Design, Fabrication and Characterization of a Microwave Resonator ...

2 Theoretical Description of a CPW Resonator2.4 The Scattering MatrixWhen dealing with high-frequency (hf) networks, voltages and currents (and the relatedimpedances and admittances) are, in contrast to dc networks, abstract. For example, whenthe geometries can be compared to the wavelength, voltage and current distributions can notbe averaged along the transmission line (see Figure 2.11 on page 21). The Scattering (S)matrix is a representation of a hf-network in accord with the ideas of incident, reflected, andtransmitted waves. It provides a complete description of the network as seen at its ports byrelating the voltage waves incident on the ports to those reflected from the ports.The S-matrix is measured directly with a network vector analyzer. The scattering coeffi-Figure 2.14: A two-port device illustrating incident, reflected and transmitted waves.cient (e.g. the reflection coefficient) is determined byS 11 = V 1−V1+The reflection parameters are defined with the other port terminated in matched loads toavoid reflection.In general one can write for a two-port network( ) ( )( )V−1S11 SV2− =12 V+1S 21 S 22 V2+ } {{ }SS 21 and S 12 are called transmission coefficients.Two special cases can be considered to simplify the scattering matrix, the reciprocal andthe lossless case. Reciprocal networks have no active devices, ferrites or plasmas. Not onlythe geometry, but also the transmission coefficients are symmetric: S 12 = S 21 . In matrixterms this can be written as [S] = [S] t .For lossless devices under test (DUTs) the power leaving the two-port must equal the totalincident power. This leads to the following equations|S 11 | 2 + |S 12 | 2 = 1|S 22 | 2 + |S 12 | 2 = 1S 11 S ∗ 12 + S 12 S ∗ 22 = 124