The test method given in this standard enables measurement not only of the intrinsic surface resistance but also the intrinsic surface reactance of HTS films, regardless of the film’s th
Measurement equipment
The schematic diagram in Figure 1 illustrates the essential equipment for microwave measurement, which includes a network analyzer system for transmission measurements, a measurement apparatus, and thermometers to monitor the temperature of the HTS films being tested.
A synthesized sweeper generates incident power that is applied to a dielectric resonator within the measurement apparatus, with the transmission characteristics displayed on the network analyzer.
The measurement apparatus is fixed in a temperature-controlled cryocooler
For measuring the Z S of HTS films, a vector network analyzer is recommended because it has better measurement accuracy than a scalar network analyzer due to its wider dynamic range.
Measurement apparatus
Figure 2 illustrates a typical measurement apparatus for the Z S of high-temperature superconducting (HTS) films on a flat substrate A beryllium copper spring presses the lower HTS film, and using a plate-type spring enhances measurement accuracy by minimizing friction and allowing smooth movement during thermal expansion and contraction of the dielectric-loaded cavity The upper HTS film is securely attached to the copper plate above with thermally conductive adhesives.
The R S is measured with the upper HTS film being in contact with the top of the Cu cavity
During the measurement of the R S, the resonator is initially cooled to the lowest temperature using a cryocooler Subsequently, it is warmed to higher temperatures with the cryocooler turned off Concurrently, the X S is measured with a small gap maintained between the upper components.
The gap distance between the HTS film and the top of the Cu cavity is predetermined at room temperature using a micrometer or a step motor connected via a PTFE rod At cryogenic temperatures, the actual gap distance is slightly longer due to thermal contraction of the PTFE This gap must be small enough to minimize radiation loss while still allowing for effective temperature control of the upper superconductor film Detailed descriptions of the dielectric resonator, including a movable top plate and a switch block for thermal connection, are illustrated in Figures 3 to 5 Additionally, procedures for controlling the temperature of the upper HTS film for X S measurements are outlined.
Each semi-rigid cable must feature a small loop at the end, aligned parallel to the HTS films to minimize unwanted TM mn0 modes Prior to measurements, it is essential to verify the coupling loops to ensure optimal coupling conditions The cables can be adjusted laterally to modify the insertion attenuation (IA), while preventing parasitic coupling to undesired cavity modes Such unwanted coupling not only diminishes the high-Q value of the TE mode resonator but also introduces uncertainty in the measured resonant frequency, complicating accurate temperature-related frequency measurements.
To effectively suppress parasitic coupling, it is essential to design dielectric resonators with resonance modes that are distinctly separated from adjacent parasitic modes Additionally, the dielectric rod must be securely fixed at the center of the bottom superconductor film using low-loss epoxy.
Vector network analyzer system Thermometer
Figure 1 – Schematic diagram for the measurement equipment for the intrinsic Z S of HTS films at cryogenic temperatures
1 polytetrafluoroethylene (PTFE) rod 7 superconductor (or metal) film
2 Cu plate 8 Be-Cu spring
3 superconductor (or metal) film 9 cold finger
5 switch for thermal connection 11 dielectric rod
Figure 2 – Schematic diagram of a dielectric resonator with a switch for thermal connection
1 acryl plate 6 dielectric rod 11 screw
2 z-axis stage 7 superconductor film 12 superconductor film
3 polytetrafluoroethylene (PTFE) screw 8 Cu plate 13 Cu plate
4 connector 9 Be-Cu spring 14 semi-rigid coaxial cable
Figure 3 – Typical dielectric resonator with a movable top plate
Figure 4 – Switch block for thermal connection
1 screw 6 Cu braid 11 Cu block
2 Cu block 7 Cu plate 12 spring
3 Cu braid 8 screw 13 Cu cavity block
4 thermal switch block 9 Cu braid 14 Cu block
Figure 5 – Dielectric resonator assembled with a switch block for thermal connection
Dielectric rods
Dielectric resonators must be engineered to ensure that the TE 021 and TE 012 modes are positioned adjacent to one another while remaining uncoupled from other TM or HE modes It is essential that the resonant frequencies of these two modes are sufficiently close to minimize measurement uncertainty in Z S, yet distinct enough to prevent any coupling Specifically, the frequency difference between the TE 021 and TE 012 modes should be less than 400 MHz, which represents approximately 1% of each resonant frequency.
