Comparison of Eddy Current Loss Calculation Techniques for Axial Flux Motors with Printed Circuit Board Windings

Abstract

In slotless machines, the winding conductors are exposed to the magnetic air gap field, which causes additional eddy current losses, thus decreasing efficiency and affecting thermal utilization. This is the case, inter alia, for axial flux motors equipped with printed circuit board windings, where the winding is made of copper–fiberglass epoxy laminations and located in the air gap. The dominant influencing factors are primarily the width of the conducting tracks and the magnetic air gap flux density and frequency. The evaluation time is a crucial constraint when calculating thousands of different designs for design space exploration or performing multi-objective optimizations. Finite element simulations can achieve very precise results, but unlike semi-analytical approximation functions, they are very time-consuming and therefore not the method of choice for design space exploration. This publication provides a comprehensive overview of a selection of different eddy current loss calculation techniques that are applicable for rectangular tracks and round wire windings. A comparison of the calculated results for a finite element simulation is presented for a slotless axial flux machine with printed circuit board windings and rectangular tracks. The calculation time consumed is also compared. The current density distribution of planar conductors of air gap windings differs from that in electrical steel sheets. In contrast to the methods based on steel sheets, only the adapted methods for conductors in air gaps offer acceptable accuracy. A recommendation is provided for the method that offers the best balance between accuracy and computation time for the early-stage design of slotless axial flux machines.

1. Introduction

Slotless machines have windings located in the air gap that reduce iron losses due to the missing stator teeth, especially at high frequencies. In the case of an ironless single inner stator and a double outer rotor configuration, for example, no stator iron losses occur. Air gap windings can be made, amongst other things, of wound copper wire that is glued to a mechanical support material, like a fiberglass epoxy-compound, or to a self-supporting part [1]. Another option is the use of printed circuit boards (PCBs), where planar conducting tracks are arranged, that have a rectangular cross-sectional shape. The tracks are made of laminated copper and fiberglass laminations, in which the conductors are structured by an etching process. The manufacturing of PCBs is well known in the electronics industry, being highly industrialized and automated at low costs. Due to the flat and discoidal shape of axial flux machines, solid PCB windings can be applied. In this case, the winding conductors are exposed to the magnetic air gap field, where a larger magnetic flux density is present compared to conductors arranged in slots, and the related eddy current losses need to be considered in the design process; see Figure 1. Axial flux machines are known for their torque density, because they utilize a large air gap surface-to-volume ratio [2]. When there is limited space, they can be advantageous due to their short axial length. Multiple axial flux machines can also be stacked to increase the power of the drive. In the case of a variable speed drive system, the power electronics of an inverter and sensorics could be mounted on the same PCB.

Eddy current losses lead to additional joule heating of the windings, which lowers the achievable current-carrying capacity at given maximum temperature constraints. They also cause a breaking torque that decreases the propulsion torque of the electrical machine, which has to be compensated by a larger machine current. The heating of the PCB must be handled with care to ensure its mechanical integrity. The glass transition temperature of the typical PCB substrate FR4 is ${\vartheta }_{\mathrm{g}}=135°\mathrm{C}$ [3] and reaches up to ${\vartheta }_{\mathrm{g}}=180°\mathrm{C}$ in the case of high-temperature boards [4], where the operating temperature is recommended to be $25°\mathrm{C}$ lower [4]. That is why it is important to consider the eddy current losses from the beginning of the design process of axial machines with PCB windings located in an air gap.

Typically, the thin copper sheets of a PCB have a thickness ranging from $18\mathsf{\mu }\mathrm{m}$ to $560\mathsf{\mu }\mathrm{m}$ [5]. Unlike round wire windings, the cross-section of the tracks can be varied locally and adapted to specific design constraints [6]. Apart from magnetic flux density and frequency, track width has a strong influence on eddy current losses. Additionally, joule heating due to eddy current losses has to be taken into account by an appropriate cooling design [7].

In the first design stage, thousands of different samples with several design parameters and constraints need to be evaluated to explore the design space. This requires sizing equations that are calculated within a short amount of time. Often, the design process of electric machines starts with more or less coarse analytical calculated predesigns and determines the eddy current losses by performing finite element analysis (FEA) with the chosen design in a second step, where FEA is very time-consuming. To accelerate the design process, (semi-) analytical expressions for determining the eddy current losses are crucial. There are some approximations that exist, mostly for radial flux machines, but there is no comparison between them regarding accuracy and time consumption to find the most appropriate one. In the table below, a comprehensive overview of different analytical calculation techniques to evaluate the eddy current losses is provided. These techniques were adapted to the conditions of slotless axial flux machines with air gap windings. Methods M3, M7, and M8 originate from different parts of electrical machines, which were transferred to tracks of PCB windings in slotless axial flux machines.

Table 1. Overview of different eddy current loss calculation methods and their respective assumptions.

