IEC 61788-7:2020

IEC 61788-7 ®
Edition 3.0 2020-03
REDLINE VERSION
INTERNATIONAL
STANDARD
colour
inside
Superconductivity –
Part 7: Electronic characteristic measurements – Surface resistance of
high‑temperature superconductors at microwave frequencies

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IEC 61788-7 ®
Edition 3.0 2020-03
REDLINE VERSION
INTERNATIONAL
STANDARD
colour
inside
Superconductivity –
Part 7: Electronic characteristic measurements – Surface resistance of

high‑temperature superconductors at microwave frequencies

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS 17.220.20; 29.050 ISBN 978-2-8322-7917-5

– 2 – IEC 61788-7:2020 RLV © IEC 2020
CONTENTS
FOREWORD . 5
INTRODUCTION . 7
1 Scope . 8
2 Normative references . 8
3 Terms and definitions . 8
4 Requirements . 9
5 Apparatus . 9
5.1 Measurement system . 9
5.2 Measurement apparatus for R . 10
s
5.3 Dielectric rods . 12
6 Measurement procedure . 13
6.1 Specimen preparation . 13
6.2 Set-up . 13
6.3 Measurement of reference level . 14
6.4 Measurement of the frequency response of resonators . 14
6.5 Determination of surface resistance of the superconductor and ε′ and tan δ

of the standard sapphire rods. 16
7 Precision and accuracy Uncertainty of the test method . 17
7.1 Surface resistance . 17
7.2 Temperature . 18
7.3 Specimen and holder support structure . 19
7.4 Specimen protection . 19
7.5 Uncertainty of surface resistance measured by standard two-resonator
method . 19
8 Test report . 19
8.1 Identification of test specimen . 19
8.2 Report of R values . 19
s
8.3 Report of test conditions . 20
Annex A (informative) Additional information relating to Clauses 1 to 8 . 21
A.1 Scope . 21
A.1.1 General . 21

A.1.2 Cylindrical cavity method [10] [17] . 21
A.1.3 Parallel-plates resonator method [18] [19] . 21
A.1.4 Microstrip-line resonance method [20] [21] . 21
A.1.5 Dielectric resonator method [22] [23] [24] [25] . 21
A.1.6 Image-type dielectric resonator method [26] [27] . 22
A.1.7 Two-resonator method [28] [29] . 23
A.2 Requirements . 23
A.3 Theory and calculation equations . 23
A.4 Apparatus . 27
A.5 Dimensions of the standard sapphire rods . 27
A.6 Dimension of the closed type resonator . 29
A.7 Precision and accuracy of the test method .
A.7 Sapphire rod reproducibility . 31
A.8 Test results . 32

A.9 Reproducibility of measurement method . 32
A.10 tan δ deviation effect of sapphire rods on surface resistance . 33
Annex B (informative) Evaluation of relative combined standard uncertainty for surface
resistance measurement . 36
B.1 Practical surface resistance measurement . 36
B.2 Determination of surface resistance of the superconductor . 37
B.3 Combined standard uncertainty . 38
B.3.1 General . 38
B.3.2 Calculation of c to c (12 GHz resonance at 20 K) . 38
2 5
B.3.3 Determination of u to u . 39
1 5
B.3.4 Combined relative standard uncertainty . 41
Bibliography . 43

