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

2 Theoretical Description of a CPW ResonatorparameterrangeSWSfHε r f T Emin 0.1 0.1 1.5 0max 5 5 50 10Table 2.1: Values where (2.14) is valid.Figure 2.7: Frequency dependent effective dielectric constant ε e f f for all designs realizedin this thesis. The cut-off frequencies are indicated by vertical lines for eachsubstrate material.2.1.6 AttenuationIn general the electric field ⃗E of a wave traveling along a transmission line can be written as⃗E = ⃗E 0 e (ωt−(α+iβ)z)where z is the direction of propagation. ω is the angular frequency of the wave. We seethat the wave travels with the phase constant β corresponding to the wavelength (imaginarypart) and that it decays with distance with the attenuation constant α (real part).The attenuation of a circuit should be as low as possible to avoid damping effects. Severalloss sources contribute to the attenuation of a CPW per unit length:α = α d + α c + α r + α s (2.15)α d represents the losses in the dielectric substrate. α c denotes the losses in the conductor10