Equivalent Circuit and Phasor Diagram of Transformers

Electrical and Automation Engineer . Specialized in LV Switchgear Design and process automation

Ideal transformer

Ideal transformer

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

1. Primary and secondary resistances are assumed to be zero. Hence there is no power loss and voltage drop in an ideal transformer.
2. Leakage flux is completely absent.
3. The permeability of the core is infinite and so the magnetizing current is zero.
4. 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 R₁ and R₂.These resistances are uniformly spread throughout the windings. These resistances give rise to the copper losses (I²R). Consider that Φ_l1 be the leakage flux caused by the MMF I₁N₁ in the primary windings and Φ_l2 be the leakage flux caused by the MMF I₂N₂ 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 R₂ and reactance X _l1 and X _l2 in series with corresponding windings. Because of the presence of these lumped quantities the induced emf s E₁ and E₂ may vary from the secondary voltages V₁ and V₂ 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) ≈( V₁ / V₂)

Real transformer

Equivalent circuit

The exciting current I₀‾ 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.

I₀‾ = 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

R₂^’ = (N1/N2)² R₂

The load voltage and currents referred to primary side are

V₂’ = (N1/N2) V₂

I₂’ = (N1/N2) I₂

Equivalent circuit referred to secondary side is as follows:

Equivalent circuit referred to secondary

The load voltage and currents referred to primary side are

V₁’ = (N1/N2) V₁

I₁’ = (N1/N2) I₁

G_i’= (N1/N2)² G_i

B_i’= (N1/N2)² B_i

X _l1^’ = (N1/N2)² X _l1

R₁^’ = (N1/N2)² R₁

Phasor diagram

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

V₁­=E₁+I₁R₁+jI₁X₁

V₂­=E₂+I₂R₂+jI₂X₁

I₁=I₂’+I₀’= I₂’+ (I_i + I_m)

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

Phasor diagram

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Reference

• Equivalent circuit and Phasor diagram of a transformer
  Equivalent circuit of a transformer is a schematic representation of a practical transformer.

© 2013 OSBERT JOEL C

Comments

EMOVON ALFRED on May 09, 2018:

This is really helpful. Thank you for the precise update.

Dharmendra Yadav on September 19, 2014:

B.tech. (EN)

Kanpur.

ashenafi on December 02, 2013:

in my opinion that is good for me.