nach oben

e+i Elektrotechnik und Informationstechnik

Erschienen in:

Open Access 08.08.2023 | Originalarbeit

Lumped model for the calculation of harmonic eddy current losses in permanent magnets for homogeneous flux distributions considering eddy current reaction flux

verfasst von: Mike Königs, Bernd Löhlein

Erschienen in: e+i Elektrotechnik und Informationstechnik | Ausgabe 5/2023

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

Abstract

This is the first of a series of papers on novel methods for the calculation of eddy current losses in permanent magnets (PMs) and the shortcomings of previously conducted analyses. Eddy current losses in PMs and their mitigation are significant factors when designing inverter-fed electric drives. Especially with the need for drives with high power density, a trend toward increased rotational speeds and, therefore, higher fundamental frequencies and inverter switching frequencies are observable. Higher harmonic frequency components of concentrated windings and fractional slot windings are widely taken into account when designing rare sintered earth magnets for permanent magnet synchronous machines (PMSMs). However, inverter-related losses are rarely simulated in Finite Element Method (FEM) co-simulations and can contribute as a major factor to the power losses in PMSMs. In the worst case, this neglection can lead to the thermal demagnetization of the permanent magnets. Segmentation provides an effective measure to limit eddy current losses. However, not every kind of segmentation proves effective in a drive application. Depending on the frequency behavior of the magnetic flux in a permanent magnet, segmentation can increase eddy current losses within the magnet compared to a non-segmented magnet. This behavior is due to the reaction to the flux caused by the eddy currents within the magnet. The reaction of the eddy currents is often neglected [1, 2], and, in other cases, the vicinity of ferromagnetic material is neglected in analytical calculations [3] but proves crucial for the exact calculation of higher frequency losses [4, 5]. In this paper, an analytical solution to calculate harmonic eddy current losses in permanent magnets, including the reaction flux with homogeneous excitation, is given.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

$A$

Area

$B$

Magnetic flux density

$\vec{E}$

Electric field strength in vectorial from

$f$

Frequency

$h$

Geometric height

$I$

Electrical current

$J$

Current density

$k$

Discrete integration variable from the magnets inside to the outside

$S$

Number of discrete eddy current paths

$K_{1}$

Mathematical expression for a constant for the continuous integration

$l$

Geometric lenght

$l_{1}$

Geometric lenght of a magnet

$l_{2}$

Geometric width of a magnet

$L$

Self inductance

$M$

Mutual inductance

$p$

Conversion variable between current path models

$P$

Active power

$R$

Resistance

$\vec{s}$

Integration pathway element

$U$

Electrical voltage

$x$

Continuous integration variable from the magnets inside to the outside

$\mu_{0}$

Permeability of the vacuum

$\mu_{r}$

Relative material permeability

$\Psi$

Magnetic flux

$\rho$

Specific resistance

1 Introduction

Eddy current losses in permanent magnets of permanent magnet synchronous machines heat the magnet material and therefore reduce the magnetic performance of the permanent magnets. If the magnet heating increases the demagnetization temperature of the PM the magnets will be permanently demagnetized. With the need for higher power densities a need for higher rotational speeds and therefore increased operating and switching frequencies has developed. This development induces challenges in the mitigation of eddy current losses as inductance decreases and higher frequency fields increase in amplitude. The knowledge of eddy current losses in PM and the evaluation of segmentation effectiveness is therefore becoming increasingly vital in the design process of electric drives. In this work the work of Ruoho [6] will be expanded to facilitate eddy current reaction fields.

2 Estimation of eddy current paths in permanent magnets

For an analytic approximation of eddy current losses in permanent magnets we need to assume some boundary conditions, in the case of the Schwarz-mirroring-principle those are only the boundaries at the edges of the magnets [4]. Another approach is the definition of geometric eddy current paths [6]. Based on the approach in [6] an expansion to the solution, including frequency dependant behavior and the reaction of the eddy currents on the flux will be given. As eddy currents tend to form more “rectangular shapes” as shown in Fig. 1 for high frequencies, such as inverter switching frequencies of 4 kHz and above, and more “round shapes” as shown in Fig. 2 for low frequencies a mathematical model that can model both variants will be utilized. The development of eddy current path forms is furthermore a function of material conductivity as widely known by the formulas for the skin depth.

