May 10, 2017

# Transformer equivalent circuit

To understand transformer equivalent circuit .. we should firstly know what is transformer you can check this topic .. if you have knowledge about what is transformer .. So, lets go to this important lecture.

We assumed  that ideal transformer windings have only reactance and have no physical resistance ,  we assumed This assumption to obtain the conversion rate of Voltages, currents and impedance of Load, But in fact there is a resistance to both primary and secondary windings since they are made of copper.

## Real Transformer :

Our interest now is to obtain the true values of currents and transmitted power by obtaining the transformer equivalent circuit . Based on this, we will take into consideration winding resistance where the primary winding resistance is symbolized by R1 and secondary winding resistance is symbolized by R2 .

We assumed that “In the ideal transformer there is no magnetic leakage” …. But in fact, the magnetic flux resulting from the flow of a current in the primary windings is not completely linked with the secondary winding, But a small part of magnetic flux (ɸ1) leaks around the primary windings  and completes its magnetic circuit through the air …So, this leaking flux linked with the primary winding and produces Self induced Electro Motive force (EMF)  which results in leakage reactance of the primary winding  X1.. such that  X1=2πfL1

Also, when the transformer is loaded and there is a current passed in the secondary winding this caused a magnetic flux (ɸ2)  Also generated, this flux causes a part of the secondary winding leaks out.

This leaking flux linked with the secondary winding and produces an Electro motive force (EMF) results in leakage reactance of the Secondary winding  X2 … such that  X2=2πfL2

Previously we assumed that : ” In the ideal transformer there is no loss of electrical power “ …  But actually Transformer has two types of lossesiron core losses  and copper winding losses

the previous figure shows:

### equivalent circuit of transformer

such that:

Xo : Magnetic reactance of the iron core

Ro : Magnetic resistance of the iron core

They represented iron core losses impedance …

Io : Current at no load condition

Ia : core loss effective current

Im : magnetizing current

From the previous figure :

We can simplify and abbreviate equivalent circuit referred to primary windings:

In the previous figure we notice Also that all the parameters in the secondary windings is moved to the primary windings  and  take values that are marked with the mark this mark (ʼ)

these values are differed from their first position So, we can calculate the new values from the following equations :

such that:

Vʼ2 = (N1/N2) * V2

Iʼ2 =  (N2/N1) * I2

Rʼ2 = R2 *  (N1/N2)²

Xʼ2 = X2 *  (N1/N2)²

By moving the parallel branch either to the primary windings  or to the secondary windings

So, equivalent circuit can be simplified as shown in the following figure

We can also ignore The parallel branch (magnetism)  to obtain the approximate equivalent circuit of the transformer

We can calculate Equivalent Resistance Req1 and equivalent Reactance  Xeq1 values Referred to primary windings from the following equations :

Req1 = R1 + Rʼ2

Xeq1 = X1 + Xʼ2

ZʼL = ZL *  (N1/N2)²

#### equivalent circuit of transformer referred to Secondary winding :

Also, Equivalent Resistance Req2 and equivalent Reactance  Xeq2 values Referred to secondary windings are calculated from the following equations :

Such that:

Req2 = Rʼ1 + R2

Xeq2 = Xʼ1 + X2

Vʼ1 = (N2/N1) * V1

Iʼ1 =  (N1/N2) * I1

Rʼ1 = R1 *  (N2/N1)²

Xʼ1= X1 *  (N2/N1)²

To understand transformer equivalent circuit .. we must give solved example

Solved Example :

A single-phase transformer with a capacity of 100 kVA, 2000/400 volts and equivalent circuit parameters

R0=500 Ω            Xo=150 Ω       R1=0.01 Ω      X1=0.03 Ω       R2=0.25 Ω       X2=0.75 Ω

It feeds a Load of 90 kVA at a voltage of 2000 volts and a power factor of 0.8 Lag

Calculate the primary voltage and current According to this figure

Solution: