**SELF INDUCTANCE**: It is the basic property of a coil due to which it resists increasing or decreasing of flux by flowing current through it.

**EXPLANATION:** Suppose a coil of wire is connected with a battery and we want to increase current as well as flux through it. It is found that an effort to increase current through it is always opposed by counter e.m.f. of self induction. In case of decreasing current through it is also opposed by same self induced e.m.f. but this time in opposite direction.

So the property due to which it opposes any increase or decrease of current or flux through it, is known as self inductance.

It is similar to the experience that it is difficult to set a heavy body into motion initially, but once it is done it is also equally difficult to stop it. Hence self inductance act as electrical inertia.

**CO-EFFICIENT OF SELF INDUCTANCE** (L): Co-efficient of self induction may be defined by three methods,

(i) The weber turns per ampere in the coil, L=Nø/I henry, where N = number of turn, I = current in ampere, ø = flux in weber.

(ii) L =µ_{0} µ_{r}AN^{2}/l henry, as flux produced in asolenoid is ø =NI/l/µ_{0} µ_{r}A. and

(iii) L = e_{L}/di/dt henry, as e_{L}= -LdI/dt.

**ENERGY STORED IN MAGNETIC FIELD**: Suppose current passes through a coil. The self induced e.m.f of the coil opposes. So energy is needed to overcome the opposition. This energy is stored in the magnetic field of the coil.

The value of the stored energy in the magnetic field is described as,

If i = instantaneous current, e = induced e.m.f. at that instant = L di/dt.

Then, work done to overcome the opposition in dt time is,

dW = ei dt = Ldi/dt x i dt =Li di

By integrating both side the energy stored in magnetic field is E = 1/2LI^{2} joules.

**MUTUAL INDUCTANCE**: The phenomenon due to which one coil causes an induced emf in the nearby coil induction when flux produced by it is changed.

**EXPLANATION**: Mutual inductance is the ability of one coil to produce an e.m.f in a nearby coil by induction, when the current in the first coil changes. By this process the second coil also may induce an emf in the first. This ability is measured by the co- efficient of mutual inductance (M).

**CO-EFFICIENT OF MUTUAL INDUCTANCE:** It also defined by three methods,

(i) M = N_{2}ø_{1}/I_{1} henry.

(ii) M = µ_{0}µ _{r}AN_{1}N_{2}/l henry.

(iii) M =e_{M}/dI_{1}/dt.

**CO-EFFICIENT OF COUPLING**: When two coil links with each other, the ratio of mutual inductance present between the coils is denoted as co-efficient of coupling. So, if two coils linked each other fully, then, co-efficient of coupling (k) is unity and if both does not link at all, then k is zero.

The value of co-efficient of coupling is k = M/ √L_{1}L_{2} ,

Where L_{1} L_{2} = co efficient of self inductance of two coil.

**INDUCTANCES IN SERIES**: In case of inductances in series, there are two conditions.

*If the two coils joined in series and their m.m.f’s are additive, i.e, in the same direction.*

Self induced emf in coil A, e_{1}=-L_{1}di/dt.

Where,L_{1} = co-efficient of self inductance of 1st coil.

Mutual induced emf in coil A, e’_{1}=-Mdi/dt.

M= co-efficient of mutual inductance.

Same way, in coil B, e_{2}= -L_{2}di/dt.

Here L_{2} = co-efficient of self inductance of 2nd coil.

So, mutual induced emf in coil B, e’_{2}=-Mdi/dt.

Total induced emf= -L_{1}di/dt-Mdi/dt-L_{2}-Mdi/dt.

= -di/dt(L_{1}+L_{2}+2M)—————(i)

Now let, L be the equivalent inductance of the single coil AB,

Then emf, e=-Ldi/dt ——————-(ii)

Equating (i) and (ii)

We get, L=L_{1}+L_{2}+2M.

*When coil are so joined that m.m.f are opposite:*

Here e_{1}=-L_{1}di/dt.

e’_{1}=+Mdi/dt.(as opposite direction)

e_{2}= -L_{2}di/dt.

e’_{2}=+Mdi/dt.