The geometric dimensions of a block-shaped permanent magnet are defined as given in Table 1:

Table 1

Geometric dimensions of the magnet

Quantity	Variable
Lenght of the magnet	$l_{1}$
Width of the magnet	$l_{2}$
Height of the magnet	$h_{m}$

Where the dimensions follow $l_{1}> l_{2}$. A mathematical model can then be found to accommodate both, the eddy current paths in Fig. 1 and Fig. 2. The length of an infinitesimal thin eddy current path can than be calculated as follows:

$$l_{\text{eddy}}=2\cdot(l_{1}-l_{2})+2\cdot x\cdot p$$

(1)

Where x is a variable denoting the distance of the investigated path from the innermost eddy current path and p denotes a variable with a frequency dependant value in between $\pi$ and 4. If $p=\pi$ then the model in Fig. 2 is fulfilled. If $p=4$ then the model in Fig. 1 is fulfilled. Furthermore, the equation for the total area enclosed by the eddy current $A_{\text{eddy}}$ which is needed to calculate the induced voltage can be given as:

$$A_{\text{eddy}}=2\cdot(l_{1}-l_{2})\cdot x+p\cdot x^{2}$$

(2)

With those equations a general equation for the eddy current losses within the magnet can be stated.

3 General eddy current calculations

Starting with the rotation of the electrical field or the inducted voltage for one eddy current path.

$$U_{\text{eddy}}=\oint\vec{E} d\vec{s}=-\frac{d\Psi}{dt}=-\int_{A}\frac{dB}{dt}dA$$

(3)

For homogeneous fields and a constant area $A_{\text{eddy}}$ the integral can be simplified further

$$u_{\text{eddy}}(t)=-\int_{A}\frac{dB}{dt}dA=-A_{\text{eddy}}\cdot dB/dt$$

(4)

With the RMS (Root mean square) vaulue:

$$U_{\text{eddy}}=-A_{\text{eddy}}\cdot 2\pi\cdot f\cdot \hat{B}$$

(5)

Based on Ohms law the Eq. (5) can be modified to be a function of the current density.

$$\begin{array}[]{l}i(t)=\frac{u(t)}{R}=\frac{u(t)\cdot A_{\text{conductor}}}{\rho\cdot l}\\ =\frac{-\int_{A}\frac{dB}{dt}dA\cdot A_{\text{conductor}}}{\rho\cdot l}\end{array}$$

(6)

$$J_{\text{eddy}}(x)=\frac{U_{\text{eddy}}}{\rho\cdot l_{\text{eddy}}}=\frac{-A_{\text{eddy}}(x)\cdot 2\pi\cdot f\cdot \hat{B}}{\rho\cdot l_{\text{eddy}}(x)}$$

(7)

Based on the knowledge of the eddy current the eddy current power dissipation can be calculated:

$$P_{\text{eddy}}=\int_{0}^{h_{m}}\int_{0}^{x}J_{\text{eddy}}(x,h)^{2}\cdot\rho\cdot l_{\text{eddy}}(x)dxdh$$

(8)

When the magnetic flux is assumed to be homogeneous in the direction of the magnet height the equation can be simplified.

$$P_{\text{eddy}}=h_{m}\cdot\int_{0}^{x}J_{\text{eddy}}(x)^{2}\cdot\rho\cdot l_{\text{eddy}}(x)dx$$

(9)

Inserting Eq. (7) in Eq. (9) results in an equation for the eddy current power dissipation in the permanent magnet without taking into account any flux reaction.

$$\begin{array}[]{l}P_{\text{eddy}}=h_{m}\cdot\int_{0}^{x}\left(\frac{-A_{\text{eddy}}(x)\cdot 2\pi\cdot f\cdot\hat{B}}{\rho\cdot l_{\text{eddy}}(x)}\right)^{2}\cdot\rho\cdot l_{\text{eddy}}(x)dx\\ =h_{m}\cdot(2\pi\cdot f\cdot\hat{B})^{2}\cdot\frac{1}{\rho}\cdot\int_{0}^{x}\frac{A_{\text{eddy}}(x)^{2}}{l_{\text{eddy}}(x)}dx\end{array}$$

(10)