Figure 1 – Schematic diagram of measurement system for temperature dependence of
R using a cryocooler . 10
s
Figure 2 – Typical measurement apparatus for R . 11
s
Figure 3 – Insertion attenuation, IA, resonant frequency, f , and half power bandwidth,
∆f, measured at T kelvin . 14
Figure 4 – Reflection scattering parameters (S and S ) . 16
11 22
Figure 5 – Term definitions in Table 4 . 18
Figure A.1 – Schematic configuration of several measurement methods for the surface
resistance . 22
Figure A.2 – Configuration of a cylindrical dielectric rod resonator short-circuited at
both ends by two parallel superconductor films deposited on dielectric substrates . 24
Figure A.3 – Computed results of the u-v and W-v relations for TE mode . 25
01p
Figure A.4 – Configuration of standard dielectric rods for measurement of R and tan δ . 26
s
Figure A.5 – Three types of dielectric resonators . 27
Figure A.6 – Mode chart to design TE resonator short-circuited at both ends by
parallel superconductor films [28] . 28
Figure A.7 – Mode chart to design TE resonator short-circuited at both ends by
parallel superconductor films [28] . 29
Figure A.8 – Mode chart for TE closed-type resonator [28] . 30
Figure A.9 – Mode chart for TE closed-type resonator [28] . 31
Figure A.10 – Temperature-dependent R of YBCO film with a thickness of 500 nm
s
and size of 25 mm square . 32
Figure A.11 – Temperature dependent R of YBCO film when R was measured three
s s
times. 33
Figure B.1 – Schematic diagram of TE and TE mode resonance . 36
011 013
Figure B.2 – Typical frequency characteristics of TE mode resonance . 37
Figure B.3 – Frequency characteristics of a resonator approximated by a Lorentz
distribution . 41

Table 1 – Typical dimensions of pairs of standard single-crystal sapphire rods for
12 GHz, 18 GHz and 22 GHz . 12
Table 2 – Dimensions of superconductor film for 12 GHz, 18 GHz, and 22 GHz . 13

– 4 – IEC 61788-7:2020 RLV © IEC 2020
Table 3 – Specifications for vector network analyzer . 17
Table 4 – Specifications for sapphire rods . 18
Table A.1 – Standard deviation of the surface resistance calculated from the results of
Figure A.11 . 33
Table A.2 – Relationship between x, defined by Equation (A.12), and y, defined by

Equation (A.13) . 34

INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
SUPERCONDUCTIVITY –
Part 7: Electronic characteristic measurements –
Surface resistance of high-temperature
superconductors at microwave frequencies

FOREWORD
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– 6 – IEC 61788-7:2020 RLV © IEC 2020
International Standard IEC 61788-7 has been prepared by IEC technical committee 90:
Superconductivity.
This third edition cancels and replaces the second edition, published in 2006. This edition
constitutes a technical revision.
This edition includes the following significant technical changes with respect to the previous
edition:
a) informative Annex B, relative combined standard uncertainty for surface resistance
measurement has been added;
b) precision and accuracy statements have been converted to uncertainty;
c) reproducibility in surface resistant measurement has been added.
The text of this International Standard is based on the following documents:
FDIS Report on voting
90/447/FDIS 90/452/RVD
Full information on the voting for the approval of this International Standard can be found in the
report on voting indicated in the above table.
This document has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts of the IEC 61788 series, published under the general title Superconductivity,
can be found on the IEC website.
The committee has decided that the contents of this document will remain unchanged until the
maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the document will be
• reconfirmed;
• withdrawn;
• replaced by a revised edition, or
• amended.
IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates
that it contains colours which are considered to be useful for the correct understanding
of its contents. Users should therefore print this document using a colour printer.

