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--- admittance_inverter [2025/04/21 18:37] – admin
+++ admittance_inverter [2025/12/27 16:05] (current) – [Definition] admin
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 =====Admittance inverter=====
-An admittance inverter changes an output admittance $Y_{out}$ to its inversely proportional value $Y_{in}$, multiplied by a value B²:
-$$Y_{in}=\frac{B^2}{Y_{out}}$$
-B is a susceptance, and is called the characteristic admittance of the inverter.
+====Definition====
-Different options exist to realize an admittance inverter, for example:
+An admittance inverter changes an output admittance $Y_{out}$ to its inversely proportional value $Y_{in}$, multiplied by a value J²:
-{{ :admittance_inverter_general_example.png?400 |}}
+$$Y_{in}=\frac{J^2}{Y_{out}}$$
-The susceptance B can be an inductance or a capacitance. For example, when we choose an inductor, B equals $\omega L$ and we get the following circuit. (Note: a negative inductance corresponds to a capacitance, i.e., a capacitor instead of a coil).
+J is a susceptance (in siemens), and is called the characteristic admittance of the inverter.
-For example, when we choose a capacitor, B equals $\omega C$ and we get the following circuit. (Note: a negative capacitance corresponds to an inductance, i.e., a coil instead of a capacitor).
+For the ABCD matrix of an admittance inverter, it holds that A=0 and D=0, with the ABCD matrix of a two-port network defined as:
+$$\begin{bmatrix}
+V_1 \\ I_1
+\end{bmatrix}
+=\begin{bmatrix}
+A & B\\ C & D
+\end{bmatrix}
+\begin{bmatrix}
+V_2 \\ -I_2
+\end{bmatrix}$$
+Since the admittance 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 admittance inverter is given by:
+$$ABCD=\begin{bmatrix}
+& -\frac{j}{J}\\-jJ & 0
+\end{bmatrix}$$
-{{ :admittance_inverter_T-l-network.png?300 |}}
+====Example: capacitive wireless power transfer coupling====
-Note that this is exactly the magnetic coupling for inductive wireless power transfer, where L is the mutual inductance!
+Different options exist to realize an admittance inverter, for example:
-The impedance matrix of the corresponding two-port network equals:
+{{ admittance_inverter_general_example.png?400 |}}
-$$\begin{bmatrix}
+For example, when we choose a capacitor, J equals $\omega C$ and we get the following circuit. (Note: a negative capacitance corresponds to an inductance, i.e., a coil instead of a capacitor).
-& j\omega L\\j \omega L & 0
+$$J=\omega C$$
+{{ immittance_inverter_pi-c-network.png?300 |}}
+Note that this is exactly the electric coupling for capacitive wireless power transfer, where C is the mutual capacitance!
+The ABCD matrix of this admittance inverter is given by:
+$$ABCD=\begin{bmatrix}
+& \frac{-j}{\omega C}\\-j \omega C & 0
 \end{bmatrix}$$
-When considering relay resonators, the ABCD matrix of the admittance inverter is relevant, given by:
+The corresponding impedance and admittance matrix equals:
-$$\begin{bmatrix}
-& j\omega L\\j \omega L & 0
+$$Z=\begin{bmatrix}
+& \frac{j}{\omega C} \\ \frac{j}{\omega C} & 0
 \end{bmatrix}$$
+$$Y=\begin{bmatrix}
+& -j\omega C\\-j \omega C & 0
+\end{bmatrix}$$
-For example, when we choose a capacitor, B equals $\omega C$ and we get the following circuit. (Note: a negative capacitance corresponds to an inductance, i.e., a coil instead of a capacitor).
+====Another example====
+Many admittance inverters exist. Another example is the following circuit:
-{{ :admittance_inverter_pi-c-network.png?300 |}}
+{{ immittance_inverter_pi-l-network.png?300 |}}
-Note that this is identical to the electric coupling for capacitive wireless power transfer.
+Here,
-$$\begin{bmatrix}
+$$J=-\frac{1}{\omega L}$$
-& \frac{-j}{\omega C}\\-j \omega C & 0
-\end{bmatrix}$$
-{{ :admittance_inverter_pi-l-network.png?300 |}}
+The ABCD matrix of the corresponding two-port network equals:
+$$ABCD=\begin{bmatrix}
+& j \omega L \\ \frac{j}{\omega L} & 0
+\end{bmatrix}$$
+The corresponding impedance and admittance matrix equals:
-{{ :admittance_inverter_T-c-network.png?300 |}}
+$$Z=\begin{bmatrix}
+& -j \omega L \\ -j \omega L & 0
+\end{bmatrix}$$
+$$Y=\begin{bmatrix}
+& \frac{j}{\omega L} \\ \frac{j}{\omega L} & 0
+\end{bmatrix}$$
+----
+<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>
-Reference: Tosic, D. V., & Potrebic, M. (2006). Symbolic analysis of immittance inverters, 14th Telecommunication Forum. Belgrade (Serbia), 21-23.
+----
+Looking for impedance inverter?
+  * [[Impedance inverter]]