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Chapter
Inductance
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
19
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
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
19
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.
This ability is called self-inductance, or simply
inductance.
19
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.
19
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.
19
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.
19
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.
19
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.
19
19--2: SelfSelf-Inductance L
 The symbol for inductance is L, for linkages of magnetic
flux.
L=
VL
di / dt
 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.
19
19--2: SelfSelf-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.
19
19--2: SelfSelf-Inductance L
Calculating the Inductance of a Long Coil
air-core
symbol
d
iron-core
symbol
(μ r = 1)
(μr >> 1)
L = μr
N 2A
l
1.26 × 10−6 H
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
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19
19--2: SelfSelf-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 mH (microhenry) = 1 × 10-6 H
1919-3: SelfSelf-Induced Voltage vL
 Formula:
vL = L
di
dt
 Induced voltage is proportional to
inductance (L).
 Induced voltage is proportional to the rate of
current change:
di
dt
1919-3: SelfSelf-Induced Voltage vL
Energy Stored in the Field
LI 2
Energy =
2
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.
19
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.
19--5: Mutual Inductance LM
19
 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 M = k L 1L 2
19--5: Mutual Inductance LM
19
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.
19--5: Mutual Inductance LM
19
 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.
19--5: Mutual Inductance LM
19
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.
19--5: Mutual Inductance LM
19
Calculating LM
 Mutual inductance increases
with higher values for
primary and secondary
inductances.
 LM = k L1  L2
where L1 and L2 are the selfinductance values of the two
coils, k is the coefficient of
coupling, and LM is the
mutual inductance.
19--6: Transformers
19
 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.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--6: Transformers
19
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.
19--6: Transformers
19
 A transformer can step up or step down the voltage
level from the ac source.
A transformer is a device that
uses the concept of mutual
inductance to step up or step
down an alternating voltage.
Primary
Secondary
Load
Step-up (VLOAD > VSOURCE)
Primary
Secondary
Load
Step-down (VLOAD < VSOURCE)
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--6: Transformers
19
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.
19--6: Transformers
19
 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%).
19--6: Transformers
19
 Voltage Ratio
1:3
Step-up (1:3)
120 V
Primary
Secondary
Step-down (3:1)
3:1
120 V
Primary
Secondary
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
VL = 3 x 120
= 360 V
Load
360 V
VL = 1/3 x 120
= 40 V
Load
40 V
19--6: Transformers
19
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.
19--6: Transformers
19
 Current Ratio
120 V
1:3
IL = 1/3 x 0.3
= 0.1 A
Primary
Secondary
Load 360 V
0.3 A
0.1 A
IS/IP = VP/VS
3:1
120 V
Primary
0.1 A
IL = 3 x 0.1
= 0.3 A
Secondary
0.3 A
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Load
40 V
19--6: Transformers
19
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%.
19--6: Transformers
19
 Transformer Efficiency
120 V
Primary
3:1
Secondary
Load
40 V
0.3 A
0.12 A
PPRI = 120 x .12 = 14.4 W
PSEC = 40 x 0.3 = 12 W
PSEC
Efficiency =
12
× 100 % = 83 %
× 100 % =
PPRI
14.4
Primary power that is lost is dissipated as heat in the transformer.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--6: Transformers
19
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 = V S x I S
 Use PS to find PP:
PP = P S
 Finally, IP can be
calculated:
IP = PP/VP
19--6: Transformers
19
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
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
19
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.
19
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.
19
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.
19
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.
19--8: Impedance Transformation
19
 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:

N
P
ZP =    ZS
 NS 
2
primary
secondary
19--8: Impedance Transformation
19
 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:
NP
=
NS
ZP
ZS
19--8: Impedance Transformation
19
Impedance Ratio
1:3
Primary
ZRATIO = 1/9
Secondary
Load = 9 W
The load on the source is 1 W.
3:1
ZRATIO = 9/1
Primary
Secondary
Load = 9 W
The load on the source is 81 W.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--8: Impedance Transformation
19
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.
19--8: Impedance Transformation
19
Internal resistance (ri) is 200 Ω.
RL is 8 Ω.
Fig. 19-20: 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.
19--8: Impedance Transformation
19
VG 2
PL = (
)  RL
ri  RL
100V
=(
) 2  8
200  8
= 1.85W
Connected as shown, the load receives 1.85 W of power.
Fig. 19-20: (b) Connecting the amplifier directly to RL.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--8: Impedance Transformation
19
Connection shown increases the power delivered to the load.
Fig. 19-20(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.
19--8: Impedance Transformation
19
Turns ratio
NP
=
NS
=
5
=
1
ZP
ZS
200
200

8
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.
19-9: Core Losses
Fig. 19-21: Cross-sectional view of iron core showing eddy currents.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19
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).
19--10: Types of Cores
19
 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
19--10: Types of Cores
19
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.
19--11: Variable Inductance
19
 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. 19-24: Methods of varying inductance. (a) Tapped coil. (b) Slider contact.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--11: Variable Inductance
19
 Figure 19-24 (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. 19-24: Methods of varying inductance. (c) Adjustable slug.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--11: Variable Inductance
19
 The symbol in Fig. 19-24 (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. 19-24: Methods of varying inductance. (d) Variometer.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--11: Variable Inductance
19
 For any method of varying
L, the coil with an arrow in
Fig. 19-24 (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. 19-24: Methods of varying inductance. (e) Symbol for variable L.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19--12: Inductances in
19
Series or Parallel
 With no mutual coupling:
 For series circuits, inductances add just like
resistances.
LT = L1 + L2 + L3 + ... + etc.
 For parallel circuits, inductances combine according
to a reciprocal formula as with resistances.
1
LEQ =
1
1
1
+
+
+ ... + etc.
L3
L1
L2
19--12: Inductances in
19
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.
19--12: Inductances in
19
Series or Parallel
The coupling depends on the coil connections and direction of
winding. Reversing either one reverses the field.
Fig. 19-28: Inductances L1 and L2 in series but with mutual coupling LM. (a) Aiding magnetic
fields. (b) Opposing magnetic fields.
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19--12: Inductances in
19
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.
19--13: Energy in Magnetic
19
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 = 1/2LI2
The factor of ½ gives the average result of I in
producing energy.
1919-14: 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.
1919-14: 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.
1919-14: Stray Capacitive and
Inductive Effects
 As shown in Fig. 19-29, 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. 19-29: 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.
1919-14: Stray Capacitive and
Inductive Effects
 As shown by the high-frequency
equivalent circuit in Fig. 19-30, a
resistor can include a small
amount of inductance and
capacitance.
 The inductance of carboncomposition resistors is usually
negligible.
 Wire-wound resistors, however,
have enough inductance to be
evident.
Fig. 19-30: High-frequency equivalent circuit of a resistor.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19
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 capacitorinductor analyzer should be
used.
Fig. 19-31: Typical LCR meter.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19
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. 19-31 (previous
slide).
 This is a handy piece of test equipment, however,
most LCR meters only measure the value of a
component.
19
19--15: Measuring and
Testing 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.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
19
19--15: Measuring and
Testing Inductors
 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.
Fig. 19-33: The internal dc resistance ri of a coil is in series with its inductance L.