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.     

Unsteady MHD flow due to non-coaxial rotations of a porous disk and a fluid at infinity

Summary An exact solution of the unsteady three-dimensional Navier-Stokes equations is derived for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field. An analytical solution of the problem is established by the method of Laplace transform, and the velocity field is presented in terms of the tabulated functions. It is found that the boundary layer thickness in the cases of suction/blowing decreases with the increase in the magnetic parameter.     

Coaxial stability of nano-bearings constructed by double-walled carbon nanotubes

How to effectively control the coaxial stability of nano-bearings has an important influence on improving the quality of nano-bearings. Some relevant problems are studied in this paper. Firstly, we investigate basic non-coaxial modes in double-wall carbon nanotubes (DWCNTs). On the basis of analysis for these non-coaxial modes, a planar continuum model is established according to the principle of homogenization. By means of this model, a?dynamic parameter λ(*) characterizing the coaxial stability of nano-bearings is determined. λ(*)?is the explicit function of the angular velocity and interlayer spacing of DWCNTs. In terms of λ(*), a criterion used to judge the coaxial stability of nano-bearings is given. Through discussing the influence of the angular velocity and interlayer spacing on the dynamic parameter λ(*), some important conclusions are drawn.     

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     

Biharmonic analysis of rectilinear plates by the boundary element method

An improved boundary element formulation (BEM) for two-dimensional non-homogeneous biharmonic analysis of rectilinear plates is presented. A boundary element formulation is developed from a coupled set of Poisson-type boundary integral equations derived from the governing non-homogeneous biharmonic equation. Emphasis is given to the development of exact expressions for the piecewise rectilinear boundary integration of the fundamental solution and its derivatives over several types of isoparametric elements. Incorporation of the explicit form of the integrations into the boundary element formulation improves the computational accuracy of the solution by substantially eliminating the error introduced by numerical quadrature, particularly those errors encountered near singularities. In addition, the single iterative nature of the exact calculations reduces the time necessary to compile the boundary system matrices and also provides a more rapid evaluation of internal point values than do formulations using regular numerical quadrature techniques. The evaluation of the domain integrations associated with biharmonic forms of the non-homogeneous terms of the governing equation are transformed to an equivalent set of boundary integrals. Transformations of this type are introduced to avoid the difficulties of domain integration. The resulting set of boundary integrals describing the domain contribution is generally evaluated numerically; however, some exact expressions for several commonly encountered non-homogeneous terms are used. Several numerical solutions of the deflection of rectilinear plates using the boundary element method (BEM) are presented and compared to existing numerical or exact solutions.     

Exact integration of surface and volume potentials

A method for exact analytical integration of potentials from sources distributed on planar and volume elements is presented. The method is based on reduction of the surface integrals to a function similar to an incomplete elliptic integral, giving the integrals in closed form as functions of geometric properties of the surface or volume element. Explicit formulae and recursions are given for the integrals, allowing the evaluation of the potential for arbitrary polynomial sources. Volume integrals are derived from the surface integrals using a simple coordinate transformation which gives the volume integral with little more effort than that required for the surface calculation.     

Current distribution calculation for a superconducting multi-layered power cable

The calculation method for the current distributions within a coaxial multi-layered power cable consisting of helically wound long superconducting tape conductors have been studied. The analytical expressions for the self- and mutual inductances of the coaxial helical long thin arch-shaped tapes are analytically and numerically compared with straight arch-shaped and flat tapes. It is confirmed that these expressions are consistent with each other. The current distributions within a coaxial 3-layered cable are calculated, using the self- and mutual inductances obtained due to the thin approximation. The effect of the non-uniform gaps between tapes in a coaxial multi-layered power cable is studied.     

多壁碳纳米管横向扰动波传播的研究

将N层碳纳米管模拟成N层连续弹性壳模型，研究横向扰动波在单个多壁碳纳米管中轴向传播问题。研究表明，当扰动频率低于多壁碳纳米管的临界频率时，振动模态是同轴的；如果扰动频率至少高于多壁碳纳米管的一个临界频率时，振动模态是非同轴的，并以不同的波速传播，或低或高于单壁连续弹性壳模型的波速。因此，多壁碳纳米管的THz高频波存在着复杂现象，振动基本是不同轴的。特别是当多壁碳纳米管THz高频波以不同的速度传播时，传播速度不仅依赖于频率，而且依赖于非同轴的振动模态。     

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.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999     

Boundary integral equations analysis of induction devices with rotational symmetry

The application of boundary integral equations (BIE's) for the analysis of linear induction devices with rotational symmetry is considered. One-dimensional Fredholm integral equations are derived for the tangential field components at the boundary of a conducting medium with constant scalar conductivity and permeability excited by a time-harmonic azimuthal current source. The important special case of a short right circular conducting cylinder (magnetic or nonmagnetic) coaxial with one or more short coils is treated in detail. The explicit form of the kernels and the numerical solution technique are presented. Numerical results are presented for typical induction heating applications where the load length as well as the coil length are finite. Results are also presented for the magnetostatic problem of finding the demagnetization factors for short magnetic rods. In each case the results are compared with published results and the accuracy and efficiency of the proposed approach are demonstrated.     

