On Eddy Current Brakes with Anisotropic Material Structures: Electromagnetic Model and Its Detailed Validation

1. Introduction

In electric vehicle applications, a wear-free eddy current brake (ECB) offers an attractive supplement to the friction brake. However, electrically excited ECBs with solid active material [1,2,3,4] suffer from low power density, limited to approximately 4 kW/kg [5], primarily due to non-uniform heat dissipation caused by the skin effect [6]. As shown in Figure 1a, the skin effect arises because eddy currents in a homogeneous, isotropic conductive material are coupled via a high-permeability magnetic circuit, allowing them to displace the primary field toward the surface.

To reduce the skin effect, the magnetic circuit must be interrupted in one direction, forcing the field to fully penetrate the material (Figure 1b). Since the relative permeability of real materials cannot drop significantly below ${\mu }_r=1$, the skin effect cannot be completely eliminated, but only reduced.

Even with anisotropic materials, the relative permeability in the direction of low magnetization still reaches ${\mu }_r=4000--6000$ at $B=1\mathrm{T}$ [7,8], so a macroscopically anisotropic structure is required. This is realized by flux-conducting steel pins surrounded by electrically conductive perforated sheets [9], as shown in Figure 2a. The gaps between the stacked sheets simultaneously serve as cooling channels, significantly increasing the free surface area in contact with the cooling medium. The torque characteristic can be tailored via the geometric parameters of the structure.

To further reduce the skin effect within the cylindrical pins, multiple thin pins can be grouped in a hexagonal cluster configuration (Figure 2b) [10]. These two variants the PIN structure and the Cluster structure are the subject of this work.

Figure 3 shows a functional illustration of a rotating ECB with the anisotropic material structure. This paper presents a 2D reluctance-network-based electromagnetic model for both structure variants and validates it experimentally against measured torque characteristics.

The material structure with cylindrical flux-conducting elements is hereinafter referred to as the PIN structure, while the material structure with flux-conducting elements in a hexagonal cluster configuration is designated the Cluster structure. Figure 3 shows an exemplary CAD design of a rotating eddy current brake with the described anisotropic material structure as a Pin structure. In this work the electromagnetic model of the proposed eddy current brake with anisotropic material structure is presented and validated in detail.

2. Electromagnetic Model

The electromagnetic model calculates the mean torque ${\overline{M}}_{\mathrm{ecb}}$ over one revolution for a given excitation current $I_{\mathrm{ex}}$ and rotational speed n. A full 3D FEM simulation of the ECB with anisotropic material structure is computationally prohibitive for design optimization. Due to the skin effect, a fine mesh is required at the pin surfaces, resulting in a mesh with a large number of elements at maximum speed. Combined with the 3000 time steps required to reach a quasi-steady state, the computation time for a single operating point exceeds several hundred hours on modern hardware [2]. A reluctance network model, by contrast, abstracts the electromagnetic domain into flux tubes, each characterized by a reluctance determined analytically or by FEM pre-computation. The resulting system of equations is small and computationally efficient, yet sufficiently accurate for torque prediction [11,12]. A hybrid approach is adopted here, in which air-space reluctances with non-uniform flux density are pre-computed by FEM [13,14]

The following simplifications are adopted: (i) the ECB is represented by a 2D reluctance model in a representative plane; (ii) flux-conducting elements form flux tubes with analytically derived reluctances; (iii) torque ripple is negligibly small. Reluctances of regions with non-uniform flux density are pre-computed by FEM. The mean torque follows from the energy balance:

$${\overline{M}}_{\mathrm{ecb}}={\displaystyle \frac{1}{2\pi n}}\left({\overline{P}}_{\mathrm{ec}}+{\overline{P}}_{\mathrm{iron}}+{\overline{P}}_{\mathrm{vent}}\right)$$

(1)

where ${\overline{P}}_{\mathrm{ec}}$ is the mean eddy-current Joule heat, ${\overline{P}}_{\mathrm{iron}}$ the mean iron losses, and ${\overline{P}}_{\mathrm{vent}}$ the ventilation losses (assumed negligible).

