Table of Contents
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:
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).
$$J=\omega C$$
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:
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:
$$Z=\begin{bmatrix} 0 & -j \omega L \\ -j \omega L & 0 \end{bmatrix}$$
$$Y=\begin{bmatrix} 0 & \frac{j}{\omega L} \\ \frac{j}{\omega L} & 0 \end{bmatrix}$$
References
- Tosic, D. V., & Potrebic, M. (2006). Symbolic analysis of immittance inverters, 14th Telecommunication Forum. Belgrade (Serbia), 21-23.
- J.S.G. Hong and M.J. Lancaster, ``M.J. Microstrip filters for RF/microwave applications,'' John Wiley and Sons.: Hoboken, NJ, USA, 2004.
Looking for impedance inverter?