Inductance calculation of twisted conductors by the broken line approximation

The mutual inductance between two skew straight thin conductors is obtained as a function of two vectors corresponding to two current carrying line segments. Based on the obtained analytical expressions for the mutual inductance, the versatile calculation method for the self- and mutual inductances of various twisted conductors is studied by means of the broken line or polygonal curve approximation. In particular, it is confirmed that this numerical calculation is consistent with the analytical calculation of the self- and mutual inductances for coaxial helical conductors for the asymptotic form of the long axial length. Furthermore, for the inductances of general twisted conductors, the similar asymptotic forms of the length dependence are obtained.     

CUDI: A model for calculation of electrodynamic and thermal behaviour of superconducting Rutherford cables

CUDI is the extended Fortran code to calculate the electrodynamic and thermal behaviour of any type of Rutherford cable subject to global and/or local variations in field, transport current, and external heat release. The internal parameters of the cable can be freely varied along the length and across the width, such as contact resistances, critical current, cooling rates etc. In this way, all the typical non-uniformities occurring in a cable, e.g. broken filaments, strand welds, cable joints, and edge degradation can be simulated. Also the characteristics of the strands in the cable can be varied from strand to strand. Heat flows through the matrix, through the interstrand contacts, and to the helium are incorporated, as well as the self-field and self- and mutual inductances between the strands. The main features and structure of the program will be discussed.     

Analysis of current distributions in a multi-laminated HTS tape conductor for solenoid coils

A multi-laminated HTS tape conductor has been recently developed for large coils. If the HTS tapes are simply laminated to form the conductor, the current distribution in the laminated tape conductor of the coil is imbalanced because of the differences among inductances of tapes. Transposition of the tapes in the conductor is effective for homogeneous current distribution, but the tape may be damaged due to the lateral bending. The solenoid coil has enough space to transpose the tapes at both ends. However, a proposed theory so far requires a restriction in the number of coil layers for homogeneous current distribution in the laminated tape conductor. It is very important to analyze current distributions in the multi-laminated tape conductor for the solenoid coil with arbitrary layers. In this paper, we apply the Maxwell integral equation to the region contoured by adjacent laminated tapes to analyze the current distributions of the tapes in an infinite solenoid coil, and demonstrate that the flux across the region is conserved as long as the tapes are not saturated, and finally induce the fundamental equations as functions of coil construction parameters, such as layer radii, laminated tape spaces, and winding pitches. We use the fundamental equations for 2-layer and 4-layer coils to verify the homogeneous current distribution of the laminated tape conductor for an arbitrary layer number. Since the flux between the tapes in the inner layer of a 2-layer coil is contributed from the outer layers, the tape space in the outer layer must be larger than that in the inner layer because of the balance between the two fluxes. Moreover, we have developed an analysis method for a finite solenoid coil.     

Circuit analysis model for AC losses of superconducting YBCO cable

Information about AC losses and electromagnetic behaviour is essential when designing superconducting cables. In this work, AC losses of coaxial YBCO cables are determined using circuit analysis based computational model tailored for the needs of the YBCO cable design work. In the equivalent circuit superconducting layers are connected in parallel, the layers have an inductive coupling between each other and AC loss within a layer generates an effective resistance. The layer currents can be solved from a set of circuit equations. The computational model takes into account that the current in the cable creates magnetic field, which generates magnetisation loss and affects strongly the critical current of the YBCO tapes. The model was applied on several coaxial superconducting YBCO cable designs, which had nominal currents of 1-10 kA (rms). Low AC loss values were predicted for these compact YBCO cable designs. For example, AC losses less than 4 W/m were predicted for 10 kA cables.     

Inductance Calculations for Circular Coils of Rectangular Cross Section and Parallel Axes Using Bessel and Struve Functions

A simple method for calculating the mutual and self inductances of circular coils of rectangular cross section and parallel axes is presented. The method applies to non-coaxial as well as coaxial coils, and self inductance can be calculated by considering two identical coils which coincide in space. It is assumed that current density is homogeneous in the coil windings. The inductances are given in terms of one-dimensional integrals involving Bessel and Struve functions, and an exact solution is given for one of these integrals. The remaining terms can be evaluated numerically to great accuracy using computer packages such as Mathematica. The method is compared with other exact methods for the coaxial case, and with a filamentary method for the non-coaxial case. Excellent agreement was found in all cases, and the exact method presented here agrees with another exact coaxial method to great numerical accuracy.     