2.1. Representative Mapping of the Geometry in 2D

In axial-flux machines such as the ECB, the pole pitch ${\tau }_{\mathrm{p}}$ varies with radius, so a representative model plane must be chosen. Assuming the torque-generating force originates equally from the inner and outer surfaces, the model diameter $d_{\mathrm{m}}$ satisfies:

$$d_{\mathrm{m}}=\sqrt{{\displaystyle \frac{d_{\mathrm{o}}^2+d_{\mathrm{i}}^2}{2}}}$$

(2)

With the number of pole pairs the model pole pitch is ${\tau }_{\mathrm{pm}}=\pi d_{\mathrm{m}}/\left(2p\right)$. To exploit anti-periodic boundary conditions, the number of flux-conducting elements per model pole pitch is rounded to the nearest integer, giving a representative element spacing $w_{\mathrm{pm}}$ and a representative element diameter that preserves the real fill factor of the flux conducting elements $k_{\mathrm{p}}$. The real and model arrangements are illustrated in Figure 4.

2.2. Reluctance Network Model

Applying Hopkinson’s law $V_{\mathrm{m}}=\mathfrak{R}\varphi$ [12] and Ampere’s circuital law to each mesh m of N reluctances, the governing equation of a mesh-flux reluctance network is:

$$\left({\displaystyle \sum _{n=1}^N}{\mathfrak{R}}_{nm}\right){\Phi }_m-{\displaystyle \sum _{n=1}^N}{\mathfrak{R}}_{nm}{\Phi }_{nm}=I_m$$

(3)

where ${\Phi }_m$ is the mesh flux of mesh m, ${\Phi }_{nm}$ the adjacent mesh flux, and $I_m$ the mesh current. Figure 5 shows the reluctance model for one pole using antisymmetric boundary conditions. The model depth equals $w_{\mathrm{pm}}$. The excitation part consists of a yoke with pole cores surrounded by excitation coils. The magnetic flux traverses the air gap (reluctance ${\mathfrak{R}}_{\delta i}$) and the flux-conducting elements (${\mathfrak{R}}_{\mathrm{a}ij}$). Tangential leakage between adjacent elements is accounted for by ${\mathfrak{R}}_{\mathrm{t}ij}$. Eddy currents $I_{\mathrm{eddy}ij}$ are induced by the time-varying fluxes, where indices i and j denote the tangential and axial position respectively.

Applying (3) to each material-structure mesh $ij$ and using Faraday’s law, the eddy current is:

$$I_{\mathrm{eddy}ij}=-{\displaystyle \frac{1}{R_j}}{\displaystyle \frac{\partial {\Phi }_{ij}}{\partial t}}=-v{\displaystyle \frac{{\Phi }_{i+1j}-{\Phi }_{i-1j}}{R_{j}2w_{\mathrm{pm}}}}$$

(4)

where the time derivative is approximated by a central difference at constant tangential velocity $v=\pi d_{\mathrm{m}}n$. The governing equation to be solved is:

$$\left({\displaystyle \sum _{n=1}^{N_{ij}}}{\mathfrak{R}}_{nij}\right){\Phi }_{ij}-{\displaystyle \sum _{n=1}^{N_{ij}}}\left({\Phi }_{nij}{\mathfrak{R}}_{nij}\right)=-v{\displaystyle \frac{{\Phi }_{i+1j}-{\Phi }_{i-1j}}{R_{j}2w_{\mathrm{pm}}}}$$

(5)

For the meshes assigned to the air gap and the excitation circuit, the general equation (3) applies with the excitation current $I_{\mathrm{ex}}$. The resulting system of equations is solved iteratively using the biconjugate gradient method [15], accounting for the nonlinear magnetization curves of the respective materials.

The total Joule heating power resulting from the eddy currents in meshes $ij$ can be calculated using equation (4) and the mesh fluxes ${\Phi }_{ij}$ determined by solving equation (5), via

$$\begin{array}{cc}\hfill {\overline{P}}_{\mathrm{ec}}& ={\displaystyle \frac{b_{\mathrm{m}}}{w_{\mathrm{pm}}}}\{2\left[{\displaystyle \sum _{j=1}^{\lfloor N_{\mathrm{sh}}/2\rfloor }}{\displaystyle \sum _{i=1}^{N_{\mathrm{p}\tau }}}{\left({\displaystyle \frac{{\Phi }_{i+1,j}-{\Phi }_{i-1,j}}{\sqrt{R_j}2w_{\mathrm{pm}}}}v\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{N_{\mathrm{p}\tau }}}{\left({\displaystyle \frac{{\Phi }_{i+1,\lceil N_{\mathrm{sh}}/2\rceil }-{\Phi }_{i-1,\lceil N_{\mathrm{sh}}/2\rceil }}{\sqrt{R_{j=\lceil N_{\mathrm{sh}}/2\rceil }}2w_{\mathrm{pm}}}}v\right)}^2\left({\displaystyle \frac{1-{(-1)}^{N_{\mathrm{sh}}}}{2}}\right)\}.\hfill \end{array}$$

