Eddy current losses in power voltage transformer open-type cores

Abstract

Purpose

The purpose of this paper is to present a methodology for calculating eddy current losses in the core of a single-phase power voltage transformer, which, unlike a standard power transformer, has an open-type core (I-type core). In those apparatus, reduction of core losses is achieved by using a multipart open-type core that is created by merging a larger number of leaner cores.

Design/methodology/approach

3D FEM approach for calculation of eddy current losses in open-type cores based on a weak AλA formulation is presented. Method in which redundant degrees of freedom are eliminated is shown. This enables faster convergence of the simulation. The results are benchmarked using simulations with standard AVA formulation.

Findings

Results using weak AλA formulation with elimination of redundant degrees of freedom are in agreement with both simulation using only weak AλA formulation and with simulation based on AVA formulation.

Research limitations/implications

The presented methodology is valid in linear cases, whereas the nonlinear case will be part of future work.

Practical implications

Presented procedure can be used for the optimization when designing the open-type core of apparatus like power voltage transformers.

Originality/value

The presented method is specifically adapted for calculating eddy currents in the open-type core. The method is based on a weak formulation for the magnetic vector potential A and the current vector potential λ, incorporating numerical homogenization and a straightforward elimination of redundant degrees of freedom, resulting in faster convergence of the simulation.

Citation

Frljić, S., Trkulja, B. and Drandić, A. (2023), "Eddy current losses in power voltage transformer open-type cores", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 42 No. 5, pp. 1039-1051. https://doi.org/10.1108/COMPEL-01-2023-0016

1. Introduction

A power voltage transformer is a power transformer with an open-type laminated core, which is particularly suitable for powering the power plant’s own consumption and for electrifying areas where there is no distribution network (Ziger et al., 2018). Power transformers with open-type core offer several additional advantages over power transformers with conventional cores. These include simpler manufacturing, enhanced robustness, explosion-proof safety, reduced weight and significantly lower ferroresonance (Ziger et al., 2014; Lapthorn and Keenan, 2015). The use of the open-type core in combination with superconducting windings is being explored to reduce losses in windings (Liew and Bodger, 2001). A comparison of an open-type core transformer and a transformer with conventional core is shown in Figure 1.

In the conventional core [Figure 1(a)], most of the magnetic flux remains within the core, resulting in negligible stray flux. Eddy currents are predominantly induced by the tangential magnetic flux on the lamination surfaces, forming narrow loops within the laminations. These localized eddy currents can be treated as a 1D phenomenon, as discussed in Dular et al. (2003) and Gyselinck et al. (1999). To account for edge effects, analytical homogenization methods are proposed in Hollaus and Schöberl (2015) and Hollaus and Schoberl (2018) using a multiscale approach with micro-shape functions. In addition, a multiscale modeling approach, incorporating both eddy currents and hysteresis, is presented in Niyonzima et al. (2013). The method described in Henrotte et al. (2015) uses algebraic approximation to determine material characteristics, with parameters obtained in the initial stage of the simulation for each finite element in the mesh.

In certain situations, the stray magnetic flux is not negligible, so in addition to the eddy currents induced by the main magnetic flux, it is also necessary to model the eddy currents induced by the stray magnetic flux, which usually flow in large loops tangential to the lamination surfaces (De Gersem et al., 2012).

In the case of the open-type core shown in Figure 1(b), all the magnetic flux from the core passes through the core/air interface and a large part of the magnetic flux is perpendicular to the lamination surfaces. Therefore, the induced planar eddy currents forming wide loops take on significant values (Bíró et al., 2005). A multiscale approach that also considers planar eddy currents using asymmetric micro-shape functions is presented in Hanser et al. (2022).

This paper presents a 3D FEM approach for calculation of eddy current losses in open-type cores based on a weak AτA formulation. The results are verified by comparison with simulations based on standard AVA formulation.

