1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

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

16. 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

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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

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

28. 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

29. 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

30. 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