(6)

The second summation term accounts for the fact that this is a half-model, and that for an odd total number $N_{\mathrm{sh}}$ of perforated sheets j, the Joule losses of the last sheet $j=\lceil N_{\mathrm{sh}}/2\rceil$ are counted at half weight. Note that the floor brackets in the upper limit of the sum over all j in the first term and the ceiling brackets at the index $j=\lceil N_{\mathrm{sh}}/2\rceil$ in the second term must be observed. The path resistances of the eddy currents between the flux-conducting elements $R_j$ are described in subSection 2.4.

2.3. Modeling of the Reluctances

For a homogeneous flux density distribution and constant cross-sectional area, a reluctance is obtained from the length of a flux path $l_{\mathrm{path}}$, its cross-sectional area $A_{\mathrm{path}}$, and the permeability of the material $\mu$.

$${\mathfrak{R}}_{\mathrm{path}}={\displaystyle \frac{l_{\mathrm{path}}}{A_{\mathrm{path}}{\mu }_r{\mu }_0}}$$

(7)

An approximately homogeneous flux density is present in the yoke and the pole core of the exciter. The relative permeability of all ferromagnetic components is determined iteratively from the measured magnetization curve of the respective material, which is extended using the saturation model from [16]. For all air-space reluctances — including the air gap and tangential leakage between flux-conducting elements — a scalar magnetic potential FEM model is used. In current-free regions, the scalar magnetic potential $\psi$ satisfies the Laplace equation $\Delta \psi =0$, which is solved numerically with prescribed potential boundary conditions ${\psi }_1$ and ${\psi }_2$. The reluctance follows directly from the resulting magnetic flux $\varphi$ as $\mathfrak{R}=({\psi }_2-{\psi }_1)/\varphi$. To calculate the air gap reluctance, it is convenient to multiply the air gap reluctance obtained by neglecting leakage fluxes by a factor, referred to here as ${\kappa }_{\mathrm{p}\delta }$, which accounts for the leakage flux.

$${\mathfrak{R}}_{\delta }={\kappa }_{\mathrm{p}\delta }{\displaystyle \frac{4\delta }{\pi d_{\mathrm{p}}^2{\mu }_0}}$$

(8)

Due to the linear behavior of the air region, the factor ${\kappa }_{\mathrm{p}\delta }$ depends only on geometric ratios. This factor is evaluated after a FEM simulation as follows.

$${\kappa }_{\mathrm{p}\delta }={\displaystyle \frac{\pi d_{\mathrm{p}}^2{\mu }_0\left({\mathfrak{R}}_{\mathrm{p}\delta }-{\mathfrak{R}}_{\mathrm{p}}\right)}{8\delta }}$$

(9)

Due to the linear behavior of the air region, the factor ${\kappa }_{\mathrm{p}\delta }$ depends only on geometric ratios. This factor is evaluated after a FEM simulation as follows:

$${\mathfrak{R}}_{\mathrm{p}\delta }={\displaystyle \frac{H(h_{\mathrm{mat}}+2\delta )}{\varphi }}$$

(10)

Figure 6a and Figure 6b illustrate the computational domain used to determine the air gap reluctance above the flux-conducting elements. The corresponding results are presented in Figure 6. Figure 6c and Figure 6d show the magnetic flux density vectors obtained from the FEM calculations. The resulting reluctance factor is plotted as a function of the ratio of air gap to pin diameter for various fill factors, for the PIN structure in Figure 7a and for the Cluster structure in Figure 7b. The reluctances of the flux-conducting elements them self’s are frequency- and field-strength-dependent due to the skin effect, and are pre-computed via a harmonic 2D FEM model [17].

