Thanks to dynamic progress in application of numerical computations started in the eighties of the 20th century, currently in analysis and design of electrical machines and electromagnetic transducers, the algorithms determining electromagnetic field in 3D space are commonly applied [1, 2]. Most standard approaches are based on a description of magnetic field by scalar Φ or vector A magnetic potentials, while the electric field is expressed by scalar V or vector T and/or T0 electric potentials. Formulation employing electric potential V is widely applied for description of conductive currents distributions in single and multiply connected conductive domains. Nevertheless, the potential V can only be employed in algorithms for numerical solving electromagnetic field leads in which the magnetic field is described by the magnetic vector potential A. Higher versatility can be achieved when using electric vector potentials. Application of these potentials allow to analyze the electromagnetic field using for description of the magnetic field scalar potential Φ as well as vector potential A. To describe the magnetic field, the electric vector potential T is commonly applied for describing currents in single connected conductive domains , while potential T0 is typically utilized for description of currents in the stranded wire coils  and in eddy currents problems in multiple connected domains (for example holes in massive conductors) [4, 5]. In practice, description of currents in multiply connected conductive domains needs incorporating both T and T0 potentials. Description of the magnetic field sources in 3D space related to the conduction currents by means of electric vector potentials has many advantages [4, 5]. The superiority of the vector potentials over the scalar electric potential formulation or direct numerical application of the Biot-Savart law is especially apparent for description of the current distribution of stranded windings of the electromagnetic transducers. The method based on solving current flow equations and the current vector potential T0 formulation has been proposed to describe the current flow in the stranded windings of complex geometry and current flow paths . In this approach, it is assumed that turn distribution inside of the winding can be treated as “naturally” (i.e. determined by solving the flow field equation) current flow paths . However, it should be noted that taking this assumption often leads to increase of heterogeneity of current density J0 in fragments of winding, in which the current changes the direction. The heterogeneity of current density leads to heterogeneity of turns density inside the coil. To mitigate this problem and ensure homogenous current density distribution, a modification of the classical formulation (CF) described in  has been proposed. It incorporates to EEM equations for the T0 formulation the coefficients γi representing the equivalent conductivity of the finite elements of the stranded winding domains . In this approach the equivalent conductivity γi of each ith facet element of electric facet network (EFN) are determined on basis of ratio of average value of the current density Jav inside the winding to the current density Je,i in the ith element of EFN. The way of calculation of coefficients γi and modification of T0 method has been explained on the example of the simple coil in Figure 1.
The way of calculation equivalent conductivity of the finite elements inside coil region has been discussed in detail in Section 2. This section also consists of equations of T0 method that were a base for formulation of EEM equations. The results of comparative analysis between CF and MF of T0 method on the chosen examples have been presented in Section 3.
2 Edge element equations for T0 method
The starting point for seeking the distribution of the turns inside the stranded coil from Figure 1 is the equations of steady state current flow . These equations, taking into account relation between vector potential T0 and current density vector J0 (i.e. J0 = curl T0), can be expressed as: (1a)
and are solved with respect to following boundary conditions:
where: γ is the material conductivity, and τ0 is the function that describes distribution of the potential T0 on the boundary surface ΓT0 (see Figure 1).
Taking into account the boundary conditions (2) in (1), the equations describing distribution of current vector potential T0 inside region of considered coil will have following form:
In general, the equations (3), obtained by means of T0 method, are solved employing numerical methods based on the space discretization. To solve (3) the 3D Edge Element Method has been applied [9, 10], for which equations (3) can be expressed in the following form:
where: i0c and i0n are the vectors describing the edge values of the potential T0 inside region (Ωc) of the coil and beyond this region (Ωn) respectively ; ks is the transposed loop matrix of EFN , and the vector i0τ describes given source of edge values of T0 on surface ΓT0. The matrices RΩc and RΩn are the coefficient matrices of the EE equations for considered regions Ωc and Ωn in the coil calculated from the following expressions:
where: wfq, wfq are interpolation functions of facet element .
