Consider two coils with inductances $L_1$ and $L_2$, coupled by mutual inducance $M$. The inductors have series resistance $R_1$ and $R_2$.

This circuit can be represented by the following equivalent circuit:

INSERT FIGURE HERE

Other equivalent circuit representations can be found here.

The impedance matrix of the circuit is given by:

\begin{align} Z = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \\ \end{bmatrix}= \begin{bmatrix} R_1+j\omega L_1 & j \omega M \\ j \omega M & R_2+j\omega L_2 \\ \end{bmatrix} \end{align}

with $\omega$ the angular frequency.

\begin{align} Y = \frac{1}{det(Z)} \begin{bmatrix} z_{22} & -z_{12} \\ -z_{21} & z_{11} \\ \end{bmatrix}= \frac{1}{R_1R_2+\omega^2(M^2-L_1L_2)+j\omega (R_1L_2+R_2L_1)} \begin{bmatrix} R_2+j\omega L_2 & -j \omega M \\ -j \omega M & R_1+j\omega L_1 \\ \end{bmatrix} \end{align}

with $det(Z)$ the determinant of Z.