Introduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
 5. Magnetic Field
 5.3. Electromagnetic Induction
 5.3.1. The General Law of Induction 5.3.2. Utilisation of the Phenomena of Induction 5.3.3. Inductance

### 5.3.1. The General Law of Induction

The phenomenon of electromagnetic induction was discovered by the English physicist Michael Farady (1791 - 1867) in 1831. The law of induction, which was named after him, is of paramount importance to electrical engineering. Together with Biot-Savart’s law (see Section 5.1.), it forms the theoretical basis of all electromagnetic phenomena and numerous technical applications such as generators, motors, transformers, measuring instruments, etc.

In Section 2.2. (see Fig. 2.7.) we have already pointed, out that a primary electromotive force is produced in a conductor loop encircling a magnetic flux when this flux changes. A distinction is made between two types:

· induction of rest
· induction of motion

The two processes can also take place at the same time; then the phenomena of induction, which will be dealt with separately below, are superimposed.

In the process of induction of rest, a conductor at rest (a conductor loop, a coil) is penetrated by a magnetic field changing as to time. This may be effected, for example, by approaching a permanent magnet to a coil and then withdrawing it in the manner shown in Fig. 5.17.

Fig. 5.17. Generation of primary electromotive force by magnetic flux variation

1 - North pole
2 - South pole

It is obvious that during the approaching of the permanent magnet the magnetic flux penetrating the coil becomes greater; during withdrawing, this flux diminishes again. It is found that, in the coil, a primary electromotive force is produced as long as the magnetic flux penetrating the coil changes. The electromotive force is the greater, the quicker the magnetic flux changes, in other words, the higher the speed of approach and the greater the number of turns in the coil. The direction of the electromotive force is derived from the law of conservation of energy. When, in the manner demonstrated by Fig. 5.17., a permanent magnet is approached to a coil, then at the side facing the magnet, a homonymous magnetic pole is formed. It exerts a repellent force on the magnet to be approached. Hence, a work has to be done against the repellent force exerted by the coil through which the induced current flows. This work is the equivalent of the electrical energy obtained. When withdrawing the magnet from the coil, the conditions are inverted. These facts were described by the Russian physicist Heinrich Friedrich Emil Lenz (1804 - 1865) and, called Lenz law, run as follows:

The induced electromotive force is always in such direction that, if it produces a current, the magnetic effect of that current opposes the flux variation responsible for both electromotive force and current.

A positive induced voltage is caused by a negative flux variation (flux reduction). Mathematically, this is expressed by a negative sign (“-”).

Consequently, as the general law of induction we have:

E = - N · DF/Dt

(5.12)

where:

 E induced electromotive force in the coil N number of turns

DF/Dt speed of variation of the magnetic flux

[E] = Wb/s = V · s/s = V

Example 5.4.

In a coil having 1200 turns, the magnetic flux varies within 50 ms from 7.5µWb to 70µWb. Calculate the value of the electromotive force induced in this coil!

Given:

N = 1200
DF/Dt = [(70 - 7.5) µWb]/50 ms

To be found:

|E|

Solution:

E = - N · DF/Dt

|E| = N · DF/Dt

|E| = 1.5V

An induction of rest also takes place when - according to the demonstration given in Fig. 5.18. - two coils are magnetically coupled and one is energised. Coil 1 (primary coil) has a number of turns expressed as K.; the current I1, passing through them can be interrupted by a switch. Coil 2 (secondary coil) has a number of turns expressed as N2 and is connected with a load.

Fig. 5.18. Induction of rest in magnetically coupled coiles

When the switch is closed, a current flows through the primary coil and produces a magnetomotive force. It drives a magnetic flux F1 whose greater part, namely kF1, also penetrates the secondary coil. The quantity k is called coefficient of coupling and indicates how many per cent of the produced magnetic flux penetrates the matching coil. In closed ferromagnetic circuits (as in the present example) k » 1 (or 100 %). Consequently, an electromotive force of is induced in the secondary coil according to equation (5.12.).

