  Introduction to Electrical Engineering - Basic vocational knowledge (Institut f³r 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).