Equivalent Circuit of Transformer

Equivalent Circuit of Transformer

In this article, we will discuss the equivalent circuit of a transformer.

For the ideal transformer, we assumed that the transformer losses were zero. Therefore, we can neglect the resistance of transformer winding and the reluctance of the transformer core.

But in the case of practical transformers, we need to consider resistance and reactance.

Equivalent Circuit of Transformer

The complete equivalent circuit of the transformer is as shown in the figure below.

complete equivalent circuit of transformer
complete equivalent circuit of transformer

In the above figure,

V1 = Supply voltage (primary winding voltage)
V2 = Load voltage (secondary winding voltage)
E1 = Induced EMF in primary winding
E2 = Induced EMF in secondary winding
I0 = No-load current
R1, X1 = Primary winding resistance and reactance
R2, X2 = Secondary winding resistance and reactance

The no-load current is divided into two parts;

  • Working component (IW)
  • Magnetizing component (Iμ)

    \[ I_0 = I_W + I_\mu \]

Excitation resistance R0 and reactance X0 is connected in parallel. The working component (IW) of the no-load current passes through the excitation resistance, and magnetizing current (Iμ) passes through the excitation reactance.

R0 = Excitation resistance = It is responsible for no-load losses.

    \[ R_0 = \frac{E_1}{I_W} \]

X0 = Excitation reactance = It is responsible for setting up the flux in the core.

    \[ X_0 = \frac{E_1}{I_\mu} \]

The induced EMF in primary E1 and secondary E2 are related by turns ratio.

    \[ \frac{E_2}{E_1} = \frac{N_2}{N_1} = K \]

It is recommended to transfer all quantities (voltage, current, and impedance) either to the primary or secondary side to simplify the calculation. Because in this condition, we need to work on one winding that will be easier.

Exact Equivalent Circuit of Transformer Referred to Primary

The following steps are used to derive an equivalent transformer circuit referred to as the primary winding.

All secondary resistances and reactances are reflected primary side. And its values can be calculated by the square of the transformation ratio (K).

    \[ R_2 = \frac{R_2}{K^2} = a^2R_2 \]

    \[ X_2 = a^2X_2 \]

    \[ Z_L = a^2Z_L \]

All voltages are reflected from secondary to primary directly as the product of transformation ratio.

    \[ V_2 = \frac{V_2}{K} = aV_2 \]

    \[ E_2 = aE_2 \]

All secondary currents are reflected in the primary by multiplying the inverse transformation ratio.

    \[ I_2 = KI_2 = \frac{I_2}{a} \]

The equivalent circuit with all secondary values is reflected on the primary side. This circuit is known as the equivalent transformer circuit referred to primary, as shown in the figure below.

Exact Equivalent Circuit of Transformer Referred to Primary
Exact Equivalent Circuit of Transformer Referred to Primary

Approximate Equivalent Circuit Referred to Primary

The no-load current I0 is usually less than 5% of the full-load primary current. The voltage drop produced by the no-load current is very small, and it isn’t essential.

Therefore, the parallel branch of R0 and X0 can be placed either before to primary impedance or after the primary impedance.

There is not much effect on magnetizing current and working current; hence, the parallel branch can be connected to the input terminals for easy understanding.

The below figure shows the approximate equivalent circuit referred to primary.

Approximate Equivalent Circuit Referred to Primary
Approximate Equivalent Circuit Referred to Primary

By approximate equivalent circuit, the transformer’s voltage regulation calculation becomes more straightforward, and it is convenient to add primary and secondary impedance reflected the primary side.

From the approximate equivalent circuit, we can simplify the equations as below.

    \[ R_{e1} = R_1 + a^2R_2 \]

    \[ X_{e1} = X_1 + a^2X_2 \]

    \[ Z_{e1} = R_{e1} + X_{e1} \]

    \[ I_1 = \frac{V_1}{ Z_{e1} +a^2Z_L} \]

Simplified Approximate Equivalent Circuit Referred to Primary
Simplified Approximate Equivalent Circuit Referred to Primary

Approximate Equivalent Circuit Referred to Secondary

Like the above method, the approximate equivalent circuit of transformer referred to secondary is derived.

In this case, we need to transfer all primary resistance and reactance to the secondary. The approximate equivalent circuit of transformer referred to the secondary is shown in the figure below.

Approximate Equivalent Circuit Referred to Secondary
Approximate Equivalent Circuit Referred to Secondary

In above figure,

    \[ R_{e2} = R_2 + \frac{R_1}{a^2} \]

    \[ X_{e2} = X_2 + \frac{X_1}{a^2} \]

    \[ Z_{e2} = R_{e2} + jX_{e2} \]

    \[ I_2 = \frac{V_2}{Z_L} = \frac{E_2}{Z_{e2}+Z_L} \]

    \[ E_2  = \frac{V_1}{a} = V_2 + I_2 Z_{e2} \]

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