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--- impedance_inverter [2025/05/04 07:48] – created kl
+++ impedance_inverter [2025/12/27 16:06] (current) – [Definition] admin
@@ Line 11: / Line 11: @@
 K is a reactance (in ohm), and is called the characteristic impedance of the inverter.
-====Examples====
+For the ABCD matrix of an impedance inverter, it holds that A=0 and D=0, with the ABCD matrix of a two-port network defined as:
+$$\begin{bmatrix}
+V_1 \\ I_1
+\end{bmatrix}
+=\begin{bmatrix}
+A & B\\ C & D
+\end{bmatrix}
+\begin{bmatrix}
+V_2 \\ -I_2
+\end{bmatrix}$$
-Different options exist to realize an admittance inverter, for example:
+Since the impedance inverter is a reciprocal network, it follows that AD-BC=1, and since A=D=0, we get: B.C=-1.
+The general ABCD matrix of an impedance inverter is given by:
+$$ABCD=\begin{bmatrix}
+& -jK\\-\frac{j}{K} & 0
+\end{bmatrix}$$
+====Example: inductive wireless power transfer coupling====
+Different options exist to realize an impedance inverter, for example:
 {{ impedance_inverter_general_example.png?400 |}}
@@ Line 19: / Line 38: @@
 For example, when we choose an inductor, K equals $\omega L$ and we get the following circuit. (Note: a negative inductor corresponds to an capacitance, i.e., a capacitor instead of a coil).
+$$K=\omega L$$
 {{ immittance_inverter_t-l-network.png?300 |}}
@@ Line 24: / Line 44: @@
 Note that this is exactly the magnetic coupling for inductive wireless power transfer, where L is the mutual inductance!
+The ABCD matrix of this impedance inverter is given by:
+$$ABCD=\begin{bmatrix}
+& -j \omega L \\ -\frac{j}{\omega L} & 0
+\end{bmatrix}$$
+The corresponding impedance and admittance matrix equals:
+$$Z=\begin{bmatrix}
+& j \omega L \\ j \omega L & 0
+\end{bmatrix}$$
+$$Y=\begin{bmatrix}
+& -\frac{j}{\omega L} \\ -\frac{j}{\omega L} & 0
+\end{bmatrix}$$
+====Another example====
+Many impedance inverters exist. Another example is the following circuit:
+{{ immittance_inverter_t-c-network.png?300 |}}
+Here,
+$$K=-\frac{1}{\omega C}$$
+The ABCD matrix of the corresponding two-port network equals:
+$$\begin{bmatrix}
+&  \frac{j}{\omega C} \\ j \omega C & 0
+\end{bmatrix}$$
+The corresponding impedance and admittance matrix equals:
+$$Z=\begin{bmatrix}
+& -\frac{j}{\omega C} \\ -\frac{j}{\omega C} & 0
+\end{bmatrix}$$
+$$Y=\begin{bmatrix}
+& j\omega C\\ j \omega C & 0
+\end{bmatrix}$$
+----
+<color #808080>**References**</color>
+  * <color #808080>Tosic, D. V., & Potrebic, M. (2006). Symbolic analysis of immittance inverters, 14th Telecommunication Forum. Belgrade (Serbia), 21-23.</color>
+  * <color #808080>J.S.G. Hong and  M.J. Lancaster, ``M.J. Microstrip filters for RF/microwave applications,'' John Wiley and Sons.: Hoboken, NJ, USA, 2004.</color>
+----
 Looking for admittance inverter?
   * [[admittance inverter]]