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When there is a change in the magnetic flux associated with an electrical circuit, an electromotive force (EMF) is induced in that circuit. Examples of the practical application of this phenomenon include the electric generator and the transformer. If an EMF is available from some source and this source is connected to an electric circuit, a current may flow and a magnetic flux would be associated with this current. This chapter summarizes various units as measures of these quantities. It also discusses the concept of magnetic hysteresis and Eddy currents. Magnetic hysteresis may be defined as the lagging of the magnetic flux behind the magnetizing force.

1.1 Revision

In earlier studies, it was shown that when there is a change in the magnetic flux associated with an electrical circuit, an e.m.f. is induced in that circuit. Examples of the practical application of this phenomenon include the electric generator and the transformer. If an e.m.f. is available from some source and this source is connected to an electric circuit, a current may flow and associated with this current would be a magnetic flux. Various units are used as measures of these quantities and some of these are summarised in the following paragraphs.

When there is a change in the magnetic flux associated with a circuit, an e.m.f. is induced in the circuit and its average value can be found by using the expression

In the International System of Units (SI), the unit of magnetic flux is the weber. If a magnetic flux Φ of 1 weber (1 Wb) is reduced uniformly to zero in a time (t) of 1 second (1 sec) whilst linking a coil having N = 1 turn, the average induced e.m.f. will be 1 volt (1 V).

Using the calculus notation, the instantaneous e.m.f. is

Magnetic flux density is measured in tesla (T) and is the magnetic flux in webers perpendicular to a surface area of 1 m2. Prior to the introduction of SI units, flux density was expressed in webers per square metre (Wb/m2).

Magnetising force is a measure of the cause which sets up a magnetic flux and it is measured in amperes per unit length of the magnetic path, and may also be expressed in ampere-turns per metre.

The product of amperes and turns is called the magnetomotive force (m.m.f.) and is measured in amperes. It may also be expressed in ampere-turns.

For a non-magnetic material, the relationship between the magnetic flux density and the magnetising force is linear (Fig. 1.1(a)).

The ratio B/H is called the permeability of free space (or magnetic space constant). Strictly speaking, the non-magnetic material should be a vacuum but for all practical purposes, the relationship is true for air and all other non-magnetic materials.

Permeability of free space, μ0 = B/H henry/metre (H/m). Its numerical value is 4π × 10–7 H/m.

For magnetic materials, the relationship between B and H may be assumed for many practical purposes to be linear at the lower values of H but ultimately magnetic saturation occurs, Fig. 1.1(b).

The number of times that the magnetic flux is increased when a magnetic material replaces the non-magnetic material in a given magnetic circuit is called the relative permeability μr of the magnetic material. Its value depends on the operating conditions and may have a typical value in the region of 1000, and being simply a ratio, it has no units.

The value of m.m.f. applied to a circuit divided by the magnetic flux set up is called the reluctance (S) of the circuit and it is measured in ampere-turns per weber (At/Wb).

Hence,

The reluctance of a magnetic circuit is a function of its length (‘l’ metres), cross-sectional area (‘a’ square metres) and the magnetic operating conditions. For non-magnetic materials its value may be found by using the expression

The expression for the average induced e.m.f. caused by a change in flux linkages can be rearranged as follows.

The constant is known as the self-inductance (L) of the winding and it is measured in henrys. Thus the average value of the induced e.m.f. can be found from the expression

where I/t is the average rate of change of current through the winding. Using calculus notation, the instantaneous value of the induced e.m.f. is

Now,

and

In words, the self-inductance is equal to the flux linkages per ampere. This implies that if a flux set up per ampere of current is constant when the current varies, as is the case with a coil wound on a non-magnetic former, the inductance is independent of the current.

For a coil wound on a magnetic core, the reluctance may be calculated from the expression

where μr, the relative permeability of the magnetic material, depends on the magnetic operating conditions.

The expression for the average induced e.m.f. resulting from a change in flux linkages associated with a circuit can be rearranged as follows:

The quantity N 2μrμ0a/l is the self-inductance (L) of the coil on this magnetic circuit. The value of the relative permeability μr is constant when the magnetic material is operating on the linear part of the magnetisation curve and has various values when operating on the non-linear part of the curve. If the current setting up the flux produces an m.m.f. large enough to enable the magnetic material to be worked on the non-linear part of the curve, an increase in the current will cause a reduction in the value of μr and therefore a decrease in the self-inductance of the circuit.

Energy is stored in a magnetic field and its value is given by

where L henrys is the self-inductance of the circuit and I amperes is the d.c. current setting up the magnetic flux.

Mutual inductance exists between two or more circuits when an e.m.f. is induced in a circuit due to a change of current in another circuit. If the current in one circuit, known as the primary, is changing, the magnetic flux set up by the current will also be changing and so there will be a change of magnetic flux linkage with both the primary and the secondary windings and an e.m.f. will be induced in each winding.

The average e.m.f. induced in a secondary winding is given by

where M is the mutual inductance (in henrys) between the windings and I/t is the rate of change of current in the primary winding.

1.2 Magnetic hysteresis

Magnetic hysteresis may be defined as the lagging of the magnetic flux behind the magnetising force. Assume that a piece of magnetic material is initially unmagnetised and that a magnetising force is increased from zero value up to some maximum value. The relationship between the flux density and the magnetising force will be of the form shown in Fig. 1.2. This is the chain line OA.

If the magnetising force is reduced to zero, the curve AC is obtained. OC is called the remanance or residual flux density, that is, the flux density remaining after the magnetising force has been reduced to zero. The magnetising force must be reversed and increased to OD in order to reduce the remanance to zero. OD is called the coercive force. Further variations of the magnetising force between the maximum values result in the relationship between corresponding values of B and H following the curve DEFGA.

The loop is called a hysteresis loop and its area in units of B and H represents the energy loss for one cycle of magnetisation. Suppose that the loop was plotted to scales of 1 cm = 100 At/m and 1 cm = 0·1 T(Wb/m2), then

Let us see what the unit represents. It can be rearranged as (Wb × At) × 1/m3. Now E = –N Φ/t; neglecting the minus sign and expressing this in words,

Hence

This gives (volts × amperes ×time) and if the time is measured in seconds, this is a measure of energy in joules. The term 1/m3 is the reciprocal of volume and so the area of a single loop represents the energy loss in joules per cubic metre of magnetic material for each cycle of magnetisation. In the example, 1 cm2 represented 10 joules per cycle per cubic metre of magnetic material. The power loss in watts is found by multiplying this figure by the number of cycles of magnetisation per second (watts = joules per second). This loss is known as the hysteresis loss and it is converted into heat and so causes a rise in the temperature of the magnetic material.

This heat loss represents a reduction in the efficiency of electrical machines....