# Self and Mutual Induction

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#### Concept of Self Induction:

• When a current flowing through a coil changes the magnetic flux linked with the coil itself changes. Due to which an induced e.m.f. is generated in the same coil. This property of coil of producing induced e.m.f. in a coil due to change in the current through the same coil is called self-induction.
• Let ø be the magnetic flux linked with the coil. At any instant, this flux is directly proportional to the current in the coil. Where L = Coefficient of self-induction

Differentiating both sides w.r.t. time t, we get By Faraday’s law of electromagnetic induction we have Thus induced e.m.f. in the coil is given by Considering magnitude only we have • Thus, the self-inductance of a coil can be defined as the induced e.m.f. in the coil itself due to a unit rate of change of current in the same coil.
• The S.I. unit of self-inductance is henry (H). • Thus the self-inductance of a coil is one henry, if an e.m.f. of 1 volt is induced in the coil when the current passing through the same coil changes at the rate of 1 ampere per second.

#### Concept of Mutual Induction:

• When two coils are placed very close to each other if the current is passed through one coil (primary) a magnetic field is set around this coil. Now the second coil (secondary)  is kept in the magnetic field created by the primary coil. Thus magnetic flux is linked with the secondary.
• When a current flowing through primary changes the magnetic flux linked with the secondary changes. Due to which an induced e.m.f. is generated in the secondary coil. This property of producing induced e.m.f. in secondary due to change in the current through the primary is called mutual induction.
• Let øs be the magnetic flux linked with secondary and ip be the current through the primary. At any instant, the magnetic flux linked with secondary is directly proportional to the current in the primary. Where M = Coefficient of mutual induction

Differentiating both sides w.r.t. time t, we get By Faraday’s law of electromagnetic induction we have Thus induced e.m.f. in the secondary coil is given by • Thus, the mutual inductance of coil can be defined as the induced e.m.f. in the secondary due to unit rate of change of current in the primary coil.
• The S.I. unit of mutual inductance is henry (H). • Thus the mutual inductance of a coil is one henry, if an e.m.f. of 1 volt is induced in the secondary coil when the current passing through the primary coil changes at the rate of 1 ampere per second.

### Transformer

• The transformer is a device which converts the alternating voltage from one value to another.
• It works on the principle of mutual induction.

#### Construction:

• A transformer consists of two sets of a coil, primary coil and secondary coil, which are well insulated from each other. The primary coil is input coil and the secondary coil is output coil. The two coils are wound on a soft iron core either one above other or on the separate arm.

#### Assumptions:

• The primary current and resistance of primary current are small.
• Same magnetic flux links both the primary and secondary.

#### Working:

• When an alternating source of voltage say VP is applied across the primary. it creates a changing magnetic flux which is linked to the secondary coil. The value of the flux linked with the coils depends on the number of turns of both the coils. Let ø be the magnetic flux per unit turn linked with the coil at time ‘t’.  Let the rate of change of flux liked with the coil be dø/dt.

The e.m.f. induced in the secondary at the instant is given by Where Ns = Number of turns of the secondary coil.

The back e.m.f. induced in the secondary at the instant is given by Where, NP = Number of turns of primary coil.

For open circuit ES =VS and EP = VP This equation is known as the equation of the transformer.

For ideal transformer,

Power Input = Power Output From equations (1) and (3) we get #### Types of Transformers:

• Depending upon the ratio of output voltage to the input voltage, transformers are classified into two types.

#### Step up transformer:

• When the output voltage is greater than the input voltage (VS > VP or NS > NP), the transformer is called as a step up transformer.

#### Step down transformer:

• When the output voltage is less than the input voltage (VS < VP or NS < NP), the transformer is called as a step down transformer.

#### Flux leakage:

• If the windings are. not proper one over the other or there are air gaps, then the magnetic flux due to primary never gets fully linked with the secondary. This loss can be minimized by winding the two windings one over the other carefully.

#### Copper Losses:

• Copper wires used for windings, both the windings possess some resistance. Hence,  some energy will be lost in the form of heat given by I²R. This loss can be minimized by reducing resistance by using thick wires.

#### Eddy Current Losses:

• The core of the transformer is made up of soft iron. The continuously changing magnetic flux produces eddy currents in it. Which tend to heat the core. Thus some energy is lost due to eddy currents in the core. These losses can be minimized by making core laminar.

#### Hysteresis Losses:

• Iron has a tendency to retain residual magnetism even if the magnetizing field is removed. Thus for each cycle, some energy is lost in destroying the residual magnetism of the previous cycle. This loss cannot be minimized and it is irrecoverable

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