30 October 2017

EMF Equation of the Single Phase Transformer

  • 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
 Фm = Maximum flux in the core ( Weber )
       = Bm × Ai
 Bm = Maximum flux density in the core ( Weber / meter2 )
  Ai = 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 ( NФm ) / dt
             α N ( dФm ) / dt
         Where
           NФm = Flux linkages
           dФm = Change in flux
               dt = Change in time
  • Therefore the average value of induced emf per turn is given by
               E = ( Фm – 0 ) / ( T / 4 )
                  = 4f Фm
  • 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 Фm
                                                             = 4.44f Фm …….( 2 )
  • RMS value of induced emf in the transformer primary winding
         E1 = 4.44f Фm× N1
             = 4.44f N1Фm……….…..( 3 )
  • Similarly the RMS value of induced emf in the transformer secondary winding
         E2 = 4.44f Фm× N2
             = 4.44f N2Фm……………( 4 )
  • As we know that Фm = Bm × Ai
        E1 = 4.44f BmAiN1………….( 5 )
        E2 = 4.44f BmAiN2………….( 6 )
From equation ( 5 ) and ( 6 )
         E1 / E2 = N1 / N2……….…..( 7 )
                    OR
        E1 / N1 = E2 / N2 = 4.44f BmAi …( 8 )
  • The induced emf per turn is same in both primary and secondary winding from equation 8.
  • Under no load condition in the ideal transformer
         V1 = E1 and
         V2 = E2
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