Quantitative analysis of ITER Poloidal Field joints through rigorous resistivity parameterization

The lap-type twin-box joints are integral components in International Thermonuclear Experimental Reactor (ITER) fusion magnets, with profound implications for magnet stability based on their electro-magnetic, thermal, and mechanical properties. Throughout the extensive R&D process, rigorous qualification tests are conducted to meet stringent standards. However, existing tests often prioritize global performance, which lack of strand-level details due to inherent limitations in test setups. Furthermore, as the referencing test facility of SULTAN falls short in replicating relevant ITER operating conditions, numerical methods that offer both accuracy and the requisite level of detail for comprehensive magnet and component analysis and development are necessary. This paper introduces the utilization of the JackPot-AC/DC code, developed at the University of Twente, as a fundamental tool for achieving strand-level precision in handling CICCs and joints, which encompasses copper and solder components. The primary focus of this study is to obtain precise input parameters, emphasizing their role in conducting a quantitative analysis using JackPot-AC/DC. The investigation centers on an ITER PF5 joint (PFJEU6), where contact resistances and AC losses were measured under parallel magnetic fields. Given the constraints in the measured results, an enhanced parameterization is performed to derive precise resistivity and solder-related parameters. Additionally, sensitivity analyses of individual parameters and cable compact configurations are thoughtfully evaluated. With the optimal input parameters acquired, systematic simulations of the joint exposed to transverse magnetic fields, mimicking SULTAN and ITER operating conditions, are processed and validated against experimental results. This research establishes a comprehensive foundation for the analysis of lap-type twin-box joints, including DC, AC, and stability properties. The outcomes will significantly contribute to advancing the understanding of the intricate behavior of these joints in the context of fusion magnet applications.

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Introduction
The magnetic confinement fusion machines, exemplified by International Thermonuclear Experimental Reactor (ITER) [1], feature a complex magnet system encompassing Toroidal Field (TF), Poloidal Field (PF), Central Solenoidal (CS), and Correction Coils (CC), all wound with Cable-In-Conduit Conductors (CICCs) [2].These CICCs, consisting of superconducting and copper strands transposed and twisted into a rope-like pattern, are strategically inserted into a stainless steel jacket [2].The pivotal role of joints in connecting individual conductors, facilitating electrical and thermal communication, is underscored by the prevalent use of the lap-type twin-box concept in PF, TF and CC coils [3].
The manufacturing process of a lap-type twin-box joint involves a meticulous sequence.The cable undergoes initial treatment, wherein the Ni or Cr coating on the cable periphery is removed, by a rotating stainless steel brush or reverse electroplating methods, and subsequently, the cable is silver plated, tinned, or left with a bare copper surface, to improve the cable-to-copper sole conductance.The cable is then compressed tightly into a bi-metallic box featuring a copper sole on one side, and the cable's outer strands are either soldered to or pressed into the copper sole.The conductor within the box is further compacted with a welded cover, resulting in the creation of a termination (half joint).By incorporating a copper shim between two terminations, a complete joint is formed through soldering or compression [4].Figure 1 provides illustrative axial-and cross-sectional views of a PF termination, with an accompanying cross-sectional view of the regular conductor for comparative analysis.
The influence of joint performances on coil stability, concerning AC loss, temperature margin, and current distribution, necessitates extensive experimental and numerical analyses [5][6][7][8][9].A crucial test campaign of the full-size ITER PF joints, conducted at the Swiss Plasma Center's SULTAN facility [10], is instrumental in enhancing our understanding of technology intricacies and ensuring product qualification [11,12].This comprehensive campaign encompasses rigorous assessments of DC, AC, and stability properties, leveraging the unique capabilities of the SULTAN facility.Despite the invaluable insights provided by these tests, the predominant focus on overall behaviors tends to overlook strand-level details crucial for an accurate assessment of electromagnetic and thermal properties.Significantly, the initiation of instability typically occurs when currents exceed the tolerances in specific strands, particularly in instances marked by severe current non-uniformity [13,14].To minimize this gap, the University of Twente introduces a numerical strand-level model named JackPot-AC/DC [13,15], capable of discerning electromagnetic and thermal behaviors in CICCs or joints in terms of steady-state, pulsed, and thermal stability analyses [5,16].
Critical to the efficacy of JackPot-AC/DC simulations are precise input parameters, with an emphasis on contact resistivities.Comparing to the estimation of averaged contact resistances between small strand bundles in the joint [17,18], for the first time, measurements of the inter-strand contact resistances from different cable stages and AC loss of a full-size ITER joint (PFJEU6) were conducted at the University of Twente [7,12].In this study, an enhanced parameterization is employed to derive precise resistivity and solder-related parameters for a comprehensive understanding of the full-size ITER joint.Subsequently, a thoughtful sensitivity analysis of the individual parameters is conducted to determine the optimal values.Measurements at the University of Twente are characterized by DC and AC magnetic fields provided by a solenoidal magnet, oriented parallel to the joint axis.This orientation stands in contrast to the SULTAN facility, where both magnetic fields are applied transversely to the joint axis [10].With the resistivity and solder-related parameters obtained from fitting the Twente experiments, joint performance is numerically assessed under both parallel and transverse magnetic field configurations, mirroring Twente and SULTAN experiments, respectively.Moreover, the model is adept at quantitatively analyzing the joint in ITER operating conditions [19], characterized by a complex magnetic scenario dominated by transverse magnetic field components.This capability forms a central focus of the present paper.