The constant $C_{1}$

$$C_{1}=h_{m}\cdot(2\pi\cdot f\cdot\hat{B})^{2}\cdot\frac{1}{\rho}$$

(11)

can furthermore be separated from the integral

$$\begin{array}[]{l}P_{\text{eddy}}=C_{1}\cdot\int_{0}^{x}\frac{A_{\text{eddy}}(x)^{2}}{l_{\text{eddy}}(x)}dx\\ =C_{1}\cdot\int_{0}^{x}\frac{\left(2\cdot(l_{1}-l_{2})\cdot x+p\cdot x^{2}\right)^{2}}{2\cdot(l_{1}-l_{2})+2\cdot x\cdot p}dx\\ =C_{1}\cdot\left[\frac{1}{2}\cdot\left(\frac{\left((l_{1}-l_{2})^{4}\cdot\log(l_{1}-l_{2}+p\cdot x)\right)}{p^{3}}-\frac{\left(x\cdot(l_{1}-l_{2})^{3}\right)}{p^{2}}\right.\right.\\ \left.\left.+\frac{\left(x^{2}\cdot(l_{1}-l_{2})^{2}\right)}{(2\cdot p)}+x^{3}\cdot(l_{1}-l_{2})+\frac{\left(p\cdot x^{4}\right)}{4}\right)\right]_{0}^{\frac{l_{2}}{2}}\end{array}$$

(12)

The integral has been computed using wolframalpha.com. With Eq. (12) the eddy current losses neglecting the reaction of the eddy currents on the flux can be calculated. As can be seen in Eq. (12) the losses do increase with the square of the flux amplitude and the frequency. This does coincide with the equations given by [7].

4 Reaction flux of eddy currents

4.1 Harmonic eddy current calculations

Every eddy current creates, based on Lenz’s law, a magnetic flux opposed to the flux of the source. In the following section, a simple magnetic circuit will be modeled and the equations for the consideration of reaction flux are given. First of all, an inductance for every possible eddy current path is calculated. This can be done utilizing a magnetic equivalent circuit. The magnetic reluctance of the airgap, the magnet, and the yoke is calculated. For high relative permeabilities $\mu_{r}$, the magnetic reluctance of the yoke can be neglected. Similar to ohmic resistances the magnetic reluctance can be calculated as follows:

$$R_{m}=\frac{1}{\mu_{0} \cdot \mu_{r}}\cdot\frac{l_{m}}{A_{m}}$$

(13)

With the length of the magnetic conductor $l_{m}$ and the cross sectional area of the magnetic conductor $A_{m}$ which is in our case equivalent to the area surrounded by the eddy current path. The inductance is a function of magnetic reluctance and the number of turns in the coil. In the case of eddy currents of permanent magnets, the number of turns will be one.

$$\begin{array}[]{l}L_{\text{eddy}}(x)=\frac{1}{\sum R_{m}}=\frac{1}{R_{m,\text{airgap}}+R_{m,\text{magnet}}+R_{m,\text{yoke}}}\\ =\frac{A_{m}(x)}{\frac{l_{m,\text{airgap}}}{\mu_{0}}+\frac{l_{m,\text{magnet}}}{\mu_{0} \cdot \mu_{\text{r,magnet}}}+\frac{l_{m,\text{yoke}}}{\mu_{0} \cdot \mu_{\text{r,yoke}}}}\end{array}$$

(14)

For a high permeability of the yoke material, a homogeneous distribution of the excitation flux, with no load flux density $B_{0}$, can be assumed. The reaction flux will be superposed onto the homogeneous flux to obtain the inhomogeneous flux patterns. The induction equation for one path can then be written as the sum of the external flux derivative, the integration of the flux components by the paths on the outside of the eddy current path, and the integration of the flux components of the inner eddy current paths.

$$\frac{dI_{\text{eddy}}}{dx}(x_{1})=\frac{\int_{{A(x_{1})}}\frac{dB}{dt}dA}{\rho\cdot\frac{l_{\text{eddy}}(x_{1})}{h_{m}}}=\frac{A(x_{1})}{\rho\cdot\frac{l_{\text{eddy}}(x_{1})}{h_{m}}}\cdot\left(\frac{dB_{0}}{dt}-\frac{d}{dt}\left(\int_{{x_{1}}}^{\frac{l_{2}}{2}}\frac{dI_{\text{eddy}}}{dx}(x)\cdot L(x)\cdot\frac{A(x_{1})}{A(x)}dx+\int_{{0}}^{x_{1}}\frac{dI_{\text{eddy}}}{dx}(x)\cdot L(x)dx\right)\right)$$

(15)

