United States

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However the off axis field magnetic involves the subtraction of elliptical integrals and second kind which is computationally expensive, hard to manipulate in equations and difficult to visualize. By doing a different binomial series expansion on the original loop integral a series solution is created, which can be simplified to a set of approximations with useful characteristics: exactly correct at along the axis, and at distance, while at the current loop itself the relative error is limited, computational simple, easy to visualize the behavior and symmetry. Figure 1 shows for the radial field the first and second order fits where the parameters are optimized to give minimize the relative peak error. Note the symmetry that occurs by expressing as function of W, allowing maximum relative errors of 0.025 for 1&lt;sup&gt;st&lt;/sup&gt; order and 2.9E-4 for second order (Figure 2 shows 2&lt;sup&gt;nd&lt;/sup&gt; order relative error in all space). For axial field (Figure 1) the first order, due to the subtraction the accuracy is only modest but the 2&lt;sup&gt;nd&lt;/sup&gt; order has a 9E-4 maximum relative error along, with zero error at the loop on the z=0 plane. The axial field relative error stays low within the loop but outside does, and stays low for any h until it falls to 1% of the z=0 plane value, then lose accuracy as z is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as z further from the reveal. Higher order approximations add more W terms and see increasing accuracy – for radial 3&lt;sup&gt;rd&lt;/sup&gt; order 1.8E-5, 4&lt;sup&gt;th&lt;/sup&gt; 2,3E-6, 5&lt;sup&gt;th&lt;/sup&gt; 4.9E-7, while axil goes as 3&lt;sup&gt;rd&lt;/sup&gt; 4.6E-5, 4&lt;sup&gt;th&lt;/sup&gt; 6E-6. Note that the simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity. The final paper will show how these approximations are created, plus more details on the error distributions of these and higher orders.

![](https://assets.underline.io/uploads/markdown_image/1/image/347293d7d5dee787ac314cbdc7b3a7ca.jpg)

Figure 1: Current loop off axis field approximations of order 1 and 2

![](https://assets.underline.io/uploads/markdown_image/1/image/6324541f77db2ddbf4cd30f9337b8df7.jpg)

Figure 2: Radial field 2nd order approximation relative error

2022 Joint MMM-INTERMAG

Current Loop Off Axis field Approximations with Excellent Accuracy and Low Computational Cost

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However the off axis field magnetic involves the subtraction of elliptical integrals and second kind which is computationally expensive, hard to manipulate in equations and difficult to visualize. By doing a different binomial series expansion on the original loop integral a series solution is created, which can be simplified to a set of approximations with useful characteristics: exactly correct at along the axis, and at distance, while at the current loop itself the relative error is limited, computational simple, easy to visualize the behavior and symmetry. Figure 1 shows for the radial field the first and second order fits where the parameters are optimized to give minimize the relative peak error. Note the symmetry that occurs by expressing as function of W, allowing maximum relative errors of 0.025 for 1st order and 2.9E-4 for second order (Figure 2 shows 2nd order relative error in all space). For axial field (Figure 1) the first order, due to the subtraction the accuracy is only modest but the 2nd order has a 9E-4 maximum relative error along, with zero error at the loop on the z=0 plane. The axial field relative error stays low within the loop but outside does, and stays low for any h until it falls to 1% of the z=0 plane value, then lose accuracy as z is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as z further from the reveal. Higher order approximations add more W terms and see increasing accuracy – for radial 3rd order 1.8E-5, 4th 2,3E-6, 5th 4.9E-7, while axil goes as 3rd 4.6E-5, 4th 6E-6. Note that the simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity. The final paper will show how these approximations are created, plus more details on the error distributions of these and higher orders.