Superconductive dc transmission lines: design study study and cost estimates

A conservative design study of a dc superconductive power line based on present day technology is presented. Power is transmitted by opposing currents in coaxial conductors, formed from helically wound tapes of composite, with a potential difference between them. Conductor temperature stability and fault current conditions are shown to be the factors which dominate the conductor cost; both are analysed. The capital cost of a cable is expressed as a function of a cable dimension, the basic cable parameters, and the material properties. This cost may be minimized with respect to cable radius. The resultant optimum magnetic fields vary little with power rating and are quite low at about 1–2 × 10⁵ A m^?1. The capital cost of such cables, exclusive of terminal costs, is less than the cost of conventional dc cables at power levels above about 1 GVA. Although extensive development would be necessary, the basic principles needed to build a dc superconductive cable are known.     

Measurements of current distributions in a multi-laminated HTS tape conductor for solenoid coils

A multi-laminated HTS tape conductor has recently been developed for large coils. If the HTS tapes are simply laminated to form the conductor, the current distribution in the laminated tape conductor of the coil is unbalanced because of the differences among all tape inductances. Transposition of the tape in the conductor is effective for homogeneous current distribution, but the tape may be damaged due to the lateral bending. In our previous paper, we proposed a new theory to analyze and control current distributions in the multi-laminated tape conductor for a solenoid coil with arbitrary layers. We applied the Maxwell integral equation to the region contoured by adjacent laminated tapes to analyze the current distributions of the tapes in the infinite solenoid coil, demonstrated that the flux across the region is conserved as long as the tapes are not saturated, and finally induced fundamental equations as functions of coil construction parameters, such as layer radius, laminated tape space, and winding pitch. In order to verify the theory, we designed two kinds of coils with homogeneous and inhomogeneous current distributions in the two-laminated tape conductor by adjusting the space between the tapes in the second layer, and fabricated them. In the case when the space between the tapes in the second layer is the same as that of the first layer, 0.31 mm in thickness, we measured the tape currents of 7:3 for the inner and outer tape of the first layer, respectively. We adjusted the space between the tapes of the second layer, 1.78 mm in thickness, while the space of the first layer remained unchanged, 0.31 mm in thickness. We obtained the homogeneous current distribution in the tape conductor. The experimental data were in good agreement with the theory.     

Layer-current waveform of coaxial multi-layer HTS cable considering the flux flow state

It is important to control the layer current distributions of coaxial multi-layer HTS cables because homogeneous layer current distribution decreases AC loss and increases the largest operational current. In a previous paper, we proposed a theory that can control current distribution based on the concept of flux conservation between two adjacent layers, and demonstrated the theory is in good agreement with experiment results. The theory was effective for an operational current less than the critical current of the cable. It is important to investigate current distribution under the condition of operational current more than the critical current of the cable because the cable experiences fault currents. We have extended the theory to treat the operational current more than the critical current by considering V-I nonlinear characteristics of HTS tapes including flux flow resistance and contact resistance between the cable and terminals. In order to verify the extended theory, we have fabricated a two-layer cable with the same twisting layer pitch, and hence caused inhomogeneous current distribution. It was observed that almost all of operational current less than the critical current flowed on the outer layer because of its lower inductance. When the operational current increased above the critical current of the second layer, the flux flow resistance appeared and distorted the current waveform with phase deviations. Finally, in the case of operational current more than the critical currents of both layers, flux flow resistance strongly affected current waveforms, and thereby the currents of both layers were determined by flux flow resistance. The extended theory simulated the layer current distribution waveforms and demonstrated good agreement with the experimental results under all operational current regions.     

