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parallel plate capacitor approximation for the capacitance per unit length, C = ϵ$_{0}$ϵ$_{r}$w/d . 

Since resonators used as detectors will nearly always be operated at the highest readout power possible before non-linear effects set in, the right side of the plot with loss tangents below 2 × 10 − 6 are the most relevant for MKIDs. The flattening of the loss tangent towards the right side of the plot is due to the high readout power generating quasiparticles in the resonator. 

The sensitivity of the MKID can be calculated by first taking noise spectra on reso- nance in both the phase ( S$_{θ}$ ) and dissipation ( S$_{D}$ ) direction at a readout power just be- low the level where the MKID becomes nonlinear. The dimensionless phase or dissipa- tion shift per quasiparticle referenced to the center of the resonance loop, ∂θ/∂N$_{qp}$ and ∂D/∂N$_{qp}$ , can be computed by taking a temperature sweep of the resonance loop and con- verting the temperature to an effective number of quasiparticles in the resonator center strip, N$_{qp}$ = 2 N$_{0}$V √ 2 πk$_{B}$T Δ$_{0}$ e − Δ$_{0}$ /k$_{B}$$^{T}$. Performing this operation on the resonator measured in Figure 3 leads to ∂θ/∂N$_{qp}$ = 8 . 2 × 10 − 7 and ∂D/∂N$_{qp}$ = 3 . 8 × 10 − 7 radians per quasiparticle. Equation 4 predicts ∂θ/∂N$_{qp}$ = 4 . 9 × 10 − 7 and ∂D/∂N$_{qp}$ = 1 . 3 × 10 − $^{7}$. The match between the predicted and measured responsivities is reasonably good, showing that the derived 