80 MHz considering reduced resonator Q at higher temperatures
To ensure precise measurement accuracy in R S and X S, it is essential to use dielectric rods with low tan δ and minimal temperature variation in dielectric constants C-cut sapphire rods are particularly recommended for accurately measuring Z S, as they exhibit a relative permittivity of ε a-b ′ = 9.28 at 77 K along the a-b plane.
Designing schemes for the standard sapphire rod are described in Annex A.4 and A.5
Table 1 presents the standard dimensions of the sapphire rod utilized in 40 GHz TE 021-mode sapphire resonators Increasing the dimensions results in lower resonant frequencies; however, larger high-temperature superconducting (HTS) films are necessary to ensure the required measurement accuracy.
Table 1 – Typical dimensions of a sapphire rod
(mm) TE 011 -mode frequency TE 012 -mode frequency TE 021 -mode frequency
Superconductor films and copper cavity
Oxygen-free high-purity copper (OFHC) will be utilized for the dielectric resonator's surrounding wall, with the cavity diameter carefully chosen to achieve the necessary measurement uncertainty Typical dimensions for OFHC cavities are also considered.
HTS films suggested for the standard sapphire rod are listed in Table 2
Table 2 – Typical dimensions of OFHC cavities and HTS films
Sapphire rod OFHC cavity HTS films diameter height diameter height diameter
Set-up
To ensure accurate measurement results, the setup of the measurement equipment must follow the configuration illustrated in Figure 1 It is crucial to maintain the measurement apparatus, standard dielectric rods, and HTS films in a clean and dry condition, as dust and high humidity can adversely impact the outcomes.
Measurement of the reference level
Before measuring the resonator Q-value as a function of temperature, it is essential to determine the full transmission power at a reference level The measurement procedure involves several steps: first, set the output power of the synthesized sweeper to a value below 10 mW, typically around 1 mW, to minimize measurement uncertainty Next, connect a reference line of semi-rigid cable between the input and output connectors, ensuring that its length matches the combined lengths of the two semi-rigid cables with loops at each end, as specified in section 5.2 Finally, measure the transmission power level across the desired frequency and temperature ranges.
Measurement of the R S of oxygen-free high purity copper
The surface resistance of oxygen-free high-conductivity (OFHC) materials forming a cavity wall must be measured as a function of temperature before assessing the surface resistance of superconductor films This measurement involves determining the loaded Q-value using a transmission method, with coupling loops positioned near the cavity's bottom or at its center for all modes For the TE 012 mode, the coupling loops should be closer to the dielectric rod due to its weaker coupling strength compared to the TE 021 mode The following outlines a method for measuring the temperature dependence of the loaded Q-value for the TE 021 mode and its corresponding unloaded Q-value.
To prepare for testing, position the standard dielectric rod at the center of the lower OFHC endplate and secure it with low-loss epoxy that does not compromise the microwave properties of the OFHC plate or the superconductor film Ensure the epoxy can be easily removed with acetone Additionally, the OFHC endplates must be larger than the HTS films being tested, and their surfaces should be thoroughly polished and cleaned prior to use.
To ensure equal under-coupling of the transmission-type resonator to both loops, connect the input and output connectors to the measurement apparatus and adjust the distance between the rod and each loop of the semi-rigid cables to be the same.
(3) Put down an upper OFHC endplate gently to touch the top of the OFHC cavity
(4) Evacuate and cool down the specimen chamber below the T C of the superconductor film to the lowest temperature
(5) Identify the TE 021 mode resonance peak of this resonator using the calculated TE 021 mode resonant frequency
Set the frequency span to display only the TE 021 resonance peak and ensure that the insertion attenuation (IA) of this mode exceeds 20 dB from the reference level at the lowest temperature Additionally, verify that IA increases with rising temperature.