INTRODUCTION
Since the discovery of some Perovskite-type Cu-containing oxides, extensive research and
development (R & D) work on high-temperature oxide superconductors (HTS) has been, and is
being, done worldwide, and its application to high-field magnet machines, low-loss power
transmission, electronics and many other technologies is in progress.
In various fields of electronics, especially in telecommunication fields, microwave passive
devices such as filters using oxide superconductors HTS are being developed and are
undergoing on-site testing [1][2] .
Superconductor materials for microwave resonators [3], filters [4],, antennas [5] and delay
lines [6] have the advantage of very low loss characteristics. The parameters of superconductor
materials needed for the design of microwave low loss components are the surface resistance,
(R ) and the temperature dependence of the R . Knowledge of this parameter is of primary
s s
importance for the development of new materials on the supplier side and for the design of
superconductor microwave components on the customer side.
Recent advances in high Tc superconductor (HTS) thin films with R R of high quality HTS films
s
s
is generally several orders of magnitude lower than that of normal metals [7] [8] [9] [10], have
which has increased the need for a reliable characterization technique to measure this property
[3,4]. Traditionally, the R of niobium or any other low-temperature superconducting material
s
was measured by first fabricating an entire three-dimensional resonant cavity and then
measuring its Q-value [11]. The R could be calculated by solving the electro-magnetic field
s
(EM) distribution inside the cavity. Another technique involves placing a small sample inside a
larger cavity. This technique has many forms but usually involves the uncertainty introduced by
extracting the loss contribution due to the HTS films from the experimentally measured total
loss of the cavity.
The best HTS samples are epitaxial films grown on flat crystalline substrates and no high-quality
films have been grown on any curved surface so far. What is needed is a technique that: can
use these small flat samples; requires no sample preparation; does not damage or change the
film; is highly repeatable; has great sensitivity (down to 1/1 000 the R of copper); has great
s
dynamic range (up to the R of copper); can reach high internal powers with only modest input
s
powers; and has broad temperature coverage (4,2 K to 150 K).
The dielectric resonator method is selected among several methods [5,6,7] to determine the
surface resistance at microwave frequencies because it is considered to be the most popular
and practical at present. Especially, the sapphire resonator is an excellent tool for measuring
the R of HTS materials [8,9] [12] [13] [14].
s
The test method given in this document can also be applied to other superconductor bulk plates
including low T materials.
c
This document is intended to provide an appropriate and agreeable technical base for the time
being to engineers working in the fields of electronics and superconductivity technology.
The test method covered in this document is based on the VAMAS (Versailles Project on
Advanced Materials and Standards) pre-standardization work on the thin film properties of
superconductors.
___________
Numbers in square brackets refer to the bibliography.

– 8 – IEC 61788-7:2020 RLV © IEC 2020
SUPERCONDUCTIVITY –
Part 7: Electronic characteristic measurements –
Surface resistance of high-temperature
superconductors at microwave frequencies

1 Scope
This part of IEC 61788 describes measurement of the surface resistance (R ) of
s
superconductors at microwave frequencies by the standard two-resonator method. The object
of measurement is the temperature dependence of R at the resonant frequency.
s
The applicable measurement range of R for this method is as follows:
s
– Frequency: 8 GHz < f < 30 GHz
– Measurement resolution: 0,01 mΩ at 10 GHz

The R data at the measured frequency, and that scaled to 10 GHz, assuming the f rule for
s
comparison, is reported.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies.
For undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC 60050-815, International Electrotechnical Vocabulary (IEV) – Part 815: Superconductivity
3 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-815 apply.
ISO and IEC maintain terminological databases for use in standardization at the following
addresses:
• IEC Electropedia: available at http://www.electropedia.org/
• ISO Online browsing platform: available at http://www.iso.org/obp
In general, surface impedance Z for conductors, including superconductors, is defined as the
s
ratio of the electric field E to the magnetic field H , tangential to a conductor surface:
t t
Z = E /H = R + jX
s t t s s
where R is the surface resistance and X is the surface reactance.
s s
4 Requirements
The R of a superconductor film shall be measured by applying a microwave signal to a dielectric
s
resonator with the superconductor film specimen and then measuring the attenuation of the
resonator at each frequency. The frequency shall be swept around the resonant frequency as
the centre, and the attenuation–frequency characteristics shall be recorded to obtain
the Q- value, which corresponds to the loss.
The target precision relative combined standard uncertainty of this method is a coefficient of
variation (standard deviation divided by the average of the surface resistance determinations)
that is less than 20 % for the measurement temperature range from 30 20 K to 80 K.
It is the responsibility of the user of this document to consult and establish appropriate safety
and health practices and to determine the applicability of regulatory limitations prior to use.
Hazards exist in this type of measurement. The use of a cryogenic system is essential to cool
the superconductors to allow transition into the superconducting state. Direct contact of skin
with cold apparatus components can cause immediate freezing, as can direct contact with a
spilled cryogen. The use of an RF generator is also essential to measure high-frequency
properties of materials. If its power is too high, direct contact to human bodies can cause an
immediate burn.
5 Apparatus
5.1 Measurement system
Figure 1 shows a schematic diagram of the system required for the microwave measurement.
The system consists of a network analyzer system for transmission measurement, a
measurement apparatus, and a thermometer for monitoring the measuring temperature.
An incident power generated from a suitable microwave source such as a synthesized sweeper
is applied to the dielectric resonator fixed in the measurement apparatus. The transmission
characteristics are shown on the display of the network analyzer. The measurement apparatus
is fixed in a temperature-controlled cryocooler.

