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=====Admittance inverter=====



====Definition====

An admittance inverter changes an output admittance $Y_{out}$ to its inversely proportional value $Y_{in}$, multiplied by a value J²:

$$Y_{in}=\frac{J^2}{Y_{out}}$$

J is a susceptance (in siemens), and is called the characteristic admittance of the inverter.

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.

====Examples====

Different options exist to realize an admittance inverter, for example:

{{ admittance_inverter_general_example.png?400 |}}

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).

{{ 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 admittance matrix of the corresponding two-port network equals:

$$Y=\begin{bmatrix}
0 & -j\omega C\\-j \omega C & 0
\end{bmatrix}$$

When considering repeater resonators, the ABCD matrix of the admittance inverter is relevant, given by:
$$ABCD=\begin{bmatrix}
0 & \frac{-j}{\omega C}\\-j \omega C & 0
\end{bmatrix}$$


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).




Note that this is identical to the electric coupling for capacitive wireless power transfer.



{{ immittance_inverter_pi-l-network.png?300 |}}


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).

{{ immittance_inverter_t-c-network.png?300 |}}


The impedance matrix of the corresponding two-port network equals:

$$\begin{bmatrix}
0 & j\omega L\\j \omega L & 0
\end{bmatrix}$$

$$\begin{bmatrix}
0 & j\omega L\\j \omega L & 0
\end{bmatrix}$$

Reference: Tosic, D. V., & Potrebic, M. (2006). Symbolic analysis of immittance inverters, 14th Telecommunication Forum. Belgrade (Serbia), 21-23.

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