DC Performance and AC Loss of Cable-in-Conduit Conductors for International Thermonuclear Experimental Reactor

DC Performance and AC Loss of Cable-in-Conduit Conductors for International Thermonuclear Experimental Reactor 1 first-author 2 1 corresp Corresponding author. E-mail address: jfang@bjtu.edu.cn Corresponding author. E-mail address: jfang@bjtu.edu.cn jfang@bjtu.edu.cn 3 School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W2Y2, Canada Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W2Y2, Canada Institute of plasma physics, Chinese academy of sciences, Hefei 230031, China Institute of plasma physics, Chinese academy of sciences, Hefei 230031, China A reliable prediction of AC loss is essential for the application of International Thermonuclear Experimental Reactor (ITER) cable-in-conduit conductors (CICCs), however, the calculation of AC loss of ITER CICCs is a cumbersome task due to the complicated geometry of the multistage cables and the extreme operating conditions in ITER. In this paper, we described the models developed for hysteresis and coupling loss calculation, which can be suitable for the construction of ITER magnetic system. Meanwhile, we compared the results of theoretical analysis with the SULTAN test result to evaluate the numerical model we used. In addition, we introduced the n -value and AC loss with transport current for CICCs based on the DC measurement results at SULTAN, which lays the foundation for the further study. AC susceptibility coupling loss hysteresis loss n -value SULTAN test 1 Introduction 1 For a large superconducting nuclear fusion experimental device, AC loss of CICCs is an indicator of the stability and practical application of large superconducting nuclear fusion experimental devices. The longitudinal magnetic properties of hard superconductors of slab-like and cylindrical shapes were first performed by Bean using the critical-state (CS) model with a constant critical current density J c [ 1 1 , 2 2 ] . The method was then introduced to determine the transport AC loss of a cylindrical hard superconductor by London [ 3 3 ] and further developed by A.M. Campbell [ 4 4 ] . Based on these pioneering works, A. Nijhuis and his colleagues provided some effective methods to analyze the electromagnetic and mechanical characterization (primarily AC loss) of ITER conductor [ 5 5 - 9 9 5-9 ] , which took into account the superconducting magnets be exposed to large and transient current and field. In addition, the influence of cyclic mechanical loading on AC loss of NbTi and Nb 3 Sn CICC was explained by Sergey A. Lelekhov [ 10 10 ] . Meanwhile, analysis of AC loss and conceptual design for Experimental Advanced Superconducting Tokamak (EAST) and Chinese Fusion Engineering Test Reactor (CFETR) were well developed based on the predecessors' work [ 11 11 - 15 15 11-15 ] . The paper was organized as follows: AC susceptibility was introduced to calculate the hysteresis loss of CICCs. A detailed explanation of the term was given in section 2 before the descriptions of the dominant types of AC loss: hysteresis loss and coupling loss. The selection of conductor and the experimental procedures in Sultan facility was presented in section 3. After comparing the results of suggested method with Sultan test result, we made a brief discussion of the application and limitation of AC susceptibility in section 4.1. An example of the further research of the n -value from the DC results of Sultan was given in section 4.3 before stating the conclusions in section 5. 2 AC Loss 1 2.1. AC susceptibility 2 The complex AC susceptibility <math id="M1"><mrow><mi>χ</mi><mo>=</mo><msup><mi>χ</mi><mo>′</mo></msup><mo>−</mo><mi>j</mi><msup><mi>χ</mi><mo>″</mo></msup></mrow></math> ( <math id="M2"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M3"><msup><mi>χ</mi><mo>″</mo></msup></math> are real and imaginary parts of <math id="M4"><mi>χ</mi></math> ) of an infinitely long superconducting strand with a round cross-section of radius a was numerically calculated with a uniform transverse ac field based on the critical-state model with a constant J c before. D-X Chen and E. Pardo made such calculations based on the critical-state model using a magnetic energy minimization (MEM) method, and the results were presented as tables and figures in Ref.