– 10 – IEC 61788-7:2020 RLV © IEC 2020
Vector network analyzer
Figure 1 – Schematic diagram of measurement system
for temperature dependence of R using a cryocooler
s
The measurement apparatus is fixed in a temperature-controlled cryocooler.
For the measurement of R for superconductor films, a vector network analyzer is recommended.
s
A vector network analyzer has better measurement accuracy than a scalar network analyser
due to its wide dynamic range. The performance requirements of the vector network analyzer
are specified in 7.1.
5.2 Measurement apparatus for R
s
Figure 2 shows a schematic of a typical measurement apparatus (closed type resonator) for
the R of superconductor HTS films deposited on a substrate with a flat surface. The upper
s
superconductor HTS film is pressed down by a spring, which is made of phosphor bronze. The
plate type spring is recommended to should be used for the improvement of measurement
accuracy uncertainty. This type of spring reduces the friction between the spring and the other
part of the apparatus, and allows the smooth movement of superconductor films due to the
thermal expansion of the dielectric rod. In order to minimize the measurement error uncertainty,
the sapphire rod and the copper ring shall be set in coaxial arranged coaxially.
Two semi-rigid cables for measuring transmission characteristics of the resonator shall be
attached on both sides of the resonator in an axial symmetrical position (φ = 0 and π, where φ
is the rotational angle around the central axis of the sapphire rod). Each of the two semi-rigid
cables shall have a small loop at the ends. The plane of the loop shall be set parallel to that of
the superconductor films in order to suppress the unwanted Transverse Magnetic Wave Modes
(TM modes). The coupling loops shall be carefully checked for cracks in the spot weld joint
mn0
that may have developed upon repeated thermal cycling. These cables can move right and left
to adjust the insertion attenuation (IA). In this adjustment, coupling of unwanted cavity modes
to the interested dielectric resonance mode shall be suppressed. Unwanted, parasitic coupling
to the other modes reduces the high Q-value of the Transverse Electro-Magnetic Mode
(TE mode) resonator. For suppressing the parasitic coupling, special attention shall be paid to

designing high-Q resonators. Two other types of resonators along with the closed type shown
in Figure 2 can be used. They are explained in A.4.

Figure 2 – Typical measurement apparatus for R
s
A reference line made of a semi-rigid cable shall be used to measure the full transmission power
level, i.e. the reference level. This cable has a length equal to the sum of the two cables of the
measurement apparatus. Semi-rigid cable with an outer diameter of 1,20 mm is recommended.

