1. 14.1
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

2. Adding a ferromagnetic ring around a wire will increase the flux by several orders of magnitude

3. The magnetic flux produced is determined by the permeability of the material present
a ferromagnetic material will increase the flux density

4. 14.3
Reluctance
In a resistive circuit, the resistance is a measure of how the circuit opposes the flow of electricity
In a resistive circuit R = V/I

5. 14.4
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.

6. When a circuit forms a single loop, the e. m. f
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

7. 14.5
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)

8. 14.6
Inductors
The inductance of a coil depends on its dimensions and the materials around which it is formed

9. The inductance is greatly increased through the use of a ferromagnetic core, for example

10. Equivalent circuit of an inductor
All real circuits also possess stray capacitance

11. Inductors in Series and Parallel
14.7
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

12. Inductors in Parallel

13. Voltage and Current Consider the circuit shown here
14.8
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 VR is initially 0
hence VL is initially V
as the inductor is energised:
I increases
VR increases
hence VL decreases
we have exponential behaviour

14. 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)

15. Sinusoidal Voltages and Currents
14.9
Sinusoidal Voltages and Currents
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

16. Energy Storage in an Inductor
14.10
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

17. 14.11
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 I1 in one circuit, induces a voltage V2 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)

18. 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

19. Coupling is particularly important in transformers
the arrangements below give a coupling coefficient that is very close to 1

20. Transformers Most transformers approximate to ideal components
14.12
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

21. 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

22. 14.13
Circuit Symbols

23. The Use of Inductance in Sensors
14.14
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

24. Linear variable differential transformers (LVDTs)

25. 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