While certainly, a solution to Eq. (14) should be possible, the mathematical effort needed seems quite large, therefore a numerical solution in which also complex currents are included is used instead. As done previously the mesh flow method will be utilized with local path resistances coupled with mutual inductances. The magnet is composed of $K$ coils arranged in a similar manner as seen in Fig. 1 and Fig. 2. A single coil is denoted with a lowercase $k$. The resistance of the coil is calculated using the following Eq. (16). An equivalent circuit for harmonic eddy current calculations is shown in Fig. 3.

$$R_{\text{eddy,k}}=\rho\frac{2\cdot(l_{1}-l_{2})+2\cdot k\cdot\frac{l_{2}}{2 \cdot K}\cdot p}{h_{m}\cdot\frac{l_{2}}{2 \cdot K}}$$

(16)

The self inductance remains as stated in Eq. (14)

$$L_{\text{eddy}}(k)=\frac{1}{\sum R_{m}}=\frac{1}{R_{m,\text{airgap}}+R_{m,\text{magnet}}+R_{m,\text{yoke}}}=\frac{\frac{k\cdot l_{2}}{K}\cdot((l_{1}-l_{2})+\frac{k\cdot l_{2}}{K})}{\frac{l_{m,\text{airgap}}}{\mu_{0}}+\frac{l_{m,\text{magnet}}}{\mu_{0} \cdot \mu_{r}}+\frac{l_{m,\text{yoke}}}{\mu_{0} \cdot \mu_{r}}}$$

(17)

The flux of a partial coil $i<k$ is completely embedded in the larger partial coil. Therefore the mutual inductance $M_{ik}$ is equal to the self-inductance of the smaller coil $L_{i}$. This is not the case for the mutual inductance from a larger coil to a smaller coil $i> k$. In this case, the mutual inductance can be stated as follows:

$$M_{ik}=L_{i}\cdot\frac{A(k)}{A(i)}$$

(18)

A matrix representation of the governing harmonic system of equations for $K=2$ coils is given beneath:

$$\begin{pmatrix}A_{1}\\ A_{2}\end{pmatrix}\cdot 2\pi f\cdot \hat{B}=\begin{pmatrix}R_{1}+j\cdot 2\pi f\cdot L_{1}&j\cdot 2\pi f\cdot L_{2}\\ j\cdot 2\pi f\cdot L_{1}\cdot\frac{A_{2}}{A_{1}}&R_{2}+j\cdot 2\pi f\cdot L_{2}\\ \end{pmatrix}\cdot\begin{pmatrix}\underline{I}_{1}\\ \underline{I}_{2}\end{pmatrix}$$

(19)

As low numbers of coils yield inaccurate results increasing the number of coils $K$ will result in an asymptotic approach to a final value. The number of required coils $K$ strongly depends on the magnet dimensions, the frequency and the conductivity of the permanent magnet. Fig. 4 is giving an example of results for different values of $K$. The parameters for this simulation are stated in Table 2. For all further analysis $K=20$ coils will be used since more coils do not result in a significant deviation.

Table 2

Geometric Dimensions

Geometry	Height	Lenght	Width
Magnet	5 mm	100 mm	40 mm
Airgap	1 mm	100 mm	40 mm
Frequency	4 kHz
Conductivity	1.6 $\upmu$$\Omega$m (NdFeB)

4.2 Impact of reaction flux on modeled losses

To review the impact of eddy current reaction flux on the eddy current losses the modeled loss behavior will be investigated. In one case the inductances used for modeling the reaction flux will be considered. In the other case, all reaction inductances are set to zero. The geometric parameters of our sample magnet as seen in Fig. 5 are given beneath. The magnet height will be 5 mm. Both length and width will be 50 mm. A sinusoidal magnetic flux with an amplitude of $\hat{B}=4\,\text{m}\text{T}$ will be used as an estimation of inverter related flux variation.

For low frequencies beneath 1 kHz a low impact of eddy current reaction flux seen in Fig. 6 is observable. This would be the case for winding harmonics or line start operation.

However, a significant amount of eddy current losses in permanent magnets is generated due to inverter-related current fluctuations. State-of-the-art inverters utilize switching frequencies of multiple kHz, e.g. 8 kHz or 16 kHz. In this frequency range the effect of eddy current reaction flux is more pronounced than for low frequencies as can be seen in Fig. 7. A neglection of eddy current reaction flux is therefore not feasible.

