Emf Equation of Transformer

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 byFlux. Since we we apply alternating voltage, flux varies sinusoidal waveform is obtained.

Sinusoidal Flux Waveform with Time

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:

Magnitude of induced emf from the faraday law    .. …. …..  (A)


E=  Induced Emf

N= Number of Primary and Secondary Winding turns

Rate of change of flux= 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;

Sinusoidal Flux Formula… …. ..  (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 Maximum allowable flux.

Mathematical Derivation of Emf Equation :

Mathematical derivation of emf equation of transformer is as follows:

Taking derivative of eq. 1  w.r.t “t”,

Transformer emf equation derivation

Transformer emf equation derivation

Or, simply we can write as:

Emf equation of Transformer derivation

Putting the value of in equation A

Emf equation of Transformer derivation

Since, we know that

Emf equation of Transformer derivation,

Hence the above equation can be written as :

Maximum induced emf equation

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 Fluxby. We can draw the fig.as:

Induced emf and flux Relation

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 :

RMS Value of Induced emf formula

Putting the value of maximum induced emf

RMS Value of Induced emf formula

 Putting Mathematics formulaand  Frequency Formulain the above equation, we have

Effective Erms value of Induced emf calculations

By simply multiplying,  Emf calculations we get

Emf equation of Transformer   … .. ….. (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:

Maximum Flux density formula

Or simply, we can write as :

Maximum flux Formula

In the above equation, Bm is the maximum allowable flux density of the transformer  core. Putting the value of  fluxMaximum allowable fluxin the general emf equation (2), we have

Emf Equation of Transformer … .. …. (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 :

Induced emf equation at the primary side

Similarly, Induced emf at the secondary side will be termed as;

Induced emf equation at the secondary side

Since the fluxMaximum allowable 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.


That’s all. Hope this will helps you.

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