Transcript Bates

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.
The ability of a conductor to induce voltage in
itself when the current changes is called selfinductance, or simply inductance.
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.
19-1: 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.
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.
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?
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:
L is the inductance in henrys
I is the current in amperes
http://www.magnet.fsu.edu/education/tutorials/java/index.html
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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 ?
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.
http://www.launc.tased.edu.au/online/sciences/Physics/Lenz%27s.html
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
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
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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
12
× 100 % = 83 %
× 100 % =
Efficiency =
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 = V S x I S
 Use PS to find PP:
PP = P S
 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.
19-12: 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
1
1
+
+
+ ... + etc.
L3
L1
L2
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.
19-15: 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.