Method	Original Motor Component	Approach	Magnetic Flux Density Component	Harmonic Content
1	Electrical steel sheets	Eddy current density distribution and homogenous magnetic flux density distribution	Magnetic flux density vector parallel to sheet surface	Working wave and correction factor for harmonic content
2	Electrical steel sheets	Eddy current density distribution and homogenous magnetic flux density distribution	Magnetic flux density vector parallel to sheet surface	Working wave amplitude
3	Motor can	Sinusoidal magnetic flux density distribution in peripheral direction and current density distribution	Only magnetic flux density vector normal to rotor surface	Fourier series coefficients, each order separately
4	Laminated-core edges after stamping and stacking	Rotor surface losses of induction machines	Only magnetic flux density vector normal to rotor surface	Fourier series coefficients, each order separately
5	Air gap windings	Eddy current density distribution and homogenous magnetic flux density distribution	Axial and tangential component of magnetic flux density	Either Fourier series coefficients, each order separately, or harmonic correction factor
6	Round and flat wire in stator slots	Analytical determination of eddy current losses with magnetic flux density from FEA for each conductor in stator slots	Normal and tangential component of magnetic flux density	Fourier series coefficients, each order separately
7	Rotor cage of induction machines	Strip model with equivalent electrical resistance and equivalent circuit	Only axial component of magnetic flux density	Fourier series coefficients, each order separately
8	Surface-mounted permanent rotor magnets	Penetration depth of eddy currents in electrically conductive material	Only axial component of magnetic flux density	Fourier series coefficients, each order separately
9	Ironless planar PCB winding	Lorentzforce and law of conservation of energy	Only axial component of magnetic flux density	Full spectrum with Parseval´s theorem
10	Round wire air gap windings in coreless axial flux machine	Current displacement with 2D partial differential equations	Only axial component of magnetic flux density	Fourier series coefficients, each order separately

presents a brief overview of the calculation techniques, their origin, the basic approach, and the handling of normal/axial and tangential components of the spatial magnetic flux density vector and their harmonic content. The results were compared to FEA regarding accuracy and time consumption. The focus is on ironless/slotless windings with a rectangular cross-sectional shape. Eddy current loss equations for round wires can be found in [2,8,9]. These techniques can be used for PCBs where the tracks are augmented with round wire welded onto a track [10].

Some of the presented techniques are also applicable to eddy current loss evaluation in permanent magnets or to slotted machine designs if the magnetic flux density at each location of the conductors in the slot is known. If the copper tracks are placed in slots, they are shielded from the air gap field. In the case of straight slot openings, the air gap field penetrates the tracks located close to the air gap. Even through air gap flux density drops significantly over the slot openings, the eddy current losses in the affected conductors cannot be neglected. Slotted designs also have a magnetic slot stray field caused by the slot conductor currents themselves, which generate eddy current losses. This is typical, e.g., for hairpin windings [11,12,13,14,15], but this is not within the scope of this paper.

2. Eddy Current Loss Calculation Techniques

2.1. Prerequisites

In the case of rotating instead of alternating flux density, the magnetic flux density is split into an axial and a tangential component, each represented by Fourier series coefficients, and they can be evaluated, e.g., using the method in [16] or finite element analysis. If not explicitly stated, calculation techniques only consider the axial flux density component. Current-carrying conductors only generate torque when they are close to the vertices of the magnetic flux density. However, eddy current losses are primarily generated when the conductors are close to the pole change domain. Thus, the current controller injects the phase current when the conductor is close to the center of the magnet, and the phase current is close to zero when the conductor is close to the pole change domain. That is why joule losses due to phase current and eddy current losses do not appear simultaneously. Eddy current losses can therefore be determined independently. In axial flux machines, a radial dependency of the magnetic flux density can occur, particularly when there is a large effective air gap and a small outer-to-inner radius ratio. Eddy current losses have to be evaluated piecewise by considering the corresponding magnetic flux density. Compared to the penetration depth, the tracks in PCB motors are so thin that current displacement can be neglected. It will be assumed that there is no repercussion on the excited magnetic field due to the eddy currents. According to [17], the specific electrical conductivity of copper depends on its temperature according to (1), where the specific electrical conductivity of copper ${\sigma }_{\mathrm{cu},{20}^{\circ }\mathrm{C}}$ at $20°\mathrm{C}$ corresponds to $58·{10}^6\mathrm{S}/\mathrm{m}$ and ${\alpha }_{\vartheta ,\mathrm{cu}}$ to approximately $0.00392{\mathrm{K}}^{-1}$.

$${\sigma }_{\mathrm{cu}}\left(\vartheta \right)={\displaystyle \frac{{\sigma }_{\mathrm{cu},{20}^{\circ }\mathrm{C}}}{1+{\alpha }_{\vartheta ,\mathrm{cu}}·\left(\vartheta -20°\mathrm{C}\right)}}$$

(1)

2.2. Method 1

In [17], the specific eddy current losses of an electrical steel sheet are determined based on a given sheet thickness and infinite in-plane dimensions. This result can be carried over to rectangular conductors by equating the sheet thickness with the conductor width $w_{\mathrm{Cu}}$ and integrating the eddy current losses over the conductor thickness $h_{\mathrm{Cu}}$. This leads to (2)–(4), where $k_1$ considers the harmonic content of the magnetic flux density and $k_2$ refers to rotating magnetic flux density with an axial and a tangential component.

$$P_{\mathrm{eddy}}=h_{\mathrm{cu}}{l}_{\mathrm{cu}}{\left(w_{\mathrm{cu}}\omega B_{\max }\right)}^2{\displaystyle \frac{\sigma }{24\rho }}{k}_1{k}_2$$

(2)

$$k_1={\displaystyle \frac{B_1^2+{\left(3B_3\right)}^2+{\left(5B_5\right)}^2+\dots }{B_{\max }^2}}={\displaystyle \frac{{\sum }_{\nu }{\left(\nu B_{\nu }\right)}^2}{B_{\max }^2}}$$

(3)

$$k_2=1+{\displaystyle \frac{B_{\tan }^2}{B_{\mathrm{ax}}^2}}$$

(4)

For axial flux motors, the conductor length is then according to (5):

$$l_{\mathrm{cu}}=r_{\mathrm{o}}-r_{\mathrm{i}}$$

(5)

