Describe the basic principle of operation of a single phase transformer and derive the emf equation.

#### Solution

When an alternating voltage π1 is applied to a primary winding, an alternating current πΌ1flows in it producing an alternating flux in the core. As per Faraday’s laws of electromagnetic induction, an emf π1 is induced in the primary winding.

`e_1=-N_1(dvarphi)/(dt)`

Where π_{1 }is the number of turns in the primary winding. The induced emf in the primary winding is nearly equal and opposite to the applied voltage π_{1}.

Assuming leakage flux to be negligible, almost the flux produced in primary winding links with the secondary winding. Hence, an emf π_{2} is induced in the secondary winding.

`e_2=-N_2(dvarphi)/(dt)`

Where π_{2 }is the number of turns in the secondary winding. If the secondary circuit is closed through the load, a current πΌ_{2} flows in he secondary winding. Thus energy is transferred from the primary winding to the secondary winding.

**EMF EQUATION.**

As the primary winding is excited by a sinusoidal alternating voltage, an alternating current flows in the winding producing a sinusoidally varying flux π in the core.

π=πππ ππππ‘

As per Faraday’s law of electromagnetic induction an emf π_{1} is induced in the primary winding.

`e_1=-N_1(dvarphi)/(dt)`

`e_1=-N_1(dvarphi)/(dt)`(πππ ππππ‘)

π_{1}=−π_{1}π_{π}ππππ ππ‘ = −π_{1}π_{π}πsin (ππ‘−90°) = 2πππ_{1}π_{π}πsin (ππ‘−90°)

Maximum value of induced emf = 2πππ_{π}π_{1}

Hence, rms value of induced emf in primary winding is given by,

`E_1=(E_(max))/sqrt2=(2pifN_1varphi_m)/sqrt2=4.44 fN_1varphi_m`

Similarly rms value of induced emf in the secondary winding is given by,

πΈ_{2}=4.44ππ_{2}π_{π}

_{Also, `E_1/N_1=E_2/N_2=4.44fvarphim`}

_{Thus emf per turn is same in primary and secondary winding and an equal emf is induced in each turn of the primary and secondary winding.}