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.     

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     

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.     

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.     

Self-inductance of air-core circular coils with rectangular cross section

Using analytic integration, accurate self-inductance expressions (requiring only single numerical integration) for air-core circular coils with rectangular cross sections (and two special cases, namely thin-wall solenoids and disk (pancake) coils) are derived from the basic mutual-energy formula of two coaxial circular current loops. Based upon these expressions, with integration singularities analytically treated, the self-inductance of any air-core coil can be precisely calculated by use of a microcomputer. Values so computed agreed, to one part in 10⁴, with those reported in the literature.     

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.     

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.     

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.     

Multiple-value inductance coils for working standards

The circuit and construction of new multiple-value toroidal inductance coils of 1 μH to 1 H for first- and second-class working inductance standards are described. Optimal relations between the dimensions of toroidal coils with different cross-section geometries providing the best Q factor for a given inductance and external coil diameter have been obtained. The results of experimental investigations are presented. Translated from Izmeritel'naya Tekhnika, No. 10, pp. 21–24, October, 1997.     

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。     

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.     

电磁感应起爆系统电流互感原理的研究

通过试验和理论分析，得出ＱＢ－１型起爆器起爆系统的实质是电流互感，为起爆器的改进和线路联接方法的改变提供了理论依据；要想获得次线线圈足够大的电流，必须保证初次级线圈匝数比Ｗ１／Ｗ２〉１。     

Two-dimensional normal zone propagation in BSCCO-2223 pancake coils

We present the results of an experimental and analytical study of two-dimensional normal zone propagation in pancake test coils, wound with silver-sheathed BSCCO-2223 tapes. Two test coils were studied in detail, one having three and the other eight layers. Each test coil was housed in an adiabatic environment whose temperature (20-70 K) was controlled and maintained by a two-stage G-M cryocooler and placed in a background field (0-6 T) generated by a Bitter magnet. With a test coil carrying a transport current (0-200 A), a local heat disturbance was applied by a heater attached to the outermost layer of the coil. The resulting electrical and thermal responses of the coil were recorded with voltage taps and thermometers attached to the coil. A normal zone propagation code was developed to accurately simulate the voltage and temperature responses of each coil for both quenching and recovering events. The code solves the nonlinear transient heat diffusion equation in two-dimensional cylindrical coordinates with a finite difference method. As an application of this code, a two-coil system, with each coil comprised of one double pancake wound with silver-sheathed BSCCO tape, was studied for its quench behaviour as one of the coils was driven normal locally. The simulation results indicate that the value of a shunt resistor connected across the terminals of each coil has a profound effect on the level of hot-spot temperature reached in the quench initiation spot.     

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.     

Evaluation of inductance for various distributions of windings onstraight ferromagnetic cores: an unusual approach

The detailed electromagnetic duality that relates small magnetic structures and small electric structures at ultralow and extremely low frequencies has recently been treated. The strict duality relationship holds provided that the structures possess similar geometry. For example, a short electric dipole antenna gapped in the middle is strongly related to the magnetic structure consisting of a short coil mounted in the middle of a rod of highly permeable material. In particular, the inductance of the latter magnetic structure can be derived directly from the capacitance between the electric dipole arms. The relationship has been thought to hold only for situations where the coil in the center of the core is very short. Here, however, we show that inductance can be evaluated even in cases where the windings are distributed beyond the center and cover relatively long portions of the ferromagnetic core. The extension of the theory described here relies on the fact that the distribution of the flux along the core is triangular-a fact derived from the similarity of the magnetic structure to a short electric dipole antenna, in which the current has a triangular profile. The triangular distribution provides a basis for evaluating the inductance of extended coils by relying on integration of the flux profile through the windings along the core. The results of the evaluation are in good agreement with laboratory measurements. The evaluation method, despite its simplicity, is based on analytical approach, unlike the many empirical methods of inductance evaluation described in the literature     

Optimization of endoluminal loop radiofrequency coils for gastrointestinal wall MR imaging

In this paper, we describe the optimization of endoluminal planar coils for high-resolution magnetic resonance imaging of gastrointestinal walls. For maximizing the coil performances, electromagnetic parameters of planar rectangular radio frequency (RF) coils were simulated using the finite element method. The eddy currents were fully computed to determine the electromagnetic losses in both wires and surrounding environment. Geometric parameters of the coils (length, conductive layer section, number of layers, and turns number) were varied. Based on simulations, five loop RF coil prototypes with planar geometry were designed to fit in a 5-mm inner diameter catheter. In the immediate vicinity of single-loop coils, the signal-to-noise ratio (SNR) decreases with the length of the coil, whereas penetration depth increases with it. The double-loop coil offers a greater penetration depth in comparison to the same length single-loop coil. The multilayer coil preserves the RF field B/sub 1/ by inducing a reduction in the electrical resistance of the conductor, therefore resulting in an increase in SNR. Experimental verifications were performed on a 1.5 T clinical scanner. Simulation results were found to be in good agreement with that of MR experiments. Developed prototypes provided a dramatic increase in SNR at the region of interest.     

Calculation of self- and mutual impedances in planar sandwichinductors

High-frequency planar magnetic components, employing thin film and thick film technology, have become important components in applications, such as filters and switching converters, due to their ease of manufacture and reliability. In a previous paper, the authors established a frequency dependent impedance formula for planar coils on a magnetic substrate that is infinitely thick. In this paper, two new impedance models are described: the first is for planar coils on a magnetic substrate of finite thickness, and the second represents a planar coil sandwiched between two substrates. The models include the electrical conductivity of the magnetic material so that the effects of eddy currents, particularly at high frequencies, are taken into account. The eddy currents reduce the inductance and increase the losses associated with the device. The new impedance formulas are derived from Maxwell's equations. Simulations were carried out on a typical device, using finite element analysis, and the results validate the new formulas. This paper establishes the frequency limitations of lossy magnetic substrates     

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.     

Measurement of angle errors of mutual inductance coils by means of an alternating current bridge

Summary In using the described method for the measurement of angle errors of mutual inductance coils, it should be kept in mind that the error is determined by a number of factors, of which eddy currents in windings and the distributed capacitance are the most important. The presence of distributed capacitance leads to a situation where, in transition from the accordant connection of windings to connection in opposition, the leakage current changes. Therefore, the measured value of the angle error can be, generally speaking, substantially different from the value obtained if the given coil is used as an actual measure of the 90° phase shift. Consequently, the described method of measuring the angle error is suitable for those mutual inductance coils where the capacitance leakage is sufficiently small or does not change to a great extent when the coil connections are interchanged.It should be remarked that this circumstance affects the measured value of the coefficient of mutual inductance to a certain extent if the measurement results are obtained by using (6).Finally, we shall make another remark regarding the difference (R₂-R₁). In deriving (7), it was assumed that the resistance of both coils remained unchanged during the measurements. As the resistance corresponding to the loss angle is small, in order that the above condition be satisfied, it is necessary, in particular, to keep the temperature conditions of the coils under careful control by not overloading them with a current which would cause appreciable heating.The requirement for the stability of resistances can be set forth in a concrete manner by assigning a certain given error caused by their instability.Denoting the change in resistance by R, we obtain accordingly:Thus, for R=1000 cps, M=0.01 h, and for =5·10^–5, we have R=3·10^–3 ohm.