# Coupled Inductors as Transformer

Figure 1 shows a pair of coupled inductors with self inductance and , magnetically coupled through coupling coefficient .

For the dot notation shown in Figure 1, the equations that relate terminal voltages and terminal current in frequency domain are given by,

(1) where is the mutual inductance between the two coupled inductors. From the previous discussion on self inductance and mutual inductance, we know that .

Solving for and from the Eq-(1)

(2) Substituting the Eq-(2) in the Eq-(1) and collecting like terms, we have

(3) Substituting for ‘ ‘ in Eq-(3), we get

(4) where The first term in Eq-(4) is due to leakage flux and the second term indicates the transformer action. If the coupling between two inductors is perfect, coupling coefficient is unity and they behave like ideal transformer with turns ratio of “1:n”.

### T model

Rearranging Eq-(1), we get

(5) From the Eq-(5), we can draw the electrical equivalent of coupled inductors or coils as shown in Figure 2. Since the equivalent circuit appears like the letter ‘T’, it called the T-model. In this model none of the inductors are magnetically coupled, hence it simplify the analysis.

### Ideal Transformer model

In circuits where isolation is required, the equivalent circuit model shown in Figure 3 is used. In this circuit ideal transformer represents mutual coupling part and by self inductors without coupling represents leakage part.

The terminal equations for the block representing leakage part, shown in Figure 3, are given by

(6) The terminal behavior of the entire equivalent circuit shown in Figure 3 resemble that of shown in Figure 1, if

(7) Substituing Eq-(7) in Eq-(6) and rearranging the equation, we get

(8) Figure 4 illustrates an equivalent circuit with T-model to represent leakage inductance and with an ideal transformer to represent coupling.

If , the last term in Eq-(8) vanishes. Using the relationship , the Eq-(8) is reduced to Eq-(9). The equivalent circuit is then shown in Figure 5.

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