2.4. Modeling of the Path Resistance

Due to the perforations in the conductive sheets, the resistance of the eddy current path depends on the opening ratio $k_{\mathrm{oa}}$ of the perforated sheets. Furthermore, in the 2D model the return path of the eddy currents at the inner and outer sides is neglected. The resistance at the inner and outer sides can be accounted for by the so-called Russell-Norsworthy factor [18]. The dependence of the effective conductivity on the fill factor of the flux-conducting elements is calculated using a FEM model of a partial segment of the eddy current path. Since the relationship between current density $\overrightarrow{j}$ and electric field strength $\overrightarrow{E}$ is linear, a resistance factor ${\kappa }_R$ is determined as a function of the fill factor. This factor indicates the ratio of the resistance of a path with perforations to a path without perforations. The resistance of a non-perforated path of width $b_{\mathrm{path}}$, length $l_{\mathrm{path}}$, thickness s, and specific resistance ${\rho }_{el}$ is given by:

$$R_0={\displaystyle \frac{l_{\mathrm{path}}{\rho }_{el}}{b_{\mathrm{path}}s}}$$

(11)

Using the FE models in Figure 8, the current $I_{res}$ through the path segment and thus the resistance can be determined for given potential boundary conditions ${\phi }_1=1\mathrm{V}$ and ${\phi }_0=0\mathrm{V}$ via

$$R={\displaystyle \frac{{\phi }_1-{\phi }_0}{I_{res}}}$$

(12)

From the FEM calculation, the resistance factor can be evaluated as

$${\kappa }_{\mathrm{R}}={\displaystyle \frac{b_{\mathrm{path}}s({\phi }_1-{\phi }_0)}{I_{res}{l}_{\mathrm{path}}{\rho }_{el}}}$$

(13)

Figure 8c shows the resistance factor as a function of the opening ratio of a perforated sheet with hexagonal perforations (red) and cylindrical perforations (blue).

The difference between the two variants at high opening ratios arises from the fact that for hexagonal perforations the resistance only tends to infinity as the opening ratio approaches 1, whereas for cylindrical perforations this already occurs at an opening ratio of $k_{oa}=\pi /\left(2\sqrt{3}\right)$. Using the resistance factor determined in this way, the resistance of the eddy current path between two flux-conducting elements of a perforated sheet with index j can be determined via

$$R_j=R_{0j}{\kappa }_{\mathrm{R}}{k}_{\mathrm{n}}$$

(14)

Since in the reluctance model the reference path width and the reference path length correspond to the spacing of the flux-conducting elements, the reference resistance, accounting for the temperature of the perforated sheet j with the specific resistance at $20{°}^{\mathrm{C}}$${\rho }_{\mathrm{el}20,j}$ and the temperature coefficient ${\alpha }_{\mathrm{el},j}$, is:

$$R_{0,j}={\displaystyle \frac{{\rho }_{\mathrm{el},j}}{s}}={\displaystyle \frac{{\rho }_{\mathrm{el}20,j}\left[1+{\alpha }_{\mathrm{el},j}{({\vartheta }_{\mathrm{sh},j}-20)}^{\circ }\mathrm{C}\right]}{s}}$$

(15)

It should be noted that not only the temperature may differ across all j, but also the material properties ${\rho }_{\mathrm{el}20}$ and ${\alpha }_{\mathrm{el}}$.

3. Experimental Investigation and Model Validation

The primary objective of the investigation is the measurement of the torque characteristics of the demonstrator using various derivatives of the stator as a function of the excitation current. The derivatives differ in particular in their size, the type of magnetically conductive elements (cylindrical or hexagonal), and the number of perforated sheets within the material structure. Figure 9a shows a stator with cylindrical flux-conducting elements (PIN structure) and Figure 9b shows a stator with flux-conducting elements in hexagonal cluster configuration (Cluster structure).

The torque is measured as a function of rotational speed at excitation currents of $I_{\mathrm{ex}}=10$–$80\mathrm{A}$ in increments of $10\mathrm{A}$. Figure 10 shows the calculated torque characteristics compared to the measured ones at various excitation currents for the ECB with pin-structure Figure 10a and for the ecb with cluster-structure Figure 10b.

For validation purposes, in addition to the excitation current, the measured air gap is incorporated into the electromagnetic model. Nevertheless, deviations can be observed that can be attributed to identifiable sources of error, as described in the following subsection “Identification of Error Sources and Error Correction”.