![](https://assets.underline.io/uploads/markdown_image/1/image/347293d7d5dee787ac314cbdc7b3a7ca.jpg)

Figure 1: Current loop off axis field approximations of order 1 and 2

![](https://assets.underline.io/uploads/markdown_image/1/image/6324541f77db2ddbf4cd30f9337b8df7.jpg)

Figure 2: Radial field 2nd order approximation relative error

technical paper

I. Introduction Stator-permanent magnet (PM) electrical machines are potential for developing as high-speed (HS) electrical machines, due to their advantages of high torque density and robust rotor structure. As one of the most promising stator-PM machines, flux-switching permanent magnet machine (FSPMM) has comparable torque density with the rotor-PM machines and easy thermal dissipation. However, the high fundamental frequency of conventional FSPMM is inevitable, making it difficult to be driven by the power electronics device. In this paper, a novel dual-step 2-phase 4/3 FSPMM is proposed and analyzed. Compared with the traditional FSPMM with the odd number of rotor poles, the proposed structure has good flux linkage waveform, non-UMF and high torque density. II. Design and Performance of the Proposed 4/3 FSPM Machine Fig. 1(a) shows the structure of the novel dual-step 4/3 FSPM machine. With using this structure, it can achieve the least number of rotor poles. The PMs have the same magnetization direction, whereas the winding directions are inverse, as shown in Fig. 1(b) and (c). The two set of windings can be connected in series, therefore it is convenient to drive the machine. The rear rotor is offset by 60° mechanically with respect to the corresponding front rotor. An appropriate gap distance between two stators is necessary to accommodate the end of inverse windings and avoid excessive magnet flux leakage. The flange and chamfering rotor is designed to reduce the cogging torque, as shown in Fig.2 (a). The performance of the novel 4/3 FSPM is obtained by using the finite element method and the parameters optimization method. Fig. 2(b) shows that the UMF of the proposed machine. It can be seen that the UMF of the proposed machine is relatively small. The peak-to-peak value of the cogging torque which varies with width of rotor flange is drawn in Fig.2 (c). The designed machine can obtain minimum cogging torque of 360.71 mNm.


![](https://assets.underline.io/uploads/markdown_image/1/image/cfeb923ad4dd15fdd6d408f889bdb4bb.jpg)

Design and Analysis of a Novel 2-Phase 4

![](https://assets.underline.io/uploads/markdown_image/1/image/c9e1a53d9fd13df6d0243077e46d6a55.jpg)

3 FSPMM with Chamfering and Flange Rotor Poles

Design and Analysis of a Novel 2-Phase 4/3 FSPMM with Chamfering and Flange Rotor Poles

Doubly salient brushless DC (DSBLDC) machine has been attracting much attention for aviation and vehicular applications, owing to its simple structure, adjustable flux, and reliability. Despite those merits, the torque ripple of DSBLDC machine limits its applications at medium-and-low speeds. The current harmonic injection (CHI) method has been proposed to reduce torque ripple, dealing with the subsequent issue, such as vibrations, acoustic noise, even starting failure. The traditional CHI choice of odd order harmonics to limit torque ripple is almost useless for the 3-phase 12/8-pole DSBLDC machine, due to the excitation and armature winding configurations, which generate its unique magnetic circuit and inductance characteristics. Firstly, A new instantaneous torque model from the perspective of harmonics has been derived. The main components of the excitation inductance are the 1st, 2nd and 4th harmonics. Only the armature current harmonics interact with the same-order inductance harmonics can contribute to the average torque, otherwise it will lead to torque harmonics. Thus, a harmonic injection algorithm based on the 2nd and 4th current harmonics is proposed to reduce the torque ripple. Fig.1(a) shows the topology of the DSBLDC machine, which is marked by the concentrated stator-field-winding and the absence of rotor excitation. Experiment platform is shown in Fig. 1(b). The instantaneous torque as shown in Fig. 1(c) can be divided into two parts: average torque and torque ripple, which is used to solve the current amplitude and phase parameters of injected current as shown in Fig.2(a). Fig.2 shows the solution based on the harmonic injection algorithm. Case I contains only the fundamental wave. The 2nd harmonic is added into Case II. Case III contains the 1st, 2nd and 4th harmonics, reducing the torque ripple from 54.84% to 44.28%. The torque waveform and its FFT results are shown in Fig. 2(b) and Fig. 2(c). Using the system diagram in Fig. 2(d) can experimentally validate the foregoing simulation, the partial result about Case I is given in Fig. 2(e).

