Effective reluctivity of an insulated magnetic spherical particle excited by an axisymmetric polar magnetic field

In this study, an electrically insulated conductive magnetic particle is homogenized. Homogenization refers to treating the material inside the domain as homogeneous and determining an appropriate material model for it. The motivation for such a study partly arises from the homogenization of soft magnetic composite (SMC) materials. More importantly, the theoretical aspects provide strong motivation. It turns out that by using orthogonal decompositions for the fields, a homogenized reluctivity for an insulated particle can be derived analytically for a variety of boundary conditions. This additional freedom in the choice of boundary conditions makes it possible to study how different boundary conditions and insulation types affect losses and other system dynamics in a more detailed manner. Since the methodology is analytical, it does not suffer from numerical instabilities such as air-gap meshing issues. Furthermore, the analytical treatment reveals certain mathematical structures that are essential for performing such homogenization.

SMC materials consist of compressed, insulated ferromagnetic powders (Shokrollahi and Janghorban 2007). Modeling the behavior of SMC materials is motivated by their commercial applications, such as inductors and transformers, which can operate at higher frequencies than laminated steels owing to their finer resistive structures. Prof. Igarashi’s group explored a semi-analytical method for modeling SMC materials (Sato and Igarashi, 2016, Sato et al., 2015, Sato and Igarashi, 2017). It is based on the homogenized permeability of a single spherical particle, which is materially homogeneous. The study assumes a specific form of boundary conditions for the magnetic field. These conditions ensure uniform magnetic excitation and prevent any currents from flowing through the boundary. It is questionable if the assumption of uniform magnetic excitation holds for the particles in SMC materials. If a particle is surrounded by other particles, questions arise if other magnetic modes are excited in the particle than just the uniform mode. Furthermore, it would be beneficial for practical applications to also take into account the insulation surrounding the particle.

In more recent studies, the same form of homogenized permeability has also been applied to the modeling of randomized materials (Sato et al., 2024). These studies help to understand the logic behind homogenization; the material model is redefined such that the particle-scale geometry can be replaced by a homogeneous material. Homogenized material models are often found by equating energy-related quantities in a unit cell (Ito and Igarashi 2013; Ren et al., 2016; Corcolle et al., 2021; Maruo and Igarashi, 2019; H. Sato and Igarashi, 2021).

Whereas some modeling approaches emphasize homogenization, others emphasize multiscale. Corcolle et al. (Corcolle et al., 2021) and Bottauscio et al. (Oriano Bottauscio et al., 2006; Bottauscio et al., 2009; Bordianu et al., 2012) have studied and applied multiscale methods. Some methods, such as those presented in Bottauscio’s papers, rely on mathematical multiscale methods. Others build on the idea that Maxwell’s equations should remain invariant under homogenization, and that the information exchanged between spatial scales should consist solely of field quantities. Consequently, the computational subproblems can be interpreted as homogenized material models. Corcolle’s work is based on this latter line of reasoning.

Kameari investigated magnetic modes in spherical regions in Japanese literature (Kameari 2015, Advanced Electromagnetic Field Analysis for Effective Design, 2016). He established relationships between each pair of magnetic h-modes and b-modes. Furthermore, he identified Cauer networks that represent these relationships.

In this study, a homogenized reluctivity for an insulated particle is derived. It admits arbitrary but azimuthally symmetric boundary conditions for the h-field in the polar direction. The homogenized reluctivity is energy-consistent, meaning that the point-wise relation between the homogenized h- and b-field carries the same power losses as what is transferred through the boundary of the particle/insulation system. Orthogonal spherical magnetic modes are exploited, and connections between the modes, their boundary conditions and homogenized quantities are provided.

The remainder of this paper is organized as follows. The treatment for a particle is provided in Section 2. The treatment is extended to include the region of insulation around the particle in Section 3. Numerical examples are provided in Section 4, and the conclusions are presented in Section 5.

We use a spherical coordinate system, as shown in Figure 1(a). The spherical coordinates $(r,θ,ϕ)$ are transformed into Cartesian coordinates $(x,y,z)$ as follows:

We denote r as the radial coordinate, $θ$ as the polar coordinate and $ϕ$ as the azimuthal coordinate. The corresponding coordinate co-frames are denoted as $dr$⁠, $dθ$ and $dϕ$⁠. They are differential 1-forms (Lindell 2004).

