E.M.F Equation of aTransformer

       When the primary winding is excited by an alternating voltage V₁, it circulates alternating current, producing an alternating flux Φ. The primary winding has N₁ number of turns. The alternating flux Φ linking with the primary winding itself induces an e.m.f in it denoted as E₁. The flux links with secondary winding through the common magnetic core. It produces induced e.m.f. E₂  in the secondary winding. This is mutually induced e.m.f. Let us derive the equations for E₁ and E₂.

       The primary winding is excited by purely sinusoidal alternating voltage. Hence the flux produced is also sinusoidal in nature having maximum value of Φ_m as show in the Fig. 1.

Fig.  1  Sinusoidal flux

       The various quantities which affect the magnitude of the induced e.m.f. are : 

                         Φ = Flux

                         Φ_m = Maximum value of flux

                         N₁ = Number of primary winding turns

                         N₂ = Number of secondary winding turns

                         f = Frequency of the supply voltage

                         E₁ = R.M.S. value of the primary induced e.m.f.

                         E₂ = R.M.S. value of the secondary induced e.m.f.

       From Faraday's law of electromagnetic induction the voltage e.m.f. induced in each turn is proportional to the average rate of change of flux.

.^..      average e.m.f. per turn = average rate of change of flux

.^..      average e.m.f. per turn = dΦ/dt

Now                 dΦ/dt = Change in flux/Time required for change in flux

       Consider the 1/4 th cycle of the flux as shown in the Fig.1. Complete cycle gets completed in 1/f seconds. In 1/4 th time period, the change in flux is from 0 to Φ_m.

.^..                  dΦ/dt = (Φ_m - 0)/(1/4f)           as dt for 1/4 th time period is 1/4f seconds

                              = 4 f  Φ_m       Wb/sec      

.^..    Average e.m.f. per turn =  4 f Φ_m   volts

       As is sinusoidal, the induced e.m.f. in each turn of both the windings is also sinusoidal in nature. For sinusoidal quantity,

       From factor = R.M.S. value/Average value = 1.11

.^..    R.M.S. value of induced e.m.f. per turn 

                        = 1.11 x 4 f Φ_m = 4.44 f Φ_m

        There are number of primary turns hence the R.M.S value of induced e.m.f. of primary denoted as is E₁,

                    E₁ = N₁ x 4.44 f  Φ_m      volts

       While as there are number of secondary turns the R.M.S values of induced e.m.f. of secondary denoted is E₂ is,

                   E₂ = N₂ x 4.44 f Φ_m         volts

       The expression of E₁ and E₂ are called e.m.f. equation of  a transformer.

       Thus e.m.f. equations are,

                  E₁ = 4.44 f Φ_m  N₁              volts             ............(1)

                  E₂ = 4.44 f  Φ_m  N₂             volts             .............(2)

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