EMF Equation of the Single Phase Transformer

EMF Equation

• When an alternating voltage V₁ is applied to the transformer primary side, an alternating flux will set up in the core which links both primary and secondary winding.  
• This will result in self induced emf ( E₁ ) sets up in the transformer primary winding and mutually induced emf ( E₂ ) sets up in the secondary winding.

Let

N₁ = Primary winding turns

N₂ = Secondary winding turns

Ф_m = Maximum flux in the core ( Weber )

      = B_m × Ai

B_m = Maximum flux density in the core ( Weber / meter² )

A_i = Core area ( Meter² )

F = Frequency of alternating supply ( Hz )

• An alternating flux increases from zero to positive maximum in the one fourth of a cycle i.e. T/4 second
• According to Faraday’s law of electro – magnetic induction, the magnitude of induced emf is directly proportional to rate of change of flux linkages.

       E α  d ( NФ_m ) / dt

          α N ( dФ_m ) / dt

Where

NФ_m = Flux linkages

dФ_m = Change in flux

    dt = Change in time

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• Therefore the average value of induced emf per turn is given by

        E = ( Ф_m – 0 ) / ( T / 4 )

           = 4f Ф_m

• As the flux varies sinusoidally, the induced emf will be sinusoidal.
• Form factor = RMS Value / Average Value

                           = 1.11 ……………………( 1 )

( The value of form factor is 1.11 only for sinusoidal waveform )

RMS value of induced emf per turn = Form factor × Average emf

        = 1.11 × 4f Ф_m

       = 4.44f Ф_m ……….….….( 2 )

RMS value of induced emf in the transformer primary winding

E₁ = 4.44f Ф_m× N₁

     = 4.44f N₁Ф_m……….…..( 3 )

Similarly the RMS value of induced emf in the transformer secondary winding

E₂ = 4.44f Ф_m× N₂

     = 4.44f N₂Ф_m……………( 4 )

As we know that Ф_m = B_m × A_i

E₁ = 4.44f B_mA_iN₁………….( 5 )

E₂ = 4.44f B_mA_iN₂………….( 6 )

From equation ( 5 ) and ( 6 )

E₁ / E₂ = N₁ / N₂…………..( 7 )

        OR

E₁ / N₁ = E₂ / N₂ = 4.44f B_mA_i ………( 8 )

Equation ( 8 ) indicates that the induced emf per turn is same in both primary and secondary winding.

No Load Condition

Under no load condition in the ideal transformer

V₁ = E₁ and

V₂ = E₂

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