Inductance Calculation Research Articles

Mutual Inductance Calculation Method of Rectangular Coils with Bilateral Bounded Single-hole Type Magnetic Medium in Wireless Power Transfer Systems

Mutual Inductance Calculation Method of Rectangular Coils with Bilateral Bounded Single-hole Type Magnetic Medium in Wireless Power Transfer Systems

CALCULATION OF POWER TRANSFORMER INDUCTANCE MATRIX

Practical calculations of electric equipment and electric networks are performed using equivalent circuits. Without taking into account saturation phenomena, electromagnetic processes in a power transformer are described by a system of linear equations, which can be represented in matrix form. The transformer inductance matrix contains self and mutual winding inductances, whose values for one phase are determined directly from the passport data containing the results of no-load and short circuit experiments. Mutual inductances between phases depending on the type and size of the magnetic core are not usually given in the transformer documentation. It does not allow to simulate the operating modes that differ from the steady state with uniform phase load.
 
 The purpose of the work is to develop an algorithm for calculating elements of the power transformer inductance matrix according to its passport data and magnetic circuit parameters.
 
 The scientific novelty lies in use of information about magnetic circuit type and size to determine elements of the inductance matrix.
 
 Materials and methods. Methods of linear electric and magnetic circuit theory are used in the work.
 
 Results. The proposed algorithm for calculation of power transformer inductance matrix includes three stages: calculation of self and mutual inductances for one phase; construction of an equivalent circuit for a magnetic circuit and calculation of magnetic fluxes in order to take into account the mutual influence of windings on various phases; assembly of the inductance matrix of three-phase transformer. A method is proposed for constructing the transformer inductance matrix with an arbitrary type of magnetic circuit, phases and windings quantity. The method involves preliminary calculation of the coefficient matrix that characterizes the magnetic circuit and depends mainly on its type and, to a less, on the ratio of geometric dimensions in rods and yokes. The coefficient matrix can be calculated in advance and used to determine the inductance matrix of a wide range of transformers.
 
 Conclusions. 1. The passport data of the transformer do not allow to model transformer operation at unbalanced load validly. 2. An algorithm for determining the inductance matrix of transformers has been developed. 3. A method for compilation of a transformer inductance matrix with an arbitrary type of magnetic circuit, the number of phases and windings is introduced. The practical significance of the algorithm is determined by the simplicity of algorithmizing, as well as by the possibility of using it in computer simulation of electrical circuits with transformers, for example, using the modified nodal analysis method.

(Digital Presentation) A Novel Method for Precise Measurement of Displacement Using Eddy Currents at High Temperature

In this paper, a novel method of measuring displacement using eddy currents at high temperature is presented. The measurement system consists of a sensing coil created on a PCB by patterns, a parallel turning circuit and a microcontroller including a signal processing unit. When the gap distance between the conductive target and the sensing coil changes, the inductance value of the coil varies by the influence of the eddy current. By detecting the change in the inductance, we have proposed a novel method to precisely measure the displacement of the coil. In order to make the optimal coil, a theoretical analysis was conducted using the inductance calculation theory, and the magnetic field distribution and inductance variation of the coil were analyzed using ANSYS Maxwell. Then, the coil was fabricated as a PCB, and the measured result value was compared with each analysis result, and the coil most suitable for the system was selected. The circuit was designed so that the characteristics of the synchronization amplification circuit in the detection system were minimized in temperature change, and the temperature characteristics were primarily corrected in hardware by adding a temperature correction circuit. The output value of the primarily corrected tuning amplification circuit improved the temperature characteristics of the sensor by secondarily correcting the temperature characteristics in firmware, and is designed to meet the IPC class 3 specification. The sensing range of the implemented measurement system was 0.0 ~ 3.8 mm, the nominal value of the fabricated sensing coil was 11.2uH, and the resonance frequency of the tuning circuit including the sensing coil was 300Khz. The operating temperature of the system is – 40 ~ 150 ℃ and the resolution is 0.15 mm within the range of 0.0 ~ 3.8 mm. From the experimental results, Comparing the change in inductance with respect to the moving distance of the measurement system and the displacement signal measured by the system, it can be seen that as the inductance increases, the displacement output increases, and as the inductance decreases, the displacement also decreases. Considering this, it was found through the system that inductance and output of displacement are correlated, and is seen operates stably at high temperature and has linearity. Figure 1

Calculation of Mutual Inductance between Two Planar Coils with Custom Specifications and Positions Using a Machine Learning Approach

Wireless power transmission systems enable the transfer of electricity between grids without the use of physical wires. Different methods are employed for wireless power transfer, each suited to different distances. Inductive coupling, the subject of this study, is typically used for shorter distances. The effectiveness of inductive coupling systems is evaluated using a parameter called mutual inductance. In the present study, an attempt is made to provide a model for calculating mutual inductance in wireless power transfer systems using a machine learning approach. To achieve this goal, finite element simulations are employed, and 64 datasets are generated from mutual inductance calculations in various scenarios. These datasets are used to train machine learning regression algorithms, including linear regression, support vector regression, decision tree regression, and artificial neural networks. The evaluation results, using performance metrics such as R-squared, mean absolute error, and root mean square error, confirm that among these four algorithms, the artificial neural network exhibits higher computational accuracy with an R-squared value of 0.950 for predicting test data.