[ 16 16 , 17 17 ]. Considering a superconducting strand placed in a uniform applied ac field <math id="M5"><mrow><msub><mi>H</mi><mi>m</mi></msub><mi>s</mi><mi>i</mi><mi>n</mi><mi>ω</mi><mi>t</mi></mrow></math> in transverse direction, we calculate its complex AC susceptibility <math id="M6"><mrow><mi>χ</mi><mo>=</mo><msup><mi>χ</mi><mo>′</mo></msup><mo>−</mo><mi>j</mi><msup><mi>χ</mi><mo>″</mo></msup></mrow></math> based on the critical-state model with a constant J c . For a superconducting strand magnetized by a transverse applied field H and starting from the zero-field cooled state along the initial magnetization curve Mini ( H) with increasing H from 0 to H m , the screening currents profiles are calculated using the procedure detailed in [ 18 18 ] . Finally, <math id="M7"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M8"><msup><mi>χ</mi><mo>″</mo></msup></math> as functions of H m are obtained from Fourier analysis in the following [ 19 19 ] : <math display="block" id="M9"><mrow><mi>m</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>H</mi><mi>m</mi></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><mo stretchy="false">(</mo><msup><mi>χ</mi><mo>′</mo></msup><mi>cos</mi><mi>n</mi><mi>ω</mi><mi>t</mi><mo>+</mo><msup><mi>χ</mi><mo>″</mo></msup><mi>sin</mi><mi>n</mi><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mstyle><msub><mi>H</mi><mi>m</mi></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></munderover><mrow><mi>Re</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msup><mi>χ</mi><mo>′</mo></msup><mo>+</mo><mi>i</mi><msup><mi>χ</mi><mo>″</mo></msup><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>n</mi><mi>ω</mi><mi>t</mi></mrow></msup><mo stretchy="false">]</mo></mrow></mstyle><mo>,</mo></mrow></math> where m ( t ) is the magnetic moment for an ac magnetic field. Hence, AC susceptibility <math id="M10"><mrow><msup><mi>χ</mi><mo>′</mo></msup><mo>+</mo><mi>i</mi><msub><msup><mi>χ</mi><mo>″</mo></msup><mrow/></msub></mrow></math> is calculated as: <math display="block" id="M11"><mrow><msup><mi>χ</mi><mo>′</mo></msup><mo>+</mo><mi>i</mi><msup><mi>χ</mi><mo>″</mo></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>π</mi><msub><mi>H</mi><mi>m</mi></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mrow><mi>m</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>n</mi><mi>ω</mi><mi>t</mi></mrow></msup><mi>d</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle><mo>.</mo></mrow></math> As we can see in Eq. (1) and (2), <math id="M12"><msup><mi>χ</mi><mo>′</mo></msup></math> = <math id="M13"><msup><mi>χ</mi><mo>″</mo></msup></math> =0 for even n . The AC susceptibility for odd n is given by: <math display="block" id="M14"><mtable columnalign="left"><mtr><mtd><msup><mi>χ</mi><mo>′</mo></msup><mo>+</mo><mi>i</mi><msup><mi>χ</mi><mo>″</mo></msup><mo>=</mo><mfrac><mn>2</mn><mrow><mi>π</mi><msub><mi>H</mi><mn>0</mn></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>π</mi></msubsup><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mi>m</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mi>θ</mi><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>n</mi><mi>θ</mi></mrow></msup><mi>d</mi><mi>θ</mi></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mn>4</mn><mrow><mi>i</mi><mi>n</mi><mi>π</mi><msub><mi>H</mi><mi>m</mi></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>H</mi><mi>m</mi></msub></mrow></msubsup><mrow><mi>exp</mi><mo stretchy="false">[</mo><mi>i</mi><mi>n</mi><mi>arccos</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>2</mn><mi>h</mi></mrow><mrow><msub><mi>H</mi><mi>m</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mfrac><mrow><mo>∂</mo><mi>m</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>h</mi></mrow></mfrac><mi>d</mi><mi>h</mi></mrow></mrow></mstyle><mo>.</mo></mtd></mtr></mtable></math> According to the above equations, <math id="M15"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M16"><msup><mi>χ</mi><mo>″</mo></msup></math> versus H m could be obtained using Matlab software. It is interesting that <math id="M17"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M18"><msup><mi>χ</mi><mo>″</mo></msup></math> versus H m / H p ( H p is the full penetration field) functions for a long cylinder in longitudinal and transverse fields are very similar after we use MEM method, so we may use the exact analytical result of the longitudinal cylinder to approximate that of the transverse cylinder strand. The expressions of <math id="M19"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M20"><msup><mi>χ</mi><mo>″</mo></msup></math> are as follows [ 17 17 ] : <math display="block" id="M21"><mrow><msup><mi>χ</mi><mo>′</mo></msup><mo>/</mo><msub><mi>χ</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>/</mo><mn>16</mn><mo>,</mo></mrow></math> <math display="block" id="M22"><mrow><msup><mi>χ</mi><mo>″</mo></msup><mo>/</mo><msub><mi>χ</mi><mn>0</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>3</mn><mi>π</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> when <math id="M23"><mrow><mi>x</mi><mo>=</mo><msub><mi>H</mi><mi>m</mi></msub><mo>/</mo><msub><mi>H</mi><mi>p</mi></msub><mo>≤</mo><mn>1</mn><mo>,</mo></mrow></math> and <math display="block" id="M24"><mtable columnalign="left"><mtr><mtd><msup><mi>χ</mi><mo>′</mo></msup><mo>/</mo><msub><mi>χ</mi><mn>0</mn></msub><mo>=</mo><mo>{</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>/</mo><mn>16</mn><mo