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

3 Designs Procedure for the ResonatorsThe calculated values are shown in Table 3.1. The ”Yale” design was adopted from reference[6]. For simplicity, values evaluated from (3.1) and (3.2) were rounded to valuesdivisible by 5. Therefore, most values in Table 3.1 are equal for Silicon and Sapphire substrates.Design 2 was only designed for Sapphire substrates.design gap width W (µm) center strip width S (µm)Yale 5 10design 1 10 20design 2 25 45Table 3.1: Description of designs with corresponding geometriesThe resonator length is evaluated from (2.24) and (2.31). Here, again an approximation(3.3) is used.ε e f f ={6.007S −0.012 for Sapphire6.853S −0.0148 for Silicon(3.3)The values in Table 3.2 result from the presented procedure. While taking the approximatedvalues for the design an exact calculation gives values slightly deviating from theapproximation.permittivity ε e f fsubstrate material permittivity ε r design approximation exact calculationYale 3.45 5.67Sapphire 10.34 design 1 3.42 5.67design 2 3.39 5.66Yale 4.06 6.45Silicon 11.8 design 1 4.02 6.45Table 3.2: Effective permittivities corresponding to the designsThe length of the resonator was chosen to get the desired resonance frquency for theresonator.3.2 Coupling CapacitancesThe coupling capacitances were adopted from the design of Frunzio [6].The undercoupled resonator on the left hand side of Figure 3.1 has a gap width g of 4 µmserving as a capacitance. The appropriate capacitance for design 1 is linearly scaled withthe ratio of the gap widths W. So a gap g of 8 µm is obtained.On the right hand side of Figure 3.1 the capacitance for the overcoupled resonator in theYale design is shown. The inner conductor is divided into two stripes of length ≈ 100 µm28