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

2.1 The Coplanar Transmission LineFigure 2.5: Configuration of the dielectric capacitance of a CBCPW.for the calculation of C CPW (2.4).That means, the moduli of the dielectric and the air-filled regions are equal. The parameterq simplifies toq =which tends to 1 2for very large heights again.K(k)K(k ′ )K(k)K(k ′ ) + K(k , (2.8)1)K(k 1 ′ )2.1.2.3 Isotropy of the DielectricThe relative permittivity ε r (used in (2.4)) and the height of the substrate H (see e.g. (2.5))depend on the isotropy of the dielectric material.In this work Silicon and Sapphire are used as substrate materials.Silicon is an isotropic material with ε r = 11.9.For Sapphire one needs to replace the relative permittivity with an effective permittivityε r = √ ε x,z · ε y = 10.34with ε y = 11.5 (c-axis) and ε x,z = 9.3 (a,b-axis) according to [3].The height has to be replaced by√εx,zH e = · H = 0.9Hε y2.1.3 Summary of Capacitance CalculationsUsing C Air and C diel , as defined in the last subsections, the phase velocity v ph and the waveimpedance Z c can be calculated:v ph =Z c ==c 0√εe f f(2.9)1C CPW v ph=2 · √εe f f ·√εe f fc 0 C CPWZ 0(K(k)K(k ′ ) + K(k 1)K(k ′ 1 ) ) (2.10)7