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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).

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

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:

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