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--- ipt:basics:resonancefrequency [2024/04/04 12:03] – admin
+++ ipt:basics:resonancefrequency [2024/04/04 17:07] (current) – [Equivalent circuit] 2a02:1812:40f:9800:b890:7399:f066:e603
@@ Line 1: / Line 1: @@
+====== 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 $R_2$.
 {{:ipt:basics:coupled_coils_with_series_resistances.png?400|}}
-Imedance matrix:
+===== Equivalent circuit =====
+This circuit can be represented by the following equivalent circuit:
+INSERT FIGURE HERE
+Other equivalent circuit representations can be found 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}
-		I_1  \\
+		R_1+j\omega L_1 & j \omega M  \\
-		I_2  \\
+		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}
-		y_{11} & y_{12}  \\
+		z_{22}  & -z_{12} \\
-		y_{21} & y_{22}  \\
+		-z_{21}  & z_{11} \\
-	\end{bmatrix}
+	\end{bmatrix}=
-	.
+        \frac{1}{R_1R_2+\omega^2(M^2-L_1L_2)+j\omega (R_1L_2+R_2L_1)}
 	\begin{bmatrix}
-		V_1  \\
+		R_2+j\omega L_2 & -j \omega M  \\
-		V_2  \\
+		-j \omega M  & R_1+j\omega L_1 \\
-	\end{bmatrix}	.
+	\end{bmatrix}
-	\label{IV-relations}
 \end{align}
+with $det(Z)$ the determinant of Z.