![](https://assets.underline.io/uploads/markdown_image/1/image/06ff416686f82ddb1613b73f4d429ccc.png)

Fig.1 topology and torque model of the 12/8-pole DSBLDC machine.


![](https://assets.underline.io/uploads/markdown_image/1/image/9986831376a4f2901d749908e3201fc4.jpg)

Fig.2 simulation and experimental results based on the harmonic injection algorithm.

Torque Ripple Suppression of Doubly Salient Brushless DC Machine With Even-order Current Harmonic Injection

INTRODUCTION The structure of doubly salient laminations and concentrated windings as well as the operation with unipolar rectangle currents and pulsating torques result in the significant radial magnetic force (RMF) in switched reluctance machines (SRMs), which contributes to the vibration and acoustic noise [1][2]. The analysis of the RMF plays an essential role in SRMs, in which the analytical method (AM) featuring with fast calculation and low computing loading, is helpful in the initial design. However, due to the severe nonlinearity of SRMs, it is of challenge to analytically predict the RMF [3]. Based on magneto-motive force (MMF)-permeance model [4], an AM is developed in this paper, and the saturations are considered via a rotor-position-independent magnetic equivalent circuit (MEC). The AM predictions will be validated by the finite element (FE) results and a prototype SRM. ANALYTICAL MODEL and RMF RESULTS The overall flowchart is shown in Fig. 1. The MMF distribution along air-gap periphery originated from the winding function and excitation, together with the air-gap permeance considering both stator/rotor saliency, are employed to predict the air-gap radial flux density. Then the RMF is calculated with Maxwell stress tensor. The magnetic saturation effect is included via a simplified MEC. The calculated RMFs along the air-gap periphery at one rotor position with single-phase excited are shown in Fig. 2(a), where both the linear and saturated conditions are included, and FE results are presented for the validation. The AM calculations agree well with FE results. Moreover, the AM-RMF waveform along the half air-gap periphery during one electric period is shown in Fig. 2(b). CONCLUSION An AM has been developed for predicting the RMF of SRMs, considering doubly salient laminations, unipolar rectangle currents and magnetic saturations. The calculation results are validated by the FE predictions. In the coming full paper, the details of the method, the flux tube of MEC and saturation factor, the validations on various SRMs with different excitations including overlapped currents, will be presented. A prototype SRM is to be tested.

![](https://assets.underline.io/uploads/markdown_image/1/image/824cbe0d375218c8beaf462fb166234b.jpg)

Fig. 1 Flowchart of AM of RMF.

![](https://assets.underline.io/uploads/markdown_image/1/image/f94362f2599ed11e8d796db183624985.jpg)

Fig. 2 RMF results along air-gap periphery.

Analytical Model of Radial Magnetic Force in Switched Reluctance Machines with Consideration of Magnetic Saturations