When the switch is opened, the magnetic flux diminishes and the voltage induced in the secondary coil inverts its direction.

Since F1 = (N1 · I1)/Rm [see equations (5.3) and (5.1)], hence,

DF1 = N1/Rm · DI1, we have

This process is called, mutual induction. The latter equation indicates that, in the event of a current variation in the primary coil, an electromotive force is induced in the secondary coil whose magnitude is proportional to the rate of current change DI1/Dt. The conditions are analogous when secondary and primary coils are exchanged. The proportionality factor is called mutual inductance M.

M = k · N1N2/Rm

(5.13)

E2 = - M · DI1/Dt

(5.14)

 M mutual inductance k coefficient of coupling N1 number of turns in coil 1 N2 number of turns in coil 2 Rm magnetic resistance E2 electromotive force induced in coil 2 M mutual inductance DI1/Dt rate of current change in coil 1

[M] = 1/A/(V · s) = V · s/A = H
[E2] = V · s/A · A/s = V

Example 5.5.

The mutual inductance of two coils is 100 mH. Calculate the electromotive force induced in the secondary coil when the current in the primary coil is uniformly changed by 500 mA within 20 ms!

Given:

M = 100 mH
DI1/Dt = 500 mA/20 ms

To be found:

E2

Solution:

E2 =- M DI1/Dt
E2 = - 100 mH · 500 mA/20 ms
E2 = - 100 ··10-3 (V·· s)/A · 25 A/s = - 2500 · 10-3 V
E2 = - 2.5 V

The general law of induction does not state anything about the origin of the magnetic flux but only the fact that, as a consequence of changes in the magnetic flux, voltages are induced in the turns of a coil encircling it. When we omit the secondary coil in Fig. 5.18., the following conditions are brought about:

When we close the switch, a current passes through the coil producing a magnetomotive force. The latter drives a magnetic flux which penetrates all turns of the coil generating an electromotive force in these turns. When the switch is opened, the magnetic field breaks down, the magnetic flux is reduced to zero. This reduction of flux will also cause an induced voltage in the coil itself. This process of production of electromotive force in the turns of the coil generating the field is called self-induction.

The magnitude of the self-induced voltage can easily be determined on the basis of the general law of induction.

E = - N · DF/Dt

Since F = NI/Rm [see equations (5.3) and (5.1)], hence
F = N/Rm · DI we have
E = - N · N/Rm · DI/Dt

The latter equation indicates that, in a current-carrying coil, an electromotive force is induced by the magnetic flux associated with the current in the coil; the magnitude of the electromotive force is proportional to the rate of current change DI/Dt.

The proportionality factor is called inductance L (more precisely self-inductance).

L = N2/Rm

(5.15)

E = - L·· DI/Dt

(5.16)

where:

 L self-inductance N number of turns of the coil Rm magnetic resistance E self-induced voltage L self-inductance DI/Dt rate of current change in the coil

[L] = (1/A)/(V · s) = (V · s)/A = H
[E] = [(V ··s)/A] · (A/s) = V

Example 5.6.

A coil has a self-inductance of 1.35 H. Calculate the self-induced voltage when the current is uniformly reduced by 5.7 mA within 20/µs.

Given:

L = 1,35 H
DI/Dt = 3,7 mA/20 µs

To be found:

E

Solution:

E = - L · DI/Dt
E = - 1,35 H · (- 3,7 mA/20 µs) = [1,35 (V·· s)/A] · (185 A/s)
E = 250 V

The correlation between mutual inductance and self-inductance results from the equations (5.13) and (5.15.)

with L = N2/Rm we have M2 = k2L1L2

(5.17)

The mutual inductance of two magnetically coupled coils is equal to the product of coupling factor times geometric mean of the self-inductances.

