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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.1 Inductance and Magnetic Fields 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 Chapter 14

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.2 Introduction 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 14.1

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.3 Electromagnetism 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 14.2

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.4 The magnitude of the field is defined by the magnetic field strength, H, where 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.5 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/m 2 ) Flux density at a point is determined by the field strength and the material present or where is the permeability of the material, r is the relative permeability and 0 is the permeability of free space

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.6 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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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.7 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 and the field strength is

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.8 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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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.9 Reluctance 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 –the units of reluctance are amperes per weber (A/ Wb) 14.3

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.10 Inductance 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. 14.4

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.11 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 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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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.12 Self-inductance 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 where L is the inductance of the coil (unit is the Henry) 14.5

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.13 Inductors The inductance of a coil depends on its dimensions and the materials around which it is formed 14.6

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.14 The inductance is greatly increased through the use of a ferromagnetic core, for example

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.15 Equivalent circuit of an inductor All real circuits also possess stray capacitance

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.16 Inductors in Series and Parallel 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 14.7

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.17 Inductors in Parallel

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.18 Voltage and Current 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 V R is initially 0 hence V L is initially V –as the inductor is energised: I increases V R increases hence V L decreases we have exponential behaviour 14.8

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.19 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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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.20 Sinusoidal Voltages and Currents Consider the application of a sinusoidal current to an inductor –from above V = L d I /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 14.9

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.21 Energy Storage in an Inductor 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 Thus, when the current is increased from zero to I 14.10

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.22 Mutual Inductance 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 I 1 in one circuit, induces a voltage V 2 in another circuit, then where M is the mutual inductance between the circuits. The unit of mutual inductance is the Henry (as for self-inductance) 14.11

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.23 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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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.24 Coupling is particularly important in transformers –the arrangements below give a coupling coefficient that is very close to 1

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.25 Transformers 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 14.12

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.26 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.27 Circuit Symbols 14.13

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.28 The Use of Inductance in Sensors 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 14.14

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.29 Linear variable differential transformers (LVDTs) –see course text for details of operation of this device

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 14.30 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 ½LI 2 When a transformer is used with alternating signals, the voltage gain is equal to the turns ratio

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