Effective material modelling for laminated iron cores with an A-formulation and circuit coupling Open Access

This work addresses the modelling of non-linear magnetic materials within the context of an eddy current problem (ECP) in electrical machines or transformers with laminated cores. Laminated iron cores are composed of insulated conducting sheets to minimise the eddy current loss (ECL). The present approach uses an A-formulation with circuit coupling of voltage-driven excitation coils for an effective material (EM) to homogenise the core. Contrary to the present work, the approach described in Hanser et al. (2024) is based on a $T,Φ$-$Φ$-formulation with current-driven coils.

The material modelling of non-oriented electrical steel sheets by a parametric magneto-dynamic model for hysteresis, is discussed in Petrun and Steentjes (2020). The EM theory, a well-established homogenisation technique, is frequently used for heterogeneous structures such as laminated iron cores (Tsukerman, 2021; Schöbinger et al., 2021).

Homogenisation techniques generally aim to preserve the physical nature of the problem (Tsukerman, 2021; Schöbinger et al., 2021). For example, early research on ECLs in non-linear sheets, both in one and two dimensions, is presented in Gyselinck et al. (1999). A time-domain homogenisation method for non-linear laminated iron cores is developed in Gyselinck et al. (2006) and applied in time-domain simulations. Another approach, which uses an anisotropic material tensor for low-frequency applications, is described in Kaimori et al. (2007). In Frljić and Trkulja (2020), a three-dimensional (3D) model is transformed into a corresponding two-dimensional (2D) slice, allowing eddy currents to be treated as 2D phenomena within laminations and enabling edge effects to be captured more accurately. A finite difference homogenisation method, incorporating rate-dependent hysteretic losses for one-dimensional cases, can be found in Bergqvist and Engdahl (2001). Losses in amorphous core transformers with anisotropic permeability, using the Bertotti model, are examined in Steinmetz et al. (2010). A homogenisation method for laminated ferromagnetic cores based on the heterogeneous multiscale method, which involves upscaling and downscaling to establish material relations between mesoscale and macroscale is presented in Niyonzima et al. (2013). A comparison of various approximation formulas for calculating ECLs and equivalent conductivities is shown in Bermudez et al. (2008).

In this work, an EM approach is used in terms of a complex-valued non-linear reluctivity $νeff(|B|)$ within a static magnetic field problem (SMFP), while preserving the ECL and the reactive power (RP) of the original ECP. This EM approach eliminates the need for time-stepping schemes (Gyselinc et al., 1999; Gyselinc et al., 2006; Kaimor et al., 2007 Bergqvist and Engdahl, 2001; Steinmetz et al., 2010; Niyonzima et al., 2013; Bermudez et al., 2008; Bíró and Preis, 1995; Hanser et al., 2022; Hollaus, 2019) and harmonic balance methods (Hollaus, 2019; Ausserhofer et al., 2007). Moreover, this method enables the laminated core in 3D problems to be treated as a bulk domain. These advantages lead to substantial reduction in computational costs while maintaining accuracy.

In Section 2, the formulation of an ECP is presented using the finite element method (FEM) and is applied to a 3D reference solution (RS) of a laminated core excited by voltage-driven coils (Hollaus, 2019; Bíró et al., 2004). The one-dimensional cell problem (CP), which represents one sheet of the laminated core, is shown in Section 3. Using the calculated ECP and RP from both the reference and homogenised CP, a complex-valued effective reluctivity $νeff(|B|)$ is defined. Unlike the approach in Schöbinger et al. (2021), which is based solely on the fundamental frequency of the fields, this solution incorporates harmonics using a time-stepping method. In contrast to the CP described in Hanser et al. (2024), where the excitation is provided by a magnetic field strength $H0$⁠, in this work an averaged magnetic flux density $Bp$ is prescribed. In comparison to the 3D RS in Section 2, this effective material is used in a SMFP, as described in Section 4. The simulation with the EM approach accurately approximates the ECL, the RP and the effective current in the steady state, while achieving a significant reduction in computational costs. In Section 5, the local distributions of ECL and RP are analysed and compared to the corresponding distributions in the reference problem. An approximation of the effective circuit currents is given in Section 6. Numerical examples presented in Section 7, which cover various saturation levels, geometric configurations and serial resistors, demonstrate the high accuracy and substantial computational savings of the proposed method. Conclusions are summarised in Section 8.

In the 3D reference problem, each sheet in the laminated core is modelled in the mesh, see Figure 1 (above).

