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

2 Theoretical Description of a CPWResonatorThe aim of this thesis is to design a resonator for frequencies in the GHz-regime fabricatedin coplanar waveguide (CPW) technique. This chapter deals first with the theory ofCPWs based on the conformal mapping method [16]. The calculation of the characteristicimpedance, capacitance and inductance of a CPW is discussed in detail. Then we brieflypresent how the use of different substrate materials influences the characteristics of the CPW.At the end of the first section, the dispersion and attenuation aspects are given.After this, effects of using a superconducting material (in our case Niobium) on theimpedance and attenuation are introduced.The third part of this chapter discusses some basic details of resonators, especially resonancefrequency, quality factor and coupling to the environment. The special cases ofmicrowave and superconducting resonators are presented. The attenuation considered in thefirst section is extended to superconductivity and the temperature dependence of resonancefrequency and quality factor of the superconducting CPW (SCCPW) is discussed.The last section explains briefly the theory of scattering parameters used in our measurements.2.1 The Coplanar Transmission LineThe CPW proposed by C. P. Wen in 1969 [22] was chosen for this thesis for its easy fabricationcompared to other types of transmission lines. The conducting layer is directly fabricatedon top of a substrate without the need for wraparounds or via holes. This means, onlya single layer technology is required, therefore, the CPW can be fabricated in-house at theWalther-Meißner-Institut. Another advantage is the very small radiation loss which wouldinfluence the quality factor of the resonator. Furthermore, the characteristic impedance isdirectly related to the ratio of the width of the inner strip to the width of the gap, so it waspossible to design various geometries easily. In this thesis three different designs were realized.Taking superconductivity into account, with the so called kinetic inductance, onecan see from Figure 2.8 in subsection 2.2.1 that the CPW is not very sensitive to the kineticinductance.This section deals with two classifications of CPWs, shown in Figure 2.1. The usedequations and further information can be found in reference [18]. A conventional CPW on adielectric substrate consists of a center strip conductor with semi-infinite ground planes oneither side. However, in a real circuit the ground planes are large but of finite extend. Theconductor backed CPW (CBCPW) has an additional ground plane at the bottom surface ofthe substrate, realized by sputtering Niobium on both sides of the substrate.3