2.3. Method 2

The calculation technique (6) with (7) according [18] is also based on the eddy current loss determination of electrical steel sheets. Ref. [18] assumes that the magnetic flux density vector is parallel to the sheet plane where the sheet has an infinite in-plane dimension and the magnetic flux density vector is homogeneously distributed and only changes with time.

$$P_{\mathrm{eddy}}=\sigma {\left({\displaystyle \frac{\pi }{2\sqrt{2}}}{w}_{\mathrm{cu}}f\widehat{B}\right)}^2{\displaystyle \frac{4}{\xi }}·{\displaystyle \frac{sinh\left(\xi \right)-sin\left(\xi \right)}{cosh\left(\xi \right)-cos\left(\xi \right)}}{V}_{\mathrm{cu}}$$

(6)

$$\xi ={\displaystyle \frac{w_{\mathrm{cu}}}{{\delta }_{\mathrm{e}}}}=w_{\mathrm{cu}}\sqrt{\pi f{\mu }_0{\mu }_{\mathrm{r}}\sigma }$$

(7)

The eddy currents cause a magnetic field that dampens the excited one. This leads to a different eddy current distribution depending on the sheet thickness and penetration depth. That is why the author of [18] differentiates between thin and thick sheets based on the sheet thickness-to-penetration depth ratio $\xi$ to simplify (6). According to [18], a thin sheet is approximately entirely penetrated by eddy currents when $\xi <2$; thus, (6) becomes (8).

$$p_{\mathrm{eddy}}={\displaystyle \frac{1}{3}}{\left(\pi f{w}_{\mathrm{cu}}\widehat{B}\right)}^2{\displaystyle \frac{\sigma }{2}}$$

(8)

and (6) becomes (9) for thick sheets if $\xi >3$ [18].

$$p_{\mathrm{eddy}}={\displaystyle \frac{w_{\mathrm{cu}}}{2}}{\widehat{B}}^2\sqrt{{\displaystyle \frac{{\pi }^3{f}^3\sigma }{{\mu }_0{\mu }_{\mathrm{r}}}}}$$

(9)

2.4. Method 3

Some electrical pump drives use canned motors. The can is located in the air gap, where the can hermetically seals the conveyed fluid from the stator and the rotor is in touch with the fluid. The can is a thin-walled sleeve made of non-magnetic steel, like Hastelloy C and Austenite 1.4571 [19]. Due to the specific electrical conductivity of these steels ($\sigma \left(\mathrm{Hastelloy}\right)=\mathrm{806,452}\mathrm{S}/\mathrm{m}$, $\sigma \left(\mathrm{Austenite}1.4571\right)=\mathrm{1,333,333}\mathrm{S}/\mathrm{m}$ [19]), eddy current losses cannot be neglected. The specific eddy current losses in the can are determined separately for each harmonic wave [20] by calculating (10) and (11).

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy},\nu }& =& {\int }_V{J}_{\mathrm{cu}}{E}_{\mathrm{cu}}\mathrm{d}V={\int }_V{E}_{\mathrm{cu}}^2\sigma \mathrm{d}V\hfill \\ & =& {\int }_V{\left(B_{\nu }r\Omega \right)}^2\sigma \mathrm{d}V\hfill \end{array}$$

(10)

$$B_{\nu }={\widehat{B}}_{\nu }cos\left(\nu p\alpha -{\phi }_{\nu }\left(t\right)\right)$$

(11)

Unlike a track, a can forms a consistent conductor in the peripheral direction, where the integral of the current density in the peripheral direction is a zero mean. Similarly to [21], the eddy current density is also the zero mean over the cross-section of the track. Thus, (10) has to be adapted so that only the AC component of the magnetic flux density in the conductor has an effect on instantaneous eddy current losses. As such, (10) becomes (12) with (11), where ${\phi }_{\nu }\left(t\right)$ corresponds to the rotor position.

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy},\nu }\left({\phi }_{\nu }\left(t\right)\right)& =& {\int }_V{p}_{\mathrm{eddy},\nu }\mathrm{d}V\hfill \\ & =& l_{\mathrm{cu}}{h}_{\mathrm{cu}}{\sigma }_{\mathrm{cu}}{\left(r\Omega \right)}^2·{\int }_{{\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}}^{{\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}}{\left(B_{\nu }-{\displaystyle \frac{1}{{\alpha }_{\mathrm{cu}}}}{\int }_{{\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}}^{{\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}}{B}_{\nu }\mathrm{d}\alpha \right)}^{2}r\mathrm{d}\alpha \hfill \end{array}$$

(12)

To receive the average eddy current losses, the instantaneous eddy current losses have to be averaged over a rotational period of the rotor, where the eddy current losses of each harmonic wave are summed up (13).

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& \sum _{\nu }{\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{P}_{\mathrm{eddy},\nu }\left(\phi \right)\mathrm{d}\phi \hfill \\ & =& \sum _{\nu }{l}_{\mathrm{cu}}{h}_{\mathrm{cu}}\sigma r^3{\Omega }^2{B}_{\nu }^2·\left({\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}-{\displaystyle \frac{2{sin}^2\left(\frac{{\alpha }_{\mathrm{cu}}}{2}\nu p\right)}{{\alpha }_{\mathrm{cu}}{\nu }^2{p}^2}}\right)\hfill \end{array}$$

(13)

$${\alpha }_{\mathrm{cu}}=2·arcsin\left({\displaystyle \frac{w_{\mathrm{cu}}}{2r_{\mathrm{mean}}}}\right)$$

(14)

2.5. Method 4

So-called surface losses can appear in the rotors of induction machines, caused by electrical bridges between the edges of electrical steel sheets after stamping and stacking [18]. These bridges are forming a conductor tangential to the rotor surface. Due to the slotted stator, the magnetic air gap flux density drops over the slot openings, which causes slot-frequent magnetic flux density waves. It is assumed that these pulsating magnetic flux density waves penetrate the rotor tangential to the rotor surface, where the equivalent sheet thickness is ${\tau }_{\mathrm{slot},\mathrm{s}}/2$. The resulting frequency $f_{\mathrm{slot}}$ can be calculated according to (15).

$$f_{\mathrm{slot}}=N_{\mathrm{slot},\mathrm{s}}·n_0$$

(15)

Thus, the eddy current losses yield (16) and (17).