2. Problem definition

Problem domain Ω consists of the core region Ω_c, air region Ω₀ and the winding region Ω_s in which the source current J_s flows, as shown in Figure 2. Region Ω_c has the characteristics of an electrically conductive ferromagnetic material. Linear characteristics of the material in Ω_c are assumed in the presented work. In addition, a sinusoidal time dependence of the source current in the low-frequency range is assumed, so the set of quasi-static Maxwell’s equations in Ω_c is:

(1) ∇×ρJ=−jωB

(2) ∇×νB=J

(3) ∇⋅B=0

(4) ∇⋅J=0,

where B represents the vector field of magnetic induction, J represents the vector field of eddy current density, ω represents the frequency and j is the imaginary unit. Parameter ρ represents the electrical resistivity of the material and is equal to the reciprocal of the electrical conductivity σ, whereas ν represents the magnetic reluctivity of the material and is equal to the reciprocal of the magnetic permeability μ.

As the emphasis in the presented work is on the calculation of losses due to eddy currents exclusively in Ω_c, region Ω_s is defaulted as nonmagnetic and electrically nonconductive as the region Ω₀.

Consequently, for electrical conductivity in Ω₀ and Ω_s, it is valid that σ = 0, and for magnetic permeability, it is valid that µ = µ₀. Therefore, the set of quasi-static Maxwell’s equations in Ω₀ and Ω_s is:

(5) ∇×ν0B=Js

(6) ∇⋅B=0

(7) ∇⋅Js=0,

where B represents the vector field of magnetic induction, J_s represents the given source current density vector field and ν₀ represents the magnetic reluctivity of the vacuum for which it is valid that ν0=μ0−1.

3. Eddy currents in laminated core

As the region Ω_c is laminated, magnetic induction vector B and the eddy current density vector J can be disassembled into components in local αβγ-coordinate system of each individual lamination, where α-direction and β-direction are tangential to the lamination surface, whereas γ-direction is perpendicular to the lamination surface, as shown in Figure 3.

By disassembling the magnetic induction B into its components, we get the sum:

(8) B=Bαβ+Bγ,

where B_αβ represents tangential and B_γ represents normal component of magnetic induction B. According to equation (1), each of the components induces eddy currents inside the laminations, i.e. it holds:

(9) ∇×ρJαβγ=−jωBαβ

(10) ∇×ρJαβ=−jωBγ,

where J_αβγ represents the density of narrow loops of eddy currents that are induced by the tangential component of the magnetic induction B_αβ, whereas J_αβ represents the density of large eddy current loops that are induced by the normal component of the magnetic induction B_γ, as presented in Figure 3.

Finally, for the total current density J it is valid:

(11) J = Jαβγ + Jαβ

If for each lamination it is true that its thickness d is much smaller than its width l and height h, i.e. d≪l and d≪h, then it holds:

(12) ∫ΩLJαβ⋅JαβγdV≈0,

where Ω_L denotes the region of one lamination sheet. Equation (12) indicates that J_αβγ and J_αβ are approximately orthogonal, so it is then possible to calculate the losses P_αβγ due to eddy currents J_αβγ separately from the losses P_αβ due to eddy currents J_αβ, i.e. the total losses P can be calculated as:

(13) P=Pαβγ+Pαβ

4. Homogenization procedure

The laminated core is made of very thin laminations insulated from each other by even thinner layers of insulation. A direct consequence of the rapidly changing material characteristics is the highly oscillatory spatial dependence of the eddy current density J_αβγ. Consequently, an exceedingly dense mesh with multiple layers of finite elements per unit thickness of a single lamination sheet is necessary to correctly model the characteristics of the material, especially the field J_αβγ, which is not permissible in a practical 3D case. To enable the application of a coarse mesh, where the finite elements are wide enough to encompass parts of several neighboring lamination sheets, homogenization is needed (Kaimori et al., 2007). In the case of low frequencies, it is a reasonable approximation to assume that B_αβ remains constant across the thickness of a lamination sheet. In that case, by using equation (9), it becomes possible to explicitly express J_αβγ in terms of B_αβ, as demonstrated in Ziger et al. (2014). Consequently, the weak formulation term associated with J_αβγ is modeled using B_αβ. As B_αβ exhibits a monotonic behavior in the region of neighboring lamination sheets, the utilization of a coarse mesh is permissible. In this paper, the contribution of J_αβγ in the 3D weak formulation will also be considered by using B_αβ, but this transformation will be determined numerically through a 2D simulation in the preprocessing stage. Therefore, similar to Ziger et al. (2014), for the homogenized magnetic reluctivity tensor ν follows ν = Re{ν} + jIm{ν}. Accordingly, the diagonal tensors that will define the characteristics of the new homogenized material are as follows:

