Chapter
19
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
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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
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.
A change in current induces an EMF that
opposes the change in current .This ability
is called self-inductance, or simply
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.
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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.
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.
19-1: 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.
19-2: Self-Inductance L
The symbol for inductance is L, for linkages of magnetic
flux.
VL
L=
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.
19-2: 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.
19-2: Self-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
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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
19-3: Self-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
19-3: Self-Induced Voltage vL
Energy Stored in the Field
LI 2
Energy =
2
Where the energy is in joules:
8L is the inductance in henrys
8I is the current in amperes
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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 M = k L 1L 2
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.
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.
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.
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19-5: Mutual Inductance LM
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
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.
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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.
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19-6: Transformers
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)
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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.
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%).
19-6: Transformers
Voltage Ratio
1:3
Step-up (1:3)
120 V
Primary
Secondary
Step-down (3:1)
3:1
120 V
Primary
Secondary
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VL = 3 x 120
= 360 V
Load
360 V
VL = 1/3 x 120
= 40 V
Load
40 V
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.
19-6: Transformers
Current Ratio
120 V
1:3
Primary
IL = 1/3 x 0.3
= 0.1 A
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
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Load
40 V
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%.
19-6: Transformers
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.
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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
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.
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
19-7: 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.
19-7: 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.
19-7: 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.
19-7: 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
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:
2
primary

N
P
ZP =   × ZS
 NS 
secondary
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:
NP
=
NS
ZP
ZS
19-8: Impedance Transformation
Impedance Ratio
1:3
Primary
ZRATIO = 1/9
Secondary
Load = 9 Ω
The load on the source is 1 Ω.
3:1
ZRATIO = 9/1
Primary
Secondary
Load = 9 Ω
The load on the source is 81 Ω.
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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.
19-8: Impedance Transformation
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.
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19-8: Impedance Transformation
VG 2
PL = (
) × RL
ri + RL
100V
=(
) 2 × 8Ω
200Ω + 8Ω
= 1.85W
Connected as shown, the load receives 1.85 W of power.
Fig. 19-20: (b) Connecting the amplifier directly to RL.
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19-8: Impedance Transformation
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.
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19-8: Impedance Transformation
Turns ratio
NP
=
NS
=
5
=
1
ZP
ZS
200Ω
8Ω
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.
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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).
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
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.
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. 19-24: Methods of varying inductance. (a) Tapped coil. (b) Slider contact.
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19-11: Variable Inductance
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.
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19-11: Variable Inductance
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.
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19-11: Variable Inductance
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.
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19-12: Inductances in
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
L1
+
1
L2
+
1
L3
+ ... + etc.
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.
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. 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
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
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.
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.
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.
19-14: 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.
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19-14: 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.
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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 capacitorinductor analyzer should be
used.
Fig. 19-31: Typical LCR meter.
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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. 19-31 (previous
slide).
This is a handy piece of test equipment, however,
most LCR meters only measure the value of a
component.
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.
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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.