– 12 – IEC 61788-7:2020 RLV © IEC 2020
In order to minimize the measurement error uncertainty, two superconductor films shall be set
to be parallel to each other. To ensure that the two superconductor films remain in tight contact
with the ends of the sapphire rod, without any air gap, both of the surfaces of the films and the
ends of the rod shall be cleaned carefully.
5.3 Dielectric rods
Two dielectric rods with the same relative permittivity, ε′, and loss factor, tan δ, preferably cut
from one cylindrical dielectric rod, are required. These two rods, standard dielectric rods, shall
have the same diameter but different heights: one has a shall have height three times longer
than the other.
It is preferable to use standard dielectric rods with low tan δ to achieve the requisite
measurement accuracy uncertainty on R . Recommended dielectric rods are single-crystal
s
–6 -7
sapphire rods with tan δ less than 10 2 × 10 at 77 K. Specifications on the sapphire rods
are described in 7.1. In order to minimize the measurement error in R of the superconductor
s
films, both ends of the sapphire rods shall be polished parallel to each other and perpendicular
to the axis. Specifications for the sapphire rods are given in Clause 7.
The diameter and the heights of the standard sapphire rods shall be carefully designed so that
the TE and TE modes do not couple to other TM, HE and EH modes, since the coupling
011 013
between TE mode and other modes causes the degradation of unloaded Q. A design guideline
for the standard sapphire rods is described in Clause A.5. Table 1 shows typical examples of
dimensions of the standard sapphire rods for 12 GHz, 18 GHz, and 22 GHz resonance.
At higher frequencies the unloaded Q value will be lower, which makes the measurement easier,
and the error will be lower. In the R measurement at 22 GHz, the required film diameter can
s
be set to 20 mm, and the measured Q is small, therefore the effect of the dielectric loss of
L
sapphire rod can be reduced.
Table 1 – Typical dimensions of pairs of standard single-crystal
sapphire rods for 12 GHz, 18 GHz and 22 GHz
Diameter Height
Frequency
d h
GHz
mm mm
Short rod (TE resonator) 11,4 5,7
Long rod (TE resonator) 11,4 17,1
Short rod (TE resonator) 7,6 3,8
Long rod (TE resonator) 7,6 11,4
Short rod (TE resonator) 6,2 3,1
Long rod (TE resonator) 6,2 9,3
Frequency Diameter, d Height, h
GHz mm mm
Short rod (TE resonator) 11,40 ± 0,05 5,70 ± 0,05
Long rod (TE resonator) 11,40 ± 0,05 17,10 ± 0,05
Short rod (TE resonator) 7,60 ± 0,05 3,80 ± 0,05
Long rod (TE resonator) 7,60 ± 0,05 11,40 ± 0,05
Short rod (TE resonator) 6,20 ± 0,05 3,10 ± 0,05
Long rod (TE resonator) 6,20 ± 0,05 9,30 ± 0,05
6 Measurement procedure
6.1 Specimen preparation
From error uncertainty estimation, the film diameter shall be about three times larger than that
of the sapphire rods. In this configuration, the reduction increase in precision uncertainty of R
s
due to the different radiation losses between TE and TE mode can be considered
011 013
negligible, given the target precision relative combined standard uncertainty of 20 %. The film
thickness shall be about three two times larger than the London magnetic-field penetration
depth value at each the maximum temperature in the measurement temperature range. If the
film thickness is much less than three two times the London magnetic-field penetration depth,
the measured R should mean the effective surface resistance.
s
Table 2 shows dimensions of the superconductor films recommended for the standard sapphire
rods of 12 GHz, 18 GHz, and 22 GHz.
Table 2 – Dimensions of superconductor film for 12 GHz, 18 GHz, and 22 GHz
Standard dielectric rod Superconductor film
Frequency Diameter Diameter Thickness

d d′
GHz µm
mm mm
12 11,4 >35
≅0,5
18 7,6 >25 ≅0,5
22 6,2 >20
≅0,5
c-cut single-crystal sapphire rod Superconductor film
Frequency Diameter, d Diameter, d′ Thickness
GHz mm mm μm
12 11,40 ± 0,05 > 35 ≅ 05,
18 7,60 ± 0,05 > 25 ≅ 05,
22 6,20 ± 0,05 > 20
≅ 05,
In case of using closed type resonators, the dimensions of the superconductor films shall also
be designed taking into account the dimension of the copper cylinder between the
superconductor films. A design guideline for the dimension of the copper cylinder of the closed
type resonator is described in Clause A.6.
6.2 Set-up
Set up the measurement equipment as shown in Figure 1. All of the measurement apparatus,
standard sapphire rods, and superconductor films shall be kept in a clean and dry state as high
humidity may degrade the unloaded Q-value. The specimen and the measurement apparatus
shall be fixed in a temperature-controlled cryocooler. The specimen chamber shall be generally
totally evacuated. The temperatures of the superconductor films and standard sapphire rods
shall be measured by a diode thermometer, or a thermocouple. The temperatures of the upper
and lower superconductor films and standard sapphire rods must be kept as close as possible
shall not differ by more than 0,5 K. This can be achieved by covering the measurement
apparatus with aluminium foil or filling the specimen chamber with helium gas.