In case of the induction of motion, a conductor (a conductor loop, a coil) is moved through a magnetic field constant as to time. This may be effected, for example, by moving a conductor loop through a homogeneous magnetic field having the magnetic flux density B, length 1 and width s in accordance with Fig. 5.19.

Fig. 5.19. Induction of motion

1 - Position of the conductor at the beginning of motion
2 - Position of the conductor after a certain period

Assume, the conductor loop is moved at a constant speed v. At the commencement of motion, the conductor loop has the position 1 and after a certain period of time Dt, position 2. During the time Dt, the distance Ds has been covered, a facht, which corresponds to a rate of motion of v = Ds/Dt. The magnetic flux encircled by the conductor loop becomes greater by the share which, in this time, enters through the area DA. Thus, the induced voltage for N turns is

E = - N · DF/Dt
DF = B · DA
DA = I · Ds

E = - N · (B · I · Ds)/Dt

v = Ds/Dt we have

E = - NBlv

(5.18)

where:

 E induced voltage N number of turns of the coil B magnetic flux density of the constant magnetic field l length of the magnetic field v rate of motion of the coil or conductor

[E] = (V · s)/m2 · m ··m/s = v

Example 5.7.

A wire is moved, at a constant speed of 12.5 cm/s across a homogeneous magnetic field, of 500 mT and 8 cm in length. Calculate the value of the voltage induced, in this wire!

Given:

N = 1 (coil with 1 turn!)
B = 500 mT
l = 8 cm
V = 12,5 cm/s

To be found:

|E|

Solution:

E = - NBlv
|E| = 1 · 0,5T · 8 cm · 12,5 cm/s
|E| = 1 · 0,5 (V·· s)/m2 · 8 ··10-2 m · 12,5 · 10-2 m/s = 50 · 10-4 V
|E| = 5 mV

According to the Lenz law, the direction of the induced electromotive force is such that the magnetic field caused by the induced current acts against the cause of origin, in this case an increase in flux. From this follows the indicated current direction. For the direction of the induced voltage or of the current driven by this voltage in the induction of motion, the right-hand rule (generator rule) holds in general.

When extending the opened right hand into the magnetic field in such a way that the field lines enter the inner palm and the spread out thumb points in the direction of motion of the conductor, then the extended fingers point in the direction of the induced electromotive force (Fig. 5.20).

### 5.3.2. Utilisation of the Phenomena of Induction

Phenomena of induction are utilised very frequently in engineering. The examples given below are a very limited selection of typical applications.

Fig. 5.20. Right-hand rule (generator rule)

· Magnetic head of magnetophone (induction of rest)

The magnetophone process is a high-grade sound storage method. The sound store is a thin (18 to 50 µm) polyester or acetate tape with a ferromagnetic film applied to it. The sound information is entered in the tape in the form of a more or less intense magnetisation in the running direction of the tape and thus stored. For the reproduction of the stored information, a magnetic replay head is required. This is a magnetic circuit with a very small air gap (1 to 100 am) which carries an induction coil. The magnetised tape is transported past the air gap. The flux caused by the individual “permanent magnets” of the tape in the ferromagnetic circuit induces a voltage proportional to the sound information in the induction coil surrounding the circuit (Fig. 5.21.). This voltage is amplified and fed to a loudspeaker.

Fig. 5.21. Reproducing head of a magnetic recorder

1 - to amplifier
2 - Iron core
3 - Magnetic tape

· Transformer (induction of rest)

The transformer is a stationary electrical machine and one of the most important components of electrical engineering. In accordance with Fig. 5.18., the transformer is provided with two coils which are galvanically separated but magnetically coupled. When a voltage is applied, to the primary coil which periodically changes as to magnitude and direction (such a voltage is called alternating voltage, see Section 7), a magnetic field is produced in both coils which also changes periodically its magnitude and direction. In an ideal case, a 100 per cent coupling is effected, i.e. the coefficient of coupling k = 1. Then the magnetic flux F1 completely penetrates the secondary coil as F2, hence, F1 = F2 = F

Fig. 5.22. Transformer

According to the law of induction, the induced voltage is directly proportional to the number of turns when the rate of flux variation is given.