(14) ρ=[ραρβργ]

(15) ν=[νανβνγ]+j[0κκ],

where individual components are calculated as:

(16) ρα=ρβ=ρKf

(17) ργ−1≈0

(18) να−1=νβ−1=ν−1Kf+ν0−1(1−Kf)

(19) νγ=νKf+ν0(1−Kf),

where K_f is filling factor of a given laminated core. It remains to determine the parameter κ using the numerical homogenization procedure.

4.1 Calculation of the parameter κ

Parameter κ is determined numerically, in the preprocessing phase, by means of a 2D simulation of eddy currents. For 2D simulation, the weak AT-formulation described in Henrotte et al. (2015) is used. The value of average induction B¯2D is given as source, and after the simulation is completed, a 2D vector field J_2D is obtained that represents eddy currents J_αβγ within cross section of i-th finite element, as can be seen in Figure 4. Finally, using the expression:

(20) κi=∫SiρJ2D2dS/∫SiωB¯2D2dS

κ_i is obtained, where index i denotes the i-th finite element of the coarse mesh, with the surface S_i, marked with dashes in Figure 4. Surface S_i represents the surface of the cross-section through the i-th finite element of the 3D structural mesh. Therefore, κ is a discrete spatial function with a constant value within the i-th FE of the 3D structural coarse mesh. As can be seen in Figure 4, the parameter κ considers edge effects. The size of the coarse mesh finite element affects the value of κ at the edges of the laminations. The value of κ coincides with the value obtained through an analytical approach presented in Ziger et al. (2014).

The region Ω_c is homogenized, i.e. Faraday’s equation (9) in 3D weak formulation will be considered indirectly via equation (15) in combination with equations (18)–(20). Also, by equations (14), (16) and (17), a new material disables currents in the γ-direction to prevent inducing currents J_αβγ. Thus, B can be used instead of B_γ in (10), i.e:

(21) ∇×ρJαβ=−jωB

On the other hand, from the combination of equations (2) and (11), it follows:

(22) ∇×νB=Jαβ+Jαβγ

noting that J_αβγ in the 3D weak formulation will be accounted for indirectly, via B, by using ν instead of ν.

5. Multipart open-type cores

As the open-type core represents a magnetic circuit with a large air gap, the magnetic voltage drop in the air is dominant. Consequently, the magnetic flux significantly enters the core perpendicular to the lamination surfaces, inducing large eddy current loops as shown in Figure 3. In the case of open-type cores whose lamination sheets have large dimensions (thickness ∼ 0.35 mm, width > 5 cm and height > 50 cm), losses due to large loops of eddy currents induced by the magnetic flux perpendicular to the lamination surface are significant. Therefore, to further reduce core losses, it is technologically feasible to build a core from a larger number of leaner cores. A cross section of one such core is shown in Figure 5(b). The two-part core shown in Figure 5(b) has lower losses than the standard one-part core shown in Figure 5(a) due to the reduction in the area of the lamination sheets.

6. Weak formulation

A formulation based on the magnetic vector potential A and the electric scalar potential V is often used to model eddy currents. However, this formulation is not desirable in the case of a multipart open-type core due to the necessity to model thin insulation between the parts of the core.