stretchy="false">)</mo><mi>arccos</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><mo stretchy="false">[</mo><mo>−</mo><mn>19</mn><mo>/</mo><mn>12</mn><mo>+</mo><mn>5</mn><mi>x</mi><mo>/</mo><mn>8</mn><mo>+</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>/</mo><mo stretchy="false">(</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>0.5</mn></mrow></msup><mo>}</mo><mo>/</mo><mi>π</mi><mo>,</mo></mtd></mtr></mtable></math> <math display="block" id="M25"><mrow><msup><mi>χ</mi><mo>″</mo></msup><mo>/</mo><msub><mi>χ</mi><mn>0</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mo>/</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>/</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>3</mn><mi>π</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> when <math id="M26"><mrow><mi>x</mi><mo>≥</mo><mn>1.</mn></mrow></math> where the demagnetizing factor-based parameter called shape susceptibility <math id="M27"><mrow><msub><mi>χ</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn></mrow></math> and <math id="M28"><mrow><msub><mi>H</mi><mi>p</mi></msub><mo>=</mo><mn>2</mn><msub><mi>J</mi><mi>c</mi></msub><mi>a</mi><mo>/</mo><mi>π</mi></mrow></math> for round cross-section of strand and J c is the critical current density. 2.2. Hysteresis loss 2 The losses in high field superconductors exposed to changing magnetic fields arise from the irreversible nature of flux pinning. As the field rises (or falls), flux moves into (or out of) the superconductor which means the individual vortices must break free of their pinning centers and move through the material. This movement is an irreversible dissipative process which generates heat. Here we follow an approximate approach, based on the following assumptions: ● the critical current density is uniform in the filament and strand cross section; ● the n -value of CICC strands is relatively so large that the critical current density can be taken to be constant for the AC susceptibility calculation; ● transport current effects are neglected and the cable is not saturated. It gives rise to an irreversible magnetization curve like the innermost loop and the energy dissipation Q (J/m 3 ) around a cycle is given by [ 4 4 ] : <math display="block" id="M29"><mrow><mi>Q</mi><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∮</mo><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>M</mi><mi>d</mi><mi>H</mi></mrow></mrow></mstyle><mo>,</mo></mrow></math> where M is the magnetization, defined as the magnetic moment per unit volume, produced by the screening currents caused by magnetic field, <math id="M30"><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow></math> is permeability of vacuum and H is the external field, if we take the AC susceptibility into consideration for the calculation of area of magnetization curve, hysteresis loss Q hys can be also determined from: <math display="block" id="M31"><mrow><msub><mi>Q</mi><mrow><mi>h</mi><mi>y</mi><mi>s</mi></mrow></msub><mo>=</mo><mi>π</mi><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>H</mi><mi>m</mi></msub><msup><mrow/><mn>2</mn></msup><msup><mi>χ</mi><mo>″</mo></msup><mo>,</mo></mrow></math> where H m is the amplitude of the applied AC magnetic field, <math id="M32"><msup><mi>χ</mi><mo>″</mo></msup></math> is the imaginary part of the AC susceptibility <math id="M33"><mi>χ</mi></math> which have discussed in section 2.1. The biggest difference between this method for hysteresis loss calculation and the classical method described in [ 4 4 ] is that AC susceptibility which could be obtained by experiment or parameterized function from Eq. (4)-(7) easily. Therefore, this method is considered to be an effective way to calculate hysteresis loss for CICCs. 2.3. Coupling loss 2 The coupling loss is originated from the inductive loops whose resistive couplings are distributed among the strands in cable or within the strands. And this loss depends on the characteristics of the cable such as twist pitch, matrix resistivity of conductor, presence of wraps, coating or solder. It can be estimated by the well-known formulas in the following [ 4 4 ] : <math display="block" id="M34"><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>τ</mi></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><msub><mi>B</mi><mi>i</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mtable><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable><mrow><mo>[</mo><mrow><msup><mrow><mtext>W/m</mtext></mrow><mtext>3</mtext></msup></mrow><mo>]</mo></mrow><mo> </mo><mo>,</mo></mrow></math> where P is the loss power per unit volume, n <math id="M35"><mi>τ</mi></math> is coupling time constant of the conductor and B i is the rate of change of field inside the conductor. For steady-state regimes, we may assume that <math id="M36"><mrow><mfrac><mrow><mo>∂</mo><msub><mi>B</mi><mi>i</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mi>B</mi><mi>e</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mrow></math> , where B e is the external changing field. But for small ripples where the induced currents screen the inside of the strand, a more detailed equation is expressed by: <math display="block" id="M37"><mrow><mfrac><mrow><mo>∂</mo><msub><mi>B</mi><mi>i</mi></msub></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>B</mi><mi>i</mi></msub></mrow><mi>τ</mi></mfrac><mo>=</mo><mfrac><mrow><msub><mi>B</mi><mi>e</mi></msub></mrow><mi>τ</mi></mfrac><mo>.