Helical Currents Research Articles

Calculation of magnetic field of the helical coils.

Formulae of magnetic field calculation of finite size helical coils [Kakuyugo Kenkyu 57 (1988) 318] are extended to the case that the cross section of the coil varies along the guiding curve of the helical coil. The systemetic error in the numerical integration is discussed in the connection with the incorrect model of helical coil current; the curl of the magnetic field may remain around the coils because of the finite number of integration points.

Toroidal Helical Fields

Using the conventional toroidal coordinate system Laplace’s equation for the magnetic scalar potential due to toroidal helical currents is solved. The potential is written as a sum of an infinite series of functions. Each partial sum represents the potential within some accuracy. The effect of the winding law is analysed in the case of small curvature. Approximate magnetic surfaces formed by toroidal helical currents flowing around a standard tokamak chamber are determined. Stability of the plasma column in this system against displacements is discussed.

Shrinkage of tokamak current channel by external ergodization

Shrinkage of the plasma current channel resulting from a non-resonant helical field has been observed by using a multi-channel magnetic probe inserted into the plasma column in the TORIUT-4M tokamak. The helical field perturbation has been produced by an m = 3, n = 1 external helical coil. When we apply the (3,1) helical coil current of around one percent of the plasma current, the plasma column shrinks a few percent in radius, and the current density at the q = 2-surface decreases significantly. These phenomena confirm that the (2,1) magnetic islands are ergodized and removed.

RESISTIVE FLUID INSTABILITY AT THE NONMONOTONOUS CURRENT PROFILE

It can be shown that the dynamics of magnetic perturbation at the nonmonoto-nous profile of a current in tokamak is qualitatively different from that of normal profiles. Behaviour of the perturbation depends on the helical magnetic currents localized mainly between the two singular surfaces. In this paper the numerical method developed to study the non-linear evolution of tearing modes is described in detail.

Cross sections with polarized spin-1/2 particles in terms of helicity amplitudes

The derivation of cross sections for collisions with polarized particles of spin 1/2 can be simplified considerably if the scattering amplitude is calculated explicitly before the transition probability is obtained by a simple squaring. The states of the polarized particles are represented as superpositions of states with definite helicity. The coefficients of the superposition relate directly to the strength of the transversal and longitudinal polarization. The helicity amplitudes are products of helicity currents for which detailed formulas have been elaborated. Two-component spinors have been used. All the contractions of vector indices are done with a new set of general formulas. The resulting cross sections show terms with separate factors due to the polarization, the energy, and the directions of the particles. Therefore, the high energy approximation can be achieved very conveniently. Applications of the described method have been performed to the scattering of electrons by electrons or positrons including the exchange of Z0 and Higgs particles.

50 Hz rotating superconducting magnet for screening studies

A superconducting rotor facility has been constructed to study the magnetic screening of superconducting AC machines. The superconducting dipolar magnet 128 mm diameter, 515 mm long, 40 Kg weight, is capable of providing a field of 1.5T at a radius of 100 mm with 525A energising current. Novel features include helical vapour-cooled current leads, anti-recirculation baffles in the vapour space, vapour dump valves and liquid nitrogen cooling.

Magnetic field due to helical currents on a torus

By means coordinate transformations on different forms of the Biot-Savasrt law, general expressions are derived for both the magnetic vector potential and the magnetic field in two systems of toroidal coordinates. For the cases of interest, it is shown that the integrations can be performed exactly in proper toroidal coordinates, whereas approximations are necessary in quasi-toroidal coordinates. Different forms of helical windings are discussed. The results are applied to the calculation of a stellarator magnetic field of arbitrary polarity l, produced by a distribution of helical current filaments on a torus. The components of the magnetic field are given in terms of infinite series. In the intermediate region, away from the external separatrices and the central region, simple analytic expressions for the magnetic surfaces are obtained.

Experiment showing a mechanical manifestation of the helicity of transport current in superconducting wires

The transport current density of the mixed state of type II wires will have the form J = z J/sub z/ + THETA J/sub THETA/ when the supercurrent exhibits a helical distribution due to the application of an external field H/sub z/. This is proved by the magnetic moment measurements of Walmsley and Timms, who observed the so-called paramagnetic component of moment M when J/sub THETA/ not equal to 0. A mechanical manifestation of the helical current was observed by combining a normal-zone propagation experiment with a capacitive technique for measuring mechanical torsion of the sample. Moreover, the torsion was observed even when H/sub z/ = 0, an effect that might be explained by the theory of Kondo and Kuroda on the helicity of transport currents in normal metals due to spiral dislocations.