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& N_{\mathrm{slot},\mathrm{r}}\left({\tau }_{\mathrm{slot},\mathrm{r}}-s_{\mathrm{slot},\mathrm{r}}\right){\left({\displaystyle \frac{{\tau }_{\mathrm{slot},\mathrm{s}}}{4}}{\displaystyle \frac{2}{\pi }}\beta k_{\mathrm{Cs}}{\displaystyle \frac{{\widehat{B}}_{\delta }}{\sqrt{2}}}\right)}^2{L}_{\mathrm{fe}}{\left(N_{\mathrm{slot},\mathrm{s}}n60{\displaystyle \frac{\mathrm{s}}{\min }}\right)}^{{\displaystyle \frac{3}{2}}}·\hfill \\ & & \sqrt{{\displaystyle \frac{{\pi }^3\sigma }{{\mu }_0{\mu }_{\mathrm{r}}}}}{k}_{\exp }\hfill \end{array}$$

(16)

$$\beta ={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\frac{2\delta }{s_{\mathrm{slot},\mathrm{s}}}}{\sqrt{1+{\left(\frac{2\delta }{s_{\mathrm{slot},\mathrm{s}}}\right)}^2}}}\right)$$

(17)

It has to be considered that for $k_{\exp }$, ref. [18] uses a value between 0.08 and 0.13, which was evaluated in experiments by Richter for a $0.5\mathrm{mm}$ iron sheet of type IV. Also, this calculation technique is based on the thick-sheet approximation introduced in Section 2.3, and the eddy current losses are proportional to $n^{{\displaystyle \frac{3}{2}}}$ instead of $n^2$.

Adapted to rectangular tracks exposed to the magnetic air gap field of a slotless axial flux machine, (16) becomes (18), where $\nu$ is the corresponding harmonic wave order and ${\tau }_{\mathrm{slot},\mathrm{s}}/2$ is the equivalent sheet thickness that corresponds to the track width.

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& \sum _{\nu }{N}_{\mathrm{cu}}\left(r_{\mathrm{o}}-r_{\mathrm{i}}\right)h_{\mathrm{Cu}}{\left({\displaystyle \frac{w_{\mathrm{cu}}{k}_{\mathrm{Cs}}{\widehat{B}}_{\nu }}{\pi \sqrt{2}}}\right)}^2·{\left(\nu pn60{\displaystyle \frac{\mathrm{s}}{\min }}\right)}^{{\displaystyle \frac{3}{2}}}\sqrt{{\displaystyle \frac{{\pi }^3\sigma }{{\mu }_0{\mu }_{\mathrm{r}}}}}·k_{\exp }\hfill \end{array}$$

(18)

2.6. Method 5

In [2], rotational magnetic air gap flux density is considered and split into an axial and tangential component, ${\widehat{B}}_{\mathrm{ax},\nu }$ and ${\widehat{B}}_{\tan ,\nu }$, respectively. Either the eddy current losses (19) are evaluated for each harmonic wave and then summed up or the eddy current losses are first evaluated for the fundamental wave according to (20) and then corrected by the distortion factor ${\eta }_{\mathrm{d}}$ in (21) [2].

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& {\displaystyle \frac{{\pi }^2\sigma }{3\rho }}{f}^2{w}_{\mathrm{cu}}^2{m}_{\mathrm{con}}\sum _{\nu }{\nu }^2\left({\widehat{B}}_{\mathrm{ax},\nu }^2+{\widehat{B}}_{\tan ,\nu }^2\right)\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& =& {\displaystyle \frac{{\pi }^2\sigma }{3\rho }}{f}^2{w}_{\mathrm{cu}}^2{m}_{\mathrm{con}}{\eta }_{\mathrm{d}}^2\left({\widehat{B}}_{\mathrm{ax},1}^2+{\widehat{B}}_{\tan ,1}^2\right)\hfill \end{array}$$

(20)

$${\eta }_{\mathrm{d}}=\sqrt{\sum _{\nu }{\displaystyle \frac{{\nu }^2\left({\widehat{B}}_{\mathrm{ax},\nu }^2+{\widehat{B}}_{\tan ,\nu }^2\right)}{{\widehat{B}}_{\mathrm{ax},1}^2+{\widehat{B}}_{\tan ,1}^2}}}$$

(21)

These expressions are valid for rectangular conductors. In the case of slotless windings made of round wire, (22) can be applied [2].