</mo></mrow></math> The general solution for B i is: <math display="block" id="M38"><mrow><msub><mi>B</mi><mi>i</mi></msub><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mo>/</mo><mi>τ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>τ</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msub><mi>B</mi><mi>e</mi></msub></mrow></mrow></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mi>t</mi><mo>/</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mi>C</mi><mo stretchy="false">]</mo><mo>.</mo></mrow></math> For applied field <math id="M39"><mrow><msub><mi>B</mi><mi>e</mi></msub><mo>=</mo><mi>Δ</mi><mi>B</mi><mi>sin</mi><mi>ω</mi><mi>t</mi></mrow></math> , <math display="block" id="M40"><mrow><msub><mi>B</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mi>Δ</mi><mi>B</mi><mo stretchy="false">(</mo><mi>sin</mi><mi>ω</mi><mi>t</mi><mo>−</mo><mi>ω</mi><mi>τ</mi><mi>cos</mi><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><msup><mi>τ</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>C</mi><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mo>/</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></math> The time constant <math id="M41"><mi>τ</mi></math> is the decay time constant of the induced currents. For monolithic conductors the time constant can be written as [ 5 5 ] : <math display="block" id="M42"><mrow><mi>τ</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo>⋅</mo><msubsup><mi>L</mi><mi>p</mi><mn>2</mn></msubsup></mrow><mrow><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup><mo>⋅</mo><msub><mi>ρ</mi><mo>⊥</mo></msub></mrow></mfrac><mo>,</mo></mrow></math> where L p is the twist pitch (or the characteristic length of a coupling current path) and <math id="M43"><mrow><msub><mi>ρ</mi><mo>⊥</mo></msub></mrow></math> is the effective electrical resistivity in the transverse direction which can be obtained from experiment. Assume one only dominant coupling loss time constant is present (inter-filament coupling loss), then the coupling loss over an extended frequency range Q cpl is given by [ 4 4 ] : <math display="block" id="M44"><mrow><msub><mi>Q</mi><mrow><mi>c</mi><mi>p</mi><mi>l</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>B</mi><mi>a</mi></msub><msup><mrow/><mn>2</mn></msup><mi>π</mi><mi>n</mi><mi>τ</mi><mi>ω</mi></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><msup><mi>τ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mtable><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable><mo stretchy="false">[</mo><msup><mrow><mtext>J/m</mtext></mrow><mtext>3</mtext></msup><mtext>·cycle],</mtext></mrow></math> where B a is the amplitude of the applied AC magnetic field and n is the shape factor and equal to 2 for round strand. Inter-strand coupling loss is generally treated analogously to inter-filament coupling loss. Therefore, the equations presented above are also used for a cable by adopting an appropriate effective time constant <math id="M45"><mrow><msub><mi>τ</mi><mrow><mi>e</mi><mi>f</mi><mi>f</mi></mrow></msub></mrow></math> , which is the sum over the N stages composing the cable of the multiple time constants n i , which based on the assumption that the coupling currents in a given stage do not interfere with the coupling currents of the other stages, detailed in [ 4 4 , 27 27 ] . The method enables an easy treatment of any number of cabling stages. Consequently, this is the most frequently used model for the calculation of inter-strand coupling loss in CIC conductors. 3 Experimental Method 1 3.1 Sultan samples 2 The selected conductor called China’s forth toroidal field (TF) Conductor (TFCN4) is based on composite Nb 3 Sn internal tin strands manufactured by Western Superconducting Technologies Company (WST). Cabling and jacketing operations are performed in collaboration between Institute of Plasma Physics Chinese Academy of Sciences (ASIPP) and Baosheng Company in China according to a layout determined by ITER TF conductor specification [ 20 20 ] .The cable layout was the same for all conductors. Every cable contains 900 Nb 3 Sn strands with 522 copper strands cabled together around a central cooling spiral. The jacket for conductor is compacted using the compacting machine with a set of rollers. The compaction was done in one single step by a set of rollers. The after compaction conductor’s outer diameter is 43.75 mm, inner diameter is 39.75 mm, so the jacket tube thickness is 2 mm. The entire sample was heat-treated according to the following temperature schedule at Centre De Recherché En Physique Des Plasmas (CRPP) Villigen, Switzerland: 210 ℃ for 50 hours, 340 ℃ for 25 hours, 450 ℃ for 25 hours, 575 ℃ for 100 hours and 650℃ for 100 hours. The rate of temperature change is 5 ℃/h. Typical cross sections of the CNTF4 strand and conductor are provided in Fig. 1 Fig. 1 . The properties of the strands and conductor used for the manufacturing of the tested Sultan samples are summarized in Table 1 Table 1 and Table 2 Table 2 . 