Neumann Integrals Research Articles

Moiré patterns of space-filling curves

It is shown on the examples of Moore and Gosper curves that two spatially shifted or twisted, preasymptotic space-filling curves can produce large-scale superstructures akin to moiré patterns. To study physical phenomena emerging from these patterns, a geometrical coupling coefficient based on the Neumann integral is introduced. It is found that moiré patterns appear most defined at the peaks of those coefficients. A physical interpretation of these coefficients as a measure for inductive coupling between radiofrequency resonators leads to a design principle for strongly overlapping resonators with vanishing mutual inductance, which might be interesting for applications in MRI. These findings are demonstrated in graphical, numerical, and physical experiments. Published by the American Physical Society 2024

A Fast 3-D Winding Inductance Estimation Method for Air-Cored Resonant Induction Machines

The air-cored resonant induction machine removes the magnetic core, so the fields produced by the windings are truly 3-D in nature. The end-windings normally regarded as a nonactive and leakage source in conventional iron-cored machines now become an active part and contribute to the torque production. Therefore, the electromagnetic modeling can no longer be reduced to a 2-D analysis, and the 3-D inductance calculation becomes a key problem. The 3-D finite element analysis (FEA) can solve the 3-D magnetic field, but, first, the validity of its solution depends on the precision in geometry modeling. In particular, representing the end-winding region avoiding conductor clashing can be very complicated. Second, 3-D FEA solutions are computationally slow, and therefore inefficient as an “internal routine” of an optimization procedure. This article proposes a fast analytic 3-D winding inductance estimation method for air-cored resonant induction machines. The approach breaks down the real coils into straight conductors and represents them by single filaments located at their centers, then uses closed-form expressions derived from Neumann integrals to calculate the coil self and coil-to-coil mutual inductances, which are then collected into winding phase self and mutual inductances. All the independent coil-pair contributions are isolated, so as to eliminate redundant calculations. Good accuracy of the calculated results is confirmed by validation against both 3-D FEA and experimental results, including winding inductance breakdown and overall machine tested performance.

Modeling of Mutual Inductance Between Two Misalignment Planar Coils in Wireless Power Transfer

Mutual inductance is an important parameter for the design and optimization of the wireless power transfer (WPT) system. In this letter, a compact analytical model for mutual inductance of two coupled planar coil is proposed. The model is derived from Neumann's integral of constant current-carrying filament with a Taylor expansion approach. It can predict mutual inductance at various lateral and angular misalignments. The analytical results show a good agreement with the 3-D electromagnetic simulation results, and the maximum error for the mutual inductance remains less than 15.1%. An experimental setup has been built, and the results reveal the proposed model can predict the mutual coupling rapidly and effectively.

Analytical Computation of End-Coil Leakage Inductance of Round-Rotor Synchronous Machines Field Winding

The computation of end-coil leakage inductances of electric machines is a challenging task due to the complicated leakage flux 3-D distribution in the winding overhang region. In this paper, the problem of computing the field-circuit leakage inductance of round-rotor synchronous machines is addressed. The proposed method is fully analytical and descends from the symbolical solution of Neumann integrals applied to the computation of self-inductance and mutual inductance combined with the method of mirror images to account for core effects. With respect to existing analytical approaches, the methodology requires neither numerical integral solutions nor discretizing the end-coil geometry into small straight elements. The accuracy of the proposed technique for computing the mutual inductance between two single-end turns is assessed against measurements on a dedicated experimental setup. The extension of the method to the computation of the entire field-circuit end-coil leakage inductance is assessed by comparing with the 3-D finite-element analysis.

Modeling of Mutual Coupling Between Planar Inductors in Wireless Power Applications

This paper presents a compact model of mutual inductance between two planar inductors, which is essential to design and optimize a wireless power transmission system. The tracks of the planar inductors are modeled as constant current carrying filaments, and the mutual inductance between individual filaments is determined by Neumann's integral. The proposed model is derived by solving Neumann's integral using a series expansion technique. This model can predict the mutual inductance at various axial and lateral displacements. Mutual coupling between planar inductors is computed by a 3-D electromagnetic (EM) solver, and the proposed model shows good agreement with these numerical results. Different types of planar inductors were fabricated on a printed circuit board (PCB) or silicon wafer. Using these inductors, wireless power links were constructed for applications like implantable biomedical devices and contactless battery charging systems. Mutual inductance was measured for each of the cases, and the comparison shows that the proposed model can predict mutual coupling suitably.