5 Eddy current power dissipation in segmented magnets

The given approach is assuming a uniform flux in the direction of the magnet height. Under those circumstances, current displacement due to proximity effects can be neglected and magnet segments can be modeled as independent magnets where S represents the number of Segments.

$$P_{\text{eddy,magnet}}=\sum_{k=1}^{S}P_{\text{eddy,segment}}$$

(21)

6 Comparison with FEM simulations

To evaluate the accuracy of the analytic model a FEM simulation using a quadratic solution approach to every element has been conducted. The effects of the eddy current reaction flux can not be neglected. Therefore the simulation geometry has to include a ferromagnetic yoke. A simple rectangular core structure has been chosen. A permanent magnet and an airgap have been inserted into the core. The geometry is shown in Fig. 8. The excitation of the system is realized utilizing a current-controlled coil. A sinusoidal current is used for the analysis.

Table 3

Geometric Dimensions

	Geometric Dimensions
Geometry	Height	Lenght	Width
Magnet	5 mm	100 mm	40 mm
Airgap	1 mm	100 mm	40 mm

The comparison of FEM simulation and analytic evaluation are given in Fig. 9. The FEM simulation has been conducted for the frequencies 100 Hz, 4 kHz, 8 kHz, 12 kHz, 16 kHz, and 20 kHz. This includes the most common inverter switching frequencies of state-of-the-art inverters. The analytical model matches the results of the FEM simulation well. It has to be noted, that for the given geometry leakage flux is neglected, which as noted in upcoming papers can have a significant effect on eddy current loss behavior.

7 Effects of segmentation on harmonic eddy current losses

Classical widely recognized approaches to calculate eddy current losses in materials [7] conclude a quadratic correlation between eddy current losses and frequency. As can be seen in Fig. 10 this does not seem to be the case with the calculation of eddy current losses in permanent magnets. Under further investigation, it is clear that the loss model pioneered by Bertotti [7, 8] still holds true. The quadratic correlation stated is dependent on constant magnetic flux. The assumption of constant magnetic flux is, even when the amplitude of the excitation current remains constant, not permissible. The magnetic flux is reduced due to the reaction flux of the eddy currents. Therefore for excitations with constant amplitudes of sinusoidal currents, it can not be assumed that a quadratic correlation between frequency and eddy current losses holds true anymore.

Furthermore Fig. 10 gives eddy current losses for multiple segmentations. Interestingly due to the reaction flux, the loss characteristic of the different segmentation numbers do widely differ. For high frequencies, such as inverter switching frequencies, a segmentation can actually increase eddy current losses within the magnet. This is further illustrated in Fig. 11.

Even though non-intuitive at the first glance this behavior absolutely makes sense. A look at power adjustment in DC circuits explains the behavior. If the resistance of the load is zero the dissipated power in the resistance is also zero. The same is true for infinite resistance. Therefore a maximum of power must be found in between. The calculation of eddy current losses in permanent magnets requires a more sophisticated approach. The concept of power adjustment can be extended into the complex plane. The taught about that makes it clear that not only the magnet itself has to be considered in order to minimize eddy current losses in permanent magnets but also the behavior of the complex source and all other complex loads such as core losses in the ferromagnetic material of the drive. Further investigations and papers will take a deeper dive into those topics.

8 Summary and outlook

In this paper, Ruohos [6] work has been expanded. The original work was limited to electrostatic computations. By expanding the approach with the methods of the complex AC-computation a solution that can account for eddy current reaction flux is given. Whilst Ruohos work was able to generate an analytical closed form solution, this is not possible with the equations in the complex form anymore. Therefore a discrete numeric approach has been utilized. It is shown that especially for high frequencies such as inverter switching frequencies of $4\,\text{k}\text{Hz}$ and above, the reaction flux can not be neglected. It has been shown that segmentation can in fact increase eddy current losses. The analytical results match numeric FEM simulations well. In further work, the trivial model will be adapted to include a coil with real inverter switched voltages to account for inverter-related effects. Furthermore, additional work will investigate the impacts on eddy current losses in permanent magnets generated by eddy current losses, hysteresis in ferromagnetic materials, air gap thickness, and leakage flux characteristic. The target is to gradually obtain a trivial model which determines which factors, currently neglected, have to be considered in order to obtain accurate results for eddy current losses in PM and segmentation effectiveness.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Vorheriger Artikel Towards an interoperability roadmap for the energy transition

Nächster Artikel Lumped model eddy current analysis of influence factors on PM segmentation effectiveness