1 1 Parameters 1 1 Value 1 1 diameter, mm 1 1 0.820-0.821 1 1 Critical current (at 4.2 K/12 T), A 1 1 230.2-265.0 1 1 Hysteresis loss(±3 T), mJ/cm 3 1 1 379.5-591.6 1 1 RRR 1 1 170-208 1 1 n value(at 4.2 K/12 T) 1 1 23.0-35.8 1 1 Cu: non-Cu 1 1 0.92-1.037 1 1 Cr-coating thickness, μm 1 1 1.29-1.58 1 1 Parameters 1 1 Value 1 1 Conductor type 1 1 120 m qualification conductor 1 1 Cable pattern 1 1 ((2SC+1Cu) ×3×5×5+core) ×6,core: 3×4 Cu 1 1 Jacket material 1 1 Stainless steel 316LN 1 1 Cable twist pitch, mm 1 1 84/145/194/308/421 1 1 Void fraction 1 1 29.35% 1 1 Central spiral outer diameter, mm 1 1 10.0×9.0 3.2. Sultan test 2 A hairpin type TF sample tested in the SULTAN facility is made of two 3.5 m parallel conductor sections (legs) which are connected with an overlapping joint [ 21 21 ] . The two legs are identified by the positions. Once a sample is completely done, it is inserted into the SULTAN facility where it is cooled down to 4.2 K. The SULTAN magnet generates a high magnet field of almost 11 T which covers 450 mm of sample length [ 22 22 ] . The sample condition is monitored by means of 24 voltage taps and 16 thermometers distributed on both legs. In general, the measurement of AC losses is based on the traditional approach, the so-called calorimetric measurement in SULTAN which is primarily devoted to qualification tests of full-size CICCs for fusion magnets. The boundary conditions of CICC sample in SULTAN test is schematically illustrated in Fig. 2 Fig. 2 [ 12 12 ] . Calorimetric method is the most direct methods and offer from medium to high sensitivity and good precision for both small samples. The AC loss, E , is measured by gas flow calorimetric method and is evaluated from the increase of the coolant temperature Δ T (Δ T = T 2- T 1, is shown in Fig. 2 Fig. 2 ), the coolant specific heat C p , the coolant mass flow rate dm/dt , AC field frequency f and the cable volume V . The AC loss per cm 3 and per cycle of CICC ( E ) can be calculated by [ 12 12 ] : <math display="block" id="M46"><mrow><mi>E</mi><mo>=</mo><mfrac><mrow><mi>Δ</mi><mi>T</mi><msub><mi>C</mi><mi>P</mi></msub><mi>d</mi><mi>m</mi><mo>/</mo><mi>d</mi><mi>t</mi></mrow><mrow><mi>f</mi><mi>V</mi></mrow></mfrac><mo>.</mo></mrow></math> The SULTAN test consists of measurements in AC and DC conditions. Both types of measurements are performed on the sample at initial condition after it is inserted into the SULTAN testing well and cooled down to 4.2 K. The total loss of the conductor can be measured from the AC test, while the transition index n and the current sharing temperature T cs are derived from I c measurement during DC test. Then a certain number of electromagnetic sinusoidal cyclic loadings are performed with ramping-up and down operating current of 68 kA in a background field of 10.78 T. 4 Results and Discussion 1 4.1. AC susceptibility 2 According to the calculation procedures described in section 2.1, <math id="M47"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M48"><msup><mi>χ</mi><mo>″</mo></msup></math> as function of critical current density J c at different amplitude of applied field <math id="M49"><mrow><mi>Δ</mi><mi>B</mi></mrow></math> are plotted in Fig. 3 Fig. 3 . As the Nb 3 Sn strands are very sensitive to the strain and experience severe loads by electromagnetic force and thermal contraction in the TF conductor, J c was obtained from test result of Pacman facility at the University of Twente [ 22 22 ] . The strain description model (DEV-model) is used to calculate the critical current of the Nb 3 Sn composites. The parameters for ITER scaling law are listed in Table 3 Table 3 [ 24 24 , 25 25 ] . The result of the I c calculation of CNTF4 strand is summarized in Fig. 4 Fig. 4 as a function of strain, temperature and magnetic field. And the critical current density J c of the strand is 5.68×10 9 A/m 2 (Cu: non-Cu = 1, critical current I c is about 1500 A at DC field B dc =2 T, T =4.2 K in SULTAN test). And for the TFCN4 strands with applied magnetic field <math id="M50"><mrow><msub><mi>B</mi><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>=</mo><mn>0.3</mn><mi>sin</mi><mi>ω</mi><mi>t</mi></mrow></math> (T) in Sultan, hence <math id="M51"><msup><mi>χ</mi><mo>′</mo></msup></math> and <math id="M52"><msup><mi>χ</mi><mo>″</mo></msup></math> are approximately -1.6942 and 0.1257. 