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.     

Noncoaxial Inductance Calculations Without the Vector Potential for Axisymmetric Coils and Planar Coils

This paper presents an exact method for calculating the mutual inductance between a general axisymmetric coil and a second planar coil consisting of either a disk coil or a planar loop of essentially arbitrary shape. The approach is based directly on the magnetic field rather than the vector potential . The paper gives detailed results for two circular loops, a circular loop and an elliptic loop, and a circular loop and an annular disk coil. The method can be extended to cover the cases where all these loops and coils are extruded in the axial direction to give the corresponding solenoids. The method is also applicable to calculations for nuclear radiation detectors.     

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.     

Inductance Calculations for Noncoaxial Coils Using Bessel Functions

A relatively simple and general method for calculating the mutual inductance and self-inductance of both coaxial and noncoaxial cylindrical coils is given. For combinations of cylindrical coils, thin solenoids, pancake coils, and simple circular loops, the mutual inductance can be reduced to a one-dimensional integral of closed form expressions involving Bessel and related functions. Coaxial and noncoaxial cases differ only by the presence of an extra Bessel factor J ₀(sp) in the noncoaxial integral, where p is the perpendicular distance separating the coil axes and s is the variable of integration. The method is related to a recently given noncoaxial generalization of Ruby's formula for a nuclear radiation source and detector system, the analogy being close but not exact. In many cases, the Bessel function integral for the inductance can be easily evaluated directly using Maple or Mathematica. In other cases, it is better to transform the integral to a more numerically friendly form. A general analytical solution is presented for the inductance of two circular loops which lie in the same plane     

Self-field degradation effect in adiabatic conditions

In multifilamentary superconducting composites mutual inductance between the filament give rise to self-field degradation effects. In this paper radial current density redistribution is first examined taking into account the spiral field effect. Then the limits of stability in adiabatic conditions imposed on these composites by self-field degradation are examined and the results of experiments carried out on short samples and non-inductive coils under adiabatic conditions discussed.     

Numerical Simulation of AC Loss in 2G High-Temperature Superconducting Coils with 2D-Axisymmetric Finite Element Model by Magnetic Field Formulation Module

In this paper, numerical simulation of the second generation (2G) high-temperature superconducting coils has been developed in the module of magnetic field formulation with 2D-axisymmetric model. The brevity of expressions and the consistency between the 2D model and 2D-axisymmetric model of this approach make it easier than the method of partial differential equations (PDEs) and magnetic fields module to simulate the high-temperature superconductors. The accuracy of this technique was certified through the comparison with the results of infinite long tape solved by the analytical equation and PDE method. Then, the simulation about a high-temperature superconducting coil with 40 turns was conducted, in which the anisotropic characteristics of 2G tapes expressed with the fitting formulation of J(B, ??) was considered. The distribution of current density and magnetic field at different time steps, and the voltage variation for each turn in the coil were studied. It could be seen that the electromagnetic quantities at the inner turns of a coil must be especially noticed. And finally, the AC loss of our model was calculated.     

Sweeping a persisting superconducting magnet with a transformer

A method for sweeping a persisting superconducting magnet is described. The field sweep is achieved by including in the superconducting loop of the magnet a coil which acts as the secondary coil of a transformer. Variation of the current in the primary coil of the transformer, controlled from outside the cryostat, causes the field-sweeping action through flux-linking with the superconducting loop. Compared by the ratio of the magnet's inductance to the transformer's inductance. The advantages of using an all-metal vacuum-tight superconducting feedthrough are discussed.     

Hybrid finite difference/finite element solution method development for non-linear superconducting magnet and electrical circuit breakdown transient analysis

The problem of non-linear superconducting magnet and electrical protection circuit system transients is formulated. To enable studying the effects of coil normalization transients, coil distortion (due to imbalanced magnetic forces), internal coil arcs and shorts, and other normal and off-normal circuit element responses, the following capabilities are included: temporal, voltage and current-dependent voltage sources, current sources, resistors, capacitors and inductors. The concept of self-mutual inductance, and the form of the associated inductance matrix, is discussed for internally shorted coils. This is a Kirchhoff's voltage loop law and Kirchhoff's current node law formulation. The non-linear integrodifferential equation set is solved via a unique hybrid finite difference/integral finite element technique.     

