# EMF Equation of Transformer & Voltage Transformation Ratio

### Transformer EMF Equation

In the previous post we are discussed about working principle of transformer.In this article the amount of emf induced in the secondary winding due to the current supplied to the primary winding can be calculated by EMF equation of the transformer.This is very essential in the designing process of any transformer.Because amount of emf induced due to mutual induction also depends on number of turns in the coil.Transformer emf equation gives the output voltage value,so that we can design a transformer as per our requirements.

### Derivation : EMF Equation of Transformer

Let,

N1 = Number of turns in primary winding

N2 = Number of turns in secondary winding

Φm = Maximum flux in the core (in Webers) = (Bm x A)

f = frequency of the AC supply (in Hz).

Flux propagates in the sinusoidal wave form.If the time period of the flux wave is 'T' sec.Then it reaches it's maximum value Φm in T/4 sec i.e.,quarter of the cycle.We know that T=1/f (f is the frequency of the sine wave).

Average rate of change of flux = Φ/(T/4)    = Φm /(1/4f)
Average rate of change of flux = 4f Φm       ....... (Webers/s).
Induced emf per in the one turn = Rate of change of flux per in the one turn
So, average emf per turn = 4f Φm   ..........(Volts).
Since,Form factor = RMS value / Average value.
RMS value of emf per turn = Form factor X Average emf per turn.
Form factor of sine wave is 1.11
Therefore, RMS value of emf per turn =  1.11 x 4f Φm = 4.44f Φm Volts.
The above equation for one turn.If there are N number of turns in the coil then emf equation of the transformer becomes,

E=4.44f N Φm Volts

Where,
f = frequency of the AC supply (in Hz).
Φm = Maximum flux in the core (in Webers) = (Bm x A)
N = Number of turns in winding.[it can be primary or secondary]

### Induced EMF Equation of Transformer In Primary,Secondary winding

RMS value of induced emf in the primary winding (Say E1) = RMS value of emf per turn X Number of turns in the primary winding

E1 = 4.44f N1 Φm          ............................. (1)

Same as above, RMS induced emf in the secondary winding (Say E2) can be given as

E2 = 4.44f N2 Φm.          ............................ (2)

By dividing eqs (1) & (2).

The above relation between primary and secondary induced voltages  is called the emf equation of transformer.We can observe emf / number of turns is same for both primary and secondary winding of a transformer.

### Voltage Transformation Ratio (K) of Transformer

From equation (3) E1/N1=E2/N2=K
Where, K = constant
This constant K is known as voltage transformation ratio.
1. If N2 > N1, i.e. K > 1, then the transformer is called step-up transformer.
2. If N2 < N1, i.e. K < 1, then the transformer is called step-down transformer
3. Based on the requirement we can use any transformer as step-up or step down by changing the supply.