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.     

Elimination of flux-transformer crosstalk in multichannel SQUID magnetometers

Multichannel SQUID magnetometers are being developed for signal-field mapping in biomagnetic experiments. A problem that becomes more serious as the number of channels is increased is the crosstalk caused by the mutual inductances between the individual sensing coils. A simple and effective method for eliminating this crosstalk is presented in this Paper. The method is based on a rearrangement of the feedback loops which causes the flux-transformer circuits to become currentless. The feasibility of the method is verified experimentally.     

Exact field calculations for asymmetrical finite-pole-tip ring heads

An exact solution is given for the fields of an infinitely permeable asymmetrical finite-pole-tip ring head. The solution is obtained by conformal mapping and is expressed in terms of elliptic integrals and elementary functions. Using the method of undetermined coefficients, it is shown how to decompose the Schwarz-Christoffel integral into elementary functions and elliptic integrals. For each given pole geometry, the constants appearing in the Schwarz-Christoffel formula are calculated once. Then the fields at various points in the poles domain are calculated using Newton's method and the formulas for the decomposition of the Schwarz-Christoffel integral.<>     

CMOS gyrator-C active transformers

A systematic approach for synthesising gyrator-C active transformers using MOS transistors is presented. The topology of gyrator-C active inductors and their characteristics are briefly reviewed first. This is followed by the development of ideal gyrator-C active transformers, where only the capacitor loads of the transconductors synthesising active transformers are considered. The self and mutual inductances of both the primary and secondary windings of active transformers are investigated in detailed. Non-ideal gyrator-C active transformers are developed with the consideration of both the resistance and capacitance loads of transconductors. The intrinsic relation between the self and mutual inductances is derived. The configuration of gyrator-C active transformers with multiple primary and secondary windings is also developed. The proposed active transformers offer large and tunable self and mutual inductances with virtually no silicon area requirement. Several practical implementations of the proposed active transformers have been realised in TSMC-0.18 mum 1.8 V CMOS technology and analysed using SpectreRF with BSIM3v3 device models. Simulation results on voltage transfer characteristics, self and mutual inductances, quality factors, stability, the effect of process variations, and noise are presented. The application of the proposed active transformers is exemplified using a 1.6 GHz active transformer quadrature oscillator.     

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.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Effects of disoriented grains on the elastic constants of directionally solidified superalloys

Two models, namely, coaxial and non-coaxial, are proposed to estimate the elastic constants of directionally solidified superalloys, which behave like transversely isotropic materials and require five independent elastic constants. Coaxial model considers each grain as an individual and obtains the averaged values. Because of the longitudinal grain structure, three independent constants are carried over from original cubic single crystal and the other two are obtainable through the averaging process. For each disoriented direction in the non-coaxial model, a lumped grain, which behaves like a transverse isotropy is proposed. By assuming the disoriented angle follows Weibull distribution, non-coaxial model obtains the expectations of compliances from probability consideration. Disoriented effect could be simulated through the parameters of Weibull distribution. Experimental off-axis Young's modulus data are compared with numerical predictions by both models. Excellent agreement is observed between coaxial model and 90° off-axis experimental data. However, the coaxial model over predicts the 45° off-axis Young's modulus, because anisotropic coupling effect is very strong in the real off-axis specimens. As non-coaxial model considers the disoriented effect, excellent agreement is observed between non-coaxial model and 45° off-axis experiment data. Disoriented grain consideration reduces the anisotropic coupling effect and predicts better to the real off-axis specimens.     

Complex potential function in elasticity theory: shear and torsion solution through line integrals

Aim of this paper is to introduce a basis formulation framed into complex analysis valid to solve shear and torsion problems. Solution, in terms of a complex function related to the complete tangential stress field, may be evaluated performing line integrals only. This basis formulation framed into elasticity problems may be a useful support for a boundary method to verify the accuracy of an approximation of function solution. The numerical applications stress the latter point and show the validity of these formulas since exact solutions may be reached for sections where the exact solution is known.     

Trigonometric Integrals for the magnetic field of the coil of rectangular cross section

Formulas are derived giving the vector potential and the magnetic field components of a general coil of rectangular cross section and constant winding density. The solution is given in a cylindrical coordinate system in terms of trigonometric integrals. The formulas presented have been cross-checked and validated against alternative expressions giving the various field components as integrals of expressions containing Bessel and Struve functions. The trigonometric integrals for the fields can be evaluated easily to several hundred significant figures using mathematical packages such as Maple or Mathematica. Alternatively, they can be evaluated with a small FORTRAN program. Sample results and field line plots obtained with the method are given, and the field of a coil of rectangular cross section is examined in some detail. A comparison with the results of a finite-element method is also given.