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**19 Inductance Chapter Topics Covered in Chapter 19**

19-1: Induction by Alternating Current 19-2: Self-Inductance L 19-3: Self-Induced Voltage vL 19-4: How vL Opposes a Change in Current 19-5: Mutual Inductance LM 19-6: Transformers © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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**Topics Covered in Chapter 19**

19-7: Transformer Ratings 19-8: Impedance Transformation 19-9: Core Losses 19-10: Types of Cores 19-11: Variable Inductance 19-12: Inductances in Series or Parallel 19-13: Energy in Magnetic Field of Inductance 19-14: Stray Capacitive and Inductive Effects 19-15: Measuring and Testing Inductors McGraw-Hill

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**19-1: Induction by Alternating Current**

Induced voltage is the result of flux cutting across a conductor. This action can be produced by physical motion of either the magnetic field or the conductor. Variations in current level (or amplitude) induces voltage in a conductor because the variations of current and its magnetic field are equivalent to the motion of the flux. Thus, the varying current can produce induced voltage without the need for motion of the conductor. The ability of a conductor to induce voltage in itself when the current changes is called self-inductance, or simply inductance.

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**19-1: Induction by Alternating Current**

Induction by a varying current results from the change in current, not the current value itself. The current must change to provide motion of the flux. The faster the current changes, the higher the induced voltage.

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**19-1: Induction by Alternating Current**

At point A, the current is zero and there is no flux. At point B, the positive direction of current provides some field lines taken here in the counterclockwise direction. Fig. 19-1: Magnetic field of an alternating current is effectively in motion as it expands and contracts with the current variations. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-1: Induction by Alternating Current**

Point C has maximum current and maximum counterclockwise flux. At point D there is less flux than at C. Now the field is collapsing because of reduced current.

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**19-1: Induction by Alternating Current**

Point E with zero current, there is no magnetic flux. The field can be considered collapsed into the wire. The next half-cycle of current allows the field to expand and collapse again, but the directions are reversed. When the flux expands at points F and G, the field lines are clockwise. From G to H and I, this clockwise field collapses into the wire.

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**19-1: Induction by Alternating Current**

Characteristics of inductance are important in: AC circuits: In these circuits, the current is continuously changing and producing induced voltage. DC circuits in which the current changes in value: DC circuits that are turned off and on (changing between zero and its steady value) can produce induced voltage.

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19-2: Self-Inductance L The symbol for inductance is L, for linkages of magnetic flux. VL is in volts, di/dt is the current change in amperes per second. The henry (H) is the basic unit of inductance. One henry causes 1 V to be induced when the current is changing at the rate of 1 A per second. L = VL di / dt

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Examples The current in an inductor changes from 12 to 16 A in 1 s. How much is the di/dt rate of current change in amperes per second? The current in an inductor changes by 50 mA in 2 µs. How much is the di/dt rate of current change in amperes per second? How much is the inductance of a coil that induces 40 V when its current changes at the rate of 4 A/s? How much is the inductance of a coil that induces 1000 V when its current changes at the rate of 50 mA in 2 µs?

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**19-2: Self-Inductance L Inductance of Coils**

The inductance of a coil depends on how it is wound. A greater number of turns (N) increases L because more voltage can be induced (L increases in proportion to N). More area enclosed by each turn increases L. The L increases with the permeability of the core. The L decreases with more length for the same number of turns, as the magnetic field is less concentrated.