I. INTRODUCTION Variable flux memory (VFM) machines were proposed to address the constant air-gap flux and thus sacrificed efficiency in high-speed operations of conventional permanent magnet (PM) machines [1]. The magnetization state (MS) of low coercive force (LCF) PMs can be intentional regulated by d-axis current pulses [2]. The properties and dimensions of LCF PMs are essential in VFM machines, which straightforwardly affect the intentional MS manipulation, torque density and unintentional demagnetization (UD) withstanding [3]. Two VFM machines with different LCF PMs, i.e. SmCo (Hc=335kA/m) and AlNiCo (Hc=112kA/m), are investigated and the guideline of PM design in VFM machines for EV tractions is proposed. II. MACHINE TOPOLOGY and PERFORMANCE Fig. 1 shows the VFM machine cross section, whose outline dimension is inherited from Prius2010 traction machine [4]. Two machine cases employing SmCo and AlNiCo as LCF PMs respectively, together with NdFeB PM to realize hybrid PM configuration for torque improvement, are presented. The NdFeB PM volumes are identical in the two cases while the LCF PM thickness is 4mm for the SmCo case but 12mm for the AlNiCo case, which guarantees an equal PM MMF. As magnetization characteristics shown in Fig. 2, the MS range is 55.11% in the AlNiCo case with the limitation of rated current (236A), which is much higher than 23.15% in the SmCo machine. However, with 2 p.u. current limitation, the MS range of the SmCo case increases to 54.07%, which is close to the AlNiCo case. In addition, the torque of the SmCo case is higher than the AlNiCo case by &gt;10%. III. CONCLUSION Although the identical initial PM MMFs are presented, the AlNiCo machine always requires lower de/re-magnetizing currents than the SmCo case. Nevertheless, the MS ranges are similar in both cases with 2 p.u. current limitation, while the SmCo machine exhibits a higher torque density. More details of the electromagnetic characteristics, LCF PM optimization, operation condition effects, and experimental validation will be demonstrated in the coming full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/9f796652d2ae704bf4bb7ae10908e29a.jpg)

Fig. 1 Cross section of VFM machine.

![](https://assets.underline.io/uploads/markdown_image/1/image/1dbcfb3ab8d3cad6ba3f64b3b37e48ff.jpg)

Fig. 2 Fundamental phase back-EMFs at 1500 r/min after different d-axis excitations.

Investigation on Low Coercive Force Magnets in Variable Flux Memory Machines for Traction Applications

Open-phase failure is typical in multiphase machines [1]. During the open-phase faults and to support the desired developed torque in a six-phase interior permanent magnet motor (IPMSM), various fault tolerant control (FTC) methods have been proposed [2]-[3]. Although the degraded torque of a faulty IPMSM can be compensated via FTC, the noise and vibration (NV) behavior of the machine may still be affected. NV characteristics of healthy and open-phase IPMSMs have been studied before [4]-[6]. However, these characteristics have not been studied in FTC mode for an open-phase IPMSM. This paper, thus, predicts the NV performance of a laboratory six-phase IPMSM in a single open-phase fault under a negative sequence current compensated FTC mode, and the results are compared with the ones under healthy condition. The IPMSM is analyzed using finite element analysis (FEA) software and the single phase fault is remedied by FTC algorithm [2]. Fig. 1(a) shows the IPMSM and its section cut. The currents in healthy and faulty conditions are shown in Fig. 1(b) while phase A is open in faulty condition. Torque comparison is shown in Fig. 1(c). As seen, an average torque of 12.5 Nm is produced under both conditions. However, as presented in Fig. 2(a), radial electromagnetic force density under faulty condition is larger than the one under healthy condition. The acceleration and deformation in Figs. 2(b) and (c) show that the largest difference between the accelerations under two conditions happens at 6,800 Hz causing the sound pressure level (SPL) of the faulty IPMSM to be 14.1% larger than the healthy IPMSM as in Fig. 2(d). Furthermore, the maximum deformation under fault is 11.8% larger than the one under healthy condition. As demonstrated, although the IPMSM under faulty condition with FTC can produce comparable torque as healthy IPMSM, machine NV aspects need attentions. Experimental results and detailed NV analysis including modal analysis will be provided in the full paper.

![](https://assets.underline.io/uploads/markdown_image/1/image/f696d21c3a89261ad880a06f12b0d39d.jpg)

Fig. 1. IPMSM characteristics. (a) The investigated IPMSM and its section cut. (b) Input six phase currents. (c) Torque under two conditions.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8ef2fc43e78a009486f87b8deae8b66.jpg)

Fig. 2. NV performance. (a) Radial EM force density. (b) Radial acceleration. (c) Deformed housings. (d) SPL.

