Instrumentation & Measurement Magazine 24-9 - 17

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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