Instrumentation & Measurement Magazine 24-9 - 15

Fig. 2. (a) Sketch of the measurement line. The Vector Network Analyzer (VNA) is linked to the dielectric loaded resonator (DR) through a >1 m long room
temperature line (between P1
and P2) and a >1 m long cryogenic and non-magnetic coaxial line (between P2
and P3
in phosphor-bronze to withstand the cryogenic and magnetic environment. (b) Transmission S-parameter resonance curves measured on a YBa2
keeping the temperature within (67±0.05) K, at selected magnetic field magnitudes between 0 T (black) and 12 T (light blue). (c) Loaded quality factor Ql
resonance frequency f0 measured through the Lorentzian fit (model shown (6)) of the S21
cylindrical sapphire single crystal (diameter 7.13(1) mm, height 4.50(1) mm).
O7-x
as the reference state, and the delicate and uncertaintysource
calibration of the DR can be avoided altogether
[25]. Obviously, this approach completely rules out the
use of superconducting cavities instead of (normal metalbased)
DR to exploit their higher Qu
). This last part of the microwave line is made
Cu3
sample,
and
(f) curves shown in (b). (d) Bitonal copper-shielded DR, loaded with a
simulations. This is the major source of systematic (scale)
effects.
The u(Qu) and u(f0
and thus higher
sensitivities, since their magnetic-dependent response
would yield a heavily field dependent background indistinguishable
from the sample magnetic dependent Z(H).
An estimation of the uncertainty linked to imperfect
thermalization during the measurements, which represents
the main uncertainty source within this differential
approach, can be provided as follows: the DR without
a SC sample, at zero field, has variations ΔQu
Δf0
~300 and
~100 kHz when the temperature changes from 5 K
to 25 K. With a thermalization within ± 0.05 K, the standard
for the measurements presented here, one can give
a rough estimate ΔQu
~0.75 and Δf0
~250 Hz, fully negligible
with respect to the measured variations due to the SC
response (Fig. 3).
◗ The geometrical factor G must be accurately determined.
However, due to the lack of R, X measurement standards,
G is not a quantity directly measurable. Thus, G
can be obtained either analytically by solving the quasistationary
e.m. field configuration in the DR for simple
geometries or through electromagnetic numerical
December 2021
) uncertainties give the major contributions
to the final measurement uncertainties u(Z) (apart from
systematic effects). Their evaluation presents a complexity
increased by the cryogenic (temperatures down to 4 K) and
magnetic (fields up to 12 T) environment where measurements
must be taken, so that the uncertainties u(R) and u(X)
cannot be expected to be as small as in a completely controlled
setup. Qu
and f0
sweep measurement around f0
parameters of the DR operated in transmission (time-domain
measurements do not bring advantages due to high fluxonmotion
losses in SCs). In less demanding environments, the
standard '-3 dB method' is often employed: from the transmission
S-parameters, the frequency of the resonance curve peak
SM
gives f0
curve Δf0,−3dB yields the loaded quality factor as Ql = f0
, and the full width half maximum of the resonance
/ Δf0,−3dB
.
In conventional measurements, the effect of the transmission
lines is dealt with by a proper calibration. In cryomagnetic
cases, this simple method does not bring sufficient accuracy
and precision. First, the microphonic noise given by the vacuum
pumping system, or by the flowing cryogenic gases,
makes the Qu
and f0
measurement based on the '-3 dB method'
highly scattered (much reduced precision). Second, the
IEEE Instrumentation & Measurement Magazine
15
are typically obtained from the frequency
of the two-port scattering S
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