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# Equivalent circuit and Phasor diagram of transformers

## Read more on transformers

- How does a transformer work?
- Parts of a Power Transformer
- Equivalent circuit and Phasor diagram of transformers
- Types of transformer
- Losses in Transformers
- Testing of transformers
- Cooling of Transformer
- Tap Changing in transformers
- What is a Buchholz relay? How does it work?
- Properties of transformer oil
- Insulating materials used in transformers
- Current transformer- Definition, Principle, Equivalent circuit, Errors and types
- Potential Transformers in detail
- How does an Autotransformer work?

## Ideal transformer

## Ideal transformer

A transformer that possess the following properties are considered to be an ideal transformer.

- Primary and secondary resistances are assumed to be zero. Hence there is no power loss and voltage drop in an ideal transformer.
- Leakage flux is completely absent.
- The permeability of the core is infinite and so the magnetizing current is zero.
- The core losses are considered to be zero.

The figure shows the diagrammatic representation of an ideal transformer. It is represented in such a way that all the above conditions are satisfied.

## Real transformer and Equivalent circuits

Transformer windings are made mainly of copper. Even though copper is a good conductor, it possess a finite resistance. Both the primary and the secondary have finite resistances R_{1} and R_{2}.These resistances are uniformly spread throughout the windings. These resistances give rise to the copper losses (I^{2}R). Consider that Φ_{l1 }be the leakage flux caused by the MMF I_{1}N_{1 }in the primary windings and Φ_{l2 }be the leakage flux caused by the MMF I_{2}N_{2 }in the secondary windings.

Both the resistance and the leakage reactance of the transformer windings are series effects and at operating frequencies, which is very low (50Hz / 60Hz) these can be regarded as the lumped parameters. Hence the transformer consists of lumped resistances R1 and R_{2 }and reactance X_{ l1} and X_{ l2} in series with corresponding windings. Because of the presence of these lumped quantities the induced emf s E_{1} and E_{2} may vary from the secondary voltages V_{1 }and V_{2} because of the small voltage drops in the winding resistances and leakage reactance.

The transformer ratio can be given by:

a= (N1/N2) = (E1/ E2) ≈( V_{1} / V_{2})

## Real transformer

## Equivalent circuit

The exciting current I_{0}‾ can be resolved into two components, whose magnetizing component I_{m}‾ creates mutual flux Φ‾ and whose core loss component I_{i}‾ provides the loss associated with alternation of flux.

I_{0}‾ = I_{m}‾ + I_{i}‾

Note: vector form is indicated by the symbol ‾.

Hence the equivalent circuit can be represented as shown in the figure.

## Equivalent circuit of transformer

Here,

G_{i }= conductance

B_{i }= Susceptance

The impendence can be now referred to in the primary side resulting in the following circuit:

Equivalent circuit referred to primary side is as follows:(core is neglected)

## Equivalent circuit referred to primary

X_{ l2}^{’} = (N1/N2)^{2} X_{ l2}

_{}

R_{2}^{’} = (N1/N2)^{2 }R_{2}

_{}

The load voltage and currents referred to primary side are

V_{2}’ = (N1/N2) V_{2}

I_{2}’ = (N1/N2) I_{2}

Equivalent circuit referred to secondary side is as follows:

## Equivalent circuit referred to secondary

The load voltage and currents referred to primary side are

V_{1}’ = (N1/N2) V_{1}

I_{1}’ = (N1/N2) I_{1}

_{}

G_{i}’= (N1/N2)^{2 }G_{i}

B_{i}’= (N1/N2)^{2} B_{i}

_{}

X_{ l1}^{’} = (N1/N2)^{2} X_{ l1}

_{}

R_{1}^{’} = (N1/N2)^{2 }R_{1}

## Phasor diagram

Applying KVL on the primary and secondary side of the equivalent circuits

V_{1}=E_{1}+I_{1}R_{1}+jI_{1}X_{1}

V_{2}=E_{2}+I_{2}R_{2}+jI_{2}X_{1}

I_{1}=I_{2}’+I_{0}’= I_{2}’+ (I_{i }+ I_{m})

Using these equations phasor diagram, can be drawn as follows:

## Phasor diagram

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