Growth estimates for modified Neumann integrals in a half-space

Our aim in this paper is to deal with the growth properties for modified Neumann integrals in a half-space of R n . As an application, the solutions of Neumann problems in it for a slowly growing continuous function are also given.

Determination of Stator End Winding Inductance of Large Induction Machines: Comparison Between Analytics, Numerics, and Measurements

Knowledge of the end winding inductance of electrical machines is decisive for calculating their operating performance. In this article, two different approaches to analytically calculate the stator end winding inductance of large induction machines are discussed. The first method is based on the exact replication of the 3D conductor geometry using serially connected straight filaments, where the inductances are calculated by solving Neumann's integral. In the second method, the end winding flux is resolved into components excited by the axial and circumferential end winding magnetomotive force, resulting in a far simpler geometrical model. In both cases, end face effects are taken into account by adopting the method of images. The analytical approaches are compared to the known analytical calculation method proposed by Alger [1]. In addition, the stator end winding inductance is computed by means of 3D finite-element analysis. Using experimental validation, it is shown that both the analytical and numerical results reasonably correlate with removed rotor inductance measurements taken for several induction machines with different rated powers and frame sizes, if the permeability of the laminated core is taken into consideration.

Turbogenerator End-Winding Leakage Inductance Calculation Using a 3-D Analytical Approach Based on the Solution of Neumann Integrals

An analytical technique that can be effectively used for the calculation of the end-winding leakage inductance of a turbogenerator has been presented. It is based on a three-dimensional geometric model of the end-winding region in which each coil is modeled as a set of serially connected straight filaments. The calculation of the mutual inductance of the end coils is based on the multiple solutions of the Neumann integral. The method also accounts for the influence of stator core iron on the end-winding leakage inductance by using the method of images. The results of the calculations have been compared with the measured values of the leakage inductance for 247-MVA turbogenerator manufactured by KONCAR Generators and Motors Inc., Zagreb, Croatia. The principle of the end-winding leakage inductance calculation described herein can be effectively used for other types of machines and windings as well.

Further generalizations of Neumann's integral

AbstractA generalization of Neumann's integral connecting the two kinds of Legendre function is obtained. It contains an extra parameter which is not a function of the parameters of the Legendre functions, unlike all previous extensions of the original formula. These extensions are shown to be particular cases of the new generalization and some further particular cases are also indicated.

Mutual Impedances of Grounded Circuits

Formulas are derived for the direct-current mutual resistance and inductance between circuits grounded at the surface of the earth. For circuits composed of straight filaments, the mutual inductance is reduced to known Neumann integrals which involve only comparatively simple expressions for the case of horizontal, coplanar conductors above, below or on the surface of the earth. Numerical values for these integrals may be readily obtained from new and accurate graphs for straight filaments which meet at a point or start from a common perpendicular. It is shown that these new results supply a useful first approximation to the actual alternating-current mutual impedance of grounded circuits, when the frequency and extent of the circuits are not larger than occur in many practical applications.

Mutual Impedances of Grounded Circuits

Formulas are derived for the direct-current mutual resistance and inductance between circuits grounded at the surface of the earth. For circuits composed of straight filaments, the mutual inductance is reduced to known Neumann integrals which involve only comparatively simple expressions for the case of horizontal, coplanar conductors above, below or on the surface of the earth. Numerical values for these integrals may be readily obtained from new and accurate graphs for straight filaments which meet at a point or start from a common perpendicular. It is shown that these new results supply a useful first approximation to the actual alternating-current mutual impedance of grounded circuits, when the frequency and extent of the circuits are not larger than occur in many practical applications.