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.     

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 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.     

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.     

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.     

Magnetoelastic modeling of circular cylindrical shells immersed in a magnetic field. Part I: Magnetoelastic loads considering finite dimensional effects

Determination of magnetoelastic loads acting on a perfectly electro-conductive circular cylindrical shell immersed in a uniform applied magnetic field is addressed. The finite dimensional effects related to the finite length and finite thickness of the shell are taken into consideration. Fourier integral method is used to derive the singular integral equations governing the distributed magnetoelastic loads. As special cases, determination of magnetoelastic loads via discarding the thickness effect are obtained from the general formulation, and the magnetoelastic loads of infinitely long shells are derived. Magnetoelastic loads on plate strips or infinite plates are also reduced from the general formulation. To the best of the authors’ knowledge, this represents the first work devoted to the analytical determination of magnetoelastic loads on circular cylindrical shells considering the finite length and thickness effects.     

Thin film transformer and its analysis by integral equation method

We propose a thin film transformer for small electronic devices, and apply the integral equation method to analyze this transformer. Both the primary and secondary coils of the film transformer are arranged coaxially on the layer and multiply laminated. The operation principal of the transformer is based on the skin effect and the mutual effect between the coils at high frequency. Because of the coaxially arranged coils, the magnetic field of the transformer can be modeled with an axisymmetric assumption. Using the model, we evaluate the electromagnetic field and calculate the lumped circuit parameters, i.e., inductance and resistance, which are compared with experimental values     

The Threshold of Phase Locking in the System of Two Multi-Junction Superconducting Loops

Coherent behavior (phase locking) in the system of two superconducting loops containing arrays of Josephson junctions and influencing each other by means of mutual inductance was numerically investigated. It is shown that under certain conditions the mutual interaction between loops can result in total phase-locking of the system. The threshold of the locking state depends on the strength of the interaction (mutual inductance). It is found that the basic behavior of the system in the phase-locked state can be explained by means of simple model of single multi-junction loop with an effective inductance which depends on mutual inductance. It is shown that synchronization in this system is possible and the mutual interaction increases the stability of phase locking in arrays.     

Embedded Ring-Type Inductors Modeling With Application to Induction Heating Systems

We derive integral expressions of the equivalent impedance of ring-type coils for induction heating systems. These expressions are obtained by integrating the analytical solution of the field generated by a circular filamentary current placed between two linear and homogeneous semi-infinite media. Constant current density is assumed for the coils, since they are wound with an appropriate Litz wire. The model also gives the magnetic field in the cross section of the windings, which is necessary to calculate the proximity losses in the cables. The solution is given in terms of Bessel function integrals, showing an improvement of the convergence with respect to the filamentary model solution. We constructed several prototypes having different turns and layers and tested them with different substrates. Our measurements confirm the validity of the analytical models.      

Computation of the External Magnetic Field, Near-Field or Far-Field, From a Circular Cylindrical Magnetic Source Using Toroidal Functions

A method is developed for computing the magnetic field from a circular or noncircular cylindrical magnetic source. A Fourier series expansion is introduced which yields an alternative to the more familiar spherical harmonic solution, Elliptic integral solution, or Bessel function solution. This alternate formulation coupled with a method called charge simulation allows one to compute the external magnetic field from an arbitrary magnetic source in terms of a toroidal expansion which is valid on any finite hypothetical external observation cylinder. In other words, the magnetic scalar potential or the magnetic field intensity is computed on a exterior cylinder which encloses the magnetic source. Also, one can compute an equivalent multipole distribution of the real magnetic source valid for points close to the circular cylindrical boundary where the more familiar spherical multipole distribution is not valid. This method can also be used to accurately compute the far field where a finite-element formulation is known to be inaccurate     