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& {\displaystyle \frac{{\pi }^2\sigma }{4\rho }}{f}^2{d}_{\mathrm{cu}}^2{m}_{\mathrm{cu}}\sum _{\nu }{\nu }^2\left({\widehat{B}}_{\mathrm{ax},\nu }^2+{\widehat{B}}_{\tan ,\nu }^2\right)\hfill \\ & =& {\displaystyle \frac{{\pi }^2\sigma }{4\rho }}{f}^2{d}_{\mathrm{cu}}^2{m}_{\mathrm{cu}}{\eta }_{\mathrm{d}}\left({\widehat{B}}_{\mathrm{ax},1}^2+{\widehat{B}}_{\tan ,1}^2\right)\hfill \end{array}$$

(22)

2.7. Method 6

The technique applied in [8] first calculates the magnetic field distribution using FEA and then determines the eddy current losses for each single conductor through a hybrid analytical expression. This technique is independent of slotted or non-slotted electrical machine design. An expression for rectangular conductors and round wire is given in (23) and (24), respectively [8].

$$P_{\mathrm{eddy}}=\sum _{\nu }{\displaystyle \frac{l_{\mathrm{cu}}{N}_{\mathrm{cu}}{w}_{\mathrm{cu}}{h}_{\mathrm{cu}}{\omega }_{\nu }^2\sigma }{24}}\left(w_{\mathrm{cu}}^2{\widehat{B}}_{\mathrm{ax},\nu }^2+h_{\mathrm{cu}}^2{\widehat{B}}_{\tan ,\nu }\right)$$

(23)

$$P_{\mathrm{eddy}}=\sum _{\nu }{\displaystyle \frac{\pi }{128}}{l}_{\mathrm{cu}}{N}_{cu}{d}_{\mathrm{cu}}^4{\widehat{B}}_{\nu }^2{\omega }_{\nu }^2\sigma$$

(24)

2.8. Method 7

Similarly to [19,22], a track can be virtually divided into N strips, each represented by an ohmic resistance (see Figure 2), where [22] uses parameters based on [23]. Tangential currents between adjacent strips are neglected. Within each strip, an electromotive force (EMF) is induced, which can be described by a complex voltage phasor according to (25) and (26).

$${\underline{U}}_{\mathrm{n},\nu }=U_{\mathrm{EMF}}{\mathrm{e}}^{j{\displaystyle \frac{np\nu {\alpha }_{\mathrm{cu}}}{N_{\mathrm{strip}}}}}={\displaystyle \frac{1}{\sqrt{2}}}\widehat{B}\Omega {\displaystyle \frac{r_{\mathrm{o}}^2-r_{\mathrm{i}}^2}{2}}{\mathrm{e}}^{j{\displaystyle \frac{np\nu {\alpha }_{\mathrm{cu}}}{N_{\mathrm{strip}}}}}$$

(25)

$$n=0,1,\dots ,N_{\mathrm{strip}}-1$$

(26)

The ohmic resistance of a single strip is given in (27).

$$R_{\mathrm{strip}}={\displaystyle \frac{r_{\mathrm{o}}-r_{\mathrm{i}}}{\sigma \frac{w_{\mathrm{cu}}}{N_{\mathrm{strip}}}{h}_{\mathrm{cu}}}}$$

(27)

Each single strip current can be calculated using mesh current analysis or superposition according to Helmholtz. In the case of two strips, the eddy current losses result in (28).

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& \sum _{\nu }{\displaystyle \frac{|{\underline{U}}_{1,\nu }-{\underline{U}}_{2,\nu }{|}^2}{2R_{\mathrm{strip}}}}\hfill \\ & =& \sum _{\nu }{\displaystyle \frac{{\widehat{B}}_{\nu }^2{\Omega }^2}{16}}{\displaystyle \frac{{\left(r_{\mathrm{o}}^2-r_{\mathrm{i}}\right)}^2}{r_{\mathrm{o}}-r_{\mathrm{i}}}}\sigma w_{\mathrm{cu}}{h}_{\mathrm{cu}}·\left(1-cos\left(p\nu {\displaystyle \frac{{\alpha }_{\mathrm{cu}}}{2}}\right)\right)\hfill \end{array}$$

(28)

In the case of more than two strips, a fictive current for each single strip is determined first, as shown in Figure 3. All voltage sources, except the EMF of the corresponding strip, are masked out, and the corresponding fictive current is calculated according to (29).

$$\begin{array}{ccc}\hfill {\underline{I}}_{\mathrm{fic},\mathrm{n},\nu }& =& {\displaystyle \frac{{\underline{U}}_{\mathrm{n},\nu }}{R_{\mathrm{strip}}·\left(1+\frac{1}{N_{\mathrm{strip}}-1}\right)}}\hfill \end{array}$$

(29)

The strip current of each strip is then evaluated by Helmholtz superposition in (30); thus, the eddy current losses result in (31).

$$\begin{array}{ccc}\hfill {\underline{I}}_{\mathrm{strip},\mathrm{n},\nu }& =& {\underline{I}}_{\mathrm{fic},\mathrm{n},\nu }+\sum _{k\ne n}{\displaystyle \frac{{\underline{I}}_{\mathrm{fic},\mathrm{k},\nu }}{N_{\mathrm{strip}}-1}}\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}& =& \sum _{\nu }\sum _n{R}_{\mathrm{strip}}{\left|{\underline{I}}_{\mathrm{strip},\mathrm{n},\nu }\right|}^2\hfill \end{array}$$

(31)

2.9. Method 8

In surface-mounted permanent magnet machines, slot-frequent harmonic flux density waves also cause eddy current losses depending on the specific conductivity of the permanent magnet material. Ref. [24] applied a technique to a permanent magnet motor with concentrated windings by solving two-dimensional partial equations. So, the eddy current losses for each single harmonic wave can be calculated according to (32), in which $f_{\nu }$ has to be applied with respect to the relative movement and direction of the harmonic wave. ${\sigma }_{\mathrm{eff}}$ describes the reduction in eddy currents, as their circular trajectory close to the magnet ends in a radial direction [24].

$$\begin{array}{c}\hfill P_{\mathrm{eddy},\nu }=\left(r_{\mathrm{o}}-r_{\mathrm{i}}\right)w_{\mathrm{cu}}{h}_{\mathrm{cu}}{\left(f_{\nu }{\widehat{B}}_{\nu }{w}_{\mathrm{cu}}\pi \right)}^2{\sigma }_{\mathrm{eff}}{\displaystyle \frac{K_{\mathrm{cu},\nu }}{6}}\end{array}$$

