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

2 Theoretical Description of a CPW ResonatorCalculations show that the upper shielding of the CPW can be neglected for H 1 ≥ 1 mm.At a box height of 1 mm, the difference between the exact value of Z c with (2.2) and theapproximations for large heights (2.3) is ∆Z CS = 8.4·10 −4 Ω which results in a ratio ∆Z CSZ CSof1.6·10 −5 .In our design, the cover of the box is 2 mm above the CPW conductor. Hence, we canassume k 1 =SS+2·W .From (2.9) or (2.10) and (2.1) the inductance L CPW can be calculated as follows:C CPW = ε e f f ·C Air( K(k1 )= 2 · ε 0 · ε e f f ·K(k1 ′ ) + K(k) )K(k ′ (2.11))1with v ph = √ (2.12)CCPW · L CPWor Z cs =√LCPW⇒ L CPW = ε 0µ 0C Air=C CPW2.1.4 Treatment of Double Layer Substratesµ 02K(k)K(k ′ ) + K(k (2.13)1)(k 1 ′ )Figure 2.6: Schematic of a CPW on a double-layer dielectric substrate.On top of the Silicon substrate which is used in this thesis, 50 nm Silicon are thermallyoxidized. In Figure 2.6 a conventional CPW on a double layer substrate is shown. The twodielectric substrate thicknesses are designated H −H 2 and H 2 with relative permittivities ε r1and ε r2 .The dielectric capacitance of the dielectric regionC diel = 2ε 0 (ε 2r − ε 1r ) K(k 2diel)K(k ′ 2diel )now is the sum of C diel for a single layer substrate (2.4) for the lower dielectric and a capacitanceanalogous to (2.4) for the upper dielectric.k 2diel is the modulus equal to the modulus of a CPW with a single layer substrate (2.5)substituting h by h 2 .8