Analytical Double-2D Frequency-Dependent Leakage Inductance Model for Litz-Wired High-Frequency Transformer

Accurate and fast calculation of frequency-dependent leakage inductance is critical for the optimization of the high-frequency transformer with Litz wire winding. The analytical method is widely used due to its high computation speed. However, its computation accuracy is limited due to the following factors. i) One-dimensional (1D) or two-dimensional magnetic field model (2D) has unsatisfying accuracy for the transformer with a rotationally asymmetric structure. ii) The skin and proximity effect factors based on the multilayer foil winding are inaccurate for the Litz wire winding in the calculation of frequency-dependent magnetic energy. This paper proposes a double-2D frequency-dependent leakage inductance model for the transformer with Litz wire winding. It uses the image method to calculate the 2D leakage magnetic field inside and outside the core window, respectively, which considers the rotationally asymmetric structure in leakage inductance calculation. The round conductor model is proposed for the frequency-dependent magnetic energy calculation. The single strand in a Litz wire is taken as the basic modeling object of the round conductor model, which is more detailed than the multilayer foil winding model for the Litz wire winding. The comparison between the experiment, FEM, state-of-the-art models and the proposed model is conducted for eight transformers with E-core, U-core, round or rectangular Litz wire winding. The proposed model shows satisfying computation accuracy in the whole frequency band. And the computational speed of the proposed model is 100 times that of the FEM.

Design Optimization of Null-Flux Superconducting Electrodynamic Suspension for Cost-Saving and Performance Improvement

The onboard superconducting magnets and null-flux coils (NFCs) on the track side are the crucial components for realizing the null-flux superconducting electrodynamic suspension (EDS). In this article, the multiobjective optimization of a null-flux EDS system was investigated to improve the performance using the nondominated sorting genetic algorithm-II (NSGA-II) method. First, a precise mathematical model of the superconducting EDS was established by adopting mutual inductance calculation method we proposed before. Then, the comprehensive sensitive analysis was carried out to analyze the influence of variables on different objectives. The ratio of levitation force to drag force (drag ratio), the ratio of levitation force to mass of NFCs (mass ratio) as well as the fluctuation of levitation force, were defined to be the optimization objectives. Third, an optimization scheme employing NSGA-II method was built, and the Pareto front was obtained. Afterward, the effectiveness of optimized results was verified by the finite element analysis. Finally, an equivalent experimental method was used to characterize the performance of a down-scale EDS system to confirm the proposed optimization methodology. To conclude, the optimal method can greatly improve the performance of EDS, and provide diverse design scheme choices.

Coupled electric circuits method applied to pulsed eddy current NDT problem

The present work concerns the study of the effect of variation of the thickness of 2D cylindrical test-piece and lift-off (the air gap between the sensor and the test-piece) on the sensor response in case of transient source current. The coupled circuit’s method employed is based on the mutual’s inductances calculation and with the association to Kirchhoff laws it yields to a transient algebraic equations system which is solved at each time step. The method is applied to the study of the pulsed eddy currents in cylindrical and axisymmetric device with electrical conductivity and geometrical dimensions variations.

Mutual inductance calculation for rectangular and circular coils with parallel axes

Mutual inductance calculation for rectangular and circular coils with parallel axes

Inductance Calculation Method Considering the Window Effect of Planarized Magnetic Core

Inductance Calculation Method Considering the Window Effect of Planarized Magnetic Core

Calculation of mutual inductance between circular and arbitrarily shaped filaments via segmentation method

In this article, two analytical formulas for the calculation of mutual inductance between a circular filament and line segment arbitrarily positioning in the space are derived by using Mutual Inductance Method (MIM) and Babic’s Method (BM), respectively. Using the fact that any curve can be interpolated by a set of line segments, a method for calculation of mutual inductance between a circular filament and filament having an arbitrary shape in the space is proposed based on the derived analytical formulas. The derived two formulas and the proposed method (Segmentation Method) were numerically validated by using FastHenry software and reference examples from the literature. In particular, the proposed method was successfully applied to the calculation of mutual inductance between the circular filament and the following special curves such as circle, circular arc, elliptic arc, ellipse, spiral, helices and conical helices. All results of the calculation are in good agreement with the reference examples.