Ou J, Liu Y, Liang D, Doppelbauer M (2019) Investigation of PM Eddy current loss in surface-mounted PM motors caused by PWM. IEEE Trans Power Electron 34(11):11253–11263. https://doi.org/10.1109/TPEL.2019.2895679CrossRef

Ede JD, Atallah K, Jewell GW, Wang JB, Howe D (2007) Effect of axial segmentation of permanent magnets on rotor loss in modular permanent-magnet Brushless machines. IEEE Trans on Ind Applicat 43(5):1207–1213. https://doi.org/10.1109/TIA.2007.904397CrossRef

Yamazaki K, Shina M, Kanou Y, Miwa M, Hagiwara J (2009) Effect of Eddy current loss reduction by segmentation of magnets in synchronous motors: difference between interior and surface types. IEEE Trans Magn 45(10):4756–4759. https://doi.org/10.1109/TMAG.2009.2024159CrossRef

Sahu R, Pellerey P, Laskaris K (2020) Eddy current loss model unifying the effects of reaction field and non-homogeneous 3‑D magnetic field. IEEE Trans Magn 56(2):7508404. https://doi.org/10.1109/TMAG.2019.2953110CrossRef

Hullmann M, Ponick B (2022) General analytical description of the effects of segmentation on Eddy current losses in rectangular magnets. ICEM 2022 – International Conference on Electrical Machines. https://doi.org/10.1109/ICEM51905.2022.9910629

Ruoho S, Santa-Nokki T, Kolehmainen J, Arkkio A (2009) Modeling magnet length in 2‑D finite-element analysis of electric machines. IEEE Trans Magn 45(8):3114–3120. https://doi.org/10.1109/TMAG.2009.2018621CrossRef

Bertotti G (1998) Hysteresis in Magnetism, 1st Edition, Academic Press

Bertotti G (1998) General properties of power losses in soft magnetic materials. IEEE Trans Magn 24(1):621–630CrossRef

Titel: Lumped model for the calculation of harmonic eddy current losses in permanent magnets for homogeneous flux distributions considering eddy current reaction flux
verfasst von: Mike Königs
Bernd Löhlein
Publikationsdatum: 08.08.2023
Verlag: Springer Vienna
Erschienen in: e+i Elektrotechnik und Informationstechnik / Ausgabe 5/2023
Print ISSN: 0932-383X
Elektronische ISSN: 1613-7620
DOI: https://doi.org/10.1007/s00502-023-01147-z

Neuer Inhalt

Bildnachweise

VDI-Icon, Profil Icon, inhalt2, Springer Professional Modul/© Springer Fachmedien Wiesbaden GmbH, Die Gewinner und Laudatoren des Sustainability Award in Automotive 2024/© Uli Regenscheit | ATZlive, Search Icon, Banner Hanser, Suresh Vittal/© Alteryx, Additiv gefertigte Teile/© Marina_Skoropadskaya | Getty Images | iStock, Warnschild "Land unter"/© Bluedesign / Fotolia, Zeitschrift Wissensmanagement Cover, PatentFit-Logo/© Springer Fachmedien Wiesbaden GmbH, ATZ-Webinar: Prototypenfreie Entwicklung durch Offline- und Driver-in-the-Loop-HiL-Tests /© (c) VI-grade, chassis.tech plus 2023/© [M] ATZlive / TÜV SÜD PRODUCT SERVICE GMBH, adäsion-Webinar-Matinee/© krystiannawrocki_ Getty Images

Quantity	Variable
Lenght of the magnet	\(l_{1}\)
Width of the magnet	\(l_{2}\)
Height of the magnet	\(h_{m}\)

Springer Professional

Abstract

Publisher’s Note

1 Introduction

2 Estimation of eddy current paths in permanent magnets

3 General eddy current calculations

4 Reaction flux of eddy currents

4.1 Harmonic eddy current calculations

4.2 Impact of reaction flux on modeled losses

5 Eddy current power dissipation in segmented magnets

6 Comparison with FEM simulations

7 Effects of segmentation on harmonic eddy current losses

8 Summary and outlook

Publisher’s Note

Weitere Artikel der Ausgabe 5/2023

Lumped model eddy current analysis of influence factors on PM segmentation effectiveness

Towards interoperable local energy communities in Austria

Use cases also exist for attackers—how to foster the concept of misuse cases

Interoperability as a key factor for digitalisation — a success story for cross-sector knowledge transfer

Multi-vector optimization scheme for distributed components in energy islands

Towards an interoperability roadmap for the energy transition

Neuer Inhalt

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.