The ratio of the primary voltage to the secondary voltage is called ratio of transformation trr and is written as

trr = U1/U2 = N1/N2

(5.19.)

In a loss-free transformer, the ratio of the voltages is equal to the ratio of turns in the coils.

According to the law of conservation of energy, the primary power must be equal to the secondary power, hence, P1 = P2.

According to Section 4.1., equation (4.3), power is written as P = UI, that is to say,

U1I1 = U2I2

U1/U2 = I2/I1

Taking equation (5.19.) into account, we have

trr = U1/U2 = I2/I1 = N1/N2

(5.19a)

In a loss-free transformer, the currents are in inverse ratio to the numbers of turns of the coils.

This shows that a given alternating voltage can be transformed into any desired, higher or lower alternating voltage by means of a transformer. Therefore, the transformer is an important connecting link between energy generator and the distribution network or between the distribution network and the consumers. In information electrical engineering, the transformer is frequently used for impedance matching. Since P1 = P2, we have inaccordance with Section 4.1.,

I12R1 = I22R2

R1 = (I2/I1)2 · R2 with trr = I2/I1 we read

R1 = trr2 R2

(5.19b)

The load resistance R2 acts on the primary with the square of the transmission ratio.

· Generator (induction of motion)

In a homogeneous magnetic field, a conductor loop or a coil is arranged. If it is turned about its central axis which is perpendicular to the field direction, then the magnetic flux penetrating the coil area varies (see Fig. 5.23.).

Fig. 5.23. Generator (principle)

1 - North pole
2 - South pole

Since, according to the law of induction, any change of the magnetic flux causes an electromotive force, a voltage is induced in the rotating coil. Its direction can be deter-minded with the help of the right-hand rule. A generator is the inversion of the motor principle described in Section 5.2. Generalising, we can say that a pivoted coil in a magnetic field is the basic design of all rotating electrical machines (motors, generators).

· Eddy-current brake

In planar conductors voltages are induced by magnetic flux variations in the same manner as in wires and coils. The induction currents associated with these induced voltages are high because the current paths in a planar conductor are closed in themselves and act as short-circuits (see Fig. 5.25.). An experiment sketched in Fig. 5.24. shows the action of such induced currents. A metal plate of copper (or of another electrically conductive material such as aluminium) is suspended in such a way that it is allowed to swing through a magnetic field kie a pendulum. In this manner, currents are induced which, according to the Lenz rule, built up magnetic fields of opposite direction and thus damp the motion. The pendulum will come to rest very quickly.

Fig. 5.24. Electromagnetic induction in planar conductors

1 - Pendulum of non-ferromagnetic conductor material
2 - Total deflection of oscillation
3 - Oscillation in the magnetic field
4 - North pole
5 - South pole

Fig. 5.25. Eddy currents in planar conductors

Fig. 5.26. Reduction of the eddy currents by means of slots in planar conductors

Fig. 5.25. shows the closed current paths in the metal surface. Because of the apparently irregular course taken by the current, these currents are called eddy currents. Eddy currents can be avoided to a great extent when fine slots are made into the metal surface as shown in Fig. 5.26. The pendulum of such a slotted metal plate will hardly be damped; the braking action and thus the eddy-current formation are cancelled to a great extent.

Eddy-current brakes operate on the above described principle. They are used now and then for the braking of rail vehicles, for damping the deflection of electrical indicator operating mechanisms, and for braking electrical machines.

In most cases, eddy currents are not desired. They occur both by induction of rest in stationary electrical machines and by induction of motion in rotating electrical machines. Because of their short-circuit character, they heat the metal mass involved, thus, uselessly doing work. These eddy-current losses must be avoided as far as possible. This is achieved by avoiding compact metal masses. This is possible by composing metallic bodies of individual sheets insulated against each other and arranged in parallel to the direction of flux. Further eddy-current losses can be avoided when using ferromagnetic materials having a small electric conductivity, for example, sheet iron alloyed with silicon or certain iron-oxide compounds.

