7.2 Basic Transformer Model

The equivalent circuit of the basic transformer model, shown in Figure 7.1, consists of two mutually coupled coils. The voltages across these coils is expressed as:

(7.1)

where L ₁₁ and L ₂₂ are the self-inductance of winding 1 and 2 respectively, and L ₁₂ and L ₂₁ are the mutual inductance between the windings.

Figure 7.1: Equivalent circuit of the two-winding transformer

In order to solve for the winding currents the inductance matrix has to be inverted, i.e.

(7.2)

Since the mutual coupling is bilateral, L ₁₂ and L ₂₁ are identical. The coupling coefficient between the two coils is:

(7.3)

Rewriting equation 7.1 using the turns ratio ( a = v ₁/ v ₂) gives:

(7.4)

This equation can be represented by the equivalent circuit shown in Figure 7.2, where

(7.5)

(7.6)

Consider a transformer with a 10% leakage reactance equally divided between the two windings and a magnetising current of 0.01 p.u. Then the input impedance with the second winding open circuited must be 100 p.u. (Note from equation 7.5, L ₁ + L ₁₂ = L ₁₁ since a = 1 in the per unit system.) Hence the equivalent in Figure 7.3 is obtained, the corresponding equation (in p.u.) being:

(7.7)

or in actual values:

(7.8)

Figure 7.2: Equivalent circuit of the two-winding transformer, without the magnetising branch

Figure 7.3: Transformer example