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Inductance and Magnetic Fields
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Chapter 14
Introduction
Electromagnetism
Reluctance
Inductance
Self-inductance
Inductors
Inductors in Series and Parallel
Voltage and Current
Sinusoidal Voltages and Currents
Energy Storage in an Inductor
Mutual Inductance
Transformers
Circuit Symbols
The Use of Inductance in Sensors
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Introduction
14.1
 Earlier we noted that capacitors store energy by
producing an electric field within a piece of dielectric
material
 Inductors also store energy, in this case it is stored
within a magnetic field
 In order to understand inductors, and related
components such as transformers, we need first to
look at electromagnetism
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Electromagnetism
14.2
 A wire carrying a
current I causes a
magnetomotive
force (m.m.f) F
– this produces a
magnetic field
– F has units of
Amperes
– for a single wire
F is equal to I
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 The magnitude of the field is defined by the
magnetic field strength, H , where
HI
l
where l is the length of the magnetic circuit
 Example – see Example 14.1 from course text
A straight wire carries a current of 5 A. What is the magnetic
field strength H at a distance of 100mm from the wire?
Magnetic circuit is circular. r = 100mm, so path = 2r = 0.628m
I
5
H 
 7.96 A /m
l 0.628
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 The magnetic field produces a magnetic flux, 
– flux has units of weber (Wb)
 Strength of the flux at a particular location is measured
in term of the magnetic flux density, B
– flux density has units of tesla (T) (equivalent to 1 Wb/m2)
 Flux density at a point is determined by the field
strength and the material present
or
B  μ0 μ r H
B  μH
where  is the permeability of the material, r is the relative
permeability and 0 is the permeability of free space
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 Adding a ferromagnetic ring around a wire will
increase the flux by several orders of magnitude
– since r for ferromagnetic materials is 1000 or more
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 When a current-carrying
wire is formed into a coil
the magnetic field is
concentrated
 For a coil of N turns the
m.m.f. (F) is given by
F  IN
and the field strength is
H  IN
l
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 The magnetic flux produced is determined by the
permeability of the material present
– a ferromagnetic material will increase the flux density
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Reluctance
14.3
 In a resistive circuit, the resistance is a measure of
how the circuit opposes the flow of electricity
 In a magnetic circuit, the reluctance, S is a measure
of how the circuit opposes the flow of magnetic flux
 In a resistive circuit R = V/I
 In a magnetic circuit
SF
Φ
– the units of reluctance are amperes per weber (A/ Wb)
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Inductance
14.4
 A changing magnetic flux induces an e.m.f. in any
conductor within it
 Faraday’s law:
The magnitude of the e.m.f. induced in a circuit is
proportional to the rate of change of magnetic flux
linking the circuit
 Lenz’s law:
The direction of the e.m.f. is such that it tends to
produce a current that opposes the change of flux
responsible for inducing the e.m.f.
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 When a circuit forms a single loop, the e.m.f. induced
is given by the rate of change of the flux
 When a circuit contains many loops the resulting e.m.f.
is the sum of those produced by each loop
 Therefore, if a coil contains N loops, the induced
voltage V is given by
V  N dΦ
dt
where d/dt is the rate of change of flux in Wb/s
 This property, whereby an e.m.f. is induced as a result
of changes in magnetic flux, is known as inductance
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Self-inductance
14.5
 A changing current in a wire causes a changing
magnetic field about it
 A changing magnetic field induces an e.m.f. in
conductors within that field
 Therefore when the current in a coil changes, it
induces an e.m.f. in the coil
 This process is known as self-inductance
V  L dI
dt
where L is the inductance of the coil (unit is the Henry)
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Inductors
14.6
 The inductance of a coil depends on its dimensions
and the materials around which it is formed
μ0 AN 2
L
l
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 The inductance is greatly increased through the use
of a ferromagnetic core, for example
μ0 μr AN 2
L
l
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 Equivalent circuit of an inductor
 All real circuits also possess stray capacitance
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Inductors in Series and Parallel
14.7
 When several inductors are connected together their
effective inductance can be calculated in the same
way as for resistors – provided that they are not
linked magnetically
 Inductors in Series
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 Inductors in Parallel
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Voltage and Current
14.8
 Consider the circuit shown here
– inductor is initially un-energised
 current through it will be zero
– switch is closed at t = 0
– I is initially zero
 hence VR is initially 0
 hence VL is initially V
– as the inductor is energised:

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
I increases
VR increases
hence VL decreases
we have exponential behaviour
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 Time constant
– we noted earlier that in a capacitor-resistor circuit the
time required to charge to a particular voltage is
determined by the time constant CR
– in this inductor-resistor circuit the time taken for the
current to rise to a certain value is determined by L/R
– this value is again the time constant  (greek tau)
 See Computer Simulation Exercises 14.1 and 14.2
in the course text
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Sinusoidal Voltages and Currents
14.9
 Consider the application of a
sinusoidal current to an inductor
– from above V = L dI/dt
– voltage is directly proportional to
the differential of the current
– the differential of a sine wave is
a cosine wave
– the voltage is phase-shifted by
90 with respect to the current
– the phase-shift is in the opposite
direction to that in a capacitor
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Energy Storage in an Inductor
14.10
 Can be calculated in a similar manner to the energy
stored in a capacitor
 In a small amount of time dt the energy added to the
magnetic field is the product of the instantaneous
voltage, the instantaneous current and the time
di
Energy added  vi dt  L idt  Li di
dt
 Thus, when the current is increased from zero to I
1 2
E  L  idt  LI
0
2
I
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Mutual Inductance
14.11
 When two coils are linked magnetically then a
changing current in one will produce a changing
magnetic field which will induce a voltage in the other
– this is mutual inductance
 When a current I1 in one circuit, induces a voltage V2
in another circuit, then
dI
V M 1
dt
2
where M is the mutual inductance between the circuits. The
unit of mutual inductance is the Henry (as for self-inductance)
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 The coupling between the coils can be increased by
wrapping the two coils around a core
– the fraction of the magnetic field that is coupled is
referred to as the coupling coefficient
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 Coupling is particularly important in transformers
– the arrangements below give a coupling coefficient that
is very close to 1
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Transformers
14.12
 Most transformers approximate to ideal components
– that is, they have a coupling coefficient  1
– for such a device, when unloaded, their behaviour is
determined by the turns ratio
– for alternating voltages
V
N
2 2
V
N
1
1
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 When used with a resistive load, current flows in the
secondary
– this current itself produces a magnetic flux which
opposes that produced by the primary
– thus, current in the
secondary reduces
the output voltage
– for an ideal transformer
V1 I1  V2 I2
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Circuit Symbols
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14.13
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The Use of Inductance in Sensors
14.14
 Numerous examples:
 Inductive proximity
sensors
– basically a coil
wrapped around a
ferromagnetic rod
– a ferromagnetic plate coming close to the coil changes
its inductance allowing it to be sensed
– can be used as a linear sensor or as a binary switch
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 Linear variable differential transformers (LVDTs)
– see course text for details of operation of this device
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Key Points
 Inductors store energy within a magnetic field
 A wire carrying a current creates a magnetic field
 A changing magnetic field induces an electrical voltage in
any conductor within the field
 The induced voltage is proportional to the rate of change of
the current
 Inductors can be made by coiling wire in air, but greater
inductance is produced if ferromagnetic materials are used
 The energy stored in an inductor is equal to ½LI2
 When a transformer is used with alternating signals, the
voltage gain is equal to the turns ratio
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