(7) Measure the TE 021 mode f 0 and the half power band width ∆f 3dB The loaded Q-value,
Q L , of the TE 021 mode resonator is given by
(8) The unloaded Q-value, Q U , shall be obtained from the Q L by at least one of the two techniques described below
The first technique is to use the IA values for obtaining the Q U from the Q L , for which
The Q U values derived from Equation (5) are applicable when the input and output couplings of the resonator are identical Fabricating coupling loops presents challenges, and the coupling factors are influenced by the loop's orientation and temperature Significant asymmetry in coupling can lead to considerable uncertainties in the coupling factor, particularly when the coupling is strong (IA ≤ 10 dB) However, for weak coupling with IA greater than 20 dB, the impact of coupling asymmetry diminishes.
Figure 6 – A typical resonance peak Insertion attenuation IA , resonant frequency f 0 and half power bandwidth ∆ f 3dB are defined
The second technique is to use reflection scattering parameters at both sides of the resonator at the resonant frequency, for which Q U is expressed by [15, 16]
S 11 and S 22 , illustrated in Figure 7, are measured in linear units of power, not relative dB η 1 and η 2 denote the input and output coupling coefficients, respectively
The technique employing reflection scattering parameters has two merits and demerits
The advantages of this technique include the elimination of the extra step required for calibrating the reference level and the ability to measure coupling values on both sides of the resonator However, its drawbacks are that it is only suitable for narrow band resonance and is constrained by the dynamic range of the network analyzer when measuring reflection coefficients.
A combination of the two techniques provides an excellent way to justify validity of the measured Q U , which is therefore recommended
(9) The surface resistance of OFHC is obtained from the measured Q U using the following relation tanδ
In the context of resonant modes, the values of ktan δ are considered negligibly small compared to the 1/Q U values, as indicated in Equation (9) The constants G T, G B, and G SW, known as geometrical factors, are determined by the distributions of electromagnetic fields within the resonator and are measured in ohms The filling factor, denoted by k, represents the ratio of time-averaged electromagnetic energy stored in the dielectric to that in the entire cavity Table 3 presents the geometrical factors and filling factors for the TE 021 and TE 012 modes of a standard sapphire resonator, utilizing dielectric constants of 9.28 and 11.3 at 77 K for the a-b plane and c-axis of the sapphire rod, respectively Further details on the calculation of the geometrical factors can be found in Annex A.3.2.
Table 3 – Geometrical factors and filling factors calculated for the standard sapphire resonator
The R S (OFHC) at the TE 012-mode resonant frequency is derived from the TE 021-mode resonant frequency using the relationship R S ∝ f^{1/2} For this calculation, the TE 012-mode resonant frequency of the resonator equipped with superconductor endplates will be utilized.
Determination of the effective R S of superconductor films and tan δ of
The loaded and unloaded Q-values of the resonator will be measured at the resonant frequencies of the TE 021 and TE 012 modes, following the procedure outlined in section 6.3, steps 1) to 8) The relationship between the measured unloaded Q-values and the effective surface resistance of the superconducting films, denoted as \( R_{Se} (SC) \), is represented by the equation involving \( \tan \delta \).
In Equation (11), α = 1 for the TE 021 mode with the resonant frequency f 1 , and α = 2 for the
The TE 012 mode resonates at frequency \( f_2 \), allowing for the scaled values of \( R_{Se2} (SC) \) and \( \tan \delta_2 \) to be derived from the relationships \( R_{Se} \propto f_2 \) and \( \tan \delta \propto f \), as outlined by the two-fluid model for low-loss dielectrics The expression for \( R_{Se1} \) is also provided.
R ef lec tion c oef fic ient f 0
Figure 7 – Reflection scattering parameters S 11 and S 22 with
Equations (10) to (18) facilitate the simultaneous measurement of the effective surface resistance of superconductor films and the tan δ of standard dielectric rods, achieving minimal uncertainty when frequencies f₁ and f₂ are closely matched In Equation (11), Rₛ₁ (OFHC) and Rₛ₂ (OFHC) represent the predetermined values for the OFHC cavity wall, determined using the methodology outlined in section 6.3.