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**Calculating the Inductance of a Long Coil**

19-2: Self-Inductance L Where: L is the inductance in henrys. μr is the relative permeability of the core N is the number of turns A is the area in square meters l is the length in meters Calculating the Inductance of a Long Coil L = l N 2A 1.26 × 10−6 H μr d air-core symbol (μ r = 1) iron-core (μr >> 1) Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-2: Self-Inductance L Typical Coil Inductance Values**

Air-core coils for RF applications have L values in millihenrys (mH) and microhenrys (μH). Practical inductor values are in these ranges: 1 H to 10 H (for iron-core inductors) 1 mH (millihenry) = 1 × 10-3 H 1 µH (microhenry) = 1 × 10-6 H

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**19-3: Self-Induced Voltage vL**

( ) di dt vL L = Formula: Induced voltage is proportional to inductance (L). Induced voltage is proportional to the rate of current change:

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**19-3: Self-Induced Voltage vL**

Energy Stored in the Field 2 LI 2 Energy = Where the energy is in joules: L is the inductance in henrys I is the current in amperes Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Examples How much is the self-induced voltage across a 4-H inductance produced by a current change of 12 A/s? The current through a 200-mH L changes from 0 to 100 mA in 2 µs. How much is vL ?

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**19-4: How vL Opposes a Change in Current**

Lenz’ Law states that the induced voltage produces current that opposes the changes in the current causing the induction. The polarity of vL depends on the direction of the current variation di. When di increases, vL has polarity that opposes the increase in current. When di decreases, vL has opposite polarity to oppose the decrease in current. In both cases, the change in current is opposed by the induced voltage.

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**19-5: Mutual Inductance LM**

Mutual inductance (LM) occurs when current flowing through one conductor creates a magnetic field which induces a voltage in a nearby conductor. Two coils have a mutual inductance of 1 H when a current change of 1A/s induces 1 V in the other coil. Unit: Henrys (H) Formula: L k M = 1 2

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**19-5: Mutual Inductance LM**

Coefficient of coupling, k, is the fraction of total flux from one coil linking another coil nearby. Specifically, the coefficient of coupling is k = flux linkages between L1 and L2 divided by flux produced by L1 There are no units for k, because it is a ratio of two values of magnetic flux. The value of k is generally stated as a decimal fraction.

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**19-5: Mutual Inductance LM**

The coefficient of coupling is increased by placing the coils close together, possibly with one wound on top of the other, by placing them parallel, or by winding the coils on a common core. A high value of k, called tight coupling, allows the current in one coil to induce more voltage in the other. Loose coupling, with a low value of k, has the opposite effect. Two coils may be placed perpendicular to each other and far apart for essentially zero coupling to minimize interaction between the coils.

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**19-5: Mutual Inductance LM**

Loose coupling Tighter coupling Unity coupling Zero coupling Fig. 19-8: Examples of coupling between two coils linked by LM. (a) L1 or L2 on paper or plastic form with air core; k is 0.1. (b) L1 wound over L2 for tighter coupling; k is 0.3. (c) L1 and L2 on the same iron core; k is 1. (d) Zero coupling between perpendicular air-core coils. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-5: Mutual Inductance LM**

Calculating LM Mutual inductance increases with higher values for primary and secondary inductances. LM where L1 and L2 are the self-inductance values of the two coils, k is the coefficient of coupling, and LM is the mutual inductance.

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19-6: Transformers Transformers are an important application of mutual inductance. A transformer has two or more windings with mutual inductance. The primary winding is connected to a source of ac power. The secondary winding is connected to the load. Fig : Iron-core power transformer. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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19-6: Transformers The transformer transfers power from the primary to the secondary. Transformer steps up voltage (to 100V) and steps current down (to 1A) Fig. 19-9: Iron-core transformer with 1:10 turns ratio. Primary current IP induces secondary voltage VS, which produces current in secondary load RL. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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19-6: Transformers A transformer can step up or step down the voltage level from the ac source. Step-down (VLOAD < VSOURCE) Primary Secondary Load Step-up (VLOAD > VSOURCE) A transformer is a device that uses the concept of mutual inductance to step up or step down an alternating voltage. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-6: Transformers Turns Ratio**

The ratio of the number of turns in the primary to the number in the secondary is the turns ratio of the transformer. Turns ratio equals NP/NS. where NP equals the number of turns in the primary and NS equals the number of turns in the secondary. The turns ratio NP/NS is sometimes represented by the lowercase letter a.

