How to derive emf equation of transformer

Emf equation of a transformer:

In this article you will learn how to derive EMF equation of transformer? How to drive turns ratio of transformer using emf equation of transformer. Let’s start with emf equation derivation of transformer. By using emf equation i will drive turns ratio of transformer. Let assume we have transformer which have following parameters:

Number of turns in primary winding of transformer = Np

Number of turns in secondary winding of transformer = Ns

Maximum amount of flux in transformer core =  Φm

                We know that flux is equal to  flux = Bm * A

Where maximum flux density and A is cross-sectional area of core in m².

Frequency of input voltage = f (c/s)

As shown in figure flux reaches from its minimum value to maximum value in fourth part of one cycle. Its mean in 1/4f seconds.

emf equation of transformer
emf equation of transformer

So rate of change of flux per second = maximum flux /1/4f = maximum flux * 4f;

 According to faraday’s law rate of change flux per second is equal to induced voltage in windings. Hence

Induced emf  = 4f* maximum flux

We also know that induced emf  =dΦm/ dt ;

So induced in each turn = 4f* maximum flux volt

If flux is of sinusoidal shape, rms value of induced emf can be calculated by multiplying average value with form factor. We know that

Form factor = rms value / average value = 1.11

So rms value of induced emf  in each turn= 1.11 * 4f* maximum flux volt  = 4.44 *4f* maximum flux volt

We know that  number of turn in primary winding of transformer is Np. By using primary turns we can calculate rms value of induced voltage in primary windings of transformer. i.e.

Rms value of induced emf or voltage in primary winding = Ep = 4.44 *4f* maximum flux * Np volt

Similarly by following above steps we can calculate rms value of induced voltage in secondary winding of transformer. i.e.

Es = 4.44 *4f* maximum flux * Ns volt

 So emf equation of transformer in primary and secondary side of transformer is :

Ep = 4.44 *4f* maximum flux * Np volt

Es = 4.44 *4f* maximum flux * Ns volt

By using these two equations we can calculate turns ratio of transformer.

Transformer turns ratio:

The ratio between secondary voltage or secondary turn and primary voltage or turns of transformer is called transformer turns ratio. We know that in case of no load induced primary emf (Ep) is equal to applied primary voltage (Vp) of transformer. Similarly  Es is equal to Vs. By using above derived emf equations of transformer:

Es / Ep = Vs / Vp = Ns /Np = k

Where K is called turns ratio of transformer and its remains constant in transformer. If K>1 transformer I called step up transformer. If K<1 transformer is called step down transformer. Similarly we can also derive equation for current with turns of transformer.

We know that in ideal transformer input power is  equal to output power.

Input KVA = output KVA

VpIp = VsIs

Vp/ Vs = Is/ Ip = 1/K

Hence currents in transformer are inversely proportional to transformer turns ratio. Let me know with your feed back through comments.

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