Calculation Of Mutual Inductance Research Articles

Calculation of Mutual Inductance based on 3D Field and Circuit Coupling Analysis for WPT System

Determination of mutual inductance in the WPT system based on the electromagnetic coupling mode is crucial for system design and efficiency optimization. The analytic formulas and experimental methods which only aim at a coupling coil with a certain type of configuration and parameters are not universal approaches. Therefore they can’t serve as a design basis for the system optimization. This paper presents a novel method based on 3D field and circuit coupling analysis to calculate mutual inductance of coupling coils with arbitrary spatial distribution for WPT system. The method can obtain maximum efficiency under the condition that the spatial distribution of coupling coils are irregular and restrained in actual physical systems. The method has made a breakthrough and provided a powerful tool for the design and optimization of electromagnetic coupling power transmission system.

Mutual Inductance and Interaction Calculation Between Conductor or Coil of Rectangular Cross Section and Parallelepiped Permanent Magnet

Mutual Inductance and Interaction Calculation Between Conductor or Coil of Rectangular Cross Section and Parallelepiped Permanent Magnet

The mutual inductance calculation between circular and quadrilateral coils at arbitrary attitudes using a rotation matrix for airborne transient electromagnetic systems

Performance testing and calibration of airborne transient electromagnetic (ATEM) systems are conducted to obtain the electromagnetic response of ground loops. It is necessary to accurately calculate the mutual inductance between transmitting coils, receiving coils and ground loops to compute the electromagnetic responses. Therefore, based on Neumann's formula and the measured attitudes of the coils, this study deduces the formula for the mutual inductance calculation between circular and quadrilateral coils, circular and circular coils, and quadrilateral and quadrilateral coils using a rotation matrix, and then proposes a method to calculate the mutual inductance between two coils at arbitrary attitudes (roll, pitch, and yaw). Using coil attitude simulated data of an ATEM system, we calculate the mutual inductance of transmitting coils and ground loops at different attitudes, analyze the impact of coil attitudes on mutual inductance, and compare the computational accuracy and speed of the proposed method with those of other methods using the same data. The results show that the relative error of the calculation is smaller and that the speed-up is significant compared to other methods. Moreover, the proposed method is also applicable to the mutual inductance calculation of polygonal and circular coils at arbitrary attitudes and is highly expandable.

Mutual inductance calculation between arbitrarily positioned rectangular filaments

Abstract. The mutual inductance between the transmitted and received coils is an important parameter in the inductively cou-pled wireless power transmission system. The relative position between the linked coils may be arbitrary according to theapplications. In this paper the analytical calculation of the mutual inductance between two arbitrarily positioned rectangu-lar filaments is proposed based on a more general expression of mutual inductance calculation of two arbitrarily positionedstraight filaments than Grover’s. The calculation results are validated by using the finite element simulation tools. The previouspublished approximation methods for square coils in the coaxial, lateral misalignment or angular misalignment situations areverified that they are accurate when the two coils’ relative distances are large.Keywords: Mutual inductance, rectangular filament, arbitrary position 1. IntroductionIn the wireless power transmission system, the power is transferred from the transmitted coil to thereceived coil through the inductively coupled method whose coupled coefficient and the transfer effi-ciency are related to the mutual inductance between the two coils [1]. In addition to the inductivelycoupled power transmission application, the mutual inductance is also a basic parameter in the applica-tions like the electromagnetic induction sensors[2], and the electrical machines [3].The mutual inductance is related to the coils’ shapes,sizes, and relative positions [4]. The commonlyused shapes of the coils are circular or rectangular/square. No matter what shapes are, the relative posi-tionwillbecoaxial,coplaner,perpendicularorevenarbitraryaccordingtotheapplications.Furthermore,in some applications the relative position will be changed with the time. A typical example is the ap-plication of the wireless power transmission for the capsule endoscope, where the received coils insidethe body are movable irregularly [5]. Although two coaxial coils are needed in some applications, thelateraland/orangularmisalignmentswillbeinevitable.Insummary,itisnecessarytoanalyzethemutualinductancein the most general situation that the two coils are arbitrarily positioned.The mutual inductance of the

