Coupled Inductors as Transformer


Figure 1 shows a pair of coupled inductors with self inductance L_1 and L_2, magnetically coupled through coupling coefficient k.

Figure 1. A simplified coupled inductor as a transformer

Figure 1. A simplified coupled inductor as a transformer

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

(1)   \begin{eqnarray*} V_1 &=& j \omega L_1 I_1 + j \omega M I_2 \\ V_2 &=& j \omega M I_1 + j \omega L_2 I_2 \end{eqnarray*}

where M is the mutual inductance between the two coupled inductors. From the previous discussion on self inductance and mutual inductance, we know that M=k\sqrt{L_1 L_2}.

Solving for I_1 and I_2 from the Eq-(1)

(2)   \begin{eqnarray*} I_1 &=& {V_1 \over j \omega L_1} - {M \over L_1}I_2 \\ I_2 &=& {V_2 \over j \omega L_2} - {M \over L_2}I_1 \\ \end{eqnarray*}

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

(3)   \begin{eqnarray*} V_1 = j\omega L_1 \left(1-{M^2 \over L_1 L_2} \right)I_1 + {M \over L_2} V_2 \end{eqnarray*}

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

(4)   \begin{eqnarray*} V_1 = \underbrace{j\omega L_1 \left(1-k^2 \right)I_1}_{\mbox{drop due to leakage inductace}} + {1 \over n} V_2 \end{eqnarray*}

where n={1 \over k} \sqrt{L_2 \over L_1}

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)   \begin{eqnarray*} V_1 &=& j \omega (L_1-M) I_1 + j \omega M (I_1+I_2) \\ V_2 &=& j \omega M (I_1+I_2) + j \omega (L_2-M) I_2 \end{eqnarray*}

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.

Figure 2. T model equivalent circuit for the coupled inductors or transformer

Figure 2. T model equivalent circuit for the coupled inductors or transformer

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)   \begin{eqnarray*} V_1 &=& j \omega L_1 I_1 + j \omega M^{'} (n I_2) \\ {V_2 \over n}&=& j \omega M^{'} I_1 + j \omega L_2^{'} (nI_2)  \end{eqnarray*}

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

(7)   \begin{eqnarray*} M^{'} &=& {M / n}\\ L_2^{'} &=& {L_2 / n^2} \end{eqnarray*}

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

(8)   \begin{eqnarray*} V_1 &=& j \omega (L_1-M^{'}) I_1 + j \omega M^{'} (I_1 + n I_2)\\   &=& j \omega \left(L_1-{M \over n}\right) I_1 + j \omega {M \over n} (I_1 + n I_2)\\ {V_2 \over n} &=& j \omega M^{'} (I_1 + nI_2) + j \omega (L_2^{'}-M^{'}) (nI_2) \\  &=& j \omega {M \over n } (I_1 + nI_2) + j \omega \left({L_2 \over {n^2}}-{M \over n}\right) (nI_2) \end{eqnarray*}

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

Figure 4. Coupled inductors model with T equivalent circuit with ideal transformer

Figure 4. Coupled inductors model with T equivalent circuit with ideal transformer

If n= L_2 / M, the last term in Eq-(8) vanishes. Using the relationship M=k\sqrt{L_1 L_2}, the Eq-(8) is reduced to Eq-(9). The equivalent circuit is then shown in Figure 5.

(9)   \begin{eqnarray*} V_1 &=& j \omega L_1(1-k^2) I_1 + j \omega {k^2 L_1} (I_1 + n I_2) \\ {V_2 \over n} &=& j \omega {k^2 L_1 } (I_1 + nI_2) \end{eqnarray*}

Figure 5. Equivalent circuit model of coupled inuductor  with an ideal transformer

Figure 5. Equivalent circuit model of coupled inuductor with an ideal transformer

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