Determination of the penetration depth
The penetration depth \$\lambda\$ of superconductor films will be measured using the sapphire resonator mentioned in section 6.1 It is essential that the temperature of the upper superconductor film can be controlled independently from the rest of the resonator.
To ensure effective performance, a gap of approximately 10 µm should be created between the upper superconductor film and the remainder of the resonator This disconnection can be verified by measuring the electrical resistance between these two components It is crucial that the gap remains sufficiently small to avoid altering the ratio of the shift in the resonant frequency, represented as ∆f₁/f₁.
The resonant frequency (\$f_1\$) for the TE 021 mode is influenced by the frequency shift (\$∆f_1\$), regardless of the presence of a gap This shift is significant enough to allow for independent temperature control of the upper superconductor film, separate from the rest of the system The measurement procedure for determining the wavelength (\$λ\$) as a function of temperature is outlined as follows.
(1) Follow steps (1) ~ (3) in 6.3 with both OFHC endplates replaced with superconductor films
To ensure proper alignment, pull the upper superconductor film upward by 10 µm using a micrometer or step motor, confirming its parallelism with the lower superconductor film Additionally, verify that the upper film is thermally isolated from the resonator The 10 µm gap, which is set at room temperature, can be adjusted via a micrometer connected to the upper film through a polytetrafluoroethylene (PTFE) rod It is important to note that the actual gap distances may increase slightly at cryogenic temperatures due to the thermal contraction of the PTFE rod.
To ensure accurate measurements, evacuate and cool the specimen chamber below the critical temperature (T C) of the superconductor film while keeping the thermal connection switch closed Verify that the temperature of the upper superconductor film matches that of the rest of the chamber at this lowest temperature.
(4) Identify the TE 021 mode resonance peak of this resonator using the calculated TE 021 mode resonant frequency
(5) Set the frequency span such that only the TE 021 resonance peak is displayed (Figure 6)
To investigate the thermal connection, open the switch and allow the temperature of the upper superconductor film to rise, while maintaining the rest at the lowest temperature Record the TE 021 mode resonant frequency as it varies with temperature.
(7) Collect the shift in the TE 021 mode resonant frequency ∆f 1 (= f 1 (T) – f 1 (T min )) as a function of temperature
(8) Determine λ from a least-square-fitting of ∆ f 1 to the following equation for the changes in the effective surface reactance of the upper superconductor film, X Se,Top ,[18]
In Equation (20), it is established that \$\gamma z_3 \approx \frac{1}{\lambda}\$ due to the condition \$\sigma_2 \gg \sigma_1\$ at temperatures not too close to \$T_C\$ The term \$G_h\$ represents the ratio of effective surface impedance to intrinsic surface impedance, as detailed in Annex A.3.1 Comprehensive derivations for Equations (16) and (17) can be found in Annex A.3.3.1 To accurately determine the fitted values of \$\lambda_0\$ and \$T_C\$, a model equation that reflects the temperature dependence of \$\lambda\$ is essential, particularly for high-\$T_C\$ superconductor films.
Determination of the intrinsic surface impedance
The intrinsic surface impedance of superconductor films at f 1 shall be obtained using the following procedure:
(1) Determine σ 2 as a function of temperature from the temperature-dependent λ as obtained in step 8 of 6.5 using the following equation of
= σ à λ ω (22) with ω 1 = 2πf 1 for temperatures lower than 2T C /3
Equation (22) should be valid for temperatures lower than 2T C /3
(2) Determine σ 1 from the least-square-fit to the following equation using σ 2 as determined in step 1 with σ 1 being the only fitting parameter for temperatures lower than 2T C /3
In Equation (23), R S and X S are expressed as follows
(3) Use the σ 1 and the σ 2 values as determined in step (2) to determine R S and X S using
(4) Determine σ 1 and σ 2 from a two-paremeters fit of σ 1 and σ 2 to the following equations for temperatures higher than 2T C /3
∆ , (26) with R S and X S in Equations (23) and (26) defined as
X S S (28) and Re(G h ) and Im(G h ) being the real and the imaginary part of G h , respectively, as described below
(5) Use the σ 1 and the σ 2 values as determined in step (4) to determine R S and X S using
Equations (27) and (28) for temperatures higher than 2T C /3
For reference, for temperatures higher than 2T C /3, λ is obtained from the following equation
7 Uncertainty of the test method
Measurement of unloaded quality factor
The intrinsic surface impedance at the TE 021-mode frequency will be determined using the temperature-dependent Q U values of both TE 021-mode and TE 012-mode dielectric resonators This involves measurements taken with the upper superconductor film in contact with the resonators, as well as the frequency shifts observed when a 10-µm gap is introduced Additionally, the film thickness will be measured using various methods A vector network analyzer, as detailed in Table 4, will be employed to capture the frequency dependence of attenuation and resonant frequency, enabling the determination of Q U with a relative standard uncertainty of 4%.