Calculation of Self Inductance and Mutual Inductance of Coils with Rectangular Cross Section

An inductance and mutual inductance of coils with rectangular cross section is given in a form of one-dimensional integral. The formula is appropriate for numerical calculation of the inductances with given accuracy

Accurate Computation of Mutual Inductance of Two Air Core Square Coils with Lateral and Angular Misalignments in a Flat Planar Surface

This paper presents an analytical approach for computing the mutual inductance between two air core square coils placed in a flat planar surface. The mutual inductance of the coils is calculated for all possible variations including lateral and angular misalignments in space. In contrast to conventional approximated formulae, the straightforward approach based on Biot-Savart principle is used and their integrals are computed numerically. The results of computed mutual inductance by analytical method are validated by finite element analysis and an experimental setup. Finally, the analysis compares the three mutual inductance calculations: an analytical method, the finite-element model and an experimental results. The values computed by three methods in all cases are in good agreement.

Analysis and force calculations for induction motors with broken bars using ABC frame model combined with 2D finite element method

In this paper, an ABC transient model of the three-phase induction motor is developed that depends on self and mutual inductance calculations based on accurate finite element analysis (FEA). This model can represent both healthy and faulty motor conditions. The rotor broken bars faults are studied thoroughly including their effects on the rotor bar currents distribution, the forces exerted on the rotor bars, the motor speed and motor torque. It was observed that broken bar faults will cause fluctuations in the motor speed and torque. It was found that the non-uniform distribution of the rotor bar currents will lead to non-uniform radial forces affecting the bars. This will increase the mechanical stresses on the rotor shaft.

2D finite element analysis and force calculations for induction motors with broken bars

In this paper an ABC transient model of the three phase induction motor is developed that depends on self- and mutual inductance calculations based on accurate finite element analysis (FEA). This model can represent both healthy and broken bars’ conditions. The effects of that type of fault on the stator current, rotor bar currents distribution, the forces exerted on the rotor bars, the motor speed and motor torque are studied. It was observed that broken bar faults will cause fluctuations in the stator peak current and in the motor speed and torque. It was found that the asymmetry in the rotor bar currents will lead to asymmetrical distribution of the radial and tangential forces affecting the bars. The asymmetry in the tangential forces causes the fluctuations in the motor torque, while the asymmetry in the radial forces will increase the mechanical stresses on the rotor shaft.

P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

In transcranial magnetic stimulation (TMS), we are interested in calculating the TMS-induced electric field (E-field), which defines the given stimulus. This is known as the forward problem, and there exist no general analytic solution for it. This is because the electric field is dictated by Maxwell’s equations, which are partial differential equations. Thus, for non-trivial boundary conditions, numerical methods are needed. General methods for solving these equations are the finite element method (FEM), where we split the volume into small polyhedrons, and the boundary element method (BEM), where we divide the system into uniform-conductance regions and split the formed boundaries into small polygons. Both of these methods can be applied for nearly arbitrary geometries but are hard to parallelize. However, often one can simplify the analysis by using a spherically symmetric head model. For example, if one is to assess the general properties of a stimulator coil, using a more complicated head model will just increase the computational cost and complicate the analysis of the results. Also when the head can be approximated being locally spherical, the spherical symmetric head model allows a significant simplification in the computations. This is because the spherical symmetry of the conducting medium allows us to compute an analytic solution for the forward problem. We will present an efficient method for computing the induced E-field. The model works for spherically symmetric head models and is based on the triangle construction, (Ilmoniemi RJ, Hamalainen MS, Knuutila J. The forward and inverse problems in the spherical model. In: Biomagnetism: Applications & Theory, H. Weinberg, G. Stroink, T. Katila (eds.), Pergamon Press, Amsterdam 1985, pp. 278–282); see Fig. 1 . Devices based on the triangle construction have been used to calibrate magnetoencephalography (MEG) and TMS systems. Now, to obtain the two tangential components (the radial component is always zero) of the induced E-field in a point, one simply has to calculate the mutual inductance between the TMS coil and two perpendicular triangular loops with one corner at the sphere center and the opposite edge tangential to the spherical surface. For the mutual-inductance calculations, the TMS coil can be modeled as a set of current-carrying filaments. Then, the mutual inductance between the coil and the triangle loops can be calculated using analytic formulas. Using the triangle construction, we developed an O (n) time and memory algorithm for the TMS forward problem, where n is the number of points where the induced E-field is computed. This is a considerable improvement over O (M2) for the BEM or O (N) for the FEM, where M is the number of surface elements, and N is the number of volume elements. (For FEM, we need much more 3D volume elements than points in our construction because we have to sample also the volume surrounding the system. For BEM, we need to sample all the 2D surfaces, also the uninteresting ones. Thus, for a small region of interest, N ≫ M ≫ n.) The developed algorithm is very simple compared to those for BEM or FEM as it does not require any tessellation. Also, because in our model the computation for the field in a point does not depend on the field in the other points, the algorithm is embarrassingly parallel, i.e., trivial to parallelize with minimal overhead.

