Introduction and definition

Voltage and current are hard to measure at high frequencies. Short and open circuits (used by definitions of most n-port parameters) are hard to realize at high frequencies. Therefore, microwave engineers work with so-called scattering parameters (S parameters), that uses waves and matched terminations (normally $50 \Omega$). This procedure also minimizes reflection problems.

A (normalized) wave is defined as ingoing wave $\underline{a}$ or outgoing wave $\underline{b}$:

$$\underline{a} = \underbrace{\dfrac{\underline{u}+\underline{Z}_0\... ...line{U}_{backward}} \cdot \dfrac{1}{\sqrt{\vert\text{Re}\underline{Z}_0)\vert}}$$ (1.1)

where $\underline{u}$ is (effective) voltage, $\underline{i}$ (effective) current flowing into the device and $\underline{Z}_0$ reference impedance. The waves are related to power in the following way.

$$P = \left( \vert\underline{a}\vert^2 - \vert\underline{b}\vert^2 \right)$$ (1.2)

Sometimes waves are defined with peak voltages and peak currents. The only difference that appears then is the relation to power:

$$P = \frac{1}{2}\cdot \left( \vert\underline{a}\vert^2 - \vert\underline{b}\vert^2 \right)$$ (1.3)

Now, characterizing an n-port is straight-forward:

$$\begin{pmatrix}\underline{b}_1\\ \vdots\\ \underline{b}_n\\ \end{... ...\cdot \begin{pmatrix}\underline{a}_1\\ \vdots\\ \underline{a}_n\\ \end{pmatrix}$$ (1.4)

One final note: The reference impedance $\underline{Z}_0$ can be arbitrary chosen. It normally is real, and there is no urgent reason to use a complex one. The definitions in equation 1.1, however, are made form complex impedances. These ones stem from [1], where they are named "power waves". These power waves are a useful way to define waves with complex reference impedances, but they differ from the waves introduced in the following chapter. For real reference impedances both definitions equal each other.