For determining the edge values (i0τ) of potential T0 on the boundary surface ΓT0, the relation between edge quantities i0τ and facet currents of EFN network (facet quantities if0 of current density vector J0, where J0curlT0) has been employed . To determine the i0τ, it is necessary to include also condition that current density in normal direction to surface ΓT0 is equal to zero :
In discussed approach the equations (7) are solved first, next obtained distribution of i0τ vector is introduced to (4) and solved. Nevertheless, to solve equations (7) (at the stage of meshing of the domain) it is necessary to choose one or higher number of cross-sections of the coil on which the current density is specified. Our experience shows that using single cross-section per coil is sufficient for most of the cases, however when modeling the coils of complex geometry of current path to achieve faster convergence of the solving equations (7), it is convenient to increase number of cross-sections. Next, basing on the selected cross-sections the right hand sides of the equations (7) are determined. In the paper, equations (7) are solved for given values of the edge quantities i0τPuPv of the potential T0 of edges PuPv. These edges of the mesh elements correspond to PiPj edges of selected cross-section of the coil (Figure 2) and lay on the surface ΓT0 (Figure 1). For example, for the cross-section illustrated in Figure 2 values i0τPuPv are determined by the following formula :
where: ic is the value of the current in coil, zc is the number of turns, lu,v and li,j are the lengths of the edges PuPv and PiPj, respectively (see Figure 2).
The presented above formulas are the same as for the CF and MF of T0 method. The difference between formulations lays only in the different way of calculation of conductivity γ of the medium. In the classical approach of T0 method, as noted in , “it is assumed that γ is constant and uniform over whole coil domain Ωc, (i.e. γ = γc), whereas in the modified method the value of the conductivity of each finite element γi of the domain (i.e. ith EFN for presented EEM formulation) is determined iteratively to achieve the homogeneity of the current density distribution inside the coil. In kth iteration, the conductivity of ith element is calculated according to the following formula:
where: are the current density in ith facet element and global current density in the coil for (k-1)th iteration, and symbol M is assumed number of refinements (iterations) of current density distribution homogenization”. It is worth to notice that the modification of the T0 method, proposed by authors, is applicable for the coils of constant cross-section of the current path along studied domain.
3 Results of case study problems
In the first stage of studies, the discussed method of description of the windings of the electrical machines has been tested on a case study problem of simple multi-turn coil shown in Figure 1. Considered region has been subdivided into tetrahedron elements. The total number of finite elements was equal to 2.500, while the number of EEM equations of 3D model was over 14.000. The EEM equations have been solved iteratively using ICCG method. The calculations have been performed using our own developed software on PC with Intel 3Ghz I7 processor and 16GB of RAM memory. The values of function τ0i,j along edge Pi,j (see Figure 2) of selected cross-section have been determined to ensure resulting (expected) current density in the coil equal to 5 A/mm2. In the case study problem, single cross-section of the coil has been used to determine the electric vector potential T0 and current density distribution inside the coil. The distributions of: (a) potential T0 inside (Ωc) and outside (Ωn) of the coil region, and (b) current density J0 inside the coil have been determined by means of the CF and the MF of T0 methods. In the CF, the distributions T0 and J0 have been obtained after single solving of equations (4), whereas in the MF, the equations (4) have been solved iteratively concerning expression (8), results have been obtained in kth iteration fulfilling the convergence criterion defined as:
where: εt represents the desired level of uniformity of the current density distribution. Conducted studies on the case study problem shows that satisfactory results have been obtained for εt equal to 10−3. The comparisons of obtained distributions of T0 and J0 have been shown in Figures 3 and 4, respectively.
Analyzing both the distributions, it can be stated that imposed modification of T0 method improves the uniformity of the current density inside the coil. For the CF of T0 method the current density in the straight parts of the coil was equal to expected 5 A/mm2, while inside the regions in which the current density vector changes the direction, the norm of J0 varied markedly from 2.70 to 7.20 A/mm2. In case of MF the norm of current density vector inside whole coil domain was in range from 4.90 to 5.08 A/mm2.
