=====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.

The general ABCD matrix of an admittance inverter is given by:
$$ABCD=\begin{bmatrix}
0 & -\frac{j}{J}\\-jJ & 0
\end{bmatrix}$$


====Example: capacitive wireless power transfer coupling====

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

{{ admittance_inverter_general_example.png?400 |}}

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).

$$J=\omega C$$

{{ 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:
$$ABCD=\begin{bmatrix}
0 & \frac{-j}{\omega C}\\-j \omega C & 0
\end{bmatrix}$$

The corresponding impedance and admittance matrix equals:

$$Z=\begin{bmatrix}
0 & \frac{j}{\omega C} \\ \frac{j}{\omega C} & 0
\end{bmatrix}$$

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

====Another example====

Many admittance inverters exist. Another example is the following circuit:

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

Here, 

$$J=-\frac{1}{\omega L}$$

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

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

The corresponding impedance and admittance matrix equals:


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

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


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<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>

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Looking for impedance inverter?
  * [[Impedance inverter]]