# EMF Equation of Transformer | Turns Ratio of Transformer

EMF equation of transformer is very important for understanding the working of the transformer.When sinusoidal voltage is applied to primary of the transformer it draws magnetizing current to set up flux in the core. The flux links to the secondary and produce EMF. We can easily derive the EMF equation of transformer  by calculating  the rate of change of the flux in one cycle of AC waveform. Induced EMF in the primary and secondary of the transformer also depends  on the turns ratio of transformer. The ratio of secondary  EMF to primary EMF is called voltage transformation ratio of the transformer. The ratio of Primary  turns to secondary turns is called transformer turns ratio(TTR).

The flux set up in the core gets linked to the primary and secondary winding of the transformer. The alternating flux set up in the core when links to turns of the primary and secondary winding induces voltage called EMF. The EMF induced in the transformer winding depends on number of turns and rate of change of the flux. The EMF induced across the primary and secondary always opposes the applied voltage.

According to Faraday’s law of electromagnetic induction, if the alternating flux links to a coil the voltage is induced in the coil. The  voltage induced in the primary and the secondary winding of the transformer is as given below. The RMS value of secondary induced EMF and primary induced EMF can be calculated using Faraday’s law of electromagnetic induction.

Ep = – Np x dФ/dt

Es = – Ns x dФ/dt

Where,
Ep        = EMF induced in the primary
Es         = EMF induced in the secondary
Np        = Number of turns in the primary winding
Ns        = Number of turns in the secondary winding
dФ/ dt  = Rate of change of the flux in the core

The minus sign shows that the voltage induced in the primary and secondary opposes the applied voltage.

## Derivation of EMF Equation Of Transformer

The derivation of the EMF equation of the transformer can be derived by calculating the rate of change of the flux in the transformer core. The flux set up in the core is sinusoidal.

The flux in the core changes from + Фm to –Фm in 1/2f  seconds.

The voltage induced in the primary is

Ep = – Np x dФ/dt  ———(1)

The flux induced in the primary is because of the sinusoidal voltage applied to primary so the flux also varies sinusoidal. The instantaneous value of the flux in the transformer is ;

Ф = Фm Sinwt   ————-(2)

Putting value of flux in the equation (1)

Ep = – Np x d/dt(Фm Sinwt)

Ep = – Np x wФm Coswt

Coswt = Sin(wt- π/2)

Ep = – Np x wФm Sin(wt- π/2)

The maximum value of voltage induced in the primary is

Ep = – Np x wФm

Ep = – Np wФm

w = 2πf

Ep(max) =  Np 2πfФm   ———–(3)

The root mean square (RMS) value of the induced voltage in the primary

Ep(RMS) = Ep(Max)/√2   ——-(4)

Putting the Ep(max) value in equation(4)

Ep(RMS) = Np 2πfФm/√2

Ep(RMS) = √2π Np fФm

Ep(RMS) = 4.44 Np fФm    ——-(5)

Similarly, the  root mean square (RMS) value of the induced voltage in the secondary is ;

Es(RMS) = 4.44 Ns fФm    ——-(6)

The general EMF Equation of the transformer is;

E= 4.44 fФm  ———–(7)

Voltage Transformation Ratio of Transformer

The ratio of the secondary turns to primary turns is known as the voltage transformation ratio of the transformer and it is denoted by letter ‘K’.

Dividing equation(6) by equation(5)

Es/Ep = 4.44 Ns fФm/ 4.44 Np fФm

ES/Ep  = Ns/Np  =K  ——–(7)

If the secondary turns are more than the primary turns, the secondary voltage will be higher than the primary voltage and the turns ratio (K) is greater than 1. The transformer having voltage transformation ratio more than 1 is called step up transformer.

If the voltage transformation ratio (K) is less than 1, the transformer is called step down transformer.

The magnetizing MMF is negligible when substantial current flows in the primary and secondary winding of the transformer. The total MMF of the primary and secondary MMF is equal to the magnetizing MMF. The transformer is a constant flux machine as long as voltage and frequency remains constant.

MMFprimary + MMFsecondary = MMFmagnetizing

When transformer is substantially loaded,  the magnetizing MMF can be neglected.

MMFprimary + MMFsecondary = 0

Np x Ip  + Ns x Is  =0

Np x Ip  = -Ns x Is

Ip/Is = – Ns/Np  ——(8)
The negative sign shows that the current in the  primary and secondary winding are in opposite direction with respect to the magnetizing current.The minus sign can be dropped for calculation of transformer transformation ratio.
From equation (7) & (8)

Es / Ep = Ip / Is  = K

Es / Ep = Ns / Np =Ip / Is = K

The voltage transformation ratio equation (7) of transformer is true if the flux in the transformer core is constant. The flux in the transformer  changes if the supply voltage and or frequency gets deviate from its designed value.

## Transformer Turns Ratio

The ratio of primary(Np) to secondary turns(Ns) of the transformer is known as transformer turns ratio,or TTR. It is denoted by letter ‘a’.

Np / Ns = a

Ns / Np = K

Np / Ns = 1/K

a = 1 / K

Es / Ep = Ns / Np =Ip / Is = K

Ep / ES = Np/ Ns = Is / Ip = 1/K

Ep /  Es = Np / Ns = Is / Ip = a

Solved Problems on transformer EMF equation

A two-winding transformer has primary winding with 300 turns and secondary winding with 10 turns. The primary winding is connected to 3300 V supply system. Calculate-

•  Secondary voltage at no load
•  Primary current when the 100-amp load connected to secondary
• Apparent power flowing in the primary and secondary circuit
• Turns ratio and voltage transformation ratio of transformer?
First transformer law:
Ep/Es = Np/Ns
Es = (Ep x Ns) /NP
Es = (3300 x 10)/ 300
Es = 110 Volts
Second transformer law:
Ip/ Is = Ns/Np
Ip = (Ns x Is)/ Np
Ip =  (10 x 100)/300
Ip = 1000/300
Ip = 3.33 Amps.
Apparent power in primary circuit  =  Ep x Ip
=  3300 x 3.33
=  10989 VA
Apparent power in secondary circuit  =  Es x Is
=   110 x 100
= 11000 VA
Transformer turns ratio(TTR)

a  = Np/Ns

a  = 300/10
a  = 30

Transformer voltage transformation ratio

K  = Es/ Ep
K  = 110/3300

K = 0.0333

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