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}$:

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

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

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

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

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

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

This document was generated by Stefan Jahn on 2007-12-30 using latex2html.