When we apply alternating( sinusoidal) voltage at the primary side of transformer, alternating flux is setup in the core of transformer. Flux produced in the core of transformer is denoted by
. Since we we apply alternating voltage, flux varies sinusoidal waveform is obtained.
In order to derive the emf equation of transformer
, we have to consider the magnitude of induced voltage. According to faraday’s law, the magnitude of induced emf is equal to the rate of change of flux and it is expressed as:
.. …. ….. (A)
Here,
E= Induced Emf
N= Number of Primary and Secondary Winding turns
= rate of change of flux within the core.
Since the flux induced is sinusoidal, hence the function of flux is a sine function. Sinusoidal flux equation can be expressed as;
… …. .. (1)
It should be noted that there is a particular value of maximum allowable flux for the transformer core. Beyond the particular value of flux, the core will saturate. To avoid the saturation of core, maximum value of flux must be less than core saturation value. Let called the maximum allowable flux as 
.
Mathematical Derivation of Emf Equation :
Mathematical derivation of emf equation of transformer is as follows:
Taking derivative of eq. 1 w.r.t “t”,
Or, simply we can write as:
Putting the value of 
in equation A
Since, we know that
,
Hence the above equation can be written as :
The above equation is the maximum value of emf induced in the transformer. From the above expression, we can say that the angle between the flux and induced emf is 
. Moreover, induced emf lags the flux by. We can draw the fig.as:
In AC system, we generally deal with RMS value. Hence the rms value or effective value of induced emf due to sinusoidal flux is obtained from the following expression.
RMS Value of Induced emf :
Putting the value of maximum induced emf
Putting 
and 
in the above equation, we have
By simply multiplying, 
we get
… .. ….. (2)
This is the general induced emf equation of transformer. If we simplify more, we know that flux density is equal to flux per unit area and it is expressed as:
Or simply, we can write as :
In the above equation, Bm is the maximum allowable flux density of the transformer core. Putting the value of fluxin the general emf equation (2), we have
… .. …. (3)
In the above equation,
N= Number of Primary and Secondary winding turns
F= Frequency of the applied voltage
Bm= Maximum allowable flux density
Ac= Cross Sectional Area of transformer core.
Hence the rms induced emf equation of transformer can be represented in terms of both equation (2) and equation (3).
Further, we can write the rms induced emf equation for both Primary and Secondary side. RMS induced emf equation for the primary side is :
Similarly, Induced emf at the secondary side will be termed as;
Since the flux linking with both primary and secondary side is same and frequency of the transformer is also constant. Hence only difference in the rms induced emf equation of primary and secondary side is the number of primary and secondary winding turns. In simple words, we can say that any particular value of desired emf can be obtained by using a suitable number of turns.