We first homogenize one spherical magnetic particle of radius $Rprt$⁠, centered around the origin of the coordinate system, as depicted in Figure 1(a). The particle is surrounded by insulation, but we do not yet include the insulating region $Rprt<r<Rins$ in the model. In this section, we consider the insulated nature of the particle by assuming such a representation of the electric field inside the particle that admits no current density through the boundary. The results of this section are used in Section 3, where the insulation, visualized in Figure 1(b), is also fully considered.

We assume that our magnetic excitations are symmetric with respect to rotations around the Cartesian z-axis. The fields expressed in spherical coordinates do not depend on the azimuthal coordinate $ϕ$⁠. We express the fields as differential forms (Lindell, 2004). This provides greater clarity in the treatment of spherical coordinates and later reveals a mathematical structure essential for homogenization.

The physics inside the particle is described by the magnetic field strength h, current density j, electric field strength e and magnetic flux density b. The fields are modeled by the equations:

where i is the imaginary unit, $ω$ is the angular frequency and $μ$ and $σ$ denote the permeability and conductivity of the particle, respectively. The material parameters are assumed to be constant within the particle. Furthermore, d is the exterior derivative, and the operator $✶$ is the Hodge star, defined by the identity $α∧✶β=<α,β>η$ for all differential forms $α$ and $β$ of equal degree, where $η$ is the volume form and $<·,·>$ is the usual inner product of forms. The Hodge star has a unique inverse. In three dimensions, one has $✶−1=✶$⁠.

Magnetodynamic equation in terms of the electric field e in the frequency domain is:

For a circumferential electric field $e=eϕdϕ$⁠, equation (7) has the solution:

where $αn$ are coefficients to be determined later by the boundary conditions, $jn$ is the nth spherical Bessel function and $Yn1$ are scalar spherical harmonic functions (see for example Abramowitz and Stegun, 1965). We denote a (closed) ball around the origin with radius R as $BR$⁠. We define the inner product $<·,·>∂BR$ and normalize the spherical harmonics as:

where * denotes the complex conjugation and the symbol $δnm=1$⁠, if $n=m$⁠, and $δnm=0$ otherwise. The inner product on the surface $∂BR$ is independent of radius R, because the spherical harmonics and $sinθ$ do not depend on the radial coordinate and $dθ$ and $dϕ$ are coordinate co-frames.

Defining a magnetic vector potential as $a=−1iωe$⁠, we have:

from which we calculate:

in which $jn′$ denotes the derivative of $jn$ with respect to the inner function.

It is well known that the b-field is homogenized by spatial averaging (Bossavit, 1995; Gyselinck, Vandevelde et al., 1999; Gyselinck, Sabariego, and Dular 2006). This step involves a decision. The b-field can indeed be averaged spatially, but as a 2-form it must be averaged against a vector frame that is considered to be spatially constant. Because we are dealing with homogenization, we are interested in the Cartesian components of the b-field. We define the homogenized flux density $bprteff$ as the spatial average of the Cartesian z-component of the b-field over the volume of the particle. The spatial average can be calculated by considering the magnetic potential a on the boundary. First, we calculate the magnetic flux $Φθ$ through a surface $Sz$ which is the intersection of the particle with a plane whose z-coordinate is constant. The surface is oriented counterclockwise around the z-axis. Its boundary is the circle $ϕ→(Rprt,θ,ϕ)$ for a fixed $θ$ and $Rprt$⁠. We have:

in which the latter integral returns simply to multiplication by $2π$ because the circumferentially symmetric harmonics $Yn1$ behave as constants in the integration.

The spatial integral of the b-field is calculated by integrating equation (13) along the Cartesian z-coordinate and turning the resulting integral back to spherical coordinates by the transformation $z=Rprt cos θ$ and $dz=−Rprt sin θdθ$ to exploit equation (9). We denote the unit vectors in x and y directions as $x^$ and $y^$⁠. We get:

from which:

follows.

