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Table of Contents
Impedance inverter
Definition
An impedance inverter changes an output impedance $Z_{out}$ to its inversely proportional value $Z_{in}$, multiplied by a value K²:
$$Z_{in}=\frac{K^2}{Z_{out}}$$
K is a reactance (in ohm), and is called the characteristic impedance of the inverter.
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: $$\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} =\begin{bmatrix} A & B\\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$$
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: $$ABCD=\begin{bmatrix} 0 & -jK\\-\frac{j}{K} & 0 \end{bmatrix}$$
Example: inductive wireless power transfer coupling
Different options exist to realize an impedance inverter, for example:
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).
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: $$ABCD=\begin{bmatrix} 0 & -j \omega L \\ -\frac{j}{\omega L} & 0 \end{bmatrix}$$
The corresponding impedance and admittance matrix equals:
$$Z=\begin{bmatrix} 0 & j \omega L \\ j \omega L & 0 \end{bmatrix}$$
$$Y=\begin{bmatrix} 0 & -\frac{j}{\omega L} \\ -\frac{j}{\omega L} & 0 \end{bmatrix}$$
Another example
Many admittance inverters exist. Another example is the following circuit:
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.
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