### 5.3.3. Inductance

· Inductance and coil

A wire usually wound on ferromagnetic core is called coil. This component stores energy at a certain current. The storage capacity for magnetic energy is called inductance of a coil.

L = N · F/I

(5.20.)

where:

 L inductance (more precisely self-inductance) F magnetic flux I current N number of turns

L = Vs/A = H

The following subunits are most frequently used:

1 mH = 1 milihenry = 10-3 H
1 µH = 1 microhenry = 10-6 H

The storage capacity of the coil is dependent on the number of turns, the dimensions and the permeability of the core. From the equations (5.5.) and 5.15.) we have

L = N2 · µ · A/I

(5.21.)

where:

 N number of turns µ permeability (material constant) A coil (core) cross-section l length of coil

Like resistors, coils can be connected in series or in parallel. In series connection according to fig. 5.27., the same current passes through the coils with the individual inductances of L1 and L2. In case of a current variation, voltage proportional to the individual inductances of the coils in induced in the latter. The equivalent inductance of this arrangement is

Lequ = L1 + L2

(5.22.)

This equation has the same structure as the equation for the determination of Requ of a series connection of resistors.

Fig. 5.27. Series connection of two coils (Lers = Lequ)

Fig. 5.28. Parallel connection of two coils (Lers = Lequ)

The parallel connection of two coils is shown in Fig. 5.28. The same voltage is applied to the two coils and the equivalent inductance is analogous to the equivalent resistance of resistors connected in parallel.

1/Lequ = 1/L1 + 1/L2

(5.23.)

From the equations (5.22.) and (5.23.), the following general statement can be derived: In a series connection of coils, the equivalent inductance is always greater than the greatest individual inductance and in a parallel connection of coils, the equivalent inductance is always smaller than the smallest individual inductance.

Example 5.8.

Two coils having the inductances of 1.5 H and 5 H have to be connected in series and then in parallel. Determine the equivalent inductances for these two types of connections!

Given:

L1 = 1.5 H
L2 = 3 H

To be found:

Lequ in series connection and in parallel connection

Solution:

series connection of L1 and L2

Lequ = L1 + L2
Lequ = 1.5 H + 3 H
Lequ = 4.5 H

parallel connection of L1 and L2

1/Lequ = 1/L1 + 1/L2 = (L2 + L1)/(L1L2)
Lequ = L1L2/(L1 + L2)
Lequ = (1.5 H · 3H)/(1.5 H + 3 H) = 4.5 H/4.5
Lequ = 1 H

· Behaviour of a coil in a direct-current circuit

A coil is connected to a direct voltage source according to Fig. 5.29. (switch position 1). At the instant of switching on (time t1), current starts flowing. The maximum current limited by R cannot flow immediately because self-induction counteracts any current change. After a short time, the current has reached a certain value and the magnetic flux the value proportional to the current. The current causes a voltage drop at the resistor R; consequently, the voltage across the coil is reduced. In the following time, the current is not allowed to rise as quickly as immediately after the instant of switching on. All this shows that, after switching on, the current first increases rapidly and then more and more slowly while the coil voltage first drops rapidly and then more and more slowly.

Fig. 5.29. Circuit for switching on and off of a direct voltage in a coil

Now, the voltage source is to be switched off from the coil (switch position 2). At the instant of switching off (time t2), the current passing through the coil is not immediately interrupted because self-induction opposes any current change. The starting change in current causes a self-induced voltage which, according to the Lena rule, is so directed that it counteracts the cause of origin. An induced current is driven in the same direction as before when the voltage source was connected. Now, the magnetic field gradually dies out and the stored magnetic energy is converted into heat energy in resistor R.