This type of homogenization filters out the first mode of the magnetic flux density. If the boundary conditions of the particle are such that they excite other magnetic modes as well, their contribution is not visible in the homogenized flux density, but it would have to be lumped into the homogenized magnetic field strength. Furthermore, the first line of equation (14) shows that the spatial average is calculated against a vector frame. A Cartesian frame is chosen because we want the homogenized fields to be expressed in Cartesian coordinates. This choice of frame essentially defines an affine connection, thereby introducing an additional mathematical structure.

The particle experiences both magnetization and eddy currents. Our goal is to express the behavior of the particle in terms of a dynamic relationship between the homogenized quantities $bprteff$ and $hprteff$⁠.

In the literature, energy balance is a key factor to be considered in homogenization (Corcolle et al., 2021). Complex Poynting’s theorem for peak-valued fields states:

where the leftmost term represents the (complex) power flux through the boundary. In the absence of displacement currents, the last term vanishes, and only the first two terms on the right-hand side contribute to the energy exchange through the boundary. If the excitation is assumed to act only in the Cartesian z-direction, the left-hand side power is equated with $iω2bprteffhprteff|Ω|$⁠, where $bprteff$ and $hprteff$ are the scalar components of homogenized fields. The quantity $hprteff$ can be solved from the energy balance.

Let us consider the Poynting flux through the boundary surface, $∂BRprt$ of the particle. For simplicity, we drop coefficient $12$ from equation (16) because it appears on both sides of the equation. Substituting the electric field (8) and tangential component of the magnetic field (12) into the Poynting flux term, we obtain:

The interpretation is that all the energy exchanges are lumped into the effective scalar modes of h and b, given by $hprt,neff$ and $bprt,neff$⁠. The modes are orthogonal in terms of Poynting flux. Family of reluctivities:

is defined for $n=1,2,…$⁠. If all the magnetic modes are excited, all the reluctivities $νprt,neff$ with $n=1,2,…$ participate in the energy exchange through the boundary of the particle. At this stage, the definitions of $hprt,neff$ and $bprt,neff$ in equation (17) need not be unique if considering only energy exchange. In fact, it suffices that the products $bprt,neffhprt,neff*$ coincide with what is stated in equation (17). Hence, the definitions of $νprt,neff$ for $n=1,2,…$ are not unique. However, we already have good reasons to define them as in equations (17) and (18). First, in Subsection 2.2, we found a formula for $bprteff$⁠, that is equal to $bprt,1eff$⁠. Second, defining the rest of the effective modes as in equation (17), the reluctivity modes in equation (18) all have a formula of the same form, differing only by the index n.

We are not fully statisfied with the formulas for $hprt,neff$ and $bprt,neff$ given in equation (17), because they contain unknown parameters $αn$ that are determined by the boundary conditions. Let us consider the tangential boundary conditions of h. The general form of the boundary values is:

Taking the inner product (9) of the boundary values, we get:

Substituting equation (20) into the definition of $hprt,meff$ in equation (17), $hprt,meff$ is simplified to:

which is completely determined by the boundary values and particle radius. Equation (21) provides the effective modes $hprt,neff$ for $n=1,2,…$ a physical meaning. Obtaining such a simple physical meaning is another reason to define $hprt,neff$ and $bprt,neff$ as in equation (17). In a straightforward manner, we obtain:

which is determined by the boundary conditions. We conclude that the effective modes $hprt,neff$ in equation (21) and $bprt,neff$ in equation (22) can be calculated from the boundary conditions. These effective modes fully determine the power exchange through the boundary of the particle.

Based on equations (15) and (17), we can state that $bprteff=bprt,1eff$⁠. Regardless of the magnetic modes that are excited, homogenization only filters out the first effective mode of the b-field. If only the first magnetic mode is excited, homogenization becomes trivial. The reluctivity of the entire system is $νprt,1eff$⁠. This special case corresponds to the uniform magnetic excitation used in literature (Sato and Igarashi 2017). If other modes are also excited, their contribution to the total energy exchange must be lumped into $hprteff$⁠. Equating equation (17) with the homogenized power $iωbprteffhprteff*|BRprt|$⁠, we obtain:

from which the homogenized magnetic field strength:

can be solved. Substituting $hprt,neff=νprt,neffbprt,neff$ and equation (21), the effective reluctivity of the particle is given by:

which is fully determined by the boundary conditions $hprttan$ of the h-field and the effective reluctivities $νprt,neff$ given in equation (18). This reluctivity is the main result of the first half of this paper. The second half of this study is dedicated to extending equation (25) to include the contribution of the insulating region around the particle.