大型真空容器(核融合装置ヘリオトロンE用)への電子ビーム溶接の適用

Heliotron E, which is a large-scaled device of Heliotron series, is in operation in Kyoto University. Target plasma parameters of this machine are as follows; plasma density 1×1020m-3, electron temperature 1 keV, ion temperature 0.8 keV and nτ 1018-1019 sec/m3. Plasma confining magnetic fields are produced by the helical coil current. To assure the accuracy of the helical coils, the high level of accuracy was required to the profile of the vacuum chamber. The vacuum chamber twists around the torus by 9.5 times and its poloidal cross section shows a race-track shape in any toroidal position. Major radius is 2200 mm and minor radius is 215.5 mm. The material is the high-strength non-magnetic steel which has low permeability (μ<1.4), high yield strength (≥40kgf/mm2) and special chemical composition (25Cr-12Ni-0.3N). The thickness of the wall is from 20 to 33 mm.To make the vacuum chamber, 190 pieces were joined together by EB welding after the hot press forming.A giant apparatus (6.5 m×6.5 m×3.5 m) was prepared and applied and a special re-focusing coil was used for the long beam welding.The profile accuracy of the vacuum chamber has been controlled as follows; major radius ±3 mm, minor radius ±4.5-1.5 mm, where minus means the inner side.X-ray and dye-penetrant quality of all welds was acceptable for ASME pressure vessel code and JIS. The laekage was less than 1.3×10-8 Torr 1/sec. The final pressure is 2×10-8 Torr.

Stress Analysis of Supporting Tube of Helical Winding in Stellarator Device

In stellarator devices, the strong magnetic forces act on the helical coils due to the interaction of the helical coil currents and the toroidal field. When the helical coils are supported by a cylindrical tube, the internal stresses in the supporting tube reach the critical value of the supporting material. The analytical results of the stress and the displacement are presented in the case of a linear stellarator with helical symmetry.

Non-circular cross-section stellarators

The magnetic surfaces and rotational transform of a stellarator with low poloidal-field wave number and elliptical minor cross-section have been computed numerically by utilizing a predictor-corrector code. The surfaces exhibit stagnation points the positions of which are related to the aspect ratio of the ellipse and the ℓ-number of the configuration. The rotational transform is less than in an equivalent stellarator with circular cross-section for the same toroidal and helical currents in the magnet coils.

Spontaneous Breakdown of Symmetry and the Gauge Invariance in a Relativistic Field Theory

The Nambu-Jona-Lasinio model in which the helicity current is coupled with a massless chiral gauge field, is discussed in the pair approximation. Our model is invariant under the constant as well as local gauge transformations. We have investigated the excitation spec­ trum of physical states in this model, under the requirement that the chiral gauge invariance is maintained at every stage. Symmetry breaking solutions are found in a noncovariant form. Contrary to previous assertions, massless fields are still present in our theory. Thus, the physical fields are a massive vector field, a massless vector field and a phase field. It is shown that the gauge transformations are carried by the phase field and the massless vector field, respectively.

The Stellarator as a Nonlinear Plasma Current Transformer

Oscillation of the helical winding current of a stellarator induces toroidal plasma currents proportional to the square of the winding current.

A Negative V'' System without Use of Toroidal Solenoid Coils

Simple helical currents of same polarity and short pitch produce an average minimum B system by superimposing a vertical magnetic field B v without use of toroidal coils. Analytically it is shown that the average minimum B is present in case of large aspect ratio and short pitch. Computations of toroidal systems, which have a moderate aspect ratio and l =3 helical windings, show that a ringless quadrupole configuration and the Taylor's field are both realized by adjusting B v . The rotational transform angle exceeds π and the mirror ratio is small.

Shear and minimum B̄ in magnetic configurations with combined helical fields

An asymmetric arrangement of the conductors of a helical stellarator winding and currents passing through them unequal in absolute value cause the formation of a spatial magnetic axis. In the vicinity of this axis the useful volume bounded by a certain surface exhibits the principal properties needed to stabilize a number of plasma instabilities – shear and minimum B̄. This paper is a detailed analytical study of asymmetric systems of this kind. It is demonstrated that the use of the second and third harmonics of a helical magnetic field in devices with a high aspect ratio makes it possible to obtain δV′/V′ between 0.08 and 0.15 for shear S* =0.05 and 0.13 and a useful volume of sufficient size. The configurations considered can be used to create closed magnetic traps for the purpose of confining a high-temperature plasma and studying the stabilizing effect on it of shear and minimum B̄, both together and separately.