Force Computation and Analysis of Single-Phase Capacitor-Start Induction Motors Using Finite Element Method at Broken Bars Conditions

This paper studies the effects of broken bars faults on the single-phase capacitor-start induction motor. A transient model for the motor is used in this study that depends on self and mutual inductance calculations based on accurate finite element method (FEM). The rotor broken bars faults are studied thoroughly including their effects on the rotor bar currents distribution, the rotor bar frequency content, the electromagnetic forces exerted on the rotor bars, the motor speed and motor torque. The electromagnetic tangential and radial forces affecting the rotor bars were calculated using finite element method (FEM). It was found that the non-uniform distribution of the rotor bar currents will lead to non-uniform radial forces affecting the bars. This will increase the mechanical stresses on the rotor shaft which may lead to eccentricity.

헬리컬 코일을 이용하는 자기 공진형 무선 전력 전송 시스템에서 새로운 상호 인덕턴스의 계산식 제안

본 논문에서는 헬리컬 코일을 이용하는 공진형 무선 전력 전송 시스템에서 새로운 상호 인덕턴스 계산식을 이용한 해석적 계산을 제안한다. 상호 인덕턴스 계산 과정 중에 중요한 변수의 look-up table을 수식화 하였다. 송 수신이 대칭인 구조에서 두 헬리컬 코일 간의 거리를 53 mm부터 500 mm까지 10 mm 간격으로 거리를 증가시키며, 공진 주파수와 삽입 손실에 대한 계산 결과와 실험 결과를 비교하였다. 실험 결과와 계산 결과의 공진 주파는 평균적으로 5.63 %의 차이를 보였다. 삽입 손실은 실험 결과와 290 mm에서 0.33 dB의 가장 작은 차이를 보였고, 평균적으로 2.25 dB의 차이를 보였다. 발룬을 사용하지 않았을 때의 실험 결과가 계산 결과와 더 근접한 결과를 보이는 것을 확인했다. In this paper, analytical calculations using a novel mutual inductance formula for a resonant wireless power transmission system using helical coils. The look-up table of critical variables during the mutual inductance calculation process was formulated. The calculation results for resonant frequency and insertion loss were compared with experimental results when the distance between the two helical coils in a structure where the transmission and reception is symmetrical was varied with 10 mm increments from 53 mm to 500 mm. On average, the resonant frequency showed a difference of 5.63 % between the experimental results and the calculation results. The insertion loss had an average difference 2.25 dB where the smallest difference of 0.33 dB occurred with 290 mm. It was found that the experimental results without using a balun were in greater agreement with the calculation results.

Analytical magnetic field distribution of slotless brushless permanent magnet motors – Part I. Armature reaction field, inductance and rotor eddy current loss calculations

Two-dimensional analytical armature field calculations are presented for slotless permanent magnet brushless motors. The proposed method is applicable to both brushless AC and DC motors with any number of phases. The problem is formulated for eight annular regions: shaft, rotor back-iron, magnets, retaining sleeve, airspace, winding, stator back-iron and exterior. A general armature current density distribution is expressed as a function of spatial angle and time for any number of phases, any current waveform and any winding distribution. The governing partial differential equations of all the regions subject to a set of boundary conditions are represented and solved analytically for both internal and external rotor motors. Self- and mutual-inductance calculations as well as expressions for the calculation of rotor eddy current losses in the magnets, retaining sleeve and rotor back-iron are also represented. In the second part of these two papers, the open-circuit magnetic field is presented for different magnetisation patterns and subsequently the electromagnetic torque is calculated. The effectiveness of the analytical model is validated by comparing the results with those obtained from finite-element analyses as well as an experimental set-up which presented in the second part of the series.