In addition to the magnetic particle, we provide a treatment for the insulation region around it. The system is illustrated in Figure 1(b). The structure of this section is the same as the structure of Section 2.

Assuming that the insulation carries permeability $μins$ and the conductivity vanishes, the governing equation for the insulation is:

which is of Laplace type. For a circumferential electric field $e=eϕdϕ$⁠, equation (26) can be solved as:

where $Cn$ and $Dn$ are coefficients to be determined from the boundary conditions on the outer boundary of the insulation and the interface conditions on the boundary between the particle and insulation. Solutions (8) and (27) differ only by the radial function. We write the magnetic vector potential as $a=−1iωe$⁠, which yields:

from which we calculate:

Let us express the coefficients $Cn$ and $Dn$ in terms of $αn$ by equating the tangential components of h in equations (12) and (30) and the normal components of b in equations (11) and (29) at the interface between the particle and insulation, where $r=Rprt$⁠. We get two equations:

and:

from which we solve:

and:

Substituting $Cn$ and $Dn$ into equations (27), (29) and (30), we obtain:

that are valid in the insulation, where $Rprt<r<Rins$⁠.

The calculation of the homogenized magnetic flux density follows the treatment described in Section 2.2. Instead of repeating the same calculations starting from the a-field (36), we can exploit solution (15). We first observe that the restriction of the a-field (36) to the outer boundary of the insulation, where $r=Rins$⁠, is:

where the coefficients $κn$ are the only difference between equations (39) and (10). Hence, the calculations are very similar to those in Section 2.2, and we show only the necessary manipulations. Following the treatment given in equation (14), we obtain:

The only differences between equations (14) and (40) are that in equation (40), the coefficient $κ1$ appears, and the integration is associated with the radius $Rins$ instead of $Rprt$⁠. From equation (40), it follows that:

Substituting the tangential parts of the e- and h-fields, equations (35) and (38), to the Poynting flux integral, we obtain:

We note that the homogenized $binseff$ coincides with $bins,1eff$⁠. Furthermore, the modes are orthogonal in terms of Poynting flux. Family of reluctivities:

is defined based on the orthogonality of the modes.

We want to express the effective modes $hins,neff$ and $bins,neff$ in terms of the boundary conditions. The general form of the boundary conditions for h at the outer boundary of the insulation is:

Taking the inner product (9) from the boundary conditions, we get:

Substituting the inner product to the definition (42) of $hins,neff$⁠, we get:

which is very much of the same form as equation (21), except that the radius $Rprt$ is replaced by $Rins$⁠. Furthermore:

Using equations (46) and (47), the losses can be directly calculated from the boundary values.

For the particle/insulation system, we found that homogenization filtered out only the first mode of the b-field. Hence, the energy contributions of the remaining modes must be lumped into the homogenized h-field.

First, we notice that the representation of losses in equation (23) in terms of the effective modes is the same for the particle/insulation -system (replacing subscripts prt with ins) as for a bare particle due to the modes being orthogonal. Second, the coefficients before the inner products in the effective modes in equation (46) do not depend on the mode indices. It follows from these two observations that Section 2.4 applies directly to the particle/insulation system just by changing the subscripts prt, referring to a bare particle, to ins, referring to a particle/insulation system. The homogenized reluctivity of the system is given by:

where $νins,neff$ are the mode-reluctivities, given in equation (43), $hinstan$ are the tangential boundary values of the h-field on the outer boundary of the insulation and $Yn1$ are the scalar spherical harmonic functions whose normalization is defined in equation (9). To use the formula, it is necessary to substitute the symbols $νins,neff$⁠, $βn$⁠, $γn$ and $νprt,neff$⁠. These can be found in equations (43), (33), (34), and (18). Equation (48) is the main contribution of this study.