Table 4 – Specifications of vector network analyzer
Type B uncertainty in frequency 1 Hz at 10 GHz Type B uncertainty in attenuation 0,1 dB
Input power limitation below 10 dBm
Measurement of loss tangent
The dielectric resonators shall be provided with the loss tangent of the dielectrics being sufficiently low The best candidate having the least loss tangent is sapphire as specified in
Table 5 with the definitions of the terms being the same as described in IEC 61788-7 Ed
2:2005 (Also see the illustration in Figure 8) The loss tangent shall be measured with a relative uncertainty to not exceed 5%
Table 5 – Type B uncertainty for the specifications on the sapphire rod
Surface roughness top and bottom surface : 10 nm side wall : 0,001 mm Perpendicularity 0,1°
Axis parallel to the c-axis within 0,5°
Perpendicularity c-axis of crystal Cylinder axis
Figure 8 – Definitions for terms in Table 5
Temperature
During testing, the measurement apparatus is cooled to the specified temperature, with cryocoolers being one of the most effective methods The resonator is placed in a vacuum and cooled via thermal conduction through metallic connections It is essential that the temperature is measured with a standard uncertainty that does not exceed the specified limits.
When measuring the R Se, it is crucial to prevent temperature gradients within the apparatus Additionally, it is important to independently control the temperature of the upper HTS film while measuring the λ.
Specimen and holder support structure
The support must ensure the specimen is adequately stabilized, maintaining parallel alignment of the two films and mechanical stability during measurements, particularly in a cryocooler across a broad temperature range Additionally, it should be linked to a micrometer that accurately measures the gap distance between the top superconductor film and the resonator, achieving a standard uncertainty of 0.5 µm.
Identification of test specimen
The test specimen must be clearly identified, including essential information such as the manufacturer's name, classification or symbol, lot number, chemical composition of both the thin film and substrate, thickness and roughness of the thin film, and the manufacturing process technique.
Report of the intrinsic Z S values
The intrinsic R S and its scaled value at 10 GHz, along with λ 0 at 0 K and X S, will be documented as functions of temperature This will include the corresponding resonant frequencies, loaded and unloaded Q-values, and the insertion attenuations for both TE 021 and TE 012 modes, as well as the film thickness.
Report of the test conditions
The test conditions to be reported include the test frequency and frequency resolution, the maximum radio frequency power of the test signal, the test temperature along with the temperature differences between the two endplates, the history of sample temperature over time, and the type of dielectric rod utilized for the measurements.
Additional information relating to Clauses 1 to 8
Establishment of the standard measurement method is needed to evaluate film quality of high-
High-temperature superconductor (HTS) films exhibit an intrinsic surface resistance of less than 0.05 mΩ and a surface reactance ranging from 10 mΩ to 15 mΩ at 10 GHz, independent of film thickness Various resonance methods for measuring surface resistance in the microwave and millimeter wave range are outlined in IEC 61788-7 A schematic diagram of the measurement system, which utilizes step motors to control the motion of the upper superconductor film on the dielectric resonator, is presented in Figure A.1, while Figure A.2 provides a detailed view of the motion stage.