Methodology for resistivity parameterization
For the full-size PFJEU6 joint, each cable consists of 1152 NbTi strands winding with a 5-stage cabling pattern of 3 × 4 × 4 × 4 × 6.The simulation of contact resistances is conducted by selecting specific pairs of strands, in total 33 strand pairs over 5 stages are selected in consideration of a representative distribution in each cable.This selection process mirrors the configuration used during the contact resistance measurements, ensuring a consistent methodology, the specific scheme is detailed in [7].Each simulation involves applying a current of 50 A to the chosen pair of strands while leaving the other strands unconnected.The resulting resistance is determined by dividing the calculated voltage between the selected pair of strands by the supplied current.
In the contact resistance measurements, a distinctive behavior is observed in the distribution across different stages, as explained later in detail in figure 3. Notably, contact resistances in the first to fourth stages exhibit close values, while experiencing a substantial increase of about 4-5 times in the fifth stage [7].Consequently, the JackPot-AC/DC model incorporates two resistivity parameters, namely interstrand ρ ss and interpetal ρ ip resistivity, to accurately characterize the specific distribution between petals.
In lap-type joints, the introduction of a copper sole introduces an additional parameter, namely the strand-to-copper sole resistivity ρ sj .To enhance electrical conductivity between the strands and the copper sole, a foil of eutectic AgSn solder is applied [20].Subsequently, during the melting process, the solder permeates the peripheral strands and fills voids along the copper sole arc region [21].The solder not only affects the contact interface between the cable and copper sole but also influences internal strands submerged in the solder.Notably,  the response of the contact resistance to electromagnetic loads is highly contingent on the presence of solder [22,23].
Incorporating the thickness ∆r s and width W of the solder in the model, depicted in figure 2, the solder is treated as a uniformly distributed layer, neglecting the cumulative effect under gravity [24].The thickness ∆r s is indirectly defined with respect to the strand diameter d = 0.730 ± 0.005 mm, where ∆r s = k•d, with k as a multiplication factor.Similarly, the width of the solder layer W is defined with respect to the cable diameter Φ, which fluctuates between 31 and 36 mm, contingent on the compaction level.
Deriving the main resistivity parameters involves fitting the measured contact resistances.However, for lap-type joints, this fitting procedure is intricate, leading to non-unique results due to the involvement of multiple parameters and their interactions.As illustrated in figure 3, the blue bars represent the average measured contact resistances of two conductors, and the three sets of yellow bars depict simulated data with different resistivity parameters, detailed in table 1.The comparison reveals that the interstrand resistance, corresponding to cable stages 1-4, is primarily influenced by the parameter ρ ss .Additionally, comparable distributions of overall contact resistances are observed in simulations #1 and #3, despite significant differences in resistivity parameters.This underscores the significant impact of either ρ ip or ρ sj on interpetal resistance in stage 5. Importantly, the comparison also highlights the inherent uncertainty in resistivity parameterization based solely on fitting contact resistances.
The consideration of parameter uncertainty underscores the pivotal influence of the strand-to-copper sole resistivity on the parameterization of joint resistivity.In essence, accounting for the impact of the copper sole is imperative to impart additional confinement.One avenue for achieving this is through the examination of cable-to-cable resistance, taking into consideration the presence of the copper sole between two cables.Employing three distinct parameter sets delineated in table 1, cable-to-cable resistances are systematically simulated and juxtaposed with experimental data, as depicted in plot (a) of figure 4. Notably, the outcomes reveal an escalation in cableto-cable resistance with variations in either ρ ip or ρ sj , as evident in simulation #1 and #3 respectively.The observed variance in  resistance across diverse strand combinations is posited to be linked to the intricacies of cabling patterns and the specific contact points between strands and the sole.
Alternatively, an exploration of the AC loss of the joint, influenced by coupling currents across the copper sole, is warranted.The overall AC loss comprises coupling current, eddy current, and hysteresis losses [25].In the context of strandlevel JackPot simulations, hysteresis and inter-filament losses are excluded, with focus solely on coupling and eddy current losses.To facilitate a meaningful comparison between simulated and measured losses, the hysteresis loss component is subtracted from the total measured AC losses.Plot (b) of figure 4 presents a comparative analysis of coupling and eddy current losses for the joint across the three parameter sets in table 1.It is discerned that losses diminish with an increase in either ρ ip or ρ sj , attributable to constraints imposed on coupling current loops across petals or cables.
Given the acknowledged impact of specific contact positions on the spread of cable-to-cable resistance and recognizing the inherent larger error associated with derived parameters from these contacts compared to those from AC loss simulations, a refined methodology for resistivity parameterization is advocated.This entails integrating constraints from both interstrand contact resistances and AC losses, with the contribution from cable-to-cable resistance serving as a comparative reference.