1 1 Parameters 1 1 Value 1 1 Ca1 1 1 47.52 1 1 Ca2 1 1 0 1 1 ε 0,a (%) 1 1 0.218 1 1 ε m (%) 1 1 -0.067 1 1 μ 0 H c2m (0)( T ) 1 1 34.22 1 1 T cm (0)( K ) 1 1 16.26 1 1 C 1( AT ) 1 1 20,823 1 1 p 1 1 0.578 1 1 q 1 1 2.211 It is worth explaining that AC susceptibility of small samples with about 1 mm diameter can be obtained by test under the comparatively high frequency condition, but the CIC conductors with large diameter cannot be tested so far. But fortunately, AC susceptibility only depends on the characteristics such as J c and the conductor diameter, so we can calculate AC susceptibility of the size of conductors (not only TFCN4 conductor) using the equations in section 2.1. 4.2. AC loss 2 For the sake of providing the result of Sultan test, we calculate AC loss under the same conditions as Sultan, i.e. the background magnetic field B dc =2 (T), the alternating field <math id="M61"><mrow><msub><mi>B</mi><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>=</mo><mn>0.3</mn><mi>sin</mi><mi>ω</mi><mi>t</mi></mrow></math> (T) and the transport current I =0 (A). So from the above analysis, main results of the calculation of hysteresis loss as function of critical current density at different <math id="M62"><mrow><mi>Δ</mi><mi>B</mi></mrow></math> based on ac susceptibility, coupling loss and total loss as function of frequency (the coupling loss constant <math id="M63"><mrow><mi>n</mi><mi>τ</mi></mrow></math> is about 122.78 ms before cycling from the well-known equations in section 2.3 and [ 4 4 , 27 27 ] .) are presented in Fig. 5 Fig. 5 and 6 6 , respectively. From Fig. 3 Fig. 3 and 5 5 , the hysteresis loss is considered to be independent of frequency due to the assumptions in section 2.2. From Fig. 5 Fig. 5 , it is obviously seen that the hysteresis loss increase firstly and then decrease with the increase of critical current density. Essentially, the conductor will be fully penetrated due to the low critical current density and then the hysteresis loss is proportional to J c in low critical current density region. On the contrary, the conductor is not penetrated when the critical current density is high and then hysteresis loss is inversely proportional to J c in high critical current density region. As to the Sultan sample, the background magnetic field is small (about 2 T) and then critical current density is high according to scaling law in table.3 table.3 , in this case, hysteresis loss is very small compared to coupling loss for most of Sultan conductors. But in the real operation environment of ITER (at above 10 T background and fast transport current), the conductor may be fully penetrated and the hysteresis loss would be high. Meanwhile, AC loss of TFCN4 is measured in the transverse field of ±0.3 T alternating with frequencies from 0.1 Hz to 1 Hz in a background field of 2 T before and after the current load cycling. AC test result per cycle for TFCN4 before and after the cycling for the whole range of frequencies is given in Fig. 7 Fig. 7 [ 28 28 ] . After cycling, the AC loss decreased significantly, which is due to the significant increase of the contact resistance in conductors with Cr coated Nb 3 Sn strands. In Cr coated strands, the contact resistance was observed to decrease during the heat treatment, due to the oxygen diffusion, and increase again when transverse forces are applied during the load cycling. The contact resistance remains high when the cable is unloaded [ 29 29 ] . Therefore, before electromagnetic load cycling is applied, AC loss of the CIC conductors was larger. It should be pointed that the coupling loss per unit volume of the conductor per cycle increases linearly with frequency and with the square of amplitude of magnetic field, which can deduce the effective coupling current time constant <math id="M66"><mrow><mi>n</mi><mi>τ</mi></mrow></math> [ 4 4 ] from the slope of the results at the low frequency range shown in Fig. 7 Fig. 7 : <math display="block" id="M67"><mrow><msub><mi>Q</mi><mi>c</mi></msub><mo>=</mo><mfrac><mrow><mi>π</mi><mi>n</mi><mi>τ</mi><mi>ω</mi><mi>Δ</mi><msup><mi>B</mi><mn>2</mn></msup></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>ω</mi><mn>2</mn></msup><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>≈</mo><mfrac><mrow><mi>π</mi><mi>n</mi><mi>τ</mi><mi>ω</mi><mi>Δ</mi><msup><mi>B</mi><mn>2</mn></msup></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mi>J</mi><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> For applied field <math id="M68"><mrow><mi>B</mi><mo>=</mo><mi>Δ</mi><mi>B</mi><mi>sin</mi><mi>ω</mi><mi>t</mi></mrow></math> , <math