Here, a formulation based on the magnetic vector potential A and the current vector potential T is used, where B = ∇ × A and J_αβ = ∇ × T due to equations (3) and (4). Regarding the J_αβγ component, it will not be directly modeled using potentials. However, as previously described, the weak formulation term related to J_αβγ is approximately proportional to the weak formulation term associated with the magnetic induction B_αβ. To achieve a symmetric system of equations, a time-primitive vector potential τ is used instead of T, where T = −∂_tτ = − jωτ. Potential τ is interpolated by edge elements N_k (Bíró, 1999). Therefore, a thin layer of insulation between the parts of the multipart core is simply modeled by setting the interface condition ∇ × τ = 0. Potential A, interpolated by edge elements N_l is used to strongly ensure the continuity of the normal component of magnetic induction at the core/air interface.

Therefore, equations (21) and (22) turn into:

(23) −jω∇×ρ∇×τ+jω∇×A=0

(24) ∇×ν∇×A+jω∇×τ=Jαβγ

In the region Ω_n = Ω₀ ∪ Ω_s instead of equation (5) it follows:

(25) ∇×ν0∇×A=∇×Ts,

where J_s = ∇ × T_s is due to equation (7).

Finally, using interpolation functions (edge elements) as weighting functions in equations (23)–(25), denoted by N_m and N_n, the weak AτA-formulation follows:

(26) ∫Ωnν0∇×A⋅∇×NmdΩ+∫Ωcν∇×A⋅∇×NmdΩ+∫Ωc∇×jωτ⋅NmdΩ=∫ΩT0⋅∇×NmdΩ

(27) ∫ΩcjωA⋅∇×NndV−∫Ωcρ∇×jωτ⋅∇×NndV=0,

where the index m represents the m-th degree of freedom for the A potential and the index n represents the n-th degree of freedom for the τ potential (Frljic et al., 2021).

6.1 Elimination of redundant degrees of freedom

As stated earlier, only the current density J_αβ is modeled directly, i.e. it holds that J_αβ = −jω∇ × τ. Therefore, it is sufficient that τ = τ_αa_α + τ_βa_β + τ_γa_γ has only a γ-component (normal component), i.e. τ = τ_γa_γ and τ_αβ = τ_αa_α + τ_βa_β = 0, where a_α, a_β and a_γ are unit vectors in α, β and γ directions according to the coordinate system in Figure 3. Also, the direction of the vector a_γ is represented with a red arrow in Figure 6. Thus, the tangential component τ_αβ is redundant and it is therefore desirable to eliminate it from the simulation.

If a structural mesh is created inside the core such that each edge of each finite element is either parallel to or perpendicular to a_γ, it is possible to identify the tangential and normal components of the vector τ with the edge degrees of freedom τ_k, where τ=∑τkNk. In that case, the product N_k · a_γ is either 0 or |N_k|, for each edge of each finite element in Ω_c.

Therefore, using the rule:

(28) Nk⋅aγ={0,    then τk=0|Nk|, then τk=τk

The redundant degrees of freedom from the matrix of coefficients in the preprocessing phase are eliminated. In Figure 6(a), the appropriate finite element and anisotropy vector a_γ, represented by the red arrow, are presented. Using equation (28), the element shown in Figure 6(b) is obtained. By eliminating redundant degrees of freedom, the convergence speed of the simulation is significantly improved.

7. Verification

The simulation will be performed on an open-type four-part core model, with anisotropy directions equal to those shown in Figure 6(a). Each of the four parts of the core has dimensions 1 cm × 1 cm × 10 cm, that is, the core has overall dimensions 2 cm × 2 cm × 10 cm. Also, each part of the core has 27 laminations with a filling factor of K_f = 0.94. That means that the entire core consists of 108 laminations. The thickness of the laminations is d = 0.35 mm, the electrical resistivity is ρ = 5 · 10⁻⁷ Ωm and magnetic reluctivity is ν = 10⁻⁴/µ₀. Field T_s is defined as the excitation. Field T_s corresponds to the density of the current J_s = 2 A/mm² flowing through the hollow cylinder (inner radius r = 2.5 cm, outer radius r = 4.5 cm and height h = 5 cm) positioned around the lower half of the core. The excitation frequency is f = 50 Hz.