(32)

$${\sigma }_{\mathrm{eff}}={\displaystyle \frac{\sigma }{1+\frac{w_{\mathrm{cu}}}{l_{\mathrm{cu}}}}}$$

(33)

$$K_{\mathrm{cu}}={\displaystyle \frac{3}{\xi }}·{\displaystyle \frac{sinh\left(\xi \right)-sin\left(\xi \right)}{cosh\left(\xi \right)-cos\left(\xi \right)}}$$

(34)

$$\xi ={\displaystyle \frac{w_{\mathrm{cu}}}{{\delta }_{\mathrm{e},\nu }}}=w_{\mathrm{cu}}·\sqrt{{\mu }_{\mathrm{cu}}{\sigma }_{\mathrm{eff}}\pi f_{\nu }}$$

(35)

2.10. Method 9

According to the law of conservation of energy, eddy current losses are equal to the mechanical braking power affecting the conductor. Each time a conductor moves relatively through a pole change domain, a braking torque pulse is generated by eddy currents. The instantaneous torque can be calculated according to (36) at the mean radius (39) by applying Parseval’s theorem, where complex Fourier series coefficients describe the magnetic air gap flux density distribution along the entire mechanical circumference [21]. The related equations were adapted to a fixed track position and moving axial magnetic flux density. ${\underline{c}}_{\mathrm{B},0}$ represents the complex Fourier series coefficients of the rotor in the starting position and ${\phi }_{\mathrm{rotor}}\left(t\right)$ is the mechanical instantaneous rotor angle according to (37) and (38). The effect of the track width is described by (40).

$$\begin{array}{ccc}\hfill P_{\mathrm{eddy}}\left(t\right)& =& F\left(t\right)·\overline{r}·\Omega \left(t\right)\hfill \\ & =& -{\displaystyle \frac{1}{3}}\left(r_o^3-r_i^3\right)\overline{r}{\sigma }_{\mathrm{Cu}}{\Omega }^2\left(t\right)h_{\mathrm{Cu}}·\sum _{k=-\infty }^{\infty }\sum _{l=-\infty }^{\infty }{\underline{c}}_{\mathrm{B},\mathrm{l}}{\underline{c}}_{\mathrm{B},\mathrm{k}}^{*}\epsilon (k,l)\hfill \end{array}$$

(36)

$${\underline{c}}_{\mathrm{B},\mathrm{l}}={\underline{c}}_{\mathrm{B},0}\left(l\right)·{\mathrm{e}}^{\mathrm{j}l{\phi }_{\mathrm{rotor}}\left(t\right)}$$

(37)

$${\underline{c}}_{\mathrm{B},\mathrm{k}}={\underline{c}}_{\mathrm{B},0}\left(k\right)·{\mathrm{e}}^{\mathrm{j}k{\phi }_{\mathrm{rotor}}\left(t\right)}$$

(38)

$$\overline{r}={\displaystyle \frac{r_{\mathrm{o}}+r_{\mathrm{i}}}{2}}$$

(39)

$$\epsilon \left(t\right)=\left\{\begin{array}{cc}{\alpha }_{\mathrm{cu}}+{\displaystyle \frac{2cos\left({\alpha }_{\mathrm{cu}}k\right)-2}{{\alpha }_{\mathrm{cu}}{k}^2}}\hfill & k=l,k\ne 0\hfill \\ \hfill & \\ {\displaystyle \frac{2sin\left(\frac{{\alpha }_{\mathrm{cu}}}{2}\left(k-l\right)\right)}{k-l}}\hfill & l\ne k,k\ne 0,\hfill \\ -{\displaystyle \frac{4sin\left(\frac{{\alpha }_{\mathrm{cu}}}{2}k\right)sin\left(\frac{{\alpha }_{\mathrm{cu}}}{2}l\right)}{{\alpha }_{\mathrm{cu}}kl}}\hfill & l\ne 0\hfill \\ \hfill & \\ 0\hfill & \mathrm{else}\hfill \end{array}\right.$$

(40)

Subsequently, the instantaneous eddy current losses have to be averaged to obtain the average eddy current losses and are then multiplied by the total number of conductors. This technique considers an arbitrary harmonic content in the magnetic flux density without accepting calculation errors by handling each harmonic separately.

2.11. Method 10

For round wire winding [9], the eddy current loss density was determined according to (41) for each harmonic flux density wave and then integrated along the wire.

$$P_{\mathrm{eddy}}=\sum _{\nu }{\displaystyle \frac{\pi }{32}}{d}_{\mathrm{cu}}^4{\widehat{B}}_{\nu }^2{\nu }^2\sigma {\Omega }^2{l}_{\mathrm{cu}}$$

(41)

3. Benchmarking/Comparison of Calculation Results

The semi-analytical eddy current loss calculation techniques presented above were compared to FEA results from Ansys Maxwell regarding computation time consumption and calculation error. The specimen used was a slotless axial flux machine with surface-mounted permanent magnets, where the magnets had a rectangular cross-sectional shape. A single radially arranged copper track was located in the center of the PCB and had a constant width in the radial direction, with the PCB and the stator yoke being separated by the thickness of the thermal interface material (TIM). The FEA model in Figure 4 shows a segment where the magnetic field is periodically continued circumferentially by defining independent and dependent boundary planes. This model is based on the axial flux machine shown in Figure 1. Based on previous studies, it is assumed that the repercussion of the eddy currents in the conducting tracks of the PCB winding on the excited field is too low and was neglected. The objective of the eddy current loss techniques is to calculate the losses for a single track and then scale them up for the entire PCB winding.