1 step motor 6 switch for thermal connection
2 polytetrafluoroethylene (PTFE) rod 7 dielectric resonator
3 polytetrafluoroethylene (PTFE) plate 8 cold finger
Figure A.1 – Schematic diagram for the measurement system
1 case frame 7 photo sensor 13 m-stage shaft guide
2 2,75” flange 8 z-stage base 14 polytetrafluoroethylene (PTFE) rod
3 rotary feed through 9 stage sensor plate 15 m-stage plate
4 step motor 10 y-stage base 16 PTFE rod
5 coupling polytetrafluoroethylene (PTFE) 11 x-stage base 17 OFHC block
6 stage shaft boss 12 m-stage base 18 OFHC block
Figure A.2 – A motion stage using step motors
Utilizing smaller dielectric rods allows for the use of thinner superconductor films However, as the size of the dielectric rod decreases, it becomes challenging to create a measurement apparatus and establish a measurement system, since the diameter of semi-rigid cables is constrained by the height of the dielectric rod Additionally, the measurement system must be designed to accommodate higher frequencies.
This test method is applicable to any superconductor having a proper model for the temperature dependence of the penetration depth
This test method is applicable for measurement temperatures below 30 K if new cooling technique allows temperatures lower than 30 K
This test method is suitable for measuring temperatures above 80 K, provided that precautions are taken to ensure the intrinsic resistance (R S) is comparable to the intrinsic reactance (X S) near the critical temperature (T C).
A.3 Theory and the measurement procedure for the intrinsic surface impedance
A.3.1 Theoretical relation between the intrinsic Z S and the effective Z S [1] 3
Figure A.3 illustrates a cross-sectional view of a dielectric resonator featuring two superconducting films of identical quality positioned at the top and bottom of a dielectric rod In a cylindrical coordinate system (ρ, φ, z), the non-zero field components of the TE 0mn mode in the k-th region, excluding k = 3, are defined by specific equations [2].
Regions 1 ~ 5 stand for a dielectric rod (1), vacuum (2), superconductor films (3), and dielectric substrates of the same kind used for the film growth (4 and 5), respectively
Figure A.3 – Cross-sectional view of a dielectric resonator
The equation \$H A k zk k k k k = k 2 (A.3)\$ describes the relationship between the electric field (H) and the magnetic field (E) in a specific region Here, \$A k\$ represents a constant for the field components in the k-th region, while \$\beta zk\$ and \$\beta k\$ denote the propagation constant and the wave number in the ρ-direction, respectively Additionally, \$q k\$ is a function of \$\beta zk z\$, and \$\Psi k\$ is a function of \$\beta k ρ\$.
For k = 1, 2, β z1 =β z2 , q 1 (β z1 z)=q 2 (β z2 z)=cos(β z1 z+ψ h ), with ψ h dependent on the symmetry of the electromagnetic field components For the TE 0mn modes, the field components in region 1 are expressed by
3 In this Annex A, numerals in square brackets refer to Clause A.8, Reference documents
1 β ρ β β φ = ωà ′ (A.6) with β 1 denoting the ρ-direction wave number and J 0 , the zeroth order Bessel function of the first kind π ψ h = p (odd n) (A.7)
2 π ψ = p + h (even n) (A.8) for the TE 0mn modes with p denoting a natural number In Equations (A.4) ~ (A.6), J 0 ’(β 1 ρ )
=dJ 0 (β 1 ρ )/d(β 1 ρ) The field components in region 2 are expressed by
= (A.12) for k 0 2 ε r2 < β 2 2 with I 0 and K 0 denoting the zeroth order modified Bessel function of the first kind and the second kind, respectively If k 0 2 ε r2 > β 2 2 , I 0 , I 0 ‘, K 0 , and K 0 ’ are replaced with J 0 ,
J 0 ‘, Y 0 , and Y 0 ’, respectively, in Equation (A.12) with Y 0 denoting the zeroth order Bessel function of the second kind
Equations for β z1 and β z2 (=β z1 ) are given as follows
The relationship \(0 < \varepsilon_r < \beta_k\) (A.15) is established, where \(k_0 = \omega_0 (\varepsilon_0 \mu_0)^{1/2}\) and \(\omega_0 = 2\pi f_0\) Here, \(\beta_1\) and \(\beta_2\) represent the cut-off wave numbers in regions 1 and 2, respectively, while \(\varepsilon_{a-b}'\) and \(\varepsilon_{r2}\) denote the relative permittivity of region 1 along the a-b plane and region 2 Additionally, a characteristic equation is derived from the boundary condition at \(\rho = a\) as illustrated in Figure A.3.