Determination of stellarator equilibria using guiding- center equations

Stellarator expansion techniques, previously used to obtain an asymptotic toroidal equilibrium for the ideal scalar-pressure fluid equations, are applied to the guiding-center model. An explicit formulation for determining the asymptotic solution is given in terms of a small parameter. This parameter measures the magnitude of the magnetic fields associated with toroidal curvature, currents in the helical conductors, plasma pressures and currents, with respect to a zero-order uniform magnetic field.

The Time-Dependent Magnetic Fields of an Injected Helical Current

The electromagnetic field as a function of time in a conducting, infinitely long cylindrical tank into which a helical current is injected is considered. The longitudinal component of the magnetic field is treated in complete detail, and a plot of magnetic field vs time is given for the circularly symmetric case with parameters closely approximating the Astron configuration.

Equilibrium of a plasma pinch in a toroidal constant magnetic field and a helical high-frequency magnetic field

The authors consider the equilibrium of a toroidal plasma pinch in a constant magnetic field on which is superposed a helical high-frequency field produced by helical currents flowing on a toroidal surface with major radius R and minor radius rk. The helical currents are proportional to exp[i(ωt – nψ – k‖ζ)], where ω is the frequency imparted by the generator, ψ the polar angle in the meridional cross-section of the torus, and ζ is the length of arc along the major circumference of the torus.The authors demonstrate that it is possible to achieve equilibrium with high-frequency field pressures less than the kinetic pressure of the plasma.An expression is derived for the displacement of the pinch in the equilibrium state as a function of high-frequency field amplitude, plasma kinetic pressure, the multipolarity n, and such geometric parameters as the radius of the plasma pinch rp, the radius of the circuit rk and the radius of the torus R.In the helical high-frequency field, the equilibrium of the plasma pinch depends to a great extent on the strength of the constant magnetic field H0, the plasma concentration N, the frequency of the variable field f = 2π/ω, and the wavelength of the helical current λ‖ = 2π/k‖.The helical high-frequency electromagnetic field within the plasma is a superposition of two wave types. When H0 < Hcrit = the first type of wave transfers to the skin, and when H0 > Hcrit. it propagates. When k‖rp ≪ 1 and k‖rp ≪ H0/Hcrit. the field of the second type of wave differs only slightly from the vacuum field.In the region where the first type of wave transfers to the skin, the displacement Δ of the plasma pinch decreases with increasing H0 and becomes zero when H0 = Hres = : with further increases in the magnetic field strength Δ becomes negative. H0 = Hres is the condition for resonance excitation of the helical high-frequency field within the plasma. The most effective use of high-frequency power for equilibrium containment of the plasma pinch within the torus is to be expected near this resonance on the side where Δ > 0.The change in the sign of Δ is a consequence of a change in the direction of the force acting on the plasma. If there are two types of wave in the plasma, there appear – in addition to the forces associated with the pressure of each individual wave -forces produced by the interaction of the currents excited by one wave type with the field of the other wave type, and vice versa.For certain parameter values, the phase relations of the fields and currents may be such that the additional forces associated with the cross interaction may work in a direction opposite to that of the forces produced by the pressure of each individual wave.In the propagation region of the first wave, i.e. where H0 > Hcrit. positive and negative values of the plasma pinch displacement alternate (i.e. there are alternating regions of stable and unstable equilibrium respectively).

Helical Fields

This paper is devoted to a study of helical fields, that is fields which are invariant under screw motions of space which move a certain helix into itself. Simple, analytic, helically invariant solutions of the Laplace equation are given and combined to describe the electrostatic field of a charged helix, and of the magnetic field of a helical electric current. A flux function ψ is introduced for solenoidal helical vector fields, and differential equations resembling Cauchy-Riemann equations are derived for the potential function φ and the flux function ψ. Certain graphical flux plotting methods are outlined and illustrated, and network analogies are suggested for solving these fields.

A point source and a vortex filament in a helical current

A point source and a vortex filament in a helical current