The rotor had 22 poles and rotated at different speeds from 0 to $6000\mathrm{r}/\min$, so the electrical fundamental frequency ranged from 0 to $1100\mathrm{Hz}$. The eddy current losses were determined at a track temperature of $100°\mathrm{C}$.

The semi-analytical eddy current loss calculation techniques were fed with the axial and tangential magnetic flux density distribution Fourier series amplitudes, determined by FEA. The axial and tangential components of the magnetic flux density distribution in the circumferential direction at the mean radius are shown in Figure 5. The tangential magnetic flux density was considered in the corresponding methods 1, 5, and 6.

Magnetic flux density varies depending on the radius (see Figure 6 and Figure 7), so the semi-analytical calculations were performed piecewise from the inner to the outer radius in six pieces. If the radial variation of the axial magnetic flux density is neglected, the calculation error will be larger.

Because eddy current losses strongly depend on track width, this was varied from $1\mathrm{mm}$ to $5\mathrm{mm}$. The resulting eddy current losses depending on speed and track width are shown in Figure 8. The parameters of the device under test are summarized in

Table 2. Parameters of the evaluated axial flux machine specimen.

Parameter	Value	Parameter	Value
Stator yoke	Hoganäs HR700	Rotor yoke	S235JR
material	1P 800 MPa	material
Magnet material	NdFeB	$w_{\mathrm{cu}}$	$1\mathrm{mm}$, $3\mathrm{mm}$, $5\mathrm{mm}$
${\widehat{B}}_{\mathrm{ax},1}$	$0.6796\mathrm{T}$	${\widehat{B}}_{\tan ,1}$	$0.2034\mathrm{T}$
${\widehat{B}}_{\mathrm{ax},3}$	$0.0823\mathrm{T}$	${\widehat{B}}_{\tan ,3}$	$0.0612\mathrm{T}$
${\widehat{B}}_{\mathrm{ax},5}$	$0.0125\mathrm{T}$	${\widehat{B}}_{\tan ,5}$	$0.0117\mathrm{T}$
$H_{\mathrm{C},\mathrm{pm}}$	$-915\mathrm{kA}$	${\mu }_{\mathrm{r},\mathrm{pm}}$	1.17409
p	11	n	$0\dots 6000\mathrm{r}/\min$
$f_{\mathrm{el},\mathrm{p}}$	$0\dots 1100\mathrm{Hz}$	$r_{\mathrm{o}}$	$60\mathrm{mm}$
$h_{\mathrm{adhesiv}}$	$0.1\mathrm{mm}$	$r_{\mathrm{i}}$	$30\mathrm{mm}$
$h_{\mathrm{cu}}$	$105\mathsf{\mu }\mathrm{m}$	${\alpha }_{\mathrm{pm}}$	$85\%$
$h_{\mathrm{pcb}}$	$1.6\mathrm{mm}$	${\vartheta }_{\mathrm{cu}}$	$100°\mathrm{C}$
$h_{\mathrm{tim}}$	$0.5\mathrm{mm}$	$\sigma \left({\vartheta }_{\mathrm{cu}}\right)$	$44.153·{10}^6\mathrm{S}/\mathrm{m}$
$h_{\mathrm{pm}}$	$3\mathrm{mm}$

. The time consumption and accuracy of the analytical computations were compared against a finite element simulation; the results are shown in

Table 3. Time consumption of the various calculation techniques.

Method	Time Consumed
FEA	3 h 29 min … 3 h 41 min
1	7.50 ms
2	17.00 ms
3	2.61 ms
4	7.00 ms
5	0.49 ms
6	0.50 ms
7a *	1.28 ms
7b *	8.14 ms
8	2.70 ms
9 **	7573.48 ms

* 7a: according to (28); 7b: according to (31) with $N_{\mathrm{strip}}=10$; ** 1000 evaluation points.

and

Table 4. Absolute and relative error of calculation technique compared to FEA depending on speed.

1 mm	1000 r/min		3500 r/min		6000 r/min
Method	Abs	Rel	Abs	Rel	Abs	Rel
	(mW)	(%)	(mW)	(%)	(mW)	(%)
1	+3.23	+96.7	+39.37	+96.0	+115.27	+95.3
2	+9.72	+291.3	+118.91	+289.9	+348.99	+288.5
3	+0.37	+11.1	+4.38	+10.7	+12.43	+10.3
4	+27.31	+818.4	+159.62	+389.1	+329.36	+272.2
5	+5.32	+159.3	+64.97	+158.4	+190.49	+157.5
6	+0.39	+11.7	+4.66	+11.4	+13.26	+11.0
7a *	−0.55	−16.6	−6.91	−16.9	−20.76	−17.2
7b *	+0.33	+10.0	+3.93	+9.6	+11.10	+9.2
8	+0.39	+11.6	+4.58	+11.2	+13.01	+10.8
9	+0.34	+10.1	+3.97	+9.7	+11.24	+9.3
3 mm	1000 r/min		3500 r/min		6000 r/min
Method	Abs	Rel	Abs	Rel	Abs	Rel
	(mW)	(%)	(mW)	(%)	(mW)	(%)
1	+80.5	+83.2	+970.4	+80.9	+2808.0	+78.6
2	+255.7	+264.4	+3111.0	+259.2	+9059.6	+253.7
3	+0.0	+0.0	−14.7	−1.2	−87.1	−2.4
4	+179	+185.1	+605.6	+50.5	+482.2	+13.5
5	+136.9	+141.5	+1661.4	+138.4	+4838.8	+135.5
6	+3.8	+3.9	+31.2	+2.6	+47.9	+1.3
7a *	−23.2	−24.0	−299.1	−24.9	−922.8	−25.8
7b *	−0.9	−0.9	−26.2	−2.2	−120.8	−3.4
8	+3.7	+3.9	+26.9	+2.2	+13.0	+0.4
9 **	−0.8	−0.8	−25.3	−2.1	−118.1	−3.3
5 mm	1000 r/min		3500 r/min		6000 r/min
Method	Abs	Rel	Abs	Rel	Abs	Rel
	(mW)	(%)	(mW)	(%)	(mW)	(%)
1	+407.6	+98.8	+4.88	+94.5	+14.03	+90.5
2	+1217.6	+295.0	+14.58	+282.1	+41.22	+265.9
3	+9.9	+2.4	+0.01	+0.2	−0.29	−1.8
4	+353.3	+85.6	−0.15	−2.9	−4.24	−27.4
5	+668.8	+162	+8.08	+156.4	+23.43	+151.2
6	+52.6	+12.7	+0.53	+10.3	+1.25	+8.1
7a *	−85.7	−20.8	−1.16	−22.5	−3.73	−24.1
7b *	+6.0	+1.4	−0.039	−0.8	−0.43	−2.8
8	+51.6	+12.5	+0.40	+7.7	+0.28	+1.8
9 **	+6.2	+1.5	−0.036	−0.7	−0.42	−2.7