0 ε r >β k (A.17) with d denoting the radius of the OFHC cylinder placed between the superconductor films as seen in Figure A.3
For k = 3, general expressions for non-zero field components can be expressed as follows
H ρ 3 =γ z 3 3 ρ γ z 3 +Γ3 γ z 3 (A.19) with Γ 3 denoting a constant and f(ρ), a function of ρ, and Hz 3 = 0 for H < H C1 with H C1 denoting the lower critical field of the superconductor films In Equations (A.18) ~ (A.19), γ z3 = (j ωà 0 σ) 1/2 with σ = ( σ 1 - j σ 2 ) denoting the complex conductivity of superconductor films
For k = 4, 5, the non-zero field components are expressed by with q 4 (β z4 z) = exp(- jβ z4 z) - Γ 4 exp(jβ z4 z), Γ 4 = (exp[-β Z4 (2t + 2l + h)]), a constant, and
Equation for β z4 is given as follows
4 β (A.25) from the boundary condition at ρ = d in region 4 ν 0m is the m-th root of J 0 (x)’ = 0
If \( k_{0} < \frac{2 \epsilon r^{4}}{\beta^{4}} \), then \( J_{0} \) is substituted with \( I_{0} \) in Equations (A.20) to (A.22) In practical scenarios, the substrate diameter \( d_{S} \) is slightly larger than \( d \), which is used in Equation (A.25) The value of \( \Gamma_{3} \) in Equations (A.18) and (A.19) is determined based on the boundary condition.
H ρ3 / E φ3 = H ρ4 / E φ4 at z = (h/2) + t, which is expressed by
A relation between β z1 and γ z3 is obtained from the boundary condition of H ρ1 /E φ1 = H ρ3 /E φ3 at z = h/2, which is given by h z z z G tan 1 h 3
− (even n) (A.28) for the TE 0mn mode, with G h expressed by
= − (A.29) with t denoting the film thickness and
Z Se of the superconductor films can be obtained from the ratio of E ϕ3 to H ρ3 at z = h/2 using
Equations (A.18), (A.19) and (A.26), which is expressed by
Since Z Se = R Se + jX Se and Z S ≡ R S + iX S , Equation (A.32) gives expressions for R Se and X Se as follows
The intrinsic \( Z_S \) can be determined when both \( \sigma_1 \) and \( \sigma_2 \) of the complex conductivity \( \sigma \) are known, as indicated in Equation (A.29) The complex correction factor \( G_h \) is characterized by its real part \( \text{Re}(G_h) \) and imaginary part \( \text{Im}(G_h) \) To solve for the nine unknown parameters \( f_0, \beta_{z1}, \beta_1, \beta_2, \beta_{z4}, \beta_4, \beta_h, \sigma_1, \) and \( \sigma_2 \) in Equations (A.13) to (A.32), a total of seven equations are required.
Equations (A.13), (A.14) (or (A.15)), (A.16) (or (A.17)), (A.23) (or (A.24)), (A.25), (A.27) (or
(A.28)), (A.30) (or (A.31), and two measured quantities of Q U and f 0 , for which the details are as follow
A.3.2 Calculation of the geometrical factors [3]
For a dielectric resonator shown in Figure A.4, the Q U is related the effective surface resistance of the superconductor films R Se (SC) and the tanδ of the dielectric rod as follows ρ z
The surface resistances of the upper and lower superconductor films, denoted as R Se,T (SC) and R Se,B (SC), along with the OFHC side wall resistance R S (OFHC), are critical in understanding power losses The corresponding power losses are represented by P C,T, P C,B, and P C,SW Additionally, W 1 and W 2 signify the time average of the stored electromagnetic energy within regions s 1 and s 2, respectively.