Sensitivity analysis of parameters
The parameterization of the joint involves the determination of five key parameters, which not only exert a discernible influence on the collective behaviors of the joint, encompassing aspects such as the distribution of contact resistances and AC losses, but also serve as indirect indicators of design and technological characteristics.These parameters encapsulate the nuanced impact of design elements such as cabling patterns, twist pitch sequences of conductors, and the application of solder layers.In order to systematically assess the quantitative impact of each individual parameter and enhance the efficacy of the parameterization procedure, a sensitivity analysis is conducted.This analysis involves the incremental adjustment of one parameter at a time, as meticulously documented in table 2. The ensuing evolution of contact resistances (A.1-E.1)and AC losses (A.2-E.2) is comprehensively evaluated and visually presented in figure 5. To elucidate overarching trends and mitigate the effects of dispersion, contact resistances are portrayed as cumulative frequency distributions, facilitating a focused examination of the overall evolution.It is imperative to note that the presented AC losses exclusively encompass coupling and eddy current losses, excluding the contribution from hysteresis loss.
The simulation results elucidate general trends wherein an increase in the parameter ρ ss tends to elevate both interstrand and interpetal resistances, concomitant with a reduction in AC losses.In contrast, the parameter ρ ip primarily influences interpetal resistance, exhibiting a consistent reduction in AC losses across the entire frequency spectrum as it increases.The influence of ρ sj on contact resistance is comparatively modest, predominantly impacting interpetal rather than interstrand resistance.Nevertheless, a noteworthy impact on AC losses is observed, with a rapid decline as ρ sj increases, particularly at higher frequencies.Regarding solder-related parameters, an increase in the solder thickness ∆r s from zero to 2.0 times the strand diameter results in a marginal decrease in interpetal resistance, while AC losses exhibit an increase, particularly at higher frequencies.Similarly, the width of the solder layer W follows a comparable trend, wherein a broader solder layer is associated with decreased interpetal resistance and an increase in AC losses.