3.0.1. Identification of Error Sources and Error Correction

As a first step in the error analysis, it is useful to calculate the torque characteristics from measured magnetic fluxes in the flux conducting elements. This allows errors of the reluctance network to be separated from errors due to incorrect resistances of the eddy current paths. The flux probes are positioned in the material-structure as shown in the picture in Figure 11. The flux density’s measured via field coil 1 and field coil 3 are shown in Figure 11 with the solid lines at an excitation current of $I_{\mathrm{ex}}=60\mathrm{A}$ and a rotational speed of $n=2000/\min$.

The mesh fluxes ${\Phi }_{ij}$ in equation 4 can be replaced by the measured fluxes in the flux-conducting elements ${\varphi }_{ij}$, since

$${\varphi }_{ij}={\Phi }_{ij}-{\Phi }_{i-1j}\mathrm{and}{\varphi }_{i+1j}={\Phi }_{i+1j}-{\Phi }_{ij}$$

(16)

Thus, equation 4 can be written as:

$$I_{\mathrm{eddy}ij}=-v{\displaystyle \frac{{\varphi }_{i+1j}+{\varphi }_{ij}}{R2w_{\mathrm{pm}}}}$$

(17)

. It then follows for equation 6 that

$$\begin{array}{cc}\hfill {\overline{P}}_{\mathrm{ec}}& ={\displaystyle \frac{b_{\mathrm{m}}}{w_{\mathrm{pm}}}}\{2\left[{\displaystyle \sum _{j=1}^{j=\lfloor N_{\mathrm{sh}}/2\rfloor }}{\displaystyle \sum _{i=1}^{i=N_{\mathrm{p}\tau }}}{\left({\displaystyle \frac{{\varphi }_{i+1j}+{\varphi }_{ij}}{\sqrt{R_j}2w_{\mathrm{pm}}}}v\right)}^2\right]+\hfill \\ & +{\displaystyle \sum _{i=1}^{i=N_{\mathrm{p}\tau }}}{\left({\displaystyle \frac{{\varphi }_{i+1j=\lceil N_{\mathrm{sh}}/2\rceil }+{\Phi }_{ij=\lceil N_{\mathrm{sh}}/2\rceil }}{\sqrt{R_{j=\lceil N_{\mathrm{sh}}/2\rceil }}2w_{\mathrm{pm}}}}v\right)}^2\left({\displaystyle \frac{1-{(-1)}^{N_{\mathrm{sh}}}}{2}}\right)\}\hfill \end{array}$$

(18)

The magnetic fluxes ${\varphi }_{ij}$ are determined by interpolating the measured fluxes at positions i of the flux-conducting elements beneath a pole. In the axial direction (depth direction) with index j, the magnetic fluxes are determined under the assumption that the magnetic flux decays exponentially. The measured fluxes ${\varphi }_{\mathrm{s}1i}$ and ${\varphi }_{\mathrm{s}3i}$ interpolated at position i (markers directly on the lines in Figure 11) are used to calculate an exponential coefficient $g_i$ with

$$g_i={\displaystyle \frac{2ln\left({\displaystyle \frac{{\varphi }_{\mathrm{s}3i}}{{\varphi }_{\mathrm{s}1i}}}\right)}{N_{\mathrm{sh}}}}$$

(19)

in order to determine the magnetic fluxes (markers in Figure 11) with

$${\varphi }_{ij}={\varphi }_{\mathrm{s}3i}{e}^{-g_i(j-1)}$$

(20)

for equation 18.

The torque is finally determined via equation 1. Figure 12 shows the directly measured torque characteristics (markers) and the torque characteristics calculated from the measured magnetic fluxes at $I_{\mathrm{ex}}=60\mathrm{A}$, $I_{\mathrm{ex}}=70\mathrm{A}$, and $I_{\mathrm{ex}}=80\mathrm{A}$ with solid lines. It can be seen that the torque characteristic calculated from the measured magnetic fluxes at $I_{\mathrm{ex}}=60\mathrm{A}$ agrees very well with the directly measured one. However, at excitation currents of $I_{\mathrm{ex}}=70\mathrm{A}$ and $I_{\mathrm{ex}}=80\mathrm{A}$, a significant deviation can be observed. Since this deviation is more pronounced at $I_{\mathrm{ex}}=80\mathrm{A}$ than at $I_{\mathrm{ex}}=70\mathrm{A}$, it can be assumed that the deviation is a consequence of heating of the perforated sheets, which is more pronounced at higher power levels.