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**19-6: Transformers The voltage ratio is the same as the turns ratio:**

VP / VS = NP / NS VP = primary voltage, VS = secondary voltage NP = number of turns of wire in the primary NS = number of turns of wire in the secondary When transformer efficiency is 100%, the power at the primary equals the power at the secondary. Power ratings refer to the secondary winding in real transformers (efficiency < 100%).

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**19-6: Transformers Voltage Ratio VL = 3 x 120 = 360 V 3:1 Primary**

Secondary Load 120 V 40 V 1:3 360 V Step-up (1:3) Step-down (3:1) VL = 1/3 x 120 = 40 V Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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19-6: Transformers Current Ratio is the inverse of the voltage ratio. (That is voltage step-up in the secondary means current step-down, and vice versa.) The secondary does not generate power but takes it from the primary. The current step-up or step-down is terms of the secondary current IS, which is determined by the load resistance across the secondary voltage.

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**19-6: Transformers Current Ratio IS/IP = VP/VS 3:1 Primary Secondary**

Load 120 V 40 V 0.1 A 0.3 A 1:3 360 V IL = 1/3 x 0.3 = 0.1 A IS/IP = VP/VS IL = 3 x 0.1 = 0.3 A Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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19-6: Transformers Transformer efficiency is the ratio of power out to power in. Stated as a formula % Efficiency = Pout/Pin x 100 Assuming zero losses in the transformer, power out equals power in and the efficiency is 100%. Actual power transformers have an efficiency of approximately 80 to 90%.

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**19-6: Transformers Transformer Efficiency Primary Secondary Load 120 V**

3:1 0.12 A 0.3 A PPRI = 120 x .12 = 14.4 W PSEC = 40 x 0.3 = 12 W 14.4 12 × 100 % = 83 % × 100 % = Efficiency = PPRI PSEC Primary power that is lost is dissipated as heat in the transformer. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-6: Transformers Loaded Power Transformer 1:6**

Calculate VS from the turns ratio and VP. Use VS to calculate IS: IS = VS/RL Use IS to calculate PS: PS = VS x IS Use PS to find PP: PP = PS Finally, IP can be calculated: IP = PP/VP 20:1

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**19-6: Transformers Autotransformers**

An autotransformer is a transformer made of one continuous coil with a tapped connection between the end terminals. An autotransformer has only three leads and provides no isolation between the primary and secondary.

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**19-7: Transformer Ratings**

Transformer voltage, current, and power ratings must not be exceeded; doing so will destroy the transformer. Typical Ratings: Voltage values are specified for primary and secondary windings. Current Power (apparent power – VA) Frequency

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**19-7: Transformer Ratings**

Voltage Ratings Manufacturers always specify the voltage rating of the primary and secondary windings. Under no circumstances should the primary voltage rating be exceeded. In many cases, the rated primary and secondary voltages are printed on the transformer. Regardless of how the secondary voltage is specified, the rated value is always specified under full load conditions with the rated primary voltage applied.

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**19-7: Transformer Ratings**

Current Ratings Manufacturers usually specify current ratings only for secondary windings. If the secondary current is not exceeded, there is no possible way the primary current can be exceeded. If the secondary current exceeds its rated value, excessive I2R losses will result in the secondary winding.

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**19-7: Transformer Ratings**

Power Ratings The power rating is the amount of power the transformer can deliver to a resistive load. The power rating is specified in volt-amperes (VA). The product VA is called apparent power, since it is the power that is apparently used by the transformer. The unit of apparent power is VA because the watt is reserved for the dissipation of power in a resistance.

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**19-7: Transformer Ratings**

Frequency Ratings Typical ratings for a power transformer are 50, 60, and 400 Hz. A power transformer with a frequency rating of 400 Hz cannot be used at 50 or 60 Hz because it will overheat. Many power transformers are designed to operate at either 50 or 60 Hz. Power transformers with a 400-Hz rating are often used in aircraft because these transformers are much smaller and lighter that 50- or 60-Hz transformers.