MUTUAL INDUCTANCE FOR AN EXPLICITLY FINITE NUMBER OF TURNS

Non coaxial mutual inductance calculations, based on a Bessel function formulation, are presented for coils modelled by an explicitly flnite number of circular turns. The mutual inductance of two such turns can be expressed as an integral of a product of three Bessel functions and an exponential factor, and it is shown that the exponential factors can be analytically summed as a simple geometric progression, or other related sums. This allows the mutual inductance of two thin solenoids to be expressed as an integral of a single analytical expression. Sample numerical results are given for some representative cases and the approach to the limit where the turns are considered to be smeared out over the solenoid windings is explored.

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.

Mutual Inductance Calculation Between Circular Filaments Arbitrarily Positioned in Space: Alternative to Grover's Formula

In this paper, we present the full derivation of a new formula for calculating the mutual inductance between inclined circular filaments arbitrarily positioned with respect to each other. Although such a formula was already proposed by Grover more than 50 years ago, the formula presented here is slightly more general and simpler to use, i.e., it involves only a sequential evaluation of expressions and the numerical resolution of a simple numerical integration. We derived the new formula using the method of vector potential, as opposed to Grover's approach, which was based on the Neumann formula. We validated the new formula through a series of examples, which are presented here. Finally, we present the relationship between the two general formulas (i.e., Grover's and our new one) explicitly (Example 12).

A novel approach to calculate inductance and analyze magnetic flux density of helical toroidal coil applicable to Superconducting Magnetic Energy Storage systems (SMES)

Abstract In this paper, formulas are proposed for the self and mutual inductance calculations of the helical toroidal coil (HTC) by the direct and indirect methods at superconductivity conditions. The direct method is based on the Neumann’s equation and the indirect approach is based on the toroidal and the poloidal components of the magnetic flux density. Numerical calculations show that the direct method is more accurate than the indirect approach at the expense of its longer computational time. Implementation of some engineering assumptions in the indirect method is shown to reduce the computational time without loss of accuracy. Comparison between the experimental measurements and simulated results for inductance, using the direct and the indirect methods indicates that the proposed formulas have high reliability. It is also shown that the self inductance and the mutual inductance could be calculated in the same way, provided that the radius of curvature is >0.4 of the minor radius, and that the definition of the geometric mean radius in the superconductivity conditions is used. Plotting contours for the magnetic flux density and the inductance show that the inductance formulas of helical toroidal coil could be used as the basis for coil optimal design. Optimization target functions such as maximization of the ratio of stored magnetic energy with respect to the volume of the toroid or the conductor’s mass, the elimination or the balance of stress in some coordinate directions, and the attenuation of leakage flux could be considered. The finite element (FE) approach is employed to present an algorithm to study the three-dimensional leakage flux distribution pattern of the coil and to draw the magnetic flux density lines of the HTC. The presented algorithm, due to its simplicity in analysis and ease of implementation of the non-symmetrical and three-dimensional objects, is advantageous to the commercial software such as ANSYS, MAXWELL, and FLUX. Finally, using the presented algorithm, magnetic flux density lines in several examples are drawn.