A comparison of single-layer coaxial coil mutual inductance calculations using finite-element and tabulated methods

Quick and accurate methods to calculate the mutual inductance of coaxial single layer coils remains important to this day in a large variety of engineering and physical disciplines. While modern finite-element electromagnetic field codes can do this accurately, the engineer often requires only a first- or second-order estimate before proceeding to the numerical analysis stage. Grover's tabular data, developed in the first half of the 20th century, remains the standard for manually calculating mutual inductance for a wide variety of coil and wire forms. This investigation reports the accuracy of mutual inductance calculations for single-layer coaxial coils based on Grover's tables when compared to estimates obtained with a finite-element electromagnetic field code (FEEFC). Since it is impractical to construct and characterize the numerous coils needed for this type of investigation, the FEEFC results are treated as actual inductance measurements. Grover reported his tabular data to be accurate within five significant digits excluding the cases when the coils are loosely coupled and when the coils are short. This investigation found Grover's tabular method to be inaccurate for loosely coupled and short coils, but also found that significant error for closely coupled coils as well. The maximum error between Grover's tabular method and the FEEFC results is 9.8%. Knowing the error associated with Grover's method and the coil geometry for which the error occurs is an important aid for the engineer and scientist.     

Superconducting magnet for K-500 cyclotron at VECC, Kolkata

K-500 superconducting cyclotron is in the advanced stage of commissioning at VECC, Kolkata. Superconducting magnet is one of the major and critical component of the cyclotron. It has been successfully fabricated, installed, cooled down to 4.2 K by interfacing with LHe plant and energized to its rated current on 30th April, 2005 producing magnetic field of 4.8 T at median plane of cyclotron. The superconducting magnet (stored energy of 22MJ) consists of two coils (α and β), which were wound on a sophisticated coil winding machine set-up at VECC. The superconducting cable used for winding the coils is multi filamentary composite superconducting wire (1.29 mm diameter) having 500 filaments of 40 μm diameter Nb-Ti in copper matrix which is embedded in OFHC grade copper channel (2.794 mm × 4.978 mm) for cryogenic stability. The basic structure of coil consists of layer type helical winding on a SS bobbin of 1475 mm ID × 1930 mm OD × 1170 mm height. The bobbin was afterwards closed by SS sheet to form the LHe chamber. The total weight of the coil with bobbin was about 6 tonne and the total length of the superconducting cable wound was about 35 km. Winding was done at very high tension (2000 PSI) and close tolerance to restrict the movement of conductor and coil during energization. After coil winding, all four coils (two each on upper and lower half of median plane of cyclotron) were banded by aluminium strip (2.7 mm × 5 mm) at higher tension (20,000 PSI) to give more compressive force after cool down to 4.2 K for restricting the movement of coil while energizing and thereby eliminating the chances of quench during ramping of current.After completion of coil winding by October, 2003, cryostat assembly was taken up in house. The assembly of cryostat (13 tonne) with support links (9 Nos.) refrigeration port, instrumentation port, helium vapour cooled current loads, etc. was completed by June, 2004. Meanwhile assembly of magnet frame was taken up and the cryostat was positioned in the magnet frame with proper alignment by August, 2004. After installation of cryostat on magnet, the cryostat was connected to the helium refrigerator/liquefier, having refrigeration capacity of 200 W and 100 l/h in liquefier mode with LN2 pre-cooling. The cryogenic delivery system supplying the liquid helium and liquid nitrogen to the superconducting magnet was successfully commissioned in November, 2004. The cool down of the cryostat to 10 K took around 8 days following which the LHe was filled in the cryostat (300 l) on 15th January, 2005. Subsequently the superconducting coils (α and β) were energized by two DC current regulated power supplies (20 V, 1000 A, 10 ppm stability) with slow and fast dump resistors connected externally across the superconducting coils for protection of coils at the time of power failure and quench.The paper describes the intricacies involved in coil winding, winding set-up, assembly of cryostat, cooling down the superconducting coils, filling by LHe and energization to rated current. The paper also highlights the operating experience of superconducting magnet and related test results.     

New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air

In this paper, we present analytic-numerical expressions for the calculation of the mutual inductance of two axisymmetric circular coils with rectangular cross section in air. This original and new method may seem complicated but it is explicit, accurate, and fast, even though all expressions are obtained by the complete elliptic integrals of the first and second kind, Heuman's lambda function, and three terms that must be solved numerically. We confirm the validity of this approach by comparing it with other approaches (filament method and previously published data). We also compare the accuracy and the computational cost of this approach and that of the filament method. All results obtained by the various approaches are in excellent agreement.     