So, we can conclude as L = L_{1}+L_{2}-2M.

Inductance in parallel: If two inductances L_{1} and L_{2} henry are connected each other by parallel and mutual field assist the separate field, then

**L = L _{1} L_{2} -M^{2}/ (L_{1}+ L_{2}-2M).**

Again when two fields oppose each other, then

**L = L**

_{1}L_{2}-M^{2}/ (L_{1}+ L_{2}+2M).**LOSSES IN MAGNETIC MATERIALS:**

When an alternating flux applied on magnetic materials, two types of losses occurs, which are (1) *Hysteresis losses and (2) Eddy current losses.*

These two losses are the total core loss.

**MAGNETIC HYSTERESIS:**

Due to reversal of magnetism, when a magnetic material dissipates energy as a loss is known as magnetic hysteresis. In this process the flux density (B) always behind the magnetizing forces (H).

**EXPLANATION:** Suppose we take an unmagnetized toroidal (iron bar) and want to magnetise it. We will see that it follows the OD curve. If we lower the m.m.f, we will see that, the flux curve follows the curve line DE and zero at L point. If the m.m.f reversed, then the following path is LM. After saturation, if we again increase the m.m.f, returned towards zero, the flux path is MN and after that it follows the path NPD. The area of this loop is a measure of energy loss during the cycle and this loss becomes as heat from that material for magnetic reversal.

**AREAS OF HYSTERESIS LOOP:** Suppose we magnetize a magnetic material, in this process, energy is spent to force molecules along a straight line. The energy spent of one cycle magnetization denotes as the area of hysteresis loop. So, we can say the area of hysteresis loop represents the net energy spent in taking a magnetic material through one cycle of magnetization.

**STEINMETZ HYSTERESIS LAW**: Hysteresis loss of a magnetic material depends on (i) the maximum flux density (B_{max}). (ii) Frequency of magnetic reversal, (iii) Volume of magnetic material.

So, hysteresis loss, W_{h} α B_{max}^{1.6} f v joule/sec or watt.

Or, W_{h} =ɳ B_{max}^{1.6} f v joule/sec or watt.

The index 1.6 is empirical and holds well in the range of B_{max} value 0.1 to 1.2 wb/m^{2}. With increasing flux density index also increases.

ɳ is constant known as Steinmetz hysteresis co- efficient as this formula was experimentally first established by Steinmetz and the law known as **Steinmetz hysteresis law.**

**HYSTERESIS CO-EFFICIENT TABLE:**

MATERIAL | HYSTERESIS CO EFFICIENT,ɳ |
---|---|

1) Cast iron | 1) 27.65 to 40.0 |

2) Iron sheet | 2) 10.05 |

3) cast steel | 3) 7.5 to 30.1 |

4) Silicon steel | 4) 1.91 |

5) Mild steel | 5) 7.5 to 22.5 |

6) Permalloy | 6) 0.25 |

**EDDY CURRENT LOSS**: When an electric current circulates within the conducting magnetic materials body itself, it is known as eddy current. In this case an e.m.f is induced inside the material itself, when an alternating magnetic field is applied to a conducting magnetic material (Faraday’s law of electromagnetic induction). For these e.m.f , the current circulates only inside the mass of magnetic material, which is responsible for producing heat as it does not work for any useful work and known as loss.

**REDUCING THE EDDY CURRENT LOSS:** Eddy current loss may be reduced by (i) splitting the core i.e, cross section of flux path by thin pieces known as lamination. (ii) By increasing high value resistance in magnetic material. (iii) By grinding the magnetic material to a powder and mix it with a binder which insulates the particles from one another.

Though flowing eddy current is a loss, still it has also some application in induction heating, as damping torque in moving coil instrument, as brake etc.

Mathematical expression of eddy current loss is,

**W _{e}= k_{e}B_{max}^{2}t^{2}f^{2}V watt.**

Where, k

_{e}= co-efficient of eddy current which depends on nature of material.

t = thickness of lamination.

V = volume of material.

f = frequency.

So,

*total iron losses = W*_{h}+W_{e}.
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