The boundary value problem (BVP) of the ECP is based on Maxwell’s equations:

where H is the magnetic field strength, B the magnetic flux density, E the electric field and J the eddy current density. With the electric conductivity $σ$⁠, the magnetic reluctivity $ν$⁠, the magnetic constitutive relation $H=νB$⁠, Ohm’s law $J=σE$ and the magnetic vector potential A, the BVP reads as:

where $B=∇×A$ and a known source current density $J0$ are used. To couple voltage-driven coils with an ECP (1), the circuit coupling equation:

where $Rs$ denotes a serial resistor, $Φ(t)$ a magnetic flux and $u0(t)$ an exciting voltage is introduced. With the unknown current $i(t)$ and a normalised current density $j0$ resulting from an impressed unit current in the coils, the right hand side in (1a) is replaced by $J0=ij0$⁠.

Applying the implicit Euler time-stepping scheme $∂tu ≈ 1Δt(u−u^)$⁠, where $Δt$ denotes a time step and the hat symbol values of the previous time instant, the weak formulation of the 3D ECP with voltage-driven coils reads as:

Find $(A,i)∈VD:={(A,i):A∈U,i∈R$ and $A×n=0$ on $ΓB}$ such that

and:

for all $(A′,i′)∈V0$⁠, where $U$ is a finite element (FE) subspace of $H(curl,Ω)$ (Hollaus, 2019; Bíró et al., 2004; Schöberl and Zaglmayr, 2005). Test functions are denoted by the prime symbol. The far boundary and the $z=0$ plane are considered as $ΓB$ for the geometry in Figure 1 (above). The exciting coils driven by the volatage are part of $Ω0$ but do not need to be modelled in the FE mesh. The non-linear magnetic reluctivity $ν(|B|)$ is treated according to Bíró and Preis (1995).

Although the relationships (4) and (5) are well known, they are given again for the sake of simplicity. To avoid the modelling of the exciting coils in the FE model, the normalised Biot–Savart field $hBS$ with:

is used for the excitation of (3a), whereby the boundary term vanishes for the given example in Section 7. The normalised Biot–Savart field is used in (3b) for the circuit coupling too. The corresponding terms:

where N denotes the number of turns, $As$ the cross section, $eφ$ the orientation and l the length of a single winding of the coil, shown in Figure 2, are derived from the field equations based on the law of induction (Bíró et al., 2004). The boundary term in (5) vanishes for the given example in Section 7.

To determine the apparent power (AP) of the 3D reference problem, the instantaneous values of the ECL per unit volume (ECL$V$⁠) and the reactive power per unit volume (RP$V$⁠):

respectively, where $ω=2πf$ denotes the angular frequency, $Ωc$ the domain of the conducting sheets, see Figure 1 (above), $Ωm$ the domain of the laminated core, i.e. the sheets including the insulating layers see Figure 1 (below), and $|Ωm|$ its volume, are calculated in every time instant. In the steady state, i.e. $A(t)=A(t+T)$⁠, the time-averaged ECL and RP over one time-period T are:

respectively, leading to the AP

where j represents the imaginary unit and the underline denotes complex values.

The one-dimensional CP, shown in Figure 3, is solved to calculate the non-linear EM$νeff(|B|)$⁠. The CP represents one sheet of the laminated core and is excited by the Dirichlet boundary conditions $A×n=−B0xez$⁠, with $n=∓ex$⁠, on the boundaries of the CP which yields an averaged flux density $B0$ in $ey$ direction. The resulting eddy currents J are oriented in the $ez$ direction.

For the one-dimensional reference CP on the interval $I=[−d/2,d/2]$⁠, see Figure 3 (above), a single component magnetic vector potential $A=u(x,t)ez$ with the unidirectional magnetic flux density $B=Bey=−∂xuey$ and the eddy current density $J=−σ∂tuez$ is used. Using the notation of (3) for the implicit Euler time-stepping scheme, the weak form reads as:

Find $u∈VD:={u∈U:u(t)=u0(t) on ∂I}$ so that:

for all $u′∈V0$ where $U⊂H1$⁠. The excitation is given by the Dirichlet boundary value $u0(t)=−Bpsin(ωt)x$ on $∂I$⁠, where $Bp$ describes the peak-value of the averaged magnetic flux density (Hollaus et al., 2024). The non-linear problem is solved according to Bíró and Preis (1995).

To determine the EM$νeff(|B|)$⁠, the ECL$V$ and RP$V$ are calculated in the reference CP. In the homogenised CP, these can be calculated analytically.