Noise and Vibration Prediction of a Six-Phase IPMSM in a Single Open-Phase Failure Under a Negative Sequence Current Compensated Fault Tolerant Control Mode

Comprehensive parameter identification and analysis are critical to develop accurate machine model. As reported in [1],[2], only two parameters can be estimated properly at the same time since the rank of the machine model is 2. However, in real-time application, machine parameters vary nonlinearly due to the change in operating conditions [3]–[5].Therefore, more accurate knowledge of motor parameters requires more undetermined coefficients in machine modelling, which results in severer rank deficiency problem. To overcome the rank deficiency and achieve a better estimation accuracy of dq-axis inductance modelling, this paper designs an experimentally testing pattern illustrated in Fig. 1 for PMSM and proposes a decoupled estimation scheme in (1) considering system uncertainties including magnetic saturation, core loss, temperature effect and voltage source inverter nonlinearity. (UqId-UdIq)/ωe+(DdIscos(θ)+DqIssin(θ))△V/ωe=IqIqLq+Id(Λ+LdId)+IqIqLqΛ/Ri (1) where variables I, U, L, λ, D with subscript ‘d’ and ‘q’ are current, voltage, inductances, flux linkage and distorted coefficients on dq-axis frame accordingly. Λ and ΔV are PM flux linkage and distorted voltage drop due to dead time effect of voltage source inverter. Ri is the core loss resistance. As mentioned previously, the proposed estimation approach takes core loss effect into account. Therefore, given different speed conditions in (1), the differential model can be achieved by subtracting (1) at the first speed condition denoted as subscript 1 from the one at the second speed condition denoted as subscript 2, which contains the information of iron loss. In the full paper, an efficient least square method incorporating decoupled estimation scheme is employed to converge the undetermined coefficients in the proposed estimation model. The machine parameters including resistance, PM flux linkage and inductances will be estimated and reported. The estimated inductances illustrated in Fig. 2 will be validated through virtual signal injection based MTPA control and more experimental results will be presented to demonstrate the effectiveness of the proposed method.

![](https://assets.underline.io/uploads/markdown_image/1/image/49bc237555843438394366800f84a779.jpg)

Fig.1.Illustration of experimental data collection

![](https://assets.underline.io/uploads/markdown_image/1/image/2df751a5afa6d8760802b04a4e37dbbd.jpg)

Fig.2.Estimated inductance maps.(a)d-axis(b)q-axis

Decoupled Estimation Scheme for PMSMs Towards Accurate Inductance Modelling

Recently, Ma et al. reported a hybrid rare-earth and ferrite permanent magnet (PM)-based spoke-type motor that can generate a comparable torque to the motor used in Prius 2010 with 60% lower cost [1]. However, the paper investigated only the motor having 10% NdFeB PM in the total PM volume on the top of the ferrite when the motor employs 3-phase integrated slot distributed winding (ISDW). This paper investigated the effect of the NdFeB PM position with different volume ratios for 3 different windings, including 3-phase (3Φ) ISDW, 3Φ fractional slot concentrated winding (FSCW), and 6Φ FSCW, on the motor performance using ANSYS Maxwell 2D. Fig. 1 shows the layout of 3 windings and 2 rotors: one with NdFeB PM on the top of the ferrite (MTop) and the other with NdFeB PM on the bottom (MBot). Fig. 2 shows a change in the maximum torque (Tm) and torque ripple (Trip) of the motor with 3 windings and 2 rotors with the height (Hm) of the NdFeB PM. The Tm of both MTop and MBot decreases as the Hm decreases for all windings. This is because the Tm is proportional to the PM volume. On the contrary, a decreasing trend of Trip with respect to an increase in the Hm is observed for both rotor types and windings except for 3Φ ISDW. Further, for the negative Hm, both Tm and Trip of the MTop and MBot show no difference for all windings. On the other hand, for the positive Hm, the MBot shows 0-100 Nm higher Tm but 0.57-18 % lower Trip than the MTop for all windings. These results suggest bottom NdFeB PM for high Tm and the top NdFeB PM for low Trip. We will discuss performance metrics such as irreversible demagnetization, mechanical analysis, and maximum speed in this paper. This work was supported in part by the NSF-I/UCRC for EV-STS under Grant number 1650564.