横向拱塔斜拉桥的拱轴线逐段设计算法研究

拱塔斜拉桥的拱轴线设计是一项复杂工作,该文提出了拱形主塔拱轴线的逐段设计算法。该算法将拱轴线离散为一系列连续直线段,根据设计指定的主拱内力状态和节段的平衡条件,自拱顶起逐点计算出拱轴线坐标。另外该算法仅要求已知斜拉索竖向索力分量,其余两个索力分量可在逐段计算拱轴线坐标的同时得到。该文以主梁弯曲应变能为目标函数对索力的竖向分量进行了研究。台江大桥主塔拱轴线算例研究表明,逐段拱轴线设计算法具有较高的精度和计算效率。     

Inductance of Bitter Coil with Rectangular Cross-section

The self-inductance of Bitter coil and mutual inductance between coaxial Bitter coils with rectangular cross-section using semi-analytical expressions based on two integrations were introduced. The current density of the Bitter coil in radial direction is inversely proportional to its radius. The obtained expressions can be implemented by Gauss integration method with FORTRAN programming. We confirm the validity of inductance results by comparing them with finite filament method and finite element method. The inductance values computed by three methods are in excellent agreement. The derived expressions of inductance of Bitter coils with rectangular cross-section allow a low computational time compared with finite filament method to a specific accuracy. The derived mutual inductance expressions can be used to accurately calculate the axial force between coaxial Bitter coils with mutual inductance gradient method.     

Comparison study of cable geometries and superconducting tape layouts for high-temperature superconductor cables

High-temperature superconductor (HTS) rare-earth-barium-copper-oxide (REBCO) tapes are very promising for use in high-current cables. The cable geometry and the layout of the superconducting tapes are directly related to the performance of the HTS cable. In this paper, we use numerical methods to perform a comparison study of multiple-stage twisted stacked-tape cable (TSTC) conductors to find better cable structures that can both improve the critical current and minimize the alternating current (AC) losses of the cable. The sub-cable geometry is designed to have a stair-step shape. Three superconducting tape layouts are chosen and their transport performance and AC losses are evaluated. The magnetic field and current density profiles of the cables are obtained. The results show that arrangement of the superconducting tapes from the interior towards the exterior of the cable based on their critical current values in descending order can enhance the cable’s transport capacity while significantly reducing the AC losses. These results imply that cable transport capacity improvements can be achieved by arranging the superconducting tapes in a manner consistent with the electromagnetic field distribution. Through comparison of the critical currents and AC losses of four types of HTS cables, we determine the best structural choice among these cables.     

A Numerical Study on AC Losses of a Double Layer Polygonal BSCCO Conductor by Finite Element Method

High-temperature superconductor (HTS) cables are candidates for power transmission cables in the near future. A cylindrical arrangement of HTS tapes for the cable has proved able to reduce the AC loss. Many studies on AC loss characteristics of HTS cables have been done, but few numerical models of the cable were verified by experiments. In this paper, a numerical model of the double-layer polygonal bismuth strontium calcium copper oxide (BSCCO) conductor is developed. Current density and magnetic field intensity distribution in the inner and outer layers are also investigated. The numerical results of the AC loss for different layer current distributions are identical with the experimental ones. Accordingly, the reliability of the numerical model is verified. By using this model, the influence of distance between the inner and outer layers, gap between two neighboring wires, and layer current distribution on AC losses of different layers is evaluated. The results show that increasing distance between layers and narrowing gap between wires are effective to reduce AC loss, while the unbalance of layer current distribution increases the AC loss of the double-layer conductor.     

Investigation on AC loss of a high temperature superconducting tri-axial cable depending on twist pitches

High temperature superconductor (HTS) cables have been intensively studied because they are more compact compared with conventional copper cables. Since it is strongly expected that the HTS cables replace conventional power lines, some HTS cables are designed, manufactured, installed in power grids and tested to demonstrate full time operation. Recently, a tri-axial cable composed of three concentric phases has been developed, because of its reduced amount of HTS tapes, small leakage field and low heat loss when compared with single phase and co-axial HTS cables. The layers inside the tri-axial cable are subject to azimuthal fields applied from inner layers and axial fields applied from outer layers with different phase from their transport currents. These out-of-phase magnetic fields should be calculated under the condition of the three phase-balanced distribution of the tri-axial cable, and thereby AC losses should be evaluated. In this paper, the AC loss in the tri-axial HTS cable consisting of one layer per phase is theoretically treated for simplicity. The AC losses in the cable are calculated as functions of the twist pitches of HTS tapes. It is found that the AC losses rapidly decrease with increasing twist pitch.     