Parameterization in parallel magnetic fields
Building upon insights garnered from sensitivity analysis, a comprehensive set of resistivity and solder-related parameters, denoted as set 1, is meticulously derived and detailed in table 3. The ensuing fitting procedure involves a rigorous comparison of interstrand and interpetal contact resistance distributions derived from both simulated and measured datasets.This analysis, presented in figure 6, reveals a notably commendable fitting behavior, attesting to the robust alignment between simulated and experimental data.
The AC loss measurements of the PFJEU6 joint were conducted at the University of Twente [7], wherein four distinct magnetic field levels were applied in parallel orientation relative to the joint axis.Following the deduction of corresponding hysteresis losses, the resultant losses are quantified and denoted as Q Exp .Utilizing the set 1 resistivity parameters, the coupling and eddy current losses of the joint Q Sim are computationally determined, which are subsequently juxtaposed against the measured data Q Exp .The outcomes of this comparative analysis are presented in figure 7, demonstrating a general and commendable agreement.In the ultimate stage of our investigation, subsequent to the meticulous satisfaction of constraints pertaining to contact resistances and AC losses, the parameters encapsulated in set 1 are judiciously deemed optimal values for the PFJEU6 joint.These selected parameters are subsequently employed for further numerical analyses, thus affirming their efficacy in encapsulating and characterizing the nuanced behavior of the joint under varied operational conditions.

Effect of cable compaction (void fraction)
Owing to the intricate interplay of contacts between strands and current transitions within joints, particularly in the context of pulsed-mode operation for PF coils, variations in resistive or inductive coupling can induce current non-uniformity, potentially compromising the critical current and temperature margin [25,26].The potential for current redistribution, especially in overloaded strands, presents a significant concern, as it may surpass critical thresholds, triggering quench phenomena [27].Consequently, the AC loss [28]and stability of the joint are intricately linked to contact resistances and their distributions, with added complexity introduced by the presence of the copper sole.
In the lap-type twin-box joint, an endeavor to enhance electrical conductivity involves a tightly compacted conductor within the joint box section.The reduction in conductor diameter from 35.3 to 31.6 mm, accompanied by a decrease in void fraction from 34% to 19%, is undertaken to achieve this objective [7].The cross-sectional depiction of conductors at different positions is elucidated in figure 1.Given the reliance on contact areas between strands for the conversion of measured contact resistances to resistivities, the compaction of the cable assumes critical importance, especially for CICCs featuring intricate twisting patterns.
In addition to the previously discussed set 1 parameters corresponding to a void fraction of 19%, the parameterization of joints with a void fraction of 34% (set 2) are performed following the same methodology as described in section 2, subsequently, set 3 parameters with the same resistivity and solder-related values as set 2 but a different void fraction of 19% are also tested, the three sets of parameters are detailed in table 3. Figure 6 illustrates the distributions of simulated contact resistances with set 2 and 3 parameters.The concordance in resistance distributions between simulations with set 1 and 2 parameters underscores that higher void fraction results in diminished contact areas between strands, leading to underestimated resistivities by approximately 60%.The validation of the impact of physical cable compaction is underscored by the comparison between simulations with set 2 and 3 parameters, where increased contact areas correlate with decreased contact resistance.
Although the nominal void fraction of set 2 parameters is 34%, assuming the conductor is inside the joint box section, they are still influenced by the copper sole and not entirely analogous to a regular conductor outside the joint box section.For a standard PF5 conductor sample (PFEU3), the AC loss is experimentally measured at the SULTAN facility [29].The resistivity parameters ρ ss and ρ ip are derived from the measured data, as seen in figure 8, these values are contrasted with the relevant average values of set 2 parameters for comparative analysis, and summarized in table 4. The results reveal that, without the transfer effect of the copper sole, the interstrand resistivity of a regular conductor increases from (0.75 ± 0.05) × 10 −12 to (20 ± 1) × 10 −12 Ωm 2 , indicating a significant ratio increase of 27 ± 3. The interpetal resistivity experiences a corresponding increase, albeit with a decreased ratio of interpetal to individual interstrand resistivity from 16 to 4.
The impact of compaction on AC loss is further assessed by subjecting the conductor sample to a transverse harmonic magnetic field with an amplitude B ac of 0.4 T and zero background field.Figure 9 illustrates the coupling loss density per cycle versus frequency, showcasing higher coupling losses in the compacted conductor with much lower contact resistivities and an earlier saturation observed at a frequency around 1 Hz.
To quantitatively characterize AC loss, the coupling loss time constant nτ is considered with: where α denotes the initial slope at low frequency of the fitting curve [30].
A comparison reveals that nτ is 44 ± 3 ms and 1560 ± 120 ms for the Not-compacted and compacted conductors, respectively.This significant ratio of 35 ± 5, compared to the interstrand resistivity ratio of 27 ± 3, underscores a pronounced correlation between interstrand contact resistance and coupling loss.

