L2 1 L3 1 LN 1 + . . . A simple example would be: L 1 = 10 mH, L2 = 5 mH, L3 = 2 mH LT = + +1 10 mH 1 5 mH 1 2 mH 1 LT = 0.8 mH 1 L T = 1.25 mH Inductive Reactance Alternating current is in a constant state of change; the effects of the magnetic fields are a continuously inducted voltage opposition to the current in the circuit. This opposition is called inductive reactance, symbolized by X L, and is measured in ohms just as resistance is measured. Inductance is the property of a circuit to oppose any change in current and is measured in henries. Inductive reactance is a measure of how much the countering emf in the circuit opposes current 12-59 Inductor + L1 L2 − variations.

The inductive reactance of a component is directly proportional to the inductance of the component and the applied frequency to the circuit. By increasing either the inductance or applied frequency, the inductive reactance likewise increases and presents more opposition to current in the circuit. This relationship is given as: X L = 2πfL Where: XL = inductive reactance in ohms f = frequency in cycles per second π = 3.1416 L = inductance In Figure 12-130, an AC series circuit is shown in which the inductance is 0.146 henry and the voltage is 110 volts at a frequency of 60 cps. Inductive reactance is determined by the following method.

X L = 2π × f × L X L = 6.28 × 60 × 0.146 X L = 55 ohm In any circuit where there is only resistance, the expression for the relationship of voltage and current is given by Ohm’s Law: I = E/R. Similarly, when there is inductance in an AC circuit, the relationship between voltage and current can be expressed as: Current = or I = V oltage E Reactance XL Where: XL = inductive reactance of the circuit in ohms I = E XL I = 110 55 I = 2 amperes In AC series circuits, inductive reactances are added like resistances in series in a DC circuit. [Figure 12-131] Thus, the total reactance in the illustrated circuit equals the sum of the individual reactances.

The total reactance of inductors connected in parallel is found the same way as the total resistance in a parallel circuit. [Figure 12-132] Thus, the total reactance of inductances connected in parallel, as shown, is expressed as: (XL)T = + +1 (XL)1 1 (XL)2 1 (XL)3 1

AC Circuits

Ohm’s Law for AC Circuits The rules and equations for DC circuits apply to AC circuits only when the circuits contain resistance alone, as in the case of lamps and heating elements. In order to use effective values of voltage and current in AC circuits, the effect of inductance and capacitance with resistance must be considered. The combined effects of resistance, inductive reactance, and capacitive reactance make up the total opposition to current flow in an AC circuit. This total opposition is called impedance and is represented by the letter Z. The unit for the measurement of impedance is the ohm.

Series AC Circuits If an AC circuit consists of resistance only, the value of the impedance is the same as the resistance, and Ohm’s Law for an AC circuit, I = E/Z, is exactly the same as for a DC circuit. In Figure 12-133, a series circuit containing a lamp with 11 ohms resistance connected across a source is illustrated. To find how much current flows if 110 volts DC is applied and how much current flows if 110 volts AC are applied, the following examples are solved: I = E R I = (where Z = R) E Z I = 110 V 11 W I = 110 V 11 W I = 10 amperes DC I = 10 amperes AC When AC circuits contain resistance and either inductance or capacitance, the impedance, Z, is not the same as the 12-60 110V AC 60 cycles A XL1 XL2 + L2 − L1 resistance, R. The impedance of a circuit is the circuit’s total opposition to the flow of current. In an AC circuit, this opposition consists of resistance and reactance, either inductive or capacitive or elements of both.

Resistance and reactance cannot be added directly, but they can be considered as two forces acting at right angles to each other. Thus, the relation between resistance, reactance, and impedance may be illustrated by a right triangle. [Figure 12-134] Since these quantities may be related to the sides of a right triangle, the formula for finding the impedance, or total opposition to current flow in an AC circuit, can be found by using the law of right triangles. This theorem, called the Pythagorean theorem, applies to any right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, the value of any side of a right triangle can be found if the other two sides are known. If an AC circuit contains resistance and inductance, as shown in Z 2 = R2 + XL2 The square root of both sides of the equation gives Z = R2 + XL2 This formula can be used to determine the impedance when the values of inductive reactance and resistance are known. It can be modified to solve for impedance in circuits containing capacitive reactance and resistance by substituting XC in the formula in place of XL. In circuits containing resistance with both inductive and capacitive reactance, the reactances can be combined, but because their effects in the circuit are exactly opposite, they are combined by subtraction: X = XL − XC or X = XC − XL (the smaller number is always subtracted from the larger) In Figure 12-135, a series circuit consisting of resistance and inductance connected in series is connected to a source of 110 volts at 60 cps. The resistive element is a lamp with 6 ohms resistance, and the inductive element is a coil with an inductance of 0.021 henry. What is the value of the impedance and the current through the lamp and the coil?