Magnetic Force Calculation Between Thin Coaxial Circular Coils in Air

We present new and fast procedures for calculating magnetic forces between thin coaxial circular coaxial coils in air. The results are expressed in semianalytical form in terms of the complete elliptical integrals of the first and second kind, Heuman's Lambda function, and a term that must be solved numerically. These expressions are accurate and simple to use for several practical applications. We also describe a comparative method based on the filament technique. We discuss the computational cost and the accuracy of two methods and compare them with already published data. Results obtained by our two approaches are in excellent agreement with each other. They can be used in industrial electromagnetic applications such as electrodynamic levitation systems, linear induction launchers, linear actuators, and coil guns.     

Test Coils Made of Tl-1223 Tapes

Silver sheathed Tl-1223 tapes were prepared by a powder-in-tube process. The critical current density of short samples was 18 kA/cm² at 77 K. Longer tapes up to 1.2 m, prepared by sequential pressing, had a critical current density of 12 kA/cm². From these tapes we have wound two coils. A solenoid coil with 5 windings was made of 8 tapes with a total length of 4.5 m. At 77 K the critical current of the coil was 23 A in the self generated magnetic field (18 Gauss at the centre of the coil). Using an iron yoke the critical current remained at 22 A while the generated magnetic field increased to 120 Gauss. A pancake coil with 15 windings, made of 5 tapes with a total length of 5 m, generated a magnetic field of 149 Gauss at the critical current of 12 A. From measurements of the critical current density of our tapes in applied magnetic fields, we conclude that coils made of Tl-1223 tapes can be used to generate higher magnetic fields at 77 K.     

Cluster test facility for superconducting tokamak toroidal magnet system development

As the step of development of superconducting tokamak toroidal magnet system, a Cluster Test system is being constructed. The test facility has two coils, called cluster test coil (CTC), in a sector position, which provides back-ground field to a test module coil (TMC). The oval-shaped TMC has a mean width of 1.5 m and a mean height of 2.0 m. And it is designed to operate at a peak field of 8 T and an average current density of 3 KA/cm²on the winding space when CTC, operated with rated current, has 7 T as a peak field with current density of 3 KA/cm². Experience on the Cluster Test will provide fabrication techniques and verifications of computer codes for future toroidal coil design principles.     

Proposal of DC shield reactor type superconducting fault current limiter

Saturated DC reactor type superconducting fault current limiter (SFCL) had been proposed two years ago. It was classified to rectifier type SFCL. The changing inductance value with the operating mode has superior characteristics to reduce voltage sag during step increase of the load current. But it has the disadvantage of its weight. In this paper, rectifier type SFCL with shielded reactor has been proposed. The reactor which has superconducting ring or tube inside its winding is substituted to the DC link of the rectifier. The configuration looks like an air core transformer with secondary short winding. When the current through the bulk shield-ring reaches to a certain level, the flux penetrates to the shield body and finite impedance appears in the primary winding. In other words, when the surface flux density exceeds its critical flux density, the flux penetrates into the bulk superconductor, and increases equivalent inductance. The equivalent transient resistance of the shield was represented as a function of exponential of the time. Using this equivalent transient resistance, the transient impedance was expressed. The transient wave analysis using EMTDC (electro-magnetic transients in DC systems) has been described. Simulated waveforms are shown considering the source inductance, the leakage inductance, the coupling coefficient and the forward voltage drop of the semiconductor. And voltage sag was also investigated with 50% step load increase.Preliminary design was also performed. The coil size and number of turns are designed to obtain adequate inductance for the current limitation, and the central magnetic field of the coils are calculated. There is optimal aspect ratio to minimize the magnetic field with restriction in outer diameter of the coil.     

Proposal of rectifier type superconducting fault current limiter with non-inductive reactor (SFCL)