– 14 – IEC 61788-7:2020 RLV © IEC 2020
6.3 Measurement of reference level
The level of full transmission power (reference level) shall be measured first. Fix the output
power of the synthesized sweeper below 10 mW because the measurement accuracy
uncertainty depends on the measuring signal level. Connect the reference line of semi-rigid
cable between the input and output connectors. Then, measure the transmission power level
over the entire measurement frequency and temperature range. The reference level can change
several decibels when temperature of the apparatus is changed from room temperature to the
lowest measurement temperature. Therefore, the temperature dependence of the reference
level must be taken into account.

Figure 3 – Insertion attenuation, IA, resonant frequency, f ,
and half power bandwidth, ∆f, measured at T kelvin
6.4 Measurement of the frequency response of resonators
The temperature dependence of the R can be obtained through the measurements of resonant
s
frequency, f , and unloaded quality factor, Q , for TE and TE resonators, which shall be
0 u 011 013
measured as follows.
a) Connect the measurement apparatus between the input and output connectors (Figure 1).
Insert the standard short sapphire rod near the centre of the lower superconductor film and
fix the distance between the rod and each of the loops of the semi-rigid cables to be equal
to each other, so that this transmission-type resonator can be under-coupled equally to both
loops. Put down the upper superconductor film gently to touch the top face of the rod. Be
careful not to damage the surface of the superconductor films by excessive pressure.
Evacuate and cool down the specimen chamber below the critical temperature.
b) Find the TE mode resonance peak of this resonator at a frequency nearly equal to the
designed value of f .
c) Narrow the frequency span on the display so that only the resonance peak of TE mode
can be shown (Figure 3). Confirm that the insertion attenuation, IA, of this mode is larger
than 20 dB from the reference level, which depends strongly on the temperature.
d) Measure the temperature dependence of f and the half power band width ∆f. The loaded
Q-value, Q , of the TE resonator is given by
L 011
f
Q = (1)
L
Δf
e) The unloaded Q-value, Q , shall be extracted from the Q by at least one of the two following
u L
techniques.
1) One technique for extracting the unloaded Q-value involves measuring the insertion
attenuation IA. Q is given by
u
Q
−IA[dB]/20
L
QA, 10
ut (2)
1 − A
t
The IA of each temperature is determined as follows. In the R measurement system
s
shown in Figure 2, a semi-rigid cable is connected instead of the measurement
apparatus, and the measurement system is short-circuited to determine the reference
level. Measure the reference level at each temperature and use it to determine IA at
each temperature.
This technique of using insertion attenuation assumes that the coupling on both sides of
the resonator is identical. The coupling loops are difficult to fabricate, the orientation of
the loop is difficult to control, and any movement of the sapphire rod during measurement
is not known. These assembly-dependent effects are also temperature dependent. This
potential asymmetry in coupling can result in large errors in calculating the coupling
factor if the coupling is strong (IA <~ 10 dB). If the coupling is weak enough (IA > 20 dB),
asymmetry in the coupling becomes less important.
2) Another technique for extracting the unloaded Q-value involves measuring the reflection
scattering parameters at the resonant frequency of both sides of the resonator.
QQ ()1+ ββ+
u L 12
(3)
1− ||S
β = (4)
| SS||+ |
11 22
1− ||S
β = (5)
| SS||+ |
11 22
In the above equations Equations (4) and (5), S and S are the reflection scattering
11 22
parameters as shown in Figure 4, and are measured in linear units of power, not
relative dB. β and β are the coupling coefficients.
1 2
This technique of using the reflection scattering parameters has two advantages. It does
not require the additional step of calibration of the reference level and it gives a
measurement of the coupling values for both sides of the resonator. This also has two
disadvantages. It only works for a narrow band resonance (which is fortunately the case)
and is limited by the dynamic range of the network analyzer in measuring the reflection
coefficients [15] [16].
A combination of the two techniques is an excellent “double” check and is therefore
recommended useful for a "double" check.
f) The f and Q measured for this short rod are denoted as f and Q for TE mode
0 u 01 u1 011
resonance. By slowly changing the temperature of the cryocooler the temperature
dependence of f and Q shall be measured.