Solution: First, the inductive reactance of the coil is computed: X L = 2π × f × L X L = 6.28 × 60 × 0.021 X L = 8 ohms inductive reactance Next, the total impedance is computed: Z = R2 + XL2 Z = 62 + 82 Z = 36 + 64 Z = 100 Z = 10 ohms impedance Then the current flow, I = E Z I = 110 10 I = 11 amperes current 12-61 XL2XL1 110 V Reactance Resistance Z R Impedance XL − XC The voltage drop across the resistance (ER) is: E R = I × R E R = 11 × 6 = 66 volts The voltage drop across the inductance (EXL) is: EXL = I × XL EXL = 11 × 8 = 88 volts The sum of the two voltages is greater than the impressed voltage. This results from the fact that the two voltages are out of phase and, as such, represent the maximum voltage.

If the voltage in the circuit is measured by a voltmeter, it is approximately 110 volts, the impressed voltage. This can be proved by the equation: E = (ER)2 + (EXL)2 E = 662 + 882 E = 4,356 + 7,744 E = 12,100 E = 110 volts In Figure 12-136, a series circuit is illustrated in which a capacitor of 200 µf is connected in series with a 10 ohm lamp. What is the value of the impedance, the current flow, and the voltage drop across the lamp? Solution: First, the capacitance is changed from microfarads to farads. Since 1 million microfarads equal 1 farad, then: 200 μf = = 0.000200 farads 200 1,000,000 XC = 1 2πfC XC = 1 6.28 × 60 × 0.000200 farads XC = 1 0.07536 X C = 13 ohms capacitive reactance To find the impedance, Z = R2 + XC2 Z = 102 + 132 Z = 100 + 169 Z = 269 Z = 16.4 ohms capacitive reactance To find the current, I = E Z I = 110 16.4 I = 6.7 amperes The voltage drop across the lamp (ER) is: E R = 6.7 × 10 E R = 67 volts The voltage drop across the capacitor (EXC) is EXC = I × XC EXC = 6.7 × 13 EXC = 86.1 volts 12-62 110 V AC 60 cycles 10 Ω 200 μf A 110V AC 60 cycles 0.021 henries A 6 Ω The sum of these two voltages does not equal the applied voltage, since the current leads the voltage. To find the applied voltage, use the following formula: ET = (ER)2 + (EXC)2 ET = 672 + 86.12 ET = 4,489 + 7,413 ET = 11,902 E T = 110 volts When the circuit contains resistance, inductance, and capacitance, the following equation is used to find the impedance: Z = R2 + (XL – XC)2 Example: What is the impedance of a series circuit, consisting of a capacitor with a reactance of 7 ohms, an inductor with a reactance of 10 ohms, and a resistor with a resistance of 4 ohms? [Figure 12-137] Solution: Z = R2 + (XL – XC)2 Z = 42 + (10 – 7)2 Z = 42 + 32 Z = 25 Z = 5 ohms Assuming that the reactance of the capacitor is 10 ohms and the reactance of the inductor is 7 ohms, then X C is greater than XL. Thus, Z = R2 + (XL – XC)2 Z = 42 + (7 – 10)2 Z = 42 + (–3)2 Z = 16 + 9 Z = 25 Z = 5 ohms Parallel AC Circuits The methods used in solving parallel AC circuit problems are basically the same as those used for series AC circuits.

Out of phase voltages and currents can be added by using the law of right triangles. However, in solving circuit problems, the currents through the branches are added since the voltage drops across the various branches are the same and are equal to the applied voltage. In Figure 12-138 , a parallel AC circuit containing an inductance and a resistance is shown schematically. The current flowing through the inductance, I L, is 0.0584 ampere, and the current flowing through the resistance is 0.11 ampere. What is the total current in the circuit? Solution: IT = IL2 + IR2 = (0.0584)2 + (0.11)2 = 0.0155 = 0.1245 ampere Since inductive reactance causes voltage to lead the current, the total current, which contains a component of inductive current, lags the applied voltage. If the current and voltages are plotted, the angle between the two, called the phase angle, illustrates the amount the current lags the voltage.