Mutual inductance calculations of non-identical n-sided planar coils with arbitrary geometry and spatial orientations

Mutual inductance calculations of non-identical n-sided planar coils with arbitrary geometry and spatial orientations

Modified Approach to Inductance Calculation of Variable Reluctance Resolver Based on Segmented Winding Function Method

Modified Approach to Inductance Calculation of Variable Reluctance Resolver Based on Segmented Winding Function Method

A Novel Analytical Model for Mutual Inductance Calculations Between Two Nonidentical N-Sided Polygonal Planar Coils Arbitrarily Positioned in 3-D Space for Wireless Power Transfer

Mutual inductance (MI) calculation is of great importance, because there are many systems that operate on the principle of inductive coupling. This paper presents a novel analytical model that calculates the MI between two non-identical n-sided polygonal planar coils located in arbitrary position in 3D space. The model depends upon misalignment of both the primary and the secondary coils of a wireless power transfer (WPT) system operating based on inductive coupling. The polygonal coils used in this model may differ from each other in terms of size, shape and turns number. The misalignment cases considered in the proposed model include lateral, angular, lateral-angular and vertical distance variation between the primary and secondary coils. Based on Stokes Theorem, instead of using the magnetic flux density vector, the magnetic vector potential (MVP) is used to reach the required magnetic flux in the calculation of the MI. A single analytical solution function that can calculate the MI between two planar polygonal coils of any shape in a simple manner and short time duration is obtained by performing difficult mathematical operations through MVP. Results obtained from the developed new model are compared with finite element analysis (FEA), experimental and other studies in the literature. The good agreement of the results of this study with the other results has proved the accuracy and validity of the proposed new analytical model.

A Compact Saturated Core Fault Current Limiter Magnetically Integrated with Decoupling Windings

Traditional saturated core fault current limiters (SCFCLs) have fast rising fault state inductance that can limit faults automatically. However, the current limiting performance of traditional SCFCLs is insufficient. In order to improve this, some novel SCFCLs use additional air core reactors, resulting in a large volume and floor area. A compact saturated core fault current limiter magnetically integrated decoupling windings (CSFCL) is proposed in this study, which has the advantages of a smaller volume and floor area. The principle of the non-orthogonal decoupling method is firstly introduced. The core body of the CSFCL is then magnetically integrated with the decoupling windings in the form of non-orthogonal decoupling. Equivalent magnetic circuit and electric circuit analysis is carried out. An inductance calculation method for decoupling windings is proposed. Following this, a 220 kV simulation model is conducted based on the finite element method. Finally, a small-capacity prototype of the CSFCL and a test platform are designed to test it. The experimental results verify the correctness and feasibility of the proposed structure.

PlanetProfile: Self‐Consistent Interior Structure Modeling for Ocean Worlds and Rocky Dwarf Planets in Python

AbstractThe open‐source PlanetProfile framework was developed to investigate the interior structure of icy moons based on collectively matching their observed properties and comparative planetology. The software relates observed and measured properties, assumptions such as the type of materials present, and laboratory equation‐of‐state (EOS) data through geophysical and thermodynamic models to evaluate radial profiles of mechanical, thermodynamic, and electrical properties, as self‐consistently as possible. We have created a Python version of PlanetProfile. In the process, we have made optimization improvements and added parallelization and parameter‐space search features to utilize fast operation for investigating unresolved questions in planetary geophysics, in which many model inputs are poorly constrained. The Python version links to other scientific software packages, including for evaluating EOS data, magnetic induction calculations, and seismic calculations. Physical models in PlanetProfile have been reconfigured to improve self‐consistency and generate the most realistic relationships between properties. In this work, we focus on the description of the software design and algorithms in detail, summarize model results for major moons across the outer solar system, and discuss new inferences about the interior structure of several bodies. The high values and narrow uncertainty ranges reported for the axial moments of inertia for Callisto, Titan, and Io are difficult to reconcile with the realistic physical and chemical approximations applied, requiring highly porous rock layers equivalent to incomplete differentiation for Callisto and Titan, and a high rock melt fraction for Io. Radial profiles for each model and comparison to prior work are provided as Zenodo archives.

Calculations of Self- and Mutual Inductances for Racetrack Coils Using Equivalent Models

This paper presents the calculations of self and mutual inductances for thin racetrack coils using a novel approach. Approximate rectangular and diamond shape models for racetrack coils are used for the analysis with the same cross-sectional area and circumference as the racetrack coils. A 3D analytical method using separation of variables method and Fourier series is developed for the magnetic fields analysis and calculation of self and mutual inductances for thin racetrack coils. The analytical method utilizes magnetic vector potential with two components to solve Maxwell differential equations. Different racetrack coils with different dimensions are considered for the validation of analytical modeling for the inductance calculations. The inductance calculations of circular coils are also presented as special cases for racetrack coils. The approaches for the mutual inductance calculation of racetrack and circular coils could be utilized for wireless power transfer and charging applications.