In the next stage, we have tested the discussed approaches on case problem coils of more complex geometry of the current path. Two coils of typical design of high voltage asynchronous motors have been examined, the coils of (a) single layer and (b) double layer winding have been considered, respectively. The region of the coil (a) was discretized with 43 794 tetrahedral elements, while the domain of the coil (b) was subdivided into 44 518 finite elements. The values of function τ0i,j have been determined to ensure resulting (expected) current density inside the coils equal to 5 A/mm2. Calculated distributions of current density J0 by means of the CF and MF of T0 method have been compared in Figures 5 and 6 for coils of single and double layer windings, respectively.
Presented results also showed the superiority of the proposed MF of T0 method over classical approach. For studied coils of single and double layer windings of the high voltage induction motors after application of modified T0 method, the norm of current density inside coils is kept in range from 4.89 to 5.11 A/mm2 (for assumed current density equal to 5 A/mm2).
Working on description of the windings of the electrical machines using the classical and modified T0 methods, the authors wonder if there is any real impact of homogeneity of current density distribution in applied numerical models on functional parameters (i.e. integral quantities as torques, forces and induced voltages – emf’s) of electrical machines. To find answer for this question we have performed calculations of functional parameters – electromagnetic torque and emfs’ waveforms of two purposely selected test permanent magnet machines. The machine (a) with classical stator and rotor cores made of ferromagnetic sheets – PMM-FS motor – (Figure 7a) and (b) double rotor motor with air-core (coreless) of stator – DRACS motor – (Figure 7b) have been examined.
First, the impact of method of description of electromagnetic field sources (by means of modified and classical T0 approaches) on the electromagnetic torque waveform has been examined. Comparisons of calculated torque waveforms have been presented in Figures 8 and 9 for PMM-FS and DRACS motors, respectively. The calculations have been conducted assuming supply of the motors by 3-phase balanced system.
For the motor of classical laminated core (PMM-FS) conducted studies showed that homogeneity of current density inside the coil has marginal impact on the torque waveform. Noted differences between average value and torque ripples were below 0.5% comparing to the results obtained by means of classical T0 method. It is not surprising; keeping in mind that ferromagnetic core focuses the magnetic field along the straight part of the coil. However, considering the coreless motor (DRACS), it should be noted that proper modeling of the current sources has remarkable impact on the average value.
Next, the influence of winding description on emfs’ waveforms has been tested. The calculations have been performed at no-load conditions (open circuit) at rated speeds of the studied motors. The comparisons of the emfs’ waveforms have been presented in Figures 10 and 11 for PMM-FS and DRACS motors, respectively.
The obtained results showed that there is practically no impact of winding description on calculated waveforms – noted differences were below 0.1%.
In the paper, the results of comparative analysis between the classical and modified approach of the description of electrical machines stranded wire windings by means of the T0 method have been presented and discussed. Based on the determined distributions of current density vector, it has been proved that application of proposed modification of the T0 method allows achieving greater homogeneity of distribution current density inside region of the considered winding.
In order to examine the impact of homogeneity of the current density distribution inside the coil, the comparative analysis of functional parameters of test permanent magnet motors has been performed. The obtained results showed that homogeneity of current density distribution has negligible impact on emfs’ and torque waveforms of motors of classical ferromagnetic cores. Therefore, in authors’ opinion, the application of unmodified T0 method of winding description is sufficient to obtain reliable results. Nevertheless, when considering more and more popular coreless machines, presented results showed that adequate modeling of the magnetic field sources has notable impact on electromagnetic torque value and ripples. Therefore, for needs of modeling of coreless electromagnetic transducers it is convenient to consider and apply modification of T0 method for the winding description.
Another potential application of modified formulation of T0 method is, in authors’ opinion, the calculation of forces acting on the end connections of the windings during the inrush and short-circuits currents of high power electrical machines.
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About the article
Published Online: 2017-12-29
Citation Information: Open Physics, Volume 15, Issue 1, Pages 918–923, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0111.
© 2017 R. M. Wojciechowski and C. Jędryczka. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0