In Figure 12-139, a 112-volt generator is connected to a load consisting of a 2 µf capacitance and a 10,000-ohm resistance in parallel. What is the value of the impedance and total current flow? Solution: First, find the capacitive reactance of the circuit: XC = 1 2πfC Changing 2 μf to farads and entering the values into the formula given: 12-63 110 V AC 60 cycles 4 Ω 10 Ω 7 Ω capacitance. = 1 2 × 3.14 × 60 × 0.000002 = or 1 10,000 0.00075360 7.536 = 1,327 XC capacitive reactance To find the impedance, the impedance formula used in a series AC circuit must be modified to fit the parallel circuit: Z = R2 + XC2 RXC = (10,000)2 + (1,327)2 10,000 × 1,327 = 0.1315 W (approximately) To find the current through the capacitance: IC = E XC IC = 110 1,327 = 0.0829 ampere To find the current flowing through the resistance: IR = E R = 110 10,000 = 0.011 ampere To find the total current in the circuit: IT2 = IR2 + IC2 IT = IL2 + IR2 = 0.0836 ampere (approximately) Resonance It has been shown that both inductive reactance: (X L = 2πfL) and capacitive reactance: XC = 1 2πfC are functions of an AC frequency. Decreasing the frequency decreases the ohmic value of the inductive reactance, but a decrease in frequency increases the capacitive reactance. At some particular frequency, known as the resonant frequency, the reactive effects of a capacitor and an inductor is equal.

Since these effects are the opposite of one another, they will cancel, leaving only the ohmic value of the resistance to oppose current flow in a circuit. If the value of resistance is small or consists only of the resistance in the conductors, the value of current flow can become very high. In a circuit where the inductor and capacitor are in series, and the frequency is the resonant frequency, or frequency of resonance, the circuit is said to be “in resonance” and is referred to as a series resonant circuit. The symbol for resonant frequency is Fn. If, at the frequency of resonance, the inductive reactance is equal to the capacitive reactance, then: XL = XC, or 2πfL = 1 2πfC Dividing both sides by 2 fL, Fn2 = 1 (2π) 2LC Extracting the square root of both sides gives: Fn = 1 2π LC Where Fn is the resonant frequency in cps, C is the capacitance in farads, and L is the inductance in henries.

With this formula, the frequency at which a capacitor and inductor is resonant can be determined. To find the inductive reactance of a circuit use: XL = 2πfL The impedance formula used in a series AC circuit must be modified to fit a parallel circuit. 12-64 2 μf110VG 10,000 Ω resistance. IL G IR5 henries 1000 Ω resistance. Z = R2 = XL2 XL To find the parallel networks of inductance and capacitive reactors, use: X = XL + XC XL + XC To find the parallel networks with resistance capacitive and inductance, use: Z = XL2 XC2 + (RXL – RXC)2 R XL XC Since at the resonant frequency XL cancels XC, the current can become very large, depending on the amount of resistance. In such cases, the voltage drop across the inductor or capacitor is often higher than the applied voltage.

In a parallel resonant circuit, the reactances are equal and equal currents flow through the coil and the capacitor. [Figure 12-140] Since the inductive reactance causes the current through the coil to lag the voltage by 90°, and the capacitive reactance causes the current through the capacitor to lead the voltage by 90°, the two currents are 180° out of phase. The canceling effect of such currents would mean that no current would flow from the generator and the parallel combination of the inductor and the capacitor would appear as infinite impedance. In practice, no such circuit is possible, since some value of resistance is always present, and the parallel circuit, sometimes called a tank circuit, acts as very high impedance.

It is also called an antiresonant circuit, since its effect in a circuit is opposite to that of a series resonant circuit, in which the impedance is very low. Power in AC Circuits In a DC circuit, power is obtained by the equation, P = EI, (watts equal volts × amperes). Thus, if 1 ampere of current flows in a circuit at a pressure of 200 volts, the power is 200 watts. The product of the volts and the amperes is the true power in the circuit. True Power Defined True power of any AC circuit is commonly referred to as the working power of the circuit. True power is the power consumed by the resistance portion of the circuit and is measured in watts. True power is symbolized by the letter P and is indicated by any wattmeter in the circuit. True power is calculated by the formula: P = I2 × Z Apparent Power Defined Apparent power in an AC circuit is sometimes referred to as the reactive power of a circuit. Apparent power is the power consumed by the entire circuit, including both the resistance and the reactance. Apparent power is symbolized by the letter S and is measured in volt-amps (V A). Apparent power is a product of the effective voltage multiplied by the effective current. Apparent power is calculated by the formula: S = I2 × Z Only when the AC circuit is made up of pure resistance is the apparent power equal to the true power. [Figure 12-141] When there is capacitance or inductance in the circuit, the current and voltage are not exactly in phase, and the true power is less than the apparent power. The true power is obtained by a wattmeter reading. The ratio of the true power to the apparent power is called the power factor and is usually expressed in percent. In equation form, the relationship is: Power Factor (PF) = 100 × watts (True Power) volts × amperes (Apparent Power) Example: A 220-volt AC motor takes 50 amperes from the line, but a wattmeter in the line shows that only 9,350 watts