display="block" id="M69"><mrow><mi>n</mi><mi>τ</mi><mo>=</mo><mfrac><mrow><mi>α</mi><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup><mi>Δ</mi><msup><mi>B</mi><mn>2</mn></msup></mrow></mfrac><mtable><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable><mrow><mo>[</mo><mi>s</mi><mo>]</mo></mrow><mo>,</mo></mrow></math> where <math id="M70"><mi>α</mi></math> is the slope of the results at the low frequency range. From Eq. (13) and (14), the <math id="M71"><mrow><mi>n</mi><mi>τ</mi></mrow></math> obtained from the initial slope of the losses curve ( f : 0.1 Hz-0.2 Hz) is about 137.21 ms. The comparison between numerical method and measurement results of AC loss for TFCN4 is shown in Fig. 8 Fig. 8 and the results show a relatively good agreement (at B dc = 2 T, T = 4.2 K, <math id="M72"><mrow><msub><mi>B</mi><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>=</mo><mn>0.3</mn><mi>sin</mi><mi>ω</mi><mi>t</mi></mrow></math> (T)). As we can see in Fig. 8 Fig. 8 , experimental results are found to be deviating from the numerical results. On the one hand, AC susceptibility is complicated at very low frequency because the hysteretic losses due to pinning and the eddy-current losses are not simply additive which was described in paper “Low-frequency AC susceptibility due to the Interplay of Loss Mechanisms”, actually, AC susceptibility at low frequency is a little larger than that of the numerical results from the Eq. (4)-(7), which leads to the experimental results is a little larger than that of numerical at low frequency; on the other hand, coupling loss saturates and subsequently decreases with frequency due to shielding of the interior of the conductor at high frequency which was also described in [ 5 5 ] . However, the saturation of conductor was not taken into account when we used the classical method of coupling loss calculation because the maximum frequency of Sultan test of CNTF4 is 1 Hz, which leads to the experimental results is a little smaller than that of numerical at high frequency. 4.3. DC results 2 The DC test is the most time consuming activity in the TF sample test program which is aimed to measure the main parameters of the conductor, such as the current sharing temperature T cs and the critical current I c ( n -value). And strand witness samples for left and right conductors are wound each onto “ITER barrels”, and are heat treated together with the CICC sections in order to verify the heat treatment by measuring the strand I c . The test of the strands for TFCN4 is carried out at CRPP, immersed in liquid helium and at applied fields between 9 and 15 T. Fig. 9 Fig. 9 illustrates the results of I c measurement and n -value versus applied field and it is obvious that Fig. 9 Fig. 9 shows a good agreement with Fig. 4 Fig. 4 which was obtained by scaling law. The index of transition or n -value data is characterized using an empirical modified power law [ 25 25 ] : <math display="block" id="M75"><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">[</mo><msub><mi>I</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow></msup><mo>.</mo></mrow></math> As previously observed in [ 30 30 ] , <math id="M76"><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow></math> is approximately a constant for all temperatures and applied strains and <math id="M77"><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow></math> only merely depends on strain properties [ 31 31 ] . From Fig. 9 Fig. 9 we find that <math id="M78"><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow></math> and <math id="M79"><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow></math> are 3.235 and 0.4122, respectively, which are similar to those of other advanced internal-tin strands [ 32 32 ] . As given in Fig. 10 Fig. 10 , n -value versus critical current correspondingly to the specimens #1 and #2 of TFCN4 was presented and the parameterized function curve was also plotted. Obviously, it shows a very nice agreement between DC test result of Sultan and the result of parameterized function curve for CNTF4 strand, and this curve would be also available for other internal-tin strands. One of the n -value applications is that AC loss of CIC conductors with transport current can be calculated. In transport current–voltage ( I-V) measurements of most CICC’s strands, I changes with V in the full penetration regime following roughly a power law (PL): <math id="M80"><mrow><mi>V</mi><mo>∝</mo><msup><mi>I</mi><mi>n</mi></msup></mrow></math> [ 33 33 ] , so that I c has to be defined as that when electric field E=V/l=E c , where l is the distance between both voltage taps and criterion E c =10 <math id="M81"><mrow><mtext>μV/m</mtext></mrow></math> (at B =4-11 T, T =4-10 K) is routinely defined. The I-V curve comes from a PL relates to E(J) , which is a characteristic of collective flux creep and expressed as: <math display="block" id="M82"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>E</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mi>c</mi></msub><mo>/</mo><msub><mi>J</mi><mi>c</mi></msub><mo stretchy="false">)</mo><msup><mrow><mrow><mo>|</mo><mrow><mi>J</mi><mo>/</mo><msub><mi>J</mi><mi>c</mi></msub></mrow><mo>|</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold" mathsize="normal"><mi>J</mi></mstyle><mo>.