Coupling loss in transverse magnetic fields
In addition to the analyses of the PF joint under a parallel magnetic field, mirroring the Twente experimental conditions and employing the set 1 parameters, we systematically evaluate the coupling and eddy current losses of the PFJEU6 joint in transverse magnetic fields analogous to the SULTAN experiments.A background DC field B dc = 3 T is applied along the x-direction, as depicted in figure 2, while the AC field, modulated with an amplitude B ac = 0.2 T, is applied transversely in both the x and y directions.The effective field lengths at the SULTAN facility are approximately 39 cm and 45 cm for the AC and DC fields, respectively [10].To account for the impact of these limited effective lengths, the coupling and eddy current losses are judiciously normalized to a volume of 1088 cm 3 , representing the combined volumes of the copper sole and strands within the 39 cm applied AC field length.This normalization enables a comprehensive and standardized assessment of losses, facilitating meaningful comparisons across varying experimental conditions.The calculated loss density per cycle across frequency is depicted in figure 10.The coupling loss time constant (nτ ) is determined through a meticulous fit of the loss density, focusing on the low-frequency range up to 20 mHz.The resulting values are 6880 ± 170 ms and 924 ± 140 ms for the AC field in the x and y directions, respectively.In the x direction, substantial coupling currents circulate in loops encompassing both conductors and their associated copper soles, contributing significantly to the total losses.Conversely, in the y direction, coupling currents are predominantly confined to individual conductors, manifesting as smaller current loops.The preeminence of eddy current losses in the copper sole translates into relatively lower overall losses.This scenario mirrors the actual SULTAN configuration measurements, the comparative results from the measured PFJEU6 joint [12] are also presented in figure 10.A notable convergence between experimental and simulated outcomes, particularly in the lowfrequency region, attests to the accuracy of the resistivity parameterization.
The joint configuration comprises three components-two conductors and a copper sole.Notably, in the y direction, the specific distribution of coupling current loops results in relatively low losses.This signifies that the coupling loss in one conductor constitutes less than half of the total losses, as evidenced by nτ < 462 ± 70 ms.A comparison with the nτ = 1560 ± 120 ms observed for the compacted conductor, as illustrated in figure 9, suggests that, despite the minimal impact of the copper sole on coupling losses, a significant portion of coupling currents is still transferred to the copper sole.This transfer effect, although limited, delays the saturation of the conductor and ultimately contributes to the enhanced stability of the joint to some extent.