* 7a: according to (28); 7b: according to (31) with $N_{\mathrm{strip}}=10$; ** 1000 evaluation points.

. The computations were carried out on a high-performance computer whose specifications are listed in

Table 5. Specifications of the high-performance computer used in the study.

Unit	Specification
CPU	AMD Ryzen Threadripper PRO 7965WX, 128 MB Cache, 24 Cores,
	48 Threads, 4.2 … 5.3 GHz
RAM	256 GB DDR5, 5200 MT/s
GPU	NVIDIA ADA 6000, 48 GB, GDDR6
NV Memory	PCIe-NVMe SSD Class 40

.

4. Conclusions

For the design space exploration of slotless axial flux machines with PCB air gap windings, it is crucial to calculate eddy current losses to ensure the thermal and mechanical integrity of the PCB winding. The thermal limit defines the current-carrying capability and thus the machine performance and utilization. Finite element analysis is known to deliver the most accurate results when designing electric machines, but it is the most time-consuming technique. There are various eddy current loss calculation methods for different machine parts, but not all of them can be applied directly to axial flux machines without modification. The most accurate and fastest method was sought and its validity checked to accelerate design space exploration with thousands of samples where accuracy is less essential than fast computations. PCB windings of slotless axial flux machines are planar and have a rectangular cross-sectional shape instead of a circular one with round wire windings. A comprehensive overview of analytical calculation techniques was presented, in which the methods were adapted to slotless axial flux machines. Time consumption and accuracy were compared to finite element simulation. Originally based on radial flux canned motors, methods M3 and M7 were reworked because the winding tracks are not a consistent conductor in theperipheral direction and are limited by their edges.

In general, the accuracy of approximations depend on the neglect of the underlying modeling, as stated in Section 2.1 and the applied discretization. Inter alia, the attenuating effect on the excitation field is neglected.

The methods M1, M2, M4, and M5, which are based on eddy current loss calculations for electrical steel sheets, were significantly overestimated and had the worst errors compared to the values evaluated through FEA. According to [2] and Table 1, method M5 is said to be applicable for air gap windings, but the approach is similar to electrical steel sheets. It must be noted that the steel sheet-based methods assume that the magnetic flux density vector is oriented parallel to the sheet and the magnetic flux density is distributed homogeneously along the steel sheet with a linear current density distribution from the center to the sheet edge, which is in fact not the case for slotless windings. In the case of slotless axial flux machines, the axial magnetic flux density is perpendicular to the stator surface, leading to a different eddy current density distribution. Method M4 has the weighting factor $k_{\exp }$ that is determined by experiments with steel sheets, but copper has a higher electrical conductivity and much lower magnetic permeability compared to electrical steel, which affects the penetration depth. In method M4, the exponent of the magnetic flux density and the frequency is 1.5 instead of 2, which leads to a further deviation. Therefore, the steel sheet-based methods are considered to be not applicable.

The track strip model of method M7, with only two strips according to (28), underestimated the eddy current losses. With a higher number of strips and thus a finer discretization, the accuracy was significantly improved. The results from the methods M3, M6, M7, M8, and M9 are very close together, with M8 being the closest to FEA in the case of larger track width and where M3 and M9 gain almost same results.

The eddy current losses are proportional to the squared magnetic flux density. But if you take the monomials of the Fourier series of the magnetic flux density as a vector, the squaring results in the dyadic product instead of the scalar product. Thus, an error occurs if you calculate each order of the Fourier series separately. Furthermore, it should be mentioned that the spatial vectors of the magnetic flux density do not only consist of the axial components. Therefore, methods M1, M5, and M6 consider a tangential magnetic flux density component. The mentioned inadequacies of the modeling of eddy current loss points are considered to cause the error depending on the track width. However, the absolute errors of M3, M6, M7 (with a sufficient number of strips), M8, and M9 are small enough to be acceptable for design space exploration.

Method M6 had the fastest computation time (0.5 ms) and M9 the worst (7573.48 ms), with rotor position discretization having a significant impact on the computation time of M9. Even if method M5 is slightly faster than M6, the calculation error of M5 is very large. For the reasons mentioned, method M8 is considered to have the best trade-off between accuracy and time consumption. Therefore, method 8 is recommended for the early-stage design of slotless axial flux machines.