The course taken by current and voltage during switching on and off is shown in Fig. 5.30. It is evident that at the instant of switching on and at the instant of switching off the coil voltage reaches its highest value rapidly and, after some time, drops to zero. The current, however, changes its value only slowly in switching. In coils, there are no sudden current changes.

Fig. 5.30. Behaviour of current and voltage in a coil when a direct voltage is being switched on and off

When switching off a coil, the following should be observed: The energy stored in the magnetic field is only maintained by a current flow. In case of an interruption (instant of switching off), the field must disappear and the energy be converted into another form of energy. An instantaneous interruption (Dt = 0) according to the law of induction leads to a high induced voltage which can attain values of such a magnitude that connected components and the insulation of the coil winding may be destroyed.

When circuits include coils, caution is imperative at any time. In switching off, dangerous overvoltages can occur. They are prevented by closing the current path for the induced current. For this purpose, a resistor, a capacitor or a semiconductor diode is connected in parallel to the coil.

The coil is a storage element. The energy stored by a coil in the form of magnetic energy is

W = L/2 · I2

(5.24.)

where:

 W energy L inductance I current

W = (V · s)/A ··A2 = V · A · s = W · s

In a magnetic field considerably higher energies can be stored than in a dielectric field (see Section 6.2.2.). Therefore, large force actions can be achieved with magnetic fields.

Example 5.9.

A coil having an inductance of L = 10 H carries a current of 5 A. Calculate the energy stored!

Given:

I = 5 A
L = 10 H

To be found:

W

Solution:

W = L/2 · I2
W = 10/2 · (V·· s)/A · 52A2 = 5·· 25 W·· s
W = 125 W·· s

Any magnetic flux variation causes an electromotive force (electromagnetic induction). It is directed, in such a way that the magnetic field caused by the induced current counteracts the cause of its origin. A distinction is made between induction of rest and induction of motion. The electromagnetic induction forms the basis of a large number of technical applications including generators, motors, transformers and measuring instruments.

When a coil carries a current, the latter is associated with a magnetic flux. When the current varies, the magnetic flux also varies inducing an electromotive force. When this takes place in another, galvanically separated coil, this is called mutual induction; when it takes place in the same coil, it is called self-induction. In any case, the magnitude of the induced electromotive force is proportional to the rate of current variation. The proportionality factor in mutual induction is called mutual inductance M, that in self-induction is called inductance L.

The characteristic circuit parameter of a coil is the inductance; its unit is henry. The equivalent inductance in series and parallel connections of coils is expressed by the equations (5.22.) and (5.23.).

Loss-free coils (in practice, tow-loss coils are only possible) allow a direct current to pass without any restriction. In switching on and off, however, a certain sluggishness is imparted to the current by the self-induced voltage, that is to say, there are no sudded current changes in a coil. When switching off a coil, very high over-voltages may occur which have to be limited in the circuit.

A current-carrying coil stores energy in the form of magnetic energy by means of which great force actions can be attained.

Questions and problems:

1. Describe in which way induced voltages are brought about!

2. In which way are self-induction and mutual induction physically related?

3. Compare generator principle and motor principle and explain the relations!

4. Explain the mode of action of an eddy-current brake with the belt of the law of induction!

5. Which property of a coil is described by inductance?

6. Explain the course taken by current and voltage in a coil when it is switchen on a off a direct voltage source!

7. Why can a very high overvoltage occur in a coil when it is switched off a voltage source? By which measures can this overvoltage be limited or avoided?

8. The inductance of a coil is 4 H. The current flowing through the coil changes uniformly by - 150 mA within 5 ms. Calculate the self-induced voltage!

9. Calculate the inductance of a coil when a self-induced voltage of 100 V is brought about with a rate of current change of 50 A/s

10. Calculate the energy that is required for the building up of the magnetic field of a coil having an inductance of 500 mH when the coil carries a current of 2 A!