Conclusion
Lap-type twin-box joints, such as those employed in ITER magnets, represent critical components where conventional qualification tests typically emphasize global properties.However, a robust research and development framework necessitates a comprehensive analysis at the strand level to gain a quantitative understanding of the diverse electrical properties inherent in these joints.In a pioneering effort, the interstrand and interpetal contact resistances of a full-size ITER PF joint were meticulously measured, marking a significant stride towards acquiring essential resistivity parameters for subsequent quantitative numerical analyses using the JackPot-AC/DC model.Through an exhaustive evaluation of their impact on resistance distribution and AC losses, an enhanced parameterization process is undertaken, yielding precise resistivity and solder-related parameters.A sensitivity analysis is conducted to discern the influence of individual parameters, culminating in the derivation of a set of optimal parameters tailored for the PFJEU6 joint.This optimized parameter set serves as the cornerstone for subsequent quantitative numerical analyses.
The varying void fractions of cables within and outside the joint box section introduce notable effects on the contact areas between strands, consequently influencing the resistivity parameterization.An in-depth assessment of the joint's performance is conducted, encompassing the intricate dynamics of cable compaction and the consequential current transfer effect to the copper sole.Utilizing the optimized resistivity parameters corresponding to different cable sections, this paper systematically evaluated the joint's performance under parallel and transverse applied magnetic field conditions.Simulation results derived from conditions mimicking SULTAN testing, guided by the input parameter analysis, exhibit commendable agreement with measured data.The systematic investigation establishes a foundational framework for an extensive analysis of lap-type twin-box joints, encompassing DC, AC, and stability properties.Moreover, it enables the predictive assessment of the joint's behavior across diverse operational scenarios, thus laying the groundwork for further advancements in the field of superconducting technology.

Figure 1 .
Figure 1.Axial and transverse cross-sectional views of a PF termination (half joint), depicting the conductor compacted within the bimetallic box.

Figure 2 .
Figure 2. Schematic illustration of a termination highlighting key dimensional parameters associated with the solder layer positioned between the strand and conductor, wherein d represents the diameter of the strand and Φ denotes the diameter of the conductor.

Figure 3 .
Figure 3. Comparative analysis depicting the variations in interstrand and interpetal contact resistances of the PF joint, simulated under diverse resistivity parameter sets.

Figure 4 .
Figure 4. (a) Comparative analysis illustrating variations in cable-to-cable resistances among selected strands; (b) comparative evaluation of AC losses of the PF joint, simulated under distinct resistivity and solder parameter sets.

Figure 5 .
Figure 5. Examination of the influence of individual resistivity and solder-related parameters, presented through the analysis of contact resistance (A.1-E.1)and coupling and eddy current losses (A.2-E.2).

Figure 6 .
Figure 6.Influence of cable compaction on interstrand and interpetal contact resistances in a joint, comparing experimental data with results from three simulations employing different resistivity parameters and void fractions.

Figure 7 .
Figure 7. Comparative analysis of the coupling and eddy current losses in a joint under a parallel applied AC field, contrasting experimental data with simulations employing parameter set 1, and accounting for variations in magnetic field conditions.

Figure 8 .
Figure 8. Coupling loss density per cycle versus frequency for a PF5 conductor, featuring experimental data obtained from the SULTAN facility and its comparison with a polynomial fitting curve.

Figure 9 .
Figure 9. Calculated coupling loss density per cycle versus frequency for both Not-compacted and compacted PF5 conductors under the influence of a transversely applied AC magnetic field.The coupling loss time constant nτ is derived from the initial slope at low frequency, offering insights into the dynamic response of the conductors to varying harmonic magnetic field conditions.

Figure 10 .
Figure 10.Comparison between simulated and measured coupling loss density per cycle versus frequency for the PF5 joint sample.The joint is exposed to an AC magnetic field applied in both the x and y directions, with an amplitude Bac = 0.2 T, and operating under zero external magnetic field conditions.

Table 1 .
Experimental parameters detailing resistivity and solder properties employed in the parameterization methods.

Table 2 .
Summary of parameters employed in the sensitivity analysis, with each case featuring a singular adjustment of a specific parameter, elucidating the nuanced effects on the contact resistance and AC loss.

Table 3 .
Summary of contact resistivity and solder-related parameters for the simulations with two cable compaction configurations, 34% and 19% void fraction in the PFJEU6 joint.d and Φ are the diameter of the strand and the cable respectively.