S-Matrix

Considering an element with n ports, we obtain for the incoming wave, ai, and the outcoming wave, bi, at port No. i:

with the voltage, Vi, the current, Ii, and the reference impedance, Zi.

$image\nports.gif$

Describing the relationship between an outcoming and an incoming wave by an S-parameter and writing these parameters in matrix form leads to the S-matrix:

$image\smatrix.gif$

In general, considering frequency domain, this gives frequency-dependent S-Parameters: Sij = Sij(f).

Note that a main diagonal element describes the reflection at the related port; the other elements are transmission factors between two ports. If the element contains only reciprocal materials and if all reference impedances are real the S-matrix is symmetric: Sij = Sji.

In practice, the S-matrix is highly relevant because of the following two aspects:

The S-matrix can be easily measured.
The S-matrix can be obtained for devices that you cannot or can only barely determine voltages and currents, e.g., for waveguides.

The relation between incoming and outcoming waves and voltages and currents given above is valid for real-valued reference impedances Zi. The extension to complex-valued reference impedances is not unique. The two most commonly used wave definitions with complex-valued reference impedances are the power waves

(where * denotes complex conjugate), and the pseudo waves

CST DESIGN STUDIO™ calculates S-Parameters based on power waves. S-Parameters based on pseudo waves may be obtained via the Calculate Pseudo Wave S matrices context menu item of any S-Parameter tree folder with complex-valued reference impedances. Both wave definitions reduce to the same definition for real-valued reference impedances.

It is worth noting that not all known properties of S-parameters with real-valued reference impedances Zi are still valid if complex-valued reference impedances Zi are used.

Power waves and S-Parameters based on power waves have the following properties:

A passive network has reflection coefficients with magnitude smaller or equal to unity, also when Zi is not real.
If Zi is not real, the scattering matrix need not be symmetric even if the network in question is reciprocal.
The reflection coefficient of an open circuit is constant: .
The reflection coefficient of a short circuit depends on Zi: .
The total accepted power at port No. i is given by .

Pseudo waves and S-Parameters based on pseudo waves have the following properties:

A passive network can have reflection coefficients with magnitude greater than unity when Zi is not real.
If Zi is not real, the scattering matrix need not be symmetric even if the network in question is reciprocal.
The reflection coefficient of an open circuit is constant: .
The reflection coefficient of a short circuit is constant: .
The total accepted power at port No. i is given by .

Further references are K. Kurokawa, "Power Waves and the Scattering Matrix", IEEE Transactions on Microwave Theory and Techniques, vol. MTT-13, pp. 194-204, 1965 (power waves) and R.B. Marks and D.F. Williams, "A General Waveguide Circuit Theory", J. Research of the National Institute of Standards and Technology, vol. 97, pp. 533-561, 1992 (pseudo waves).