A Novel Approach for Calculations of Helical Toroidal Coil Inductance Usable in Reactor Plasmas

In this paper, formulas are proposed for the self- and mutual-inductance calculations of a helical toroidal coil by direct and indirect methods at superconductivity conditions. The direct method is based on the Neumann's equation, and the indirect approach is based on the toroidal and the poloidal components of the magnetic flux density. Numerical calculations show that the direct method is more accurate than the indirect approach at the expense of its longer computational time. Implementation of some engineering assumptions in the indirect method is shown to reduce the computational time without loss of accuracy. Comparison between the experimental, empirical, and numerical results for inductance, using the direct and the indirect methods, indicates that the proposed formulas have high reliability. It is also shown that the self-inductance and mutual inductance could be calculated in the same way, provided that the radius of curvature is greater than 0.4 of the minor radius and that the definition of the geometric mean radius in the superconductivity conditions is used. Plotting contours for the magnetic flux density and the inductance show that the inductance formulas of the helical toroidal coil could be used as the basis for coil optimal design. Optimization target functions such as maximization of the ratio of stored magnetic energy with respect to the volume of the toroid or the conductor's mass, the elimination or the balance of stress in certain coordinate directions, and the attenuation of leakage flux could be considered.

On the Optimum Design of Air-Cored Tesla Transformers

Tesla coil is an electrical device capable of developing high potentials ranging from a few hundreds of kilovolts up to several megavolts; the voltage is produced as AC, with a typical frequency of 50–400 kHz. Tesla coils, which are typically operated in pulsed mode, have strong coupling with pulse widths varying from some nanoseconds up to several hundreds of microseconds according to the particular application. Due to their capabilities, Tesla Transformers have found various applications such as in high voltage Subnanosecond Pulsers [1], aging of insulators and high frequency solid insulation testing [2]. The prominent effect of coupling on the performance of Tesla coils necessitates the investigation of mutual inductance (M) calculation methods. Analytical methods have been applied to calculate M for Tesla coils with rectangular cross-section conductors [3, 4]. The closed-form expressions of these methods result in complete elliptic integral of first and second kind and Neumann’s Lambda function, which makes the expression to seem complicated for users. The method proposed in this paper is a combined analytical and numerical method based on magnetic vector potential (A)and Neumann’s method which can be easily comprehended and convenient to be used [5]. Besides, the configuration role in optimizing M and voltage gain are investigated.

High-Speed and High-Accuracy Method of Mutual-Inductance Calculations

While Grover's tabular method to compute the mutual inductance of coaxial single-layer coils is convenient and fast when implemented on computer, its use has mostly been restricted to zeroth- and first-order calculations, since its accuracy is not well defined or quantified. This paper defines and quantifies the accuracy of Grover's tabular method for single-layer coaxial coils and shows that interpolation between the tabulated values is the primary factor limiting its accuracy. The results of this paper are used to develop an improved tabular-solution method that has accuracy approaching that of a finite-element field-code analysis. The analysis compares three mutual-inductance calculation methods for coaxial single-layer coils: a finite-element numerical method, an analytical method involving the numerical elliptic-integral evaluation, and Grover's tabular method. The finite-element numerical solution and the analytic solution are the most accurate and comparable to each other, while the tabular method is the least accurate. In terms of speed, however, the tabular method is the fastest while the finite-element method is the slowest. The interpolation techniques used in the analysis include cubic spline, cubic polynomial, polynomial, and linear. In comparison to finite-element numerical calculations, the improved tabular method developed in this paper decreases the computation time a factor of 4 times 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> while retaining an accuracy of at least four significant digits. The improved tabular method can be applied to similar-type problems that rely on the evaluation of elliptic integrals in their solution.

Mutual Inductance Calculation of Movable Planar Coils on Parallel Surfaces

Recent developments of the wireless battery charging platform have prompted requirements to investigate the mutual inductance between a movable planar coil and the fixed planar coil on the charging platform. The wireless battery charging platform must allow the load to be placed anywhere on the charging surface. Therefore, the relative position between the movable energy-receiving coil and the energy-transmitting coils on the charging platform must be flexible. In this paper, an extended formula is proposed for the mutual inductance calculation for two coaxial or noncoaxial planar spiral windings sandwiched between two <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">double-layer</i> substrates. It can quickly determine the mutual coupling of two planar windings that can have different relative positions and distance between them. This new calculation method provides a new and useful tool for determining the mutual inductance of a movable planar coil and the fixed planar coil on the wireless battery charging platform. The theory has been favorably tested and compared with practical measurements and also with finite-element analysis.