According to this hypothesis, this error can potentially be corrected by accounting for the temperature dependence of the specific resistance of the perforated sheets. In the correction calculation, the measured temperature of the perforated sheets is therefore incorporated into the calculation of the eddy current path resistances.

Furthermore, it is clearly visible in Figure 10b that the critical rotational speed of the calculated torque characteristics is significantly higher than that of the measured ones, even at low excitation currents. This can be attributed to the fact that the resistance of the current paths of the perforated sheets in the experimental demonstrator is lower than assumed in the model. The reason for this becomes apparent in Figure 13b, which shows the FEM models used to analyze the error. In the model, it is assumed that the current paths at the edge of the perforated sheets have the same cross-section as between the flux-conducting elements in the center, as shown in Figure 13b.

In the real design, however, potential perforations that would extend beyond the outer diameter are not realized. As a result, the current path at the edge is on average significantly wider, leading to a lower resistance of the eddy current paths, as shown in Figure 13a. This circumstance ultimately leads to a reduction of the Russell-Norsworthy factor, which, as described in Section 2.4, accounts for the resistance that the eddy currents must overcome at the edges.

The correction factor for this reduction is determined using the FEM models shown in Figure 13. In the FEM models with periodic boundary conditions of a 60-degree segment, each flux-conducting element at position i in the polar coordinate system is assigned a flux change $u_i$, which describes a cosine function in the tangential direction with angle ${\phi }_i$, amplitude $\widehat{u}$, and periodicity of the pole pair number p.

$$u_i=\widehat{u}cos\left(p{\phi }_i\right)$$

(21)

The plane of symmetry corresponds to an angle of zero. Since the FEM software used allows only a limited number of voltage sources as boundary conditions, a 60-degree segment is simulated and a pole pair number of 6 is chosen in order to exploit the periodic boundary conditions. The resulting current density at the edges and in the center of the computational domain is evaluated as a result. Figure 14 shows the current density in the normal direction along the evaluation paths (edge and center) corresponding to Figure 13a (real perforated sheet) with blue lines and with red lines for the evaluation paths in Figure 13b (perforated sheet corresponding to the model). The correction value $k_{\mathrm{n},\mathrm{corr}}$ for the Russell-Norsworthy factor can be determined by taking the ratio of the total currents along the evaluation paths of the model sheet $I_{\mathrm{mod}0}+I_{\mathrm{mod}30}$ to the total currents of the real sheet $I_{\mathrm{real}0}+I_{\mathrm{real}30}$. Since the integral of the positive current density must equal the integral of the negative current density along an evaluation path, the total current is determined via the integral of the absolute value of the current density, which is then halved. The correction factor for the Russell-Norsworthy factor is thus:

$$k_{\mathrm{n},\mathrm{corr}}={\displaystyle \frac{{\int }_{s=d_{\mathrm{i}}/2}^{d_{\mathrm{o}}/2}|j_{\mathrm{mod}0}|\mathrm{d}s+{\int }_{s=d_{\mathrm{i}}/2}^{d_{\mathrm{o}}/2}\left|j_{\mathrm{mod}30}\right|\mathrm{d}s}{{\int }_{s=d_{\mathrm{i}}/2}^{d_{\mathrm{o}}/2}|j_{\mathrm{real}0}|\mathrm{d}s+{\int }_{s=d_{\mathrm{i}}/2}^{d_{\mathrm{o}}/2}\left|j_{\mathrm{real}30}\right|\mathrm{d}s}}$$

(22)

From the evaluation of equation 22, a correction value for the Russell-Norsworthy factor of $k_{\mathrm{n},\mathrm{corr}}=0.85$ is obtained. The mean resistance of the eddy current paths $R_j$ of the real perforated sheets is therefore 15% lower than assumed in the model.

