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====== Coupled coils with series resistance ====== 
===== Circuit =====

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

===== Equivalent circuit =====

This circuit can be represented by the following equivalent circuit:

INSERT FIGURE HERE

===== Description as a two-port network =====

==== Impedance matrix ====

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.

==== Admittance matrix ====

\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^2L_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.

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