Mutual Inductance and Force Calculations Between Coaxial Bitter Coils and Superconducting Coils with Rectangular Cross Section

Mutual inductance and force calculations between coaxial Bitter coils and superconducting coils with rectangular cross section in a hybrid magnet system using derived semi-analytical expressions based on two integrations were performed. The mutual inductance and force calculations are based on the assumption of the uniform current density distribution in superconducting coils. The current density distribution of a Bitter coil in radial direction, however, is inversely proportional to the radius of the Bitter coil. The influence of the current density redistribution caused by a cooling hole and an inhomogeneous temperature distribution of Bitter coil of a water-cooled magnet was not considered. The obtained expressions can be implemented by Simpson’s integration with FORTRAN programming. We confirm the validity of mutual inductance calculation by comparing it with a filament method, and give the accuracy of two methods. The mutual inductance values computed by two methods are in excellent agreement. The derived semi-analytical expressions of mutual inductance allow a low computational time compared with filament method to a specific accuracy. The force is derived by multiplying the currents of the two coils by their mutual inductance gradient.     

Computation of the magnetic field in massive conductor systems

Some analytical expressions are reported for the calculation of the magnetic induction and the vector potential in iron-free media due to slab-shaped elements in which a current flows uniformly or linearly distributed in one direction. It is seen that no matter what current distribution functions and conductor geometries are considered, they can be approximated by a series of slabs in which the current distributions are represented by a sum of linear and constant distributions. The algorithms can also be used to calculate the magnetic field in more complicated geometries and to evaluate the self- and mutual-inductance coefficients in systems with massive conductors     

AC Loss of a Multi-Layer per Phase Tri-Axial HTS Cable with?Balanced Current Distribution

Recently, high-temperature superconductor (HTS) cables have been widely studied because of their compactness and high power capacity compared to conventional copper cables. In HTS cables, AC loss is an important issue since large losses reduce the efficiency of the power line. Among HTS cables, tri-axial cable is under intensive investigation recently, since it has a smaller amount of HTS tapes, small leakage fields and small heat loss in leak when compared with the three single-phase cables. For realizing high current capacity, more than one layer is required for each phase; therefore AC loss of the multi-layer tri-axial HTS cable should be carefully examined. In the tri-axial cable, different phase currents produce the out-of-phase magnetic fields on the other phase layers. In case of multi-layer arrangement, net magnetic fields on layer surfaces may exceed the penetration field of the HTS tape. Therefore in this paper, we analyze the AC loss of a tri-axial HTS cable which is composed of two layers per phase. Here, we treat the tri-axial cable which consists of two different longitudinal segments and thus satisfies balanced phase and homogeneous current distribution condition by controlling twist pitch and length of separate segments.     

An improved method for magnetic identification and localization of cracks in conductors

A SQUID magnetometer can be used to measure the magnetic field produced by flaws in a two-dimensional, conducting plate carrying a current. Identification of the flaw-induced magnetic field is difficult because of the large magnetic field associated with the edges of the plate and the current in the leads that connect the plate to the power supply. We have developed a technique by which the wire and edge fields can be cancelled prior to mapping the magnetic field. In this technique, a similar unflawed conducting sheet is placed adjacent to the flawed plate, with a connection between the sheet and the plate at one edge, and with the opposite edges of the sheet and of the plate connected to the two conductors of a coaxial cable. Thus, an applied current will flow along one conductor of the cable, across the cancelling sheet, cross into the flawed plate, return along the plate, and then return to the power supply along the other conductor of the coaxial cable. As a result of this geometry, there is no magnetic field from the lead-in wires because they are coaxial, and the magnetic field due to the edges of the plate is cancelled by the opposing magnetic field of the edges in the adjacent sheet. The extent of cancellation is determined primarily by the separation between the plate and the cancelling sheet, by the thickness of the plate, and by macroscopic inhomogeneities in their electrical conductivities.     

On the interaction of magnetic fields and magnetic shields

Some general mathematical expressions have been derived for the induced self and mutual inductances of the wires inside a cylindrical ferromagnetic shell. An application of these results is given. It is shown that the effect of the ferromagnetic shell on the inductances of the multiconducting wires is strongly geometry dependent; the closer the wires to the magnetic shell, the stronger the effect. For wires in the center of the cylinder, no significant result will be caused by the presence of the magnetic shell.