Emf equation of transformer derivation

        When  a   alternating  or   sinusoidal voltage is applied to primary winding of transformer , flux sets up in the core is also alternating in nature.i.e the function of flux us sinusoidal (sine wave).

     Let,

        •ϕm be the maximum flux in Weber

        •f be the supply frequency in Hz

        •N1 is the number of turns in the primary winding

        •N2 is the number of turns in the secondary winding

Φ is the flux per turn in Weber

As shown in fig.,

      Flux increases from it's zero value to it's maximum value øm in one quarter of cycle i.e. in 1/4f seconds.

Average rate of change of flux is given by ,

                dø / dt

        =Øm/ [ 1/ 4f]

       =4*f*øm       Wb/s

Now, as the rate of change of flux per turn is known as emf in volts .

         Average emf / turn 

        =4*f*øm  volts   .........(1)

Form factor=RMS value/average value

                       =1.11 . 

 

Therefore, 

      RMS value of emf /turn 

      =1.11 * average value of emf /turn 

      =1.11* 4*f*øm           ...........from (1)

      =4.44*f*øm    volts   ............(2)

Now, RMS value of induced emf in whole primary winding,

        E1=RMS value of emf /turn *no. Of primary turns 

         E1=4.44*f*øm*N1       ...... (3)

Also RMS value of induced emf in whole secondary winding,

         E2=RMS value of emf /turn *no. Of secondary turns 

         E2=4.44*f*øm*N2       .......(4)

 

Equations (3) and (4) , are emf equations of transformer. 

• dividing equation (3) by equation (4)

     E1/E2 =N1/N2 

     From above equation ,we can say that magnitudes of E1 and E2 depend upon number of turns of primary and secondary respectively. 

    If N2>N1,then E2>E1 (or V2>V1) and we get a step-up transformer.

    If N2<N1,then E2<E1 (or V2<V1) and we get a step-down transformer. 

  Voltage transformation ratio :

     Dividing equation (4) by equation (3)

     E2/E1 =N2/N1 = K 

     K is called is voltage transformation ratio.