01 u1
g) After the temperature dependence measurement of f and Q is finished, the measurement
01 u1
apparatus shall be heated up to room temperature.
h) Then, replace the TE resonator in the apparatus with the TE resonator at room
011 013
temperature, cool down the apparatus to a temperature lower than the critical temperature,
and measure the temperature dependence of f and Q of its TE resonance mode,
0 u 013
denoted as f and Q , in a similar way to the TE resonator case. When the length of
03 u3 011
=
==
– 16 – IEC 61788-7:2020 RLV © IEC 2020
the sapphire rod of the TE resonator is precisely three times longer than that of the TE
013 011
resonator, the f of the TE resonator must coincide with f of the TE resonator. If
03 013 01 011
carefully designed, the difference between f and f is usually very small (<~ 0,25 %). We
01 03
can treat as f = f = f in the calculations of 6.5 consider that the f in equation (9) can
0 01 03 0
be replaced by f or f
01 03.
Figure 4 – Reflection scattering parameters (S and S )
11 22
6.5 Determination of surface resistance of the superconductor and ε′ and tan δ of the
standard sapphire rods
Calculate the temperature dependence of the surface resistance R of the superconductor films,
s
and ε’ and tan δ of the standard sapphire rods using the temperature dependence of f , Q ,
01 u1
f , and Q from Equations (6), (7) and (8).
03 u3
Calculate the temperature dependence of the R of the superconductor films, and ε′ and tan δ
s
of the standard sapphire rods using the temperature dependence of Q and Q . Q and Q

u1 u3 u1 u3
can be calculated using equations (1) and (2). If the measured temperatures for Q appear to
u3
be slightly different from those for Q , interpolation can be used for preparing adjusted
u1
temperature values for the former at the same temperatures for the later. Equations (6) to (12)
are established in the open type cavity, but when the superconductor film size is three times
larger than sapphire rod diameter, these equations can be used for closed type cavity.

′ 
30π ×3 2h εW+ 1 1
R − (6)


s

31−+1 WQ Q
( )
u1 u3
λ0

λ
ε' = + +1 (7)
(uv )

πd

W
1+

′
ε
tan δ = −
 (8)

(3 −1) QQ
u3 u1

where
=
c
λ = (9)
f
2 2
Ju() Kv( )Kv( ) −
Kv( )
1 02
W =
(10)
K (v) J ()u − J ()u J ()u
1 1 02
 
 
λ0
2  πd 
   
=  -1
v   (11)
λ  
 0 
2h
 
 0 
 


3λ

2 πd 0

=  −1
v (11)
 
λ
0 2h
0

uv
( ) ( )
J K
0 0
u =−v
(12)
(uv) ( )
J K
1 1
In the equations, λ is the free space resonant wavelength, c is the velocity speed of light in
vacuum (c = 2,997 9 × 10 m/s), h is the height of the short standard dielectric rod. The value
2 2
u is given by the transcendental Equation (12) using the value of v , where J (u) is the Bessel
n
function of the first kind, and K (v) is the modified Bessel function of the second kind. The
n
derivations of the equations are described in Clause A.3.
Generally the thermal expansion coefficient of the rods must be known to determine the
temperature dependence of their sizes. However, the thermal expansion effect of the sapphire
rods can be neglected for the target precision of the relative standard uncertainty of R (20 %).
s
It is noted that the measured R means the effective surface resistance if the film thickness is
s
not much larger less than two times the temperature-dependent penetration depth.
7 Precision and accuracy Uncertainty of the test method
7.1 Surface resistance
The surface resistance (R ) shall be determined from the Q-value measured with a dielectric
s
resonator technique.
A vector network a
...