</mo></mrow></math> Based on (16), the transport loss Q is calculated by: <math display="block" id="M83"><mrow><mi>Q</mi><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mi>E</mi><mi>J</mi><mi>d</mi><msub><mi>V</mi><mi>t</mi></msub></mrow></mrow></mstyle><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>c</mi></msub><mo>/</mo><msub><mi>J</mi><mi>c</mi></msub><mo stretchy="false">)</mo></mrow></mrow></mstyle><mrow><mo>|</mo><mrow><msup><mi>J</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>/</mo><msub><mi>J</mi><mi>c</mi></msub><msup><mrow/><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>|</mo></mrow><mi>d</mi><msub><mi>V</mi><mi>t</mi></msub><mrow><mo>[</mo><mi>W</mi><mo>]</mo></mrow><mo>.</mo></mrow></math> As we know, logarithm is introduced to extract the index- n in Eq. (20): <math display="block" id="M84"><mrow><mi>lg</mi><mo stretchy="false">(</mo><mi>E</mi><mo>/</mo><msub><mi>E</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mi>lg</mi><mo stretchy="false">(</mo><mi>I</mi><mo>/</mo><msub><mi>I</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow></math> Then, <math display="block" id="M85"><mrow><mi>n</mi><mo>=</mo><mfrac><mrow><mi>lg</mi><mi>E</mi><mo>−</mo><mi>lg</mi><msub><mi>E</mi><mi>c</mi></msub></mrow><mrow><mi>lg</mi><mi>I</mi><mo>−</mo><mi>lg</mi><msub><mi>I</mi><mi>c</mi></msub></mrow></mfrac><mo>,</mo></mrow></math> where E and I are obtained from the measurement results based on four probe method, and n -value of internal-tin strands can also be obtained by the above parameterized function curve. The transport current loss of low temperature superconductor (LTS) with power-law is described in Eq. (21), on the one hand, this method is further work because lots of work need to be solved such as the inhomogeneous distribution of critical current density based on Norris model for a round superconducting strand, on the other hand, it is also a worthwhile work because the existed numerical formulas for calculating hysteresis loss and coupling loss of LTS are complicated and time consuming, so it would be convenient if this method is also available for calculation of AC loss of LTS. As for the CIC conductors, the result of the above equations for transport current loss calculation is part of the total loss. In consideration of the coupling current existing filaments and Cu-matrix, it is supposed that the result of Eq. (17) is including hysteresis loss and part of coupling loss of CICC and further relative research will be done in near future. 5 Conclusion 1 The transverse complex AC susceptibility <math id="M86"><mrow><mi>χ</mi><mo>=</mo><msup><mi>χ</mi><mo>′</mo></msup><mo>−</mo><mi>j</mi><msup><mi>χ</mi><mo>″</mo></msup></mrow></math> of an infinitely superconducting strand with radius a was numerically calculated, and the new model of hysteresis loss calculation for CICCs with AC susceptibility was developed. Coupling loss with classical method, hysteresis loss with AC susceptibility and total loss of ITER CNTF4 were numerically calculated. Also, AC susceptibility can be obtained by experiment or parameterized function easily, which makes the hysteresis loss calculation faster. The fourth China TF conductor sample (TFCN4) for Phase II has been tested in the SULTAN facility which contains AC and DC measurements. 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DOI: 10.1088/0953-2048/21/10/105016 http://doi.org/10.1088/0953-2048/21/10/105016 Critical current scaling laws for advanced Nb3Sn superconducting strands for fusion applications with six free parameters Bruzzone P . The index n of the voltage-current curve in the characterization and specification of technical superconductors . Physica C , 2004 , 401 : 7 - 14 . DOI: 10.1016/j.physc.2003.09.005 http://doi.org/10.1016/j.physc.2003.09.005 The index n of the voltage-current curve in the characterization and specification of technical superconductors The authors would like to thank A. Devred, A. Vostner and C. Y. Gung from ITER Organization and D-X Chen from Universitat Autonoma de Barcelona for helpful discussion. We are thankful to University of Twente crew for strand preparation work and testing and ASIPP and CRPP personnel for providing some pictures and test results of TFCN4 conductor. 10.1007/s41365-016-0061-2 Supported in part by Ministry of Science and Technology of China under Grant 2014GB105001. 2015-08-21 2015-10-09 2015-10-16 2016-05-14 2016-06-20 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0061-2 http://dx.doi.org/10.1007/s41365-016-0061-2 . 2016 Nuclear Science and Techniques 03 27