**Subject :**Electrical Machines I (DC Machines)

**Unit :**Transformer

## EMF Equation of transformer

**EMF Equation of transformer**

EMF Equation of transformer in electrical power transformer, one alternating electrical source is applied to the primary winding and due to this, magnetizing current flows through the primary which produces alternating flux in the core of transformer. This flux likes with both primary and secondary windings. As this flux is alternating in nature there must be a rate of change of flux. According to Faraday's law of electromagnetic induction if any coil or conductor links with any changing flux, there must be an induced emf in it. As the electric current source to primary, is sinusoidal, the flux induced by it will be also sinusoidal. Hence the function of flux may be considered as a sine function. Mathematically derivative of that function will give a function for rate of change of flux linkage with respect to time. This later function will be a cosine function since d(sinθ)/dt = cosθ.

So if we derive the expression for rms value of this cosine wave and multiply it with number of turns of the winding we will easily get the expression for rms value of induced emf of that winding. In this way we can easily derive the emf equation of transformer.

Let, T is number of turns in a winding, Φm is the maximum flux in the core in Wb.As per Faraday's laws of electromagnetic Induction,emf,

e = − T.dφ/dt

Where φ is the instantaneous alternating flux and represented as,

φ = Φmsin2πft

Hence, e = d(Φmsin2πft)/dt ⇒e = − TΦm2πf cos2πft

As the maximum value of cos2πft is 1, the maximum value of induced emf e is, em = T Φm2πf

To obtain the rms value of induced counter emf, divide this maximum value of e by √2. Then, E = 2π/√2 X ΦmfT Volts

⇒E = 4.44ΦmfT Volts (Since, 2π/√2 = 4.44)

This is EMF equation of transformer

If E1 & E2 are primary and secondary emfs and T1 & T2 are primary and secondary emfs then, voltage ratio or turns ratio of transformer is,

E1 / E2 = 4.44ΦmfT1 / 4.44ΦmfT2 = T1 / T2

⇒E1 / E2 = T1 / T2

**Transformation Ratio of Transformer:-**

This constant is called transformation ratio of transformer , if T2>T1, K>1, then the transformer is step up transformer. If T2<T1, K<1, then the transformer is step down transformer.

**Voltage Ratio of Transformer:-**

This above said ratio is also known as voltage ratio of transformer if it is expressed as ratio of the primary and secondary voltages of transformer.

**Turns Ratio of Transformer:-**

As the voltages in primary and secondary of transformer is directly proportional to number of turns in the respective winding, the transformation ratio of transformer is sometime expressed in ratio of turns and referred as turns ratio of transformer.