Practically all electric power machinery (e.g. transformers, generators, motors) depend for their operation upon the magnetism produced by the magnetic circuits. The closed path followed by magnetic flux is called a magnetic circuit (Fig. 1) just as the closed path followed by current is called an electric circuit. There is a lot of similarity between the magnetic and electric circuits, and will be shown later on during discussion.

A magnetic circuit consists of a structure composed for the most part of high permeability magnetic material. The presence of high permeability material causes the magnetic flux to be confined to the paths defined by the structure, much as currents are confined to the conductors of an electric circuit. A simple example of a magnetic circuit is shown in Fig. 1. The core is assumed to be composed of magnetic material whose permeability is much greater than that of the surrounding air. The core is of uniform cross-section and is excited by a winding having N turns and carrying a current of I amperes. This winding develops a magnetic field in the core, as illustrated in the figure. The magnetic field can be visualized in terms of flux lines, which form closed loops interlinking with the winding. According to basic law of magnetic field, called the Ampere’s circuital law (sometimes referred to as Ampere’s work law) the line integral of **H** around a closed path is equal to the net current enclosed by that path i.e.

……(1)

The above law is very comprehensive and provides a basis for calculation of magnetic circuits and makes it possible in some cases (such as in the cases of a long current carrying conductor or a long solenoid) to determine readily the strength of the magnetic field.

## Magnetic Circuit Concepts.

The flux producing ability of the coil in Fig. 1 or of a coil on any other magnetic circuit is proportional to the number of turns N and the current I. The product NI is called the magnetomotive force (mmf) and determines the amount of flux developed in the magnetic circuit.

MMF of the magnetic circuit is the magnetic potential difference that tends to force flux around the magnetic circuit and is analogous to the electromotive force (emf) in an electric circuit.

MMF = NI ampere-turns ….. (2)

The resulting flux in the magnetic circuit, besides being dependent on the mmf, is also a function of the opposition of the iron to carrying flux. This opposition is called the reluctance S of the magnetic circuit.

As in the case with resistance in the electric circuit, reluctance S of the magnetic circuit is directly proportional to length *l*, inversely proportional to cross-sectional area *a*, and dependent on the nature of material of the magnetic circuit. The reluctance of the magnetic circuit is given as

ampere-turns/weber ……(3)

when the flux is constant over the length and uniform over the area. The quantity expresses the property of the magnetic material called its permeability. Permeability is a measure of the receptiveness of the material of having magnetic flux developed in it. For free space, the permeability to equals H/m in the SI system. The relative permeability of magnetic material may range up to thousands.

Total flux developed in the circuit is given as

webers ……(4)

The above Eq. (4) is sometimes referred to as *Ohm’s law* for the magnetic circuit. It serves to emphasize the mathematical analogy between the magnetic circuit and the electric circuit. Analogous quantities in the two circuits are listed below.

Magnetic circuits differ from electric circuits in one important respect. The reluctance of the magnetic circuit containing iron or ferromagnetic material depends upon the flux carried by it. With the increase influx, a large change in mmf is required to develop the same change in flux.

Determination of Ampere-Turns. In any magnetic circuit, flux created is given as

Of

= ampere-turns [/latex] ……(5)

Hence for determination of AT for a magnetic circuit (i) First find field strength H in each part of the magnetic circuit (ii) find the length of various parts of magnetic circuit (iii) find the number of ampere-turns required for the various parts of magnetic circuit from the relation AT = H*l* where *l* is the length of the part in meters and lastly, (iv) find total number of ampere-turns for the whole series magnetic circuit by adding ampere-turns determined for various paths in magnetic circuit.

As both permeability and reluctance may change from One operating condition to another. Hence direct numerical application of the reluctance concept and Eqs. (4) and (5) is rare. For quantitative analysis, graphical methods are generally used because they can easily be adapted to the non-linearity involved.

## Magnetic Circuits with Air-Gaps.

Energy-conversion devices which incorporate a moving element have necessarily air gaps in their magnetic circuits. Air-gaps are also provided in the magnetic circuits to avoid Saturation. A magnetic circuit with an air gap is shown in Fig. 2. An air-gap is nothing else but a volume of air between two magnetic surfaces. The length of air gap *l _{g}* equals the distance between the two magnetic surfaces. The area of x-section of any one of the surfaces gives the air-gap area

*a*. When the air-gap length

_{g}*l*is much smaller than the dimensions of the adjacent core faces, the magnetic flux is constrained essentially to reside in the core and the air gap and is continuous throughout the magnetic circuit.

_{g}Thus, the configuration shown in Fig. 2 can be analyzed as a magnetic circuit with two series components, a magnetic or iron core of permeability and mean length *l _{i}* and an air-gap of permeability and length

*l*. Since the permeability of air is constant, the air-gap is a linear part of the magnetic circuit and the flux density in the air-gap is proportional to the mmf across the air-gap. The necessary mmf is calculated separately for the air-gap and the iron portions and then added to determine the total mmf.

_{g}## Composite Magnetic Circuits.

Consider a circular ring made from different materials of lengths *l _{1}*,

*l*and

_{2}*l*, cross-sectional areas

_{3}*a*,

_{1}*a*and

_{2}*a*and relative permeability , and respectively with a cut of length

_{3}*l*known as air-gap. The total reluctance is the arithmetic sum of individual reluctances as they are joined in series.

_{g}

Or Total ampere-turns required

= Sum of ampere-turns required for individual parts of the magnetic circuit.

## Parallel Magnetic Circuits.

In series circuits, all parts of the magnetic circuit carry same flux and total ampere-turns required to create a given flux is the arithmetic sum of the ampere-turns required for individual parts of the circuit.

But if the various paths of the magnetic circuit are in parallel, as shown in Fig. 4 the ampere-turns required for the combination is equal to the ampere-turns required to create the given flux in one path.

For example, for the circuit shown in Fig. 4 paths ABCD and AFED are in parallel, so ampere-turns required to create flux in path ABCD is equal to ampere-turns required to create flux in path AFED and also equal to the ampere-turns required for both of the paths.

Hence total ampere-turns required for magnetic circuit shown in Fig. 4.

= AT for path DA + AT for path ABCD

= AT for path DA + AT for path AFED

## Magnetic Leakage and Fringing.

Leakage flux is the flux, which follows a leaking path, as shown in Fig. 5. Flux in the air gap is known as *useful flux* which is utilized for various useful purposes. For the purpose of calculations, the iron is supposed to carry whole of the flux throughout its entire length.

The ratio of total flux (flux in the iron path) to the useful flux (flux in the air) is known as *leakage factor*.

i.e.

or

It is also seen from Fig. 5 that the useful flux passing across the gap tends to bulge outwards, thereby increasing the effective area of gap and reducing the flux density in the gap. This effect is referred to as *fringing*; and the longer the air gap, the greater is the fringing.

In electrical machines (such as in generators and motors), magnetic leakage is undesirable as it causes increase in weight (not decrease in their power efficiency) and cost of manufacture. Though magnetic leakage cannot be avoided completely but can be reduced to the minimum by placing the magnetizing or exciting coils as close as possible to the air gap or to the points in the magnetic circuit where the flux is to be utilized for useful purposes.

The value of leakage factor for modern electric machinery is about 1.2.