![](https://assets.underline.io/uploads/markdown_image/1/image/eb8c2b729ce69c6e48fb43468a1ec197.jpg)

Fig. 1. Stator view: (a) 3Φ ISDW, (b) 3Φ FSCW, and (c) 6Φ FSCW; Rotor view with (d) MTop and (e) MBot with (f) detailed description of the magnet.

![](https://assets.underline.io/uploads/markdown_image/1/image/3acedc1513aba35e89496d5cd5980874.jpg)

Fig. 2. (a) Maximum torque and (b) torque ripple of motor having 3Φ ISDW, 3Φ FSCW, and 6Φ FSCW and NdFeB top and bottom on the ferrite.

Performance Evaluation of Hybrid Spoke-type Ferrite-NdFeB Permanent Magnet Motor

1 INTRODUCTION Permanent magnet synchronous motors (PMSMs) are increasingly popular in various industrial applications due to their high power density and high efficiency. In high performance servo applications, it is essential to reduce the cogging torque to improve the positional accuracy. One of the most widely used methods is step skewing with uniform magnets. However, it suffers from unbalanced axial electromagnetic force[1]. Herringbone skew method can eliminate the unbalanced axial electromagnetic force because of the symmetrical structure, while it leads to higher cogging torque and torque ripple than the conventional step skew method[2]. In this paper, a nonuniform step skew method is proposed. The PMs of the nonuniform step skew motor have various thicknesses and various skew angles, and show an axisymmetric distribution meanwhile. The proposed method not only reduces the cogging torque to a greater extent than the uniform skew method, but also completely eliminates the unbalanced axial electromagnetic forces. 2 THE METHOD AND RESULTS Fig. 1 shows the rotor structure of the nonuniform step skew method. The 3D structure of motor with nonuniform step skew is shown in Fig. 1(a), and Fig. 1(b) shows the nonuniform distribution of magnet poles. Fig. 2 shows the FEM analysis results of cogging torque and output torque. Compared to the conventional, the uniform step skew and the herringbone skew motors, cogging torque of the nonuniform step skew motor is reduced by 86%, 44%, 80% and the torque ripple is reduced by 45%,15%,18%, without additional reduction in output torque. In addition, due to the symmetrical structure of PMs, the unbalanced axial electromagnetic force can be eliminated. 3 CONCLUSION This paper has proposed a nonuniform step skew method for cogging torque reduction in PMSMs. Based on FEM analysis results, the motor with nonuniform step skew has less cogging torque and torque ripple than the conventional, the skew and the herringbone skew motors, without more sacrifice of torque density. Moreover, the unbalanced axial electromagnetic force can be completely eliminated with the symmetrical structure.

![](https://assets.underline.io/uploads/markdown_image/1/image/39f6670df5abf00ab325198d12b169af.jpg)

Fig. 1. Rotor structure of the nonuniform step skew motor

![](https://assets.underline.io/uploads/markdown_image/1/image/eae04882046acc25709fb5dcdec79eac.jpg)

Fig. 2. Cogging torque and output torque

A Nonuniform Step Skew Method for Cogging Torque Reduction in Permanent Magnet Synchronous Motor

