Instrumentation & Measurement Magazine 24-9 - 16

Fig. 3. (a), (b), (c): Loaded quality factor Ql
1.2 T at 16.4 GHz (full symbols) and 26.6 GHz (empty symbols). (b) and (d): Vortex motion resistivity obtained from Ql
and relative variation of the resonance frequency Δf0 / f0,ref measured on a YBCO thin film (S1) at ~40 K and up to
measurements. (a) and (b):
and Δf0
/ f0,ref
Data obtained with the '-3dB method' (no uncertainty bar reported); (c) and (d): Data obtained through the fitting procedure (when not visible, uncertainty bars are
contained in the size of the markers). (e), (f) and (g): Comparison of the characteristic frequency fc
, thermal creep factor χ and flux-flow resistivity ρff
the vortex motion resistivity as determined with the '-3 dB method' (orange symbols) and from the fitting procedure (green symbols).
cross-coupling between the DR ports or the presence of higher
order electromagnetic modes make the resonance curve asymmetric,
and third, it is impossible to perform a full calibration
of the whole transmission line at cryogenic temperatures (see
Measurement Uncertainties, below). Thus, unavoidable background
contributions are added to the Lorentzian response to
form the S(f) measurements. For these reasons, different f0
Qu
and
measurement strategies, based on fitting algorithms, are
actively developed to overcome these issues [28]. In the case
presented here, the transmission S-parameter, either S12
or S21
S21 f






SM


1 i2Ql
ff0
f0
where SC is a frequency independent complex term used to
take into account both cross-coupling between the ports of
the DR and transmission line backgrounds, and where α and β
are real parameters used to include the frequency dependent
phase delay. The fit is performed on the real and imaginary
parts of S21
using standard least squares curve fitting approaches
based on the Levenberg-Marquardt algorithm on
16
C
,
acquired with a Vector Network Analyzer (VNA), is fitted with
the modified model given by:
Se

,
i  f
(6)
complex valued data and yields f0
and the loaded quality factor
Ql. Finally, from the diameters of the Q-circles and lossy-circles
traced by S11 and S22
on the complex plane, both the lossless and
lossy components of the two coupling factors are determined
to obtain the overall coupling factor β through the transmission-mode
Q-factor (TMQF) technique presented in [29],
whence Qu
= (1 + β)Ql
[23]. We note that changes in β need to be
monitored, because of the thermal contraction/expansion of
the metal enclosure, cables, and loops.
In the next subsections, the performances of the measurement
system and technique developed at Roma Tre
University are detailed in terms of measurement sensitivity
and uncertainty.
Measurement Sensitivity
From (4) and (5), the sensitivity functions are obtained:
2

c1
  

c2
f0
  


uu ,
QQ
RG
f
0,ref
.
XG
2
small G are required. Since G 4πfW H d where W is the
Thus, to increase the measurement sensitivity, large Qu
 
2

IEEE Instrumentation & Measurement Magazine
December 2021
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