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**19-8: Impedance Transformation**

Transformers are often used to change or transform a secondary load impedance to a new value as seen by the primary. The secondary load impedance is said to be reflected back into the primary and is called a reflected impedance. The reflected impedance of the secondary may be stepped up or down. An equation for the reflected impedance is: Z N P S = æ è ç ö ø ÷ 2 secondary primary

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**19-8: Impedance Transformation**

If the turns ratio NP/NS is less than 1, ZS will be stepped down in value. Transformer impedance matching is related to the turns ratio: N Z P S =

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**19-8: Impedance Transformation**

Impedance Ratio The load on the source is 1 W. Primary Secondary Load = 9 W 1:3 ZRATIO = 1/9 The load on the source is 81 W. Primary Secondary Load = 9 W 3:1 ZRATIO = 9/1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-8: Impedance Transformation**

Impedance Matching for Maximum Power Transfer Transformers are used when it is necessary to achieve maximum transfer of power from a generator to a load when the generator and load impedances are not the same. This application of a transformer is called impedance matching.

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**19-8: Impedance Transformation**

Internal resistance (ri) is 200 Ω. RL is 8 Ω. Fig : Transferring power from an amplifier to a load RL. (a) Amplifier has ri = 200 Ω and RL = 8 Ω. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-8: Impedance Transformation**

Connected as shown, the load receives 1.85 W of power. Fig : (b) Connecting the amplifier directly to RL. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-8: Impedance Transformation**

Connection shown increases the power delivered to the load. Fig (c): Using a transformer to make the 8-Ω RL appear like 200 Ω in the primary. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-8: Impedance Transformation**

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**19-9: Core Losses Eddy Currents**

Eddy currents are induced in the iron core of an inductor or transformer. Eddy currents raise the temperature of the core. Wasted power is dissipated as heat. Losses increase with frequency.

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19-9: Core Losses Fig : Cross-sectional view of iron core showing eddy currents. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-9: Core Losses Hysteresis losses**

The hysteresis losses result from the additional power needed to reverse the magnetic field in magnetic materials in the presence of alternating current. The greater the frequency, the more hysteresis loss. Air-core coils Air has practically no losses from eddy currents or hysteresis. The inductance for small coils with an air core is, however, limited to low values (e.g. mH or μH).

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19-10: Types of Cores Losses can be reduced by using a laminated core or a powered-iron core. The type of steel used can reduce hysteresis losses. The most common types of insulation are: Laminated core Powdered iron core Ferrite core

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**19-10: Types of Cores Laminated core**

A shell-type core formed with a group of individual laminations. Each laminated section is insulated by a very thin coating of iron oxide. Powdered iron Consists of individual insulated granules pressed into one solid form called a slug. Ferrite core Synthetic ceramic materials that are ferromagnetic.

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**19-11: Variable Inductance**

The inductance of a coil may be varied by one of several methods. For larger coils: More or fewer turns can be used by connection to one of the taps on the coil. A slider contacts the coil to vary the number of turns used. Fig : Methods of varying inductance. (a) Tapped coil. (b) Slider contact. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-11: Variable Inductance**

Figure (c) shows the schematic symbol for a coil with a slug of powdered iron or ferrite. Usually, an arrow at the top means that the adjustment is at the top of the coil. Fig : Methods of varying inductance. (c) Adjustable slug. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-11: Variable Inductance**

The symbol in Fig (d) is a variometer, which is an arrangement for varying the position of one coil within the other. The total inductance of the series-aiding coils is minimum when they are perpendicular. Fig : Methods of varying inductance. (d) Variometer. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-11: Variable Inductance**

For any method of varying L, the coil with an arrow in Fig (e) can be used. The variac is an autotransformer with a variable tap to change the turns ratio. The output voltage in the secondary can be varied from 0 to approximately 140 V, with a 120-V, 60Hz input. Fig : Methods of varying inductance. (e) Symbol for variable L. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-12: Inductances in Series or Parallel**

With no mutual coupling: For series circuits, inductances add just like resistances. For parallel circuits, inductances combine according to a reciprocal formula as with resistances. LT = L1 + L2 + L etc. LEQ = 1 etc. + L3 L2 L1

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**19-12: Inductances in Series or Parallel**

Series Coils for LM LM depends on the amount of mutual coupling and on whether the coils are connected series-aiding or series-opposing. Series-aiding means that the common current produces the same direction of magnetic field for the two coils. The series-opposing connection results in opposite fields.