A finite element method (FEM) is widely used in electromagnetic design and analysis, though a long calculation time and a large computer capacity are generally required. To overcome this issue, the authors proposed a reluctance network analysis (RNA) [1], which is suitable for fast calculation of various electric machines’ characteristics, including transformers and motors. However, previous researches on the RNA were limited to closed magnetic circuits. To expand the application scope of the RNA for a wireless power transfer (WPT) and a particle accelerator, which have open magnetic circuits, this paper presents a novel RNA model for open magnetic circuits with the concept of the Kelvin transformation [2], [3], which has been utilized in the open boundary problems in the FEM. Fig. 1(a) – (c) show the proposed RNA model for open magnetic circuits in case of 2-D and 3-D axisymmetric analysis. As shown in Fig. 1(a), an analytical region including the point at infinity is separated into interior and exterior regions. Then, both regions are divided into multiple elements represented by unit magnetic circuits as shown in Fig. 1(b) and (c). Each reluctance can be determined by dimensions of the element and permeability of the material. In the quadrant regions, dimensions of an element lr, lq, lz, lj, lr’, lq’, lz’, and lj’ can be expressed by the following equations: lr = |r1-r2| (1) lq = r1|q1-q3| = r3|q1-q3| (2) lz = const. (3) lj = p(r1+ r2)sin(|q1+q3|/2) (4) lr’ = |a2/r1’- a2/r2’| (5) lq’ = a2/r1’|q1’-q3’| = a2/r3’|q1’-q3’| (6) lz’ = const. (7) lj’ = p(r1’ + r2’)sin(|q1’+q3’|/2) (8) On the other hand, in the rectangular regions, dimensions of an element are lx, ly, lz, lj, lx’, ly’, lz’, and lj’. Among them, lx and lx’ can be expressed by the following equations: lx = |x1-x2| = |r1-r2| (9) lx’ = |x1’-x2’| = |a2/r1’- a2/r2’| (10) Fig. 2(a) - (c) show specifications of an analytical object and simulation results, respectively. Here, flowing flux density on a red solid line is compared. It is understood that calculated values of the RNA and FEM are in good agreement. Also, the RNA results hardly change even if the value of a changes within the range of the necessary and sufficient spatial resolution.

![](https://assets.underline.io/uploads/markdown_image/1/image/5443a706e016bc514b6319e51c6d51e5.jpg)


![](https://assets.underline.io/uploads/markdown_image/1/image/252ae9506a33d0dae7adcde2c8161a91.jpg)

A Novel Reluctance Network Analysis Applicable for Open Magnetic Circuits

In this work, we present an analytical solution of the magnetic field for a uniformly magnetized cylinder tile (= cylinder volume element, see Fig. 1). The derivation is based on the method of magnetic surface charge densities for arbitrary Cartesian magnetization directions and cylinder tile geometries. In contrast to previous works [1,2], all resulting expressions have been reduced to analytical forms containing only elementary operations, trigonometric functions and elliptic integrals. All special cases (singularities and indefinite expressions) are discussed, allowing an exact evaluation of the magnetic field (up to numerical errors) at any point in space. The results are validated by comparison with finite elements solutions and numerical integration. It is further demonstrated how vectorized implementation of all special functions enables effective computational times in few tens of microseconds on simple mobile CPUs (e.g. Intel i5-8365U). An efficient implementation has been integrated in version 4 of the open-source Python-package Magpylib [3]. Possible applications of the derivations are the analytical study of the Halbach cylinder [4] (see Fig. 2), rotary encoder geometries [5,6], electrical machines [7], or any other magnetic geometry consisting at least partially of cylinder tiles. Furthermore, since a full cylinder can be completely subdivided into cylinder tiles, it allows the study of demagnetization effects while maintaining the rotational symmetry using the magnetostatic moment method [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/bc3e0d7162c81197735b0ceeae63edec.jpg)

Geometry of cylinder tile with parametrization in cylindrical coordinates. Six surfaces contribute to the magnetic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/3b7aea7898e7ab1db5d67b69b6e94d45.jpg)

a) Cross-section through a Halbach cylinder array consisting of twelve cylinder tiles with different magnetization directions. The magnetic field is evaluated along the dashed blue line. b) Field for different numbers of subdivided cylinder tiles as a function of the radial coordinate r.

Current Loop Off Axis field Approximations with Excellent Accuracy and Low Computational Cost

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