A rectifier type superconducting fault current limiter (SFCL) with non-inductive reactor has been proposed. The concept behind this SFCL is the appearance of high impedance during non-superconducting state of the coil. In a hybrid bridge circuit, two superconducting coils connected in anti-parallel: a trigger coil and a limiting coil. Both the coils are magnetically coupled with each other and have same number of turns. There is almost zero flux inside the core and therefore the total inductance is small during normal operation. At fault time when the trigger coil current reaches to a certain level, the trigger coil changes from superconducting state to normal state. This super-to-normal transition of the trigger coil changes the current ratio of the coils and therefore the flux inside the reactor is no longer zero. So, the equivalent impedance of both the coils increased thus limits the fault current. We have carried out computer simulation using EMTDC and observed the results. A preliminary experiment has already been performed using copper wired reactor with simulated super-to-normal transition resistance and magnetic switches. Both the simulation and preliminary experiment shows good results. The advantage of using hybrid bridge circuit is that the SFCL can also be used as circuit breaker. Two separate bridge circuit can be used for both trigger coil and the limiter coil. In such a case, the trigger coil can be shutdown immediately after the fault to reduce heat and thus reduce the recovery time. Again, at the end of fault when the SFCL needs to re-enter to the grid, turning off the trigger circuit in the two-bridge configuration the inrush current can be reduced. This is because the current only flows through the limiting coil. Another advantage of this type of SFCL is that no voltage sag will appear during load increasing time as long as the load current stays below the trigger current level.     

The cryogenic system for ITER CC superconducting conductor test facility

This paper describes the cryogenic system of the International Thermonuclear Experimental Reactor (ITER) Correction Coils (CC) test facility, which consists of a 500 W/4.5 K helium refrigerator, a 50 kA superconducting transformer cryostat (STC) and a background field magnet cryostat (BFMC). The 500 W/4.5 K helium refrigerator synchronously produces both the liquid helium (LHe) and supercritical helium (SHe). The background field magnet and the primary coil of the superconducting transformer (PCST) are cooled down by immersing into 4.2 K LHe. The secondary Cable-In-Conduit Conductor (CICC) coil of the superconducting transformer (SCST), superconducting joints and the testing sample of ITER CC are cooled down by forced-flow supercritical helium. During the commissioning experiment, all the superconducting coils were successfully translated into superconducting state. The background field magnet was fully cooled by immersing it into 4.2 K LHe and generated a maximal background magnetic field of 6.96 T; the temperature of transformer coils and current leads was reduced to 4.3 K; the inlet temperature of SHe loop was 5.6 K, which can meet the cooling requirements of CIC-Conductor and joint boxes. It is noted that a novel heat cut-off device for High Temperature Superconducting (HTS) binary current leads was introduced to reduce the heat losses of transformer cryostat.     

Electromagnetic Forces of High-T c Superconducting Coated Conductor Coils Subjected to Sinusoidal Traveling Magnetic Fields

In order to optimize the structure of high temperature superconducting coils for linear motor applications, three separate coils with different shapes made of Re-BCO coated conductor were studied: circular shaped single pancake, circular shaped double-pancake and racetrack shaped single-pancake. The thrust and vertical forces of the three coils above a conventional flat linear three-phases winding were investigated experimentally. With the aid of the experimentally obtained values, it was found that single-pancake coil in the shape of a racetrack was the best selection for a flat single-sided linear motor system. Studies were also made on the frequency characteristics of the vertical force of the racetrack shaped single-pancake coil.     

Study on cross section of high temperature superconducting coil

It is in particular of importance for HTS coils to secure a larger central magnetic field and/or a large stored energy with shorter length of HTS tapes. The critical current of an HTS tape depends on both the flux density and the flux angle against tapes. From this point, the performance improvement of HTS coils is taken into account with an analytical model. The minimum volume coil derived from the Fabry Factor constant curve is taken concerning the original coil shape, which is often employed in low temperature superconducting coils. The coil critical current was analyzed in consideration of the anisotropic properties of the tape.The electric field of HTS tapes in the coil was calculated at the coil critical current and the high electric field portion were cut out. The optimal coil cross section is obtained by iterating this calculation process. As a result, the critical current and the stored energy density of the coil were improved. The stored energy density increased about 17% and the central magnetic field was almost kept constant regardless of 19% reduction of HTS tapes, as compared with the original coil with the rectangular cross section.     

Numerical treatment of the quenching process in superconducting magnet systems

A computer program for testing coil and protection design of a composed magnet system is presented. Small high field magnets consisting of two uniaxial cylindrical coils of different superconducting materials (eg NbTi and V₃Ga or Nb₃Sn) are considered, each coil may be subdivided into several sections which are protected by parallel resistors. Quench propagation due to thermal conduction and also due to rapid current increase (important in inductively coupled systems) is taken into account by means of solving a one-dimensional thermodiffusion equation. Field and temperature dependence of the critical current for every layer of the coils, is taken into account. The program calculates the time dependence of currents, external and internal voltages, resistances of the sections and of the radial temperature distribution. Calculations are compared with experimental investigations of three different systems, the results agree with experimental values concerning current decay and propagation velocity.