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=====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).

$$K=\omega L$$

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 impedance inverters exist. Another example is the following circuit:

Here,

$$K=-\frac{1}{\omega C}$$

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

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

The corresponding impedance and admittance matrix equals:

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

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

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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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