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InductanceInductance

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

ChapterChapter1919

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 19Topics 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

19-1: Induction by 19-1: Induction by Alternating CurrentAlternating 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.


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.


Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Fig. 19-1: Magnetic field of an alternating current is effectively in motion as it expands and contracts with the current variations.

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.


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.


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.


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.

19-2: Self-Inductance19-2: Self-Inductance L 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

ExamplesExamples

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?


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.


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 2A1.26 × 10−6 Hμr

dair-coresymbol

μ r = 1)

iron-coresymbol

μr >> 1)



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

19-3: Self-Induced Voltage 19-3: Self-Induced Voltage vvLL

( )di

dtvL L=

Formula:

Induced voltage is proportional to inductance (L).

Induced voltage is proportional to the rate of

current change: ( )di

dt

19-3: Self-Induced Voltage 19-3: Self-Induced Voltage vvLL

Energy Stored in the Field

2LI 2

Energy Where the energy is in joules:L is the inductance in henrysI is the current in amperes


http://www.magnet.fsu.edu/education/tutorials/java/index.html

ExamplesExamples

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 ?

19-4: How 19-4: How vvLL Opposes Opposes a Change in Currenta 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.

http://www.launc.tased.edu.au/online/sciences/Physics/Lenz%27s.html

19-5: Mutual Inductance 19-5: Mutual Inductance LLMM

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 L LM 1 2


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.


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.



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.

Loose coupling Tighter coupling Unity coupling Zero coupling


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.

1 2= k L L

19-6: Transformers19-6: 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. 19-11: Iron-core power transformer.



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.

The transformer transfers power from the primary to the secondary.

Transformer steps up voltage (to 100V) and steps current down (to 1A)


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)

Primary Secondary Load


A transformer is a device that uses the concept of mutual inductance to step up or step down an alternating voltage.


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.


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%).


Voltage Ratio

3:1

Primary Secondary Load120 V 40 V

1:3



Step-up (1:3)

Step-down (3:1)

VL = 3 x 120 = 360 V

VL = 1/3 x 120 = 40 V


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.


Current Ratio

3:1


0.1 A0.3 A

1:3

120 V Primary Secondary Load 360 V

0.3 A0.1 A


IS/IP = VP/VS

IL = 1/3 x 0.3 = 0.1 A

IL = 3 x 0.1 = 0.3 A


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%.


Transformer Efficiency


3:1

0.12 A0.3 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.


Loaded Power Transformer

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

1:6

20:1


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.

19-7: Transformer Ratings19-7: 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


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.


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.


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.


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.

19-12: Inductances in 19-12: Inductances in Series or ParallelSeries 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 + L3 + ... + etc.

LEQ = 1

+ ... + etc.++L3

1

L2

1

L1

1

19-13: Energy in Magnetic 19-13: Energy in Magnetic Field of InductanceField 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.

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

The most common trouble in coils is an open winding.

As shown in Fig. 19-32, an ohmmeter connected across the coil reads infinite resistance for the open circuit.


Fig. 19-32: An open coil reads infinite ohms when its continuity is checked with an ohmmeter.

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

Fig. 19-33: 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. 19-33, 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.