With the solution of (9), the instantaneous values of the ECL$V$ and the RP$V$⁠:

where $Ic=[−dc/2,dc/2]$ is the domain of the conducting sheet, are calculated at each time instant, respectively. A selected set of instantaneous ECL$V$s and RP$V$s is shown in Figure 4. The curves are computed using the measured BH-curve shown in Figure 5 (blue) for various peak values $Bp$ and $f=50$ Hz. The measured BH-curve represents the commutation curve of the ferromagnetic material M400-50A. The time-averaged ECL$V$ and RP$V$ are calculated in the steady state according to (7).

For the specific case of the homogenised one-dimensional CP, see Figure 3 (below), the magnetic flux density $B¯(x)=Bp$ is constant. With the material relation $H¯=νeff(|B|)B¯$⁠, the apparent power per unit volume (AP$V$⁠):

where the superscript * denotes complex conjugate, is determined analytically.

Using the EM, the ECL and the RP can be calculated in an equivalent problem without having to model individual layers of the laminated core. By equating the AP$V$s:

of the reference CP (8) and the homogenised CP (11), the EM:

for a given excitation $Bp$ is derived. The complex BH-curves of the effective reluctivities shown in Figure 5 with their real (orange) and imaginary (green) components are based on the measured BH-curve (blue), a frequency $f=50$ Hz (above) or $f=1$ kHz (below) and the parameters $σ$⁠, $dc$ and $d0$ from the numerical example are selected, see Section 7.

The complex-valued SMFP using $νeff(|B|)$ and voltage-driven coils of the 3D problem with homogenised core, see Figure 1 (below):

where $I_$ and $U_0$ are complex phasors and the exciting coils are part of $Ω0$⁠, is solved to approximate the averaged ECL and the averaged RP.

Considering (4) and (5), the regularised weak form of the problem with a homogenised core and the effective material reads as:

Find $(A¯,I¯)∈VD(Ω):={(A¯,I¯):A¯∈U,I¯∈C,A_×n=0$ on $ΓB}$ such that:

and

for all $(A¯′,I¯′)∈V0$⁠, where $U$ is a FE subspace of $H(curl,Ω)$⁠. The parameter $ε ≈ 10$ is introduced for a unique solution.

With the magnetic flux density $B¯=∇×A¯$ and the magnetic field strength $H¯=νeff(|B|)B¯$⁠, the effective AP$V$ of the homogenised core $Ωm$⁠:

is calculated. To reconstruct the ECL$V$ and the RP$V$⁠, the AP$V$ is separated by:

into the approximated ECL$V$$P˜=Re(S¯eff)$ and the approximated RP$V$$Q˜=Im(S¯eff)$⁠, where the tilde symbol indicates values based on the EM.

The ECL and RP in the reference problem (8) are equivalent to the approximated ECL and RP in the homogenised problem (17). Consequently, the ECL$V$ and RP$V$ distributions of the reference problem, averaged over one time-period T and across each i-th sheet in the normal direction, here the z-direction:

correspond to those in the homogenised problem.

Based on the solution of the homogenised problem, the averaged AP$V$ distribution in the i-th sheet

is determined, which can be separated into the averaged ECL$V$ and RP$V$ distribution:

respectively.

The effective value of the current in the reference problem:

is approximated by the effective current:

resulting from the calculations with the EM and the homogenised core.