Figure A.4 – A diagram for simplified cross-sectional view of a dielectric resonator tanδ
(A.35) with Q C and Q d (= ktanδ) denoting the conductive quality factor and the dielectric quality factor, respectively Q C is also expressed by
The surface resistances of the upper and lower superconductor films, denoted as \( R_{Se,T} \) and \( R_{Se,B} \), along with the OFHC side wall resistance \( R_S \), are critical in determining the power losses in the system The geometrical factors \( G_T \), \( G_B \), and \( G_{SW} \) correspond to these resistances, while the total power losses are represented as \( P_{C,T} \), \( P_{C,B} \), and \( P_{C,SW} \) Specifically, \( P_{C,T} \) is the sum of \( P_{C,T1} \) and \( P_{C,T2} \), and \( P_{C,B} \) is the sum of \( P_{C,B1} \) and \( P_{C,B2} \), where \( P_{C,T1} \) and \( P_{C,B1} \) refer to power losses in regions 1 and 2 of the upper and lower superconductor films, respectively The power losses \( P_{C,T1} \), \( P_{C,T2} \), \( P_{C,B1} \), \( P_{C,B2} \), and \( P_{C,SW} \) are calculated using specific expressions.
Also, in Equation (A.36), W = W 1 + W 2 with W 1 and W 2 for the time average of the stored electromagnetic energy inside region 1 and 2, respectively, with dz d d ' E
In Equations (A.37) ~ (A.41), H ρ1 , H ρ2 , Hz 2 , E φ1 and E φ2 are as expressed in Equations (A.4)
~ (A.16) (or A.17) with β z1 = n π/h, ε 0 denoting the permittivity of vacuum, and ε a-b ‘, the relative permittivity along the a-b plane of the dielectric
The geometrical factors \( G_T \), \( G_B \), and \( G_{SW} \) are calculated using Equations (A.36) to (A.41) with \( k = \frac{W_1}{W_1 + W_2} \) Table A.1 presents the geometrical and filling factors for the standard sapphire resonator, utilizing dielectric constants of 9.28 and 11.3 at 77 K for the a-b plane and the c-axis of the sapphire rod, respectively.
Table A.1 – Geometrical factors and filling factors calculated for the standard sapphire resonator
A.3.3 Procedure for determining the intrinsic Z S [1, 3, 4]
A.3.3.1 Determination of the intrinsic penetration depth[2, 5]
For a dielectric resonator with superconductor films placed at the top and the bottom, the change in the temperature-dependent resonant frequency is expressed as follows
The resonant frequency at the measured temperature is denoted as \$f_0(T)\$, while \$\varepsilon' \$ represents the temperature-dependent relative permittivity of the dielectric rod within the dielectric resonator The parameters \$a\$ and \$h\$ refer to the radius and height of the rod, respectively Additionally, \$f(\varepsilon')\$ is a function of the relative permittivity, and \$g_i(l_i)\$ is a function of \$l_i\$, which indicates the temperature-dependent dimensions of both the dielectric rod and the copper cylinder situated between the superconductor.
When the temperature of the upper superconductor film fluctuates while the rest of the resonator maintains a constant temperature, the variations in the fundamental frequency \( f_0 \) with respect to temperature can be exclusively linked to changes in the top surface's superconducting energy gap \( X_{Se,Top} \) This relationship is expressed by the equation \(-\frac{1}{f_0} \left(\frac{\partial f_0}{\partial T}\right) = \frac{1}{2G T} \left(\frac{\partial X_{Se,Top}}{\partial T}\right)\).
In Equation (A.43), the term \(\Delta f_0(T)\) is defined as \(f_0(T) - f_0(T_{\text{min}})\), where \(T_{\text{min}}\) represents the minimum temperature of the upper superconductor film Additionally, \(\Delta X_{\text{Se Top}} = X_{\text{Se Top}}(T) - X_{\text{Se Top}}(T_{\text{min}})\) Given that \(f_0(T_{\text{min}}) \gg \Delta f_0(T)\) for the sapphire resonator, the equation can be simplified accordingly.
Meanwhile, since γ z3 ≅ 1/λin Equation (A.29) and X S >> R S for σ 2 >> σ 1 at temperatures not too close to T C , X Se = [Re(G h ) X S + Im(G h ) R S ] ≅ Re(G h ) X S from Equation (A.34) Therefore,