A closed path followed by magnetic flux is known as a **magnetic circuit**. In a magnetic circuit, flux starts from one point and finishes at the same point. A magnetic circuit usually consists of magnetic materials having high permeability such as iron, soft-steel, etc. since they offer small opposition to magnetic flux.

The magnetic flux is usually produced by an electric current through a solenoid (having a large number of turns. There are three types of magnetic circuits:

- Series Magnetic Circuit
- Parallel Magnetic Circuit
- Series-parallel Magnetic Circuits

# Series Magnetic Circuit

A magnetic circuit that has many parts of different dimensions and materials connected in series is called a series magnetic circuit.

Consider a composite magnetic circuit consisting of three parts having different lengths, area of cross-sections and relative permeabilities. It also has an air-gap.

In this case, the reluctance of each part will be different depending upon the dimensions and relative permeabilities of that part. The total reluctance will be the sum of the reluctance of individual parts. The same flux will flow through complete circuits.

## Parallel Magnetic Circuit

A magnetic circuit that has more than one path for the magnetic flux is called a **parallel magnetic circuit**. It is similar to a parallel electric circuit that has more than one for electric current to flow.

Consider a parallel magnetic circuit shown in Figure. A current-carrying coil is wound on the central limb. The flux set up by this coil is divided at A into two paths i.e.

- Flux ɸ
_{1 }passes along the path AFEB, - Flux ɸ
_{2}passes along the path ADCB

It is clear that ɸ = ɸ_{1} + ɸ_{2}

Here the two magnetic paths AFEB and ADCB are in parallel, the total MMF required for this parallel circuit will be equal to the MMF required for anyone of the above said paths.

## Magnetic Flux Density (B)

The flux per unit area is defined as the *magnetic flux density*. It is measured in a plane perpendicular to flux.

Magnetic Flux Density,** B = φ ****÷ ****A**

Units: Weber per meter square (**Wb/m ^{2}**) or

**tesla(T).**

## Magnetic Field Intensity

The magnetic field strength or magnetic field intensity is given by MMF per unit length of the magnetic circuit.

Magnetic Field Intensity,** H = (N I) **

**÷**

**l**

where N = Number of turns of magnetizing coil

*I*= Current through the coil

l = length of magnetic material in meters

Units:

**AT/m.**

The magnetic field intensity is also known as magnetic field strength or magnetizing force.

## Permeability

The ability of a material to carry the magnetic lines of flux is known as *permeability* of that material.

The magnetic lines of force can pass through high permeability materials like iron, steel, very easily. Low permeability materials like wood etc. don’t allow the flux lines to pass through them easily.

**Absolute Permeability**: It is the ratio of flux density (B) in a particular medium to the magnetic field strength (H) which produces magnetic flux density. It is denoted by µ.

Absolute Permeability,** µ = µ _{o }µ_{r}**

Units: Henry/meter

**(H/m)**

**Permeability of the Air/Space/Vacuum**: If a magnet is kept in air or vacuum, then the ratio of flux density (B) and magnetic field strength (H) is defined as the permeability of free space. It is denoted by µ

_{o}.

*Permeability of Free Space*,

**µ**

_{o}= 4π x 10^{-7}H/m**Relative Permeability**: The ratio of permeability of material to the permeability of vacuum or air is known as

*relative permeability*.

Relative Permeability,

**µ**

_{r}= µ ÷ µ_{o}**It has no units**.

The relative permeability of vacuum, air and all non-magnetic materials is 1. The relative permeability of all the magnetic materials is very high. For example, the relative permeability of permalloy (nickel 78% and iron 22%) is about 50000.

## Magneto-motive Force (MMF)

The *magneto-motive force* is the driving force which produces the magnetic flux. The magnetic field intensity (H) is decided by MMF.

Magneto-motive Force, **MMF = N I**

where N = Number of turns of magnetizing coil

*I*= Current through the coil

Units: Ampere Turns (

**AT**)

**Reluctance (S)**: It is opposition offered to the flow of magnetic flux by the magnetic material.

Unit:

**AT/Wb**

Reluctance, S = l ÷ (µ x a)

where l = length of the magnetic path in meters.

a = area of the cross-section of magnetic path in meter square.

µ = absolute permeability of medium in H/m.

= µ

_{o}µ

_{r}

Therefore, Reluctance,

**S = l ÷ (µ**

_{o}µ_{r}a)The reluctance is also given by the ratio of the MMF and the amount of flux produced.

i.e. Reluctance, S = MMF ÷ flux

Reluctance,

**S = (N**

*I*) ÷ φ**Permeance**: The

*permeance*of a material represents the ease with which magnetic flux can be produced in that material. It is reciprocal of reluctance. Its unit is Wb/AT or henry.

## Magnetic Susceptibility

The magnetization of a material is proportional to the field and the proportionality constant is called the magnetic susceptibility. In other words, it can be defined as the ratio of magnetization **M** to magnetizing force **H** i.e.

Magnetic Susceptibility, **X _{m} = M/H**

It is a dimensionless quantity. It is a measure of how easily a material is magnetized in a magnetizing field. Its value for vacuum is zero as there can be no magnetization in the vacuum.

We can classify materials in terms of

**X**. Materials with positive values of

_{m}**X**are paramagnetic and those with negative values of

_{m}**X**are diamagnetic. For ferromagnetic materials,

_{m}**X**is positive and very large.

_{m}## Leakage Factor in Magnetic Circuit (λ)

The part of the total magnetic flux which flows through the magnetic circuit is called useful magnetic flux. However, the magnetic flux which does not completely pass through the magnetic path, but partially passes through the air is called *leakage magnetic flux*.

Mathematically, φ_{total} = φ_{useful} + φ_{leakage}

The ratio of total flux produced to the useful flux is called leakage factor or leakage coefficient.

Leakage factor, λ = φ_{total }/ φ_{useful}

The value of *leakage factor* is always greater than unity. Typical values of leakage factor are from 1.12 to 1.25. In the magnetic circuits, the magnetic leakage can be minimized by placing the exciting coils as close as possible to the points where the flux is to be utilized.

## Fringing in Magnetic Circuit

The magnetic lines of force repel each other while passing through a non-magnetic material. Due to this when the flux lines cross the air gap, they tend to bulge outwards. This effect is known as *fringing*.

The effect of fringing is to make the effective air gap area larger than that of magnetic path and consequently, the flux density in the air gap is reduced. The effect of fringing depends upon the length of the air gap. To minimize fringing, the air gap length is kept as small as possible. The effect of fringing can be neglected if air gap length is very small as compared to its width.