Cylindrical phase change approximation with effective thermal diffusivity

No exact, general, solution exists for phase change in a cylindrical geometry. In fact, even approximate solutions are rare and limited in applicability. The use of the effective thermal diffusivity concept has allowed a closed form approximate solution to be generated for phase change around a circular cylinder in an infinite medium. The effective diffusivity method permits solutions to be found for phase change problems merely by solving the usually linear, zero latent heat problem analogous to the phase change problem. Phase change problems are often intractable with the usual mathematical methods. The cylindrical formulae given here are shown to be of acceptable accuracy, for most engineering purposes, over a wide range of parameters. No other simple, closed form, approximation is known for the cylindrical system. Although the accuracy of the effective diffusivity method has been demonstrated for the cylindrical geometry, application to other geometries must be verified.     

Generation of uniform magnetic fields with electrically alterablefield direction in a geometric plane

For the convenience of construction and operation of a well-defined uniform magnetic field source, a structure of two orthogonal sheet current loops (TOSCLs) is proposed. The distributions of the magnetic field uniformity inside the TOSCL are investigated. Self and mutual inductances of the TOSCL structure are studied. Physical factors influencing the current amplitudes are discussed. A main advantage of the TOSCL is its simplicity of construction, especially for dimensions exceeding 0.5 m on a side, when compared with circular coils of the same comparable size. The H-field distribution in the TOSCL is easier to determine than in any other uniform magnetic field generator. Physical factors such as self inductance and frequency do affect the amplitudes and phases of the feeding currents. It is suggested that the sizes of the two orthogonal sheet loops be nearly the same so that the current phases and H-field direction can be easily handled. When both of the two loops are fed by currents, the influence of frequency and self-inductance on the H-field direction can be negligible. However, no matter whether either or both loops are used, this influence on the H-field strength should be noted, especially when the frequency exceeds 300 kHz     

A Method for Calculating the Mutual Inductance of a Coil System Using a Model of an Axially Magnetized Cylinder

Technical Physics Letters - A method for calculating the mutual inductance of a system of tightly wound coils through the energy of axially magnetized cylinders is proposed. Formulas for...     

空心线圈电感的计算与实验分析

 介绍了体内微机电系统空心线圈电感的计算,主要对非同轴的线圈互感计算进行详细研究,分析了初次级线圈主要几何参数如轴向和径向距离、夹角、线圈半径等对互感的影响.计算结果表明：轴向距离和径向距离增大,互感均减小,但轴向距离对互感的影响要大得多;夹角越大,互感减小,夹角为90°时,互感几乎为0;当线圈半径变化时,互感有个最大值.同时给出一种三轴移动测试平台,可以快捷精确地调节线圈间的轴向径向距离和夹角,在此基础上建立了互感测试系统,测试结果与计算值是相符合的.     

Cylindrical poiseuille flow of a Fermi liquid

The integral equation for cylindrical Poiseuille flow of a general excitation gas is derived for diffuse scattering at the wall. For a degenerate Fermi gas or liquid the flow through a circular cylindrical tube is determined numerically by a variational method.     

铁心对线圈不同连接方式下电感的影响

以交流伏安法作为电感的测量手段,在保持线圈相对位置不变的前提下,对有无铁心的情况分别进行测量,对比研究了线圈不同连接方式下电感的变化规律,分析了铁心对电感的作用和影响。结果表明:采用不同的连接方式,总回路中的电感值不同,四个线圈反串的连接方式是最合适的;铁心对线圈的互感产生决定性影响,不同电流下的电感值不同。     

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.     

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.     

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.     

Inductance of magnetically shielded air-core coils of circular cross section

An expression for the inductance of shielded cylindrical air-core solenoids of arbitrary diameter to length ratio is developed. Three shielding configurations are investigated: 1) the use of a coaxial-concentric magnetic shield, 2) the coaxial magnetic shield with conducting end plates, 3) the coaxial magnetic shield with conducting end rings. Case 2) is solved in closed form by separation of variables, and from this result a method is given for predicting the length of the coaxial magnetic shield required to contain the field for the three cases.