**EMF Equation Of the Transformer**

**Introduction**

Magnitude of the **induced EMF** (or Voltage) in a transformer can be found by **EMF equation of the transformer**. When a source of alternating current (AC) is applied to the primary winding of the transformer which is known as **magnetizing current**, it produces alternating flux in the core of a transformer.

The produced alternating flux in the primary of the transformer gets linked with the secondary winding of the transformer by **mutual induction** as it is alternating flux in nature, there must be a rate of change of flux according to **Faraday’s law of electromagnetic induction** which states that if a conductor or coil links with any changing flux, there must be an induced emf in it.

The same happens in transformer as well as induction motor as induction motor is fundamentally a transformer.

**Transformer EMF Equation**

Now lets know how to find the magnitude of the *inducted EMF in a transformer by EMF equation of the transformer.*

Lets,

**N**_{1 }= Number of turns in primary windings.

**N _{2}** = Number of turns in second windings.

**Φ**_{m }= Maximum flux in the core in Webbers.

**Φ**_{m=}B_{m}.A,

*f* = Frequency of A.C input in H_{z.}

Click image to enlarge

As shown in fig above- flux increases from its zero value to maximum value **Φ _{m}** in one quarter of the cycle i.e. in

**¼ second**.

Average rate of change of flux = [ **Φ**_{m /} (**¼ f**.)]

**= 4 f Φ_{m} Wb/s or volt**

Now rate of change of flux per turn means **induced e.m.f in volts**.

**Average e.m.f /per turn** = **4***f***Φ _{m }**volt.

If flux** Φ _{m }varies sinusoidally, then r.m.s value of induced .e.m.f is obtained by multiplying the average value with form factor. **

**Form factor = r.m.s value / Average value = 1.11**

r.m.s value of e.m.f/turn = **1.11. 4***f*** Φ _{m= }4.44f Φ_{m }**volt

now r.m.s value of the **induced e.m.f in the whole primary winding**.

= ( **induced e.m.f/turn**) x **number of primary turns**

**E _{1} = 4.44 x f x N_{1} Φ_{m }**

_{……….. (i)}

_{ }**E _{1} = 4.44 x f N_{1 }B_{m }A … [**

*as*

**(Φ**

_{m}= B_{m}A)]Similarly, **r.m.s value of the e.m.f. induced in secondary** is,

**E _{2} = 4.44 x f N_{2} Φ_{m }**

_{……….. }

_{(ii)}

_{ }**E _{2} = 4.44 x f N_{2 }B_{m }A. **

**… [**

*as*(Φ_{m}= B_{m}A)]It’s seen from *(i)* and *(ii)* that: **EMF Equation of the Transformer =**

** E _{1 }/ N_{1}= E_{2 }/ N_{2} = 4.44 x f Φ_{m. }**

_{…… (iii)}

** _{ }**It means that

**e.m.f / turn**

**is the same in both the primary and secondary windings in the transformer**like flux in Primary and Secondary Winding of the Transformer is same.

Moreover, we already know that from the power equation of the transformer, i.e, in ideal Transformer (there are no losses in transformer) on no-load,

**V _{1 }= E_{1} **

and

**E _{2 }= V_{2} **

Where,

**V**supply voltage of primary winding_{1}=**E**= terminal voltage induced in the secondary winding of the transformer._{2}

You may also read: Transformers Fire Protection System – Causes, Types & Requirements

**Voltage Transformation Ratio (K)**

As we have derived from the above EMF equation of the transformer _{(iii);}

**E _{1 }/ N_{1}= E_{2 }/ N_{2} = K_{
}**

Where,

**K = Constant**

The constant **“K”** is known as **voltage transformation ratio**.

- If N
_{2}> N_{1}, i.e. K > 1, then the transformer is called step-up transformer. - If N
_{2}< N_{1}, i.e. K < 1, then the transformer is called step-down transformer.

Where,

**N _{1} **= Primary number of turns of the coil in a transformer.

**N _{2 }**= Secondary number of turns.

_{ }

- You may also read: Current Transformers (CT) – Types, Characteristic & Applications

**Power Equation of the Transformer**

We have already discussed it in our previous post which can be seen it here.

You may also read:

- Transformer Efficiency, All day Efficiency & Condition for Maximum Efficiency
- Can we operate a 60HZ Transformer on 50Hz Supply Source and Vice Versa?
- How to Calculate the Rating of Transformer in kVA (1 Phase and 3 Phase)?

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It is really easy to understand the emf equation of a transformer. It is even easier to understand with the neat and clean diagram. Just like this post <a href="http://www.electricaleasy.com/2014/03/electrical-transformer-basic.html" rel="nofollow"> Electrical Transformer</a>. Thanks for the post. 🙂

How we find numerical question in this site.please tell me

Wonderful job. Keep it up, but add some MCQ question.