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**19-12: Inductances in Series or Parallel**

The coupling depends on the coil connections and direction of winding. Reversing either one reverses the field. Fig : Inductances L1 and L2 in series but with mutual coupling LM. (a) Aiding magnetic fields. (b) Opposing magnetic fields. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-12: Inductances in Series or Parallel**

To calculate the total inductance of two coils that are series-connected and have mutual inductance, LT = L1 + L2 ± 2LM The mutual inductance LM is plus, increasing the total inductance, when the coils are series-aiding, or minus when they are series-opposing to reduce the total inductance.

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**19-13: Energy in Magnetic Field of Inductance**

The magnetic flux of current in an inductance has electric energy supplied by the voltage source producing the current. The energy is stored in the field, since it can do the work of producing induced voltage when the flux moves. The amount of electric energy stored is Energy = ε = ½ LI2 The factor of ½ gives the average result of I in producing energy.

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**19-14: Stray Capacitive and Inductive Effects**

Stray capacitive and inductive effects can occur in all circuits with all types of components. A capacitor has a small amount of inductance in the conductors. A coil has some capacitance between windings. A resistor has a small amount of inductance and capacitance.

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**19-14: Stray Capacitive and Inductive Effects**

A practical case of problems caused by stray L and C is a long cable used for rf signals. If the cable is rolled in a coil to save space, a serious change in the electrical characteristic of the line will take place. For twin-lead or coaxial cable feeding the antenna input to a television receiver, the line should not be coiled because the added L or C can affect the signal.

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**19-14: Stray Capacitive and Inductive Effects**

As shown in Fig , a coil has distributed capacitance Cd between turns. Each turn is a conductor separated from the next turn by an insulator, which is the definition of capacitance. The potential of each turn is different from the next, providing part of the total voltage as a potential difference to charge Cd. This results in an equivalent circuit. The L is inductance and Re is its internal ac resistance. Fig : Equivalent circuit of an RF coil. (a) Distributed capacitance Cd between turns of wire. (b) Equivalent circuit. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-14: Stray Capacitive and Inductive Effects**

As shown by the high-frequency equivalent circuit in Fig , a resistor can include a small amount of inductance and capacitance. The inductance of carbon-composition resistors is usually negligible. Wire-wound resistors, however, have enough inductance to be evident. Fig : High-frequency equivalent circuit of a resistor. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-15: Measuring and Testing Inductors**

Many DMMs are capable of measuring the value of a capacitor, but few are capable of measuring the value of an inductor. When it is necessary to measure the value of an inductor, a capacitor-inductor analyzer should be used. Fig : Typical LCR meter. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-15: Measuring and Testing Inductors**

The capacitor-inductor analyzer can also test the quality (Q) of the inductor by using a ringing test. Another test instrument that is capable of measuring inductance L, capacitance C, and resistance R, is an LCR meter. A typical LCR meter is shown in Fig (previous slide). This is a handy piece of test equipment, however, most LCR meters only measure the value of a component.

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**19-15: Measuring and Testing Inductors**

The most common trouble in coils is an open winding. As shown in Fig , an ohmmeter connected across the coil reads infinite resistance for the open circuit. Fig : An open coil reads infinite ohms when its continuity is checked with an ohmmeter. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**19-15: Measuring and Testing Inductors**

Fig : The internal dc resistance ri of a coil is in series with its inductance L. A coil has dc resistance equal to the resistance of the wire used in the winding. As shown in Fig , the dc resistance and inductance of a coil are in series. Although resistance has no function in producing induced voltage, it is useful to know the dc coil resistance because if it is normal, usually the inductance can also be assumed to have its normal value.

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