When the primary winding is excited by an alternating voltage V

_{1}, it circulates alternating current, producing an alternating flux Φ. The primary winding has N_{1 }number of turns. The alternating flux Φ linking with the primary winding itself induces an e.m.f in it denoted as E_{1}. The flux links with secondary winding through the common magnetic core. It produces induced e.m.f. E_{2 }in the secondary winding. This is mutually induced e.m.f. Let us derive the equations for E_{1 }and E_{2}. 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 N

_{1 }= Number of primary winding turns N

_{2 }= Number of secondary winding turns f = Frequency of the supply voltage

E

_{1 }= R.M.S. value of the primary induced e.m.f. E

_{2 }= 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 E

_{1}, E

_{1 }= N_{1 }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 E

_{2 }is, E

_{2 }= N_{2 }x 4.44 f Φ_{m }volts The expression of E

_{1 }and E_{2 }are called e.m.f. equation of a transformer. Thus e.m.f. equations are,

E

_{1 }= 4.44 f Φ_{m }N_{1 }volts ............(1) E

_{2 }= 4.44 f Φ_{m }N_{2 }volts .............(2)

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