A single-phase transformer is used as a numerical example. The geometry of the laminated core is shown in Figure 1 (above), using all symmetries. The dimensions $x0=156$ mm$,x1=94$ mm$,x2=334.5$ mm$,y0=250$ mm$,y1=94$ mm$,y2=334.5$ mm and $z2=334.5$ mm are used, see Figure 1 (below). With the thickness of one conducting sheet $dc=0.5$ mm, the filling factor $kF=dc/(dc+d0)=0.95$ and the number of sheets in the core of $N=20$ or $N=184$⁠, the height of the core is given by $zm=12(Ndc+(N−1)d0)=5.25$ mm or $zm ≈ 48.41$ mm, respectively. The discretisation of the used mesh for the RS is sketched in Figure 6. For the approach with the EM and homogenised core with the same discretisation in the xy-plane and only a single element for the homogenised core in z-direction is used. The excitation of the problem is considered by the Biot-Savart field of four voltage-driven symmetric cylindrical coils with 60 turns each. The inner and outer radius of the coils are selected with $ri,1=81$ mm, $ro,1=84$ mm, $ri,2=88$ mm and $ro,2=91$ mm, the length of the coil as $l=192$ mm. The electric conductivity of iron is selected with $σc=2.08·106$ S/m and the frequency $f∈{50,103}$ Hz. To regularise the problem, a small electric conductivity $σ0<1$ is assumed in air. Because the penetration depth at a frequency of $f=103$ Hz is approximately half the thickness of the sheet, magnetic field diffusion becomes significant. Consequently, the magnetic field cannot be assumed to remain uniform across the sheet’s thickness. The arrangement of the core with the coils exhibits three planes of symmetry. Handmade structured hexahedral FE meshes are used. The same discretisation by FEs in the x, y-plane is used for the reference model and the model with the homogenised core to ensure a fair comparison. One period of time is discretised in 50 instants and 2.5 periods are considered in the simulations. The voltage-driven coils are excited by $u0(t)=−Upramp(ωt)sin(ωt)x$⁠, shown in Figure 7, where the function $ramp(ωt)$ enables a smooth increase of the exciting voltage $u0(t)$ and to achieve the steady state rapidly. The RS is computed to verify the results obtained by the approach using the homogenised core with the EM.

To achieve different saturation levels of the core, the coils are excited with voltages of different peak values $Up$⁠. The ECL$V$⁠, the RP$V$ and the approximated currents for the excitations with $Up∈{0.5,1,3,5,10}$ V, frequencies $f∈{50,103}$ Hz and a series resistor of $Rs=100$$μΩ$ are shown in Table 1 for the brute-force RS the homogenised model using the EM. The non-sinusoidal curves of the currents, shown in Figure 8, validate that a wide range of saturation levels is covered with these peak values. In addition to the simulations with 20 sheets, simulations with 184 sheets are also shown. Relative errors $εP=100%(U−U˜)/U$ with $U∈{P,Q,Ieff}$ in the range of $εP ≈ 1%$ for the ECL$V$⁠, $εQ ≈ 1%$ for the RP and $εI ≈ 1%$ for the effective currents are achieved for all simulations with $f=50$ Hz and $εP ≈ 1.5%,εQ ≈ 1.5%$ and $εI ≈ 1%$ for $f=1$ kHz, respectively. The reduction of the computational costs is shown in Table 2. In detail, for $f=50$ Hz a speed-up of approximately 300 is achieved for the problem with 20 sheets and 5,000 for the problem with 184 sheets. The reduction of the computational cost for $f=1$ kHz is similar.

Simulations with a peak value of $Up=5$ V, 20 sheets and different serial resistors $Rs∈{0.1,1,3,10,100}$$Ω$ are shown in Figure 9.

The distributions of the time and sheet averaged ECL$V$ and RP$V$ are shown in Figures 10 and 11 for $Up=5$ V, $Rs=100$$μΩ$ and 20 sheets in the upper most sheet. A comparison of the magnetic flux density distributions for $Up=10$ V is shown in Figure 12.

All simulations are carried out using Netgen/NGSolve (Schöberl, 2024).

A novel method for calculating ECL and RP in laminated magnetic cores using an EM was introduced. This approach significantly reduces computational costs while preserving high accuracy. It involves a complex-valued SMFP for the homogenised cores, coupled with a serial resistor and excited by voltage-driven coils. This formulation eliminates the need for time-stepping or harmonic balance methods and avoids modelling each individual sheet separately. Simulations of 3D cores demonstrate that the EM approach provides reliable results, with errors about 1%, 1% and 1% at $f=50$ Hz for the ECL, the RP and the effective circuit current, respectively, for various setups, including different saturation levels of the magnetic material and numbers of sheets. The errors are increasing to about 1.5%, 1.5% and 1% for a frequency $f=1$ kHz, respectively. The method achieves remarkable computational efficiency, with speed-ups of about 300 for a numerical problem with 20 sheets and 5, 000 for 184 sheets compared to conventional FEM simulations.

The local distributions of ECL, RP and the magnetic flux density were found to be in excellent agreement with reference distributions, validating the accuracy and applicability of the method.

With the CP presented, the introduced model can only take into account eddy currents in the cross section of the sheet induced by the main magnetic field. Large eddy current loops due to perpendicular stray fields or edge-effect eddy currents are not considered in this model.

In conclusion, the proposed EM-based homogenisation technique offers a powerful and efficient tool for analysing magnetic cores in electrical machines and transformers. It significantly reduces computational costs while providing accurate estimations of ECL, RP and the effective current, making it highly valuable for both research and industrial applications.