The solid lines in Figure 12 show the torque characteristics calculated from the measured magnetic fluxes, accounting for the measured temperature of the perforated sheets (indicated by the colored markers) and the correction value of the Russell-Norsworthy factor. The outliers at $n=1000/\min$ and in the range from $n=4000/\min$ to $n=5000/\min$ are attributable to errors in the measurement of the magnetic flux. Figure 15 illustrates, using the torque characteristic at an excitation current of $I_{\mathrm{ex}}=70\mathrm{A}$, the separate influence of the correction of the Russell-Norsworthy factor and the correction of the resistance through the temperature effect of the perforated sheets. Neglecting the temperature effect leads to a lower torque above the critical rotational speed, while neglecting the correction of the Russell-Norsworthy factor results in a shift of the calculated torque characteristic toward higher rotational speeds.

The uncertainty range due to undetermined thermal parameters is determined in subSection 3.1.

3.1. Determination of the Uncertainty Range of the Overall Models due to Undetermined Thermal Parameters

Since the model validation is carried out accounting for the measured temperature, the uncertainty range is investigated here on the basis of the coolant inlet temperature and the heat transfer coefficient to the coolant.

In the following analysis, the torque characteristic at an excitation current of $I_{\mathrm{ex}}=60\mathrm{A}$ is calculated for various combinations of undetermined thermal parameters. A distinction is made between a high coolant temperature of ${\vartheta }_{\mathrm{cf}}=100{°}^{\mathrm{C}}$ in combination with a low heat transfer coefficient for forced convection of ${\alpha }_{cf}=2000\mathrm{W}/\square \mathrm{m}/\mathrm{K}$ [19], and a coolant temperature of ${\vartheta }_{\mathrm{cf}}=20{°}^{\mathrm{C}}$ in combination with an infinitely high heat transfer coefficient.

For this analysis, the overall model is extended with a simplified thermal model of the material structure. In each iteration step for solving the nonlinear reluctance network, the temperature of the perforated sheets is calculated based on the inlet temperature, the heat transfer coefficient, and the current Joule heat of the respective perforated sheet, along with the resulting electrical conductivity, which feeds into equation in the next iteration step.

Figure 16 and Figure 17 show the simulated torque characteristics accounting for the bounds of the coolant temperature and heat transfer for the ECB with Cluster structure (Figure 16) and for the ECB with PIN structure (Figure 17).

At a coolant temperature of $\vartheta =20{°}^{\mathrm{C}}$ and a heat transfer coefficient of ${\alpha }_{\mathrm{cf}}=\infty$, the torque characteristic is slightly compressed along the rotational speed axis compared to the simulation accounting for the measured temperature, while at a coolant temperature of $\vartheta =100{°}^{\mathrm{C}}$ and a heat transfer coefficient of ${\alpha }_{\mathrm{cf}}=2000\mathrm{W}/\square \mathrm{m}/\mathrm{K}$, the torque characteristic is stretched along the rotational speed axis. The resulting uncertainty range is indicated by the shaded area. Since the coolant inlet temperature can vary during operation, this uncertainty range must be taken into account in future design work.

4. Conclusion

This paper presents a computationally efficient 2D reluctance network model for electrically excited eddy current brakes with anisotropic material structure, covering both the PIN structure with cylindrical flux-conducting elements and the Cluster structure with flux-conducting elements in hexagonal cluster configuration. The model accounts for the nonlinear magnetization behavior of the flux-conducting elements, the frequency- and field-strength-dependent reluctances determined by FEM, and the temperature-dependent path resistance of the perforated sheets. Experimental validation against measured torque characteristics of two demonstrators shows good agreement between simulation and measurement after applying two corrections: a correction of the Russell-Norsworthy factor to account for the geometry of the real perforated sheets at the inner and outer edges ($k_{\mathrm{n},\mathrm{corr}}=0.85$ for the Cluster structure, $k_{\mathrm{n},\mathrm{corr}}=0.95$ for the PIN structure), and a temperature correction based on the measured sheet temperatures. In particular, the critical rotational speed is accurately reproduced by the corrected model. Residual deviations of up to 15% at high excitation currents and low speeds are attributed to uncertainties in the magnetization curve of the flux-conducting elements. The uncertainty range of the overall model due to undetermined thermal parameters, namely the coolant inlet temperature and the heat transfer coefficient, was quantified and must be considered in future design applications. The model provides a suitable basis for the design and optimization of eddy current brakes with anisotropic material structure.