Fast Calculation of Coil Launcher Based on Current Filament Method

High-speed coil launcher with a large number of coils is hard to calculate and is time-consuming. To realize the design and optimization of high-speed coil launcher, this article studies the simulation method of multistage coil launcher. At present, the calculation methods of multistage coil launcher mainly include the finite-element method (FEM) and the current filament method (CFM). The CFM converts the launcher system into a circuit model and its computational efficiency is much higher than that of FEM. In this article, based on CFM of the coil launcher, combined with the structural characteristics of the coil launcher and the electromagnetic field theory, an inductance calculation method and an iterative algorithm suitable for the CFM are proposed. The proposed inductance calculation method can reduce redundant calculations. The decomposition of induction matrix is used to speed up the solving efficiency of the differential equations. Taking the 40-stage coil launcher as an example, the improved calculation method takes only about 63 s to complete, and the calculation efficiency is much higher than that of the FEM, which needs 58 612 s (about 16 h) to get results.

Static magnetic field amplification in high-T c superconducting flux transformer equipped with multi-turn coil prepared using REBCO tape conductor

In this study, a high critical temperature (high-T c) superconducting flux transformer is prepared using a commercially available REBa2Cu3O y (REBCO) high-T c superconducting (HTS) tape conductor. The device configuration includes a 10 cm diameter pickup coil and a 3 cm diameter multi-turn input coil, which are formed by the “cut-and wind” method. The static magnetic field transfer efficiency between both coils has been estimated using inductance calculation of the multi-turn coils. The transfer efficiency measured is close to the calculation, and the field amplification of the transferred magnetic field compared to the applied field to the pickup coil is confirmed in the device which equipped with the multi-turn input coil. The magnitude of this magnetic field amplification has achieved to 125% of the applied field. This result suggests that the HTS flux transformer proposed in this study is effective in the sensitivity enhancement of the magnetometer for static magnetic fields.

A novel method for calculating the mutual inductance between two perpendicular coils in wireless power transfer

SummaryThe perpendicular coil is increasingly applied in some wireless power transfer (WPT) systems where space position is limited. The existing method of calculating the mutual inductance between perpendicular coils is mainly applied to rectangular and circular coils. However, the practical rectangular coil often has rounded corners, so it unavoidably introduces systematic errors. Concerning this issue, this paper proposed an analytical method to calculate the mutual inductance between mutually perpendicular round‐corner rectangular (RCR) coils. The proposed method is based on a second‐order vector potential (SOVP) model where the mutual inductance between the coils is described as a double integral formula. In this formula, a novel shape and position function is introduced. This shape and position function describes not only RCR coils but also rectangular (including square), circular, and runway ones. A series of experiments were used to validate the proposed method. According to the consistency validation, the proposed method can be more generalized for calculating the mutual inductance of rectangular and circular coils. Through experimental comparison, the proposed method computes the mutual inductance between the perpendicular coil in excellent agreement with the measured ones. Therefore, the proposed method unifies the mutual inductance calculation of mutually perpendicular circular, rectangular (square), runway, and RCR coils and can provide an important basis for designing WPT systems using the above coils.

Improved methods for stator end winding leakage inductance calculation

PurposeCalculating the stator end-winding leakage inductance, taking into account the rotor, is difficult due to the irregular shape of the end-winding. The end-winding leakage may distribute at the end of the active part and the fringing flux of the air gap. The fringing flux belongs to the main flux but goes into the end-winding region. Then, not all the magnetic flux occurring in the end region is the end-winding leakage flux. The purpose of this paper was to find a method to accurately separate the leakage from the total flux, taking into account the rotor.Design/methodology/approachIn this paper, two methods based on energy calculation are presented. Both methods require the assumption that the machine is symmetrical. The first method depends on the total leakage inductance and the machine’s active region length. The second method is based on the energy stored in the end region of the machine. In this case, removing the energy produced by the fringing flux of the air gap is necessary. The model should have a volume-closing fringing flux to remove the part of energy belonging to the end of the air gap.FindingsThe method presented in the paper does not require rotor removal. The values of the end-winding leakage inductance computed based on the proposed method were compared with values computed using the method with the removed rotor. The computations show that the proposed method is closest to the results from the method presented in the literature. Results obtained in the first method present that rotor influence on the value of end-winding leakage inductance exists. The model of the stator end-winding described in the paper is general. Therefore, the proposed methods are suitable for calculating the end-winding leakage inductance of other electric machines.Originality/valueThe method presented in the paper considers the rotor in end-winding leakage inductance calculation. It is not necessary to remove the rotor as in the similar method presented in the literature. The authors elaborated a parametric model with a volume-closing fringing flux to remove the part of energy belonging to the end of the air gap. The authors also elaborated their 3D model of the machine winding for calculations in Opera 3D.