total energy stored in the DR, H the RF magnetic field component parallel to the surface of the sample and Ω the sample surface, small G means large surface of the sample exposed to the H . Finally, f0 ~1.8×104 is normally chosen to satisfy the need to characterize the sample at a specific frequency. As an example, for one of the resonators of the present work, at low T one has Qu c1 ~ −1.2 × 105 Ω−1 , f0 ~ 15 GHz and G ~ 2.7 × 103 and c2 ~ 1.1 × 107 Hz Ω−1 . Measurement Uncertainties The uncertainties u(R) and u(X) are evaluated from (4) and (5) with the standard uncertainty propagation procedure. These uncertainty sources are discussed here. u(Ql ) and u(f0 Ω, from which approximate Ql ≈ Qu. The value of β is monitored at each measuring temperature. Finally, the u(G) uncertainty is evaluated through Monte Carlo e.m. simulations of the DR, randomly varying all the dimensions and e.m. properties of the components, assuming as gaussian all their distributions. The simulation results allow to assess u(G)/G = 1%. All these sources of uncertainties are then combined to obtain u(Z). ) are obtained from the fitting procedure in a standard fashion from the numerically evaluated Jacobian matrix and the fit residuals. The relative standard uncertainties so evaluated are ~0.02% for Ql and ~5×10−9 for f0 ) and u(f0 quire that the frequency span of S21(f) around ƒ0 . Such low uncertainties reis optimized. We found through experimental results and numerical simulations that u(Ql ) cannot be minimized together. Best compromises lie in the frequency span in the range 4-6 of the full width half-maximum of |S21 (f)|. In cryogenic measurements, however, a major contribution to the uncertainties u(Ql) and u(f0 ) evaluation comes from the difficulty in performing the vector calibration of the whole transmission line, since thermal gradients are typically irreproducible and no microwave calibration standards are available for cryogenic T and high H fields operations. Actually, customized in-situ cryogenic calibration systems were designed [30]-[32]. However, these methods are not certified and traceable calibration systems (indeed, the only existing standard [24] about the measurement of SC R does not even address this point for X), they rely on components whose operation is certified only down to −50 °C, and no information is present on their behavior in high magnetic fields. Moreover, they are based on bulky components and thus not suitable for the typical dimensions of cryostats for high magnetic fields, thus there is the need to deal with uncalibrated measurements (at least for the cryogenic portion of the line). Without a full calibration, the VNA cannot provide uncertainties on the measured S-parameters. To estimate u(Qu ) and u(f0 a room-temperature test system with similar Ql and f0 ), we developed and operating frequencies, thus evaluating experimentally the contribution on uncertainties given by the lack of the calibration from the discrepancies of the Ql measurements obtained by applying (or not) the line calibration. We obtained experimental standard deviations of 2.2 % for Ql Ql > 104 and 0.1×10−7 for f0 ) and u(f0 ). (in the range). These are then treated as uncertainties and combined with the previous ones to obtain the combined standard uncertainty on u(Ql The u(β) was estimated for curves with SNR > 10 and β > 0.05, to be u(β/β) < 3% with the implemented TMQF algorithm [29]. This uncertainty, combined with u(Ql agates to u(Qu). To reduce the contribution of u(β) on u(Qu ), prop), the coupling of the resonator is set to the lowest possible value that yields a detectable signal (i.e., β < 0.001) in order to December 2021 Experimental Results In this section, sample measurements of the surface impedance on superconducting materials of largest technological interest for the advancement of research in fundamental experiments, such as Nb3 Sn (Tc ≈ 18 K) and YBa2 , fc Cu3O7-x (Tc ≈ 91 K), are shown and discussed, together with the extraction of the vortex motion parameters ρff and χ. Having to extract three observables, a minimum of three independent measurements is required. Alternatively, one can fit the data to (1) with a suitable number of desired quantities either neglected (typically, χ = 0) or taken as fit parameters [12], [13], but direct extraction from measurements is a more satisfying approach. This means that measurements of Z = R + iX performed with a resonating technique at two different frequencies (at least) should be used, considering that (1) includes complex quantities. As an example, we show measurements of the surface impedance on S1 sample. We developed a DR operating on two different transverse electric modes, the TE011 and TE021 at ~16.4 GHz at ~26.6 GHz. We use the analytical and numerical methods described above to obtain the loaded quality factor Ql and the resonance frequency f0 as a function of the applied magnetic field at fixed temperature, as shown in Fig. 3. In particular, we compare the results as obtained from the standard '-3 dB method' to the fitting of the resonance curves with the modified Lorentzian curve method described in the previous section. The precision of the '-3 dB method' is limited by the electronic noise and the finite frequency steps of the acquisition as well visible from the Δf0 /f0,ref measurement shown in Fig. 3a and Fig. 3c. Moreover, an evident systematic effect adds to the Ql those non-idealities which are de-embedded from the Ql measurement since the '-3 dB method' is affected by measurement by the fit procedure. The measurements shown in Fig. 3 are obtained by setting the VNA output power at -7 dBm, 8001 acquisition points and 10 kHz of intermediate frequency (IF) bandwidth. The RF power is kept low enough to ensure to remain in the linear region of the investigated Z. Regarding the choice of the number of points and the IF, there is a tradeoff between the S(f) curves acquisition rate, the needed points density near the resonance needed for the fitting procedure and the noise. Thus, these parameters are optimized, depending on the characteristics of the measurement system and the measurement requirements in terms of noise and speed. The sample Z is then obtained through (4) and (5). The next step is to determine the link between Z and the sample ρ. In electromagnetically thick samples, Z i IEEE Instrumentation & Measurement Magazine 0 ; in thin SC films with thickness t < min(δ, λ), with δ the skin penetration 17

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