7.1 2D simulation in the preprocessing phase

The simulation is performed over a cross section of a laminated core using a weak 2D AT-formulation (Frljic and Trkulja, 2021). It is sufficient to carry out the simulation only over the part of the 2D domain that is covered by at least two finite elements of the 3D structural coarse mesh, provided that suitable boundary conditions are used, i.e. one element must contain edge effects and the other must be without edge effects. An example of simulation over a part of the 2D domain is shown in Figure 4(a), where the simulation surface covers four finite elements of the coarse mesh.

After the simulation was performed, two different values for the parameter κ were obtained using equation (20).

For elements that belong to the narrow edge of the laminations, it is valid that κ = 5.69, due to the appearance of edge effects. For the remaining elements, the value of this parameter is κ = 6.39, as shown in Figure 7.

The density of the 3D structural coarse mesh is 100 FEs per cross section of one part of the core, as seen in Figure 7(b), and in total 40,000 hexahedral FEs in the core and approximately 170,000 FEs in the rest of the domain. The density of the mesh in the 2D simulation is about 800 triangular FEs per surface of one finite element of the coarse mesh.

7.2 3D simulation results

A comparison of three different types of 3D FEM simulations for the calculation of eddy current losses over a given four-part open-type core model was made. The first type of simulation is the AVA-type which uses the standard weak AVA-formulation. The second type of simulation is AτA*-type, which is based on the weak AτA-formulation [equations (26) and (27)]. The third type of simulation is the AτA-type which also uses the AτA-formulation [equations (26) and (27)] but with the elimination of redundant degrees of freedom of the current vector potential τ. For the sake of comparability of the results, all three simulations were performed on the same mesh. The hardware used was an i3-3120M (2.5 GHz) CPU and 12 GB of RAM. Table 1 shows a comparison of three types of simulation. The amount of total losses P is equal in all three cases. AτA*-type and AτA-type show an identical value of losses P_αβγ and P_αβ which means that the elimination of redundant degrees of freedom does not affect the simulation result. The deviation between AVA-type and AτA-type is larger for P_αβ losses and smaller for P_αβγ losses, which is expected because both formulations are based on the magnetic vector potential A. We can see that the duration of the simulation is slightly better for AτA-type than for AVA-type and significantly better than for the AτA*-type owing to the elimination of redundant degrees of freedom.

The distribution of total losses P obtained using AτA-type and AVA-type simulations are depicted in Figure 8.

8. Conclusion

Calculation of eddy current losses requires taking into account differences between the open-type core and the conventional core. As shown in Figure 8, using a multipart core reduces the size of the eddy current loops created by magnetic flux directed perpendicular to the surface of the lamination sheet. This lowers core losses.

The particularity of the geometry of the open-type core favors the use of a weak AτA-formulation. Presented approach, with the elimination of redundant DOFs significantly improves the convergence speed. Numerical homogenization offers a simple way of taking into account edge effects for eddy currents induced by the magnetic flux tangential to the lamination surfaces. Comparison of the results shows good agreement with other possible approaches. Proposed method, which yields accurate results, can easily be implemented and used for optimization when designing open-type cores.

Figures

Figure 1.

Comparison between an open-type core and a conventional core

Figure 2.

Problem domain with associated subdomains

Figure 3.

Eddy currents in a laminated medium

Figure 4.

Calculation of the parameter κ in the preprocessing phase

Figure 5.

Cross section comparison of two different open-type laminated cores

Figure 6.

Example of a multipart (four-part) open-type core

Figure 7.

Discrete distribution of parameter κ values

Figure 8.

Distribution of total core losses

Table 1.

Comparison of three types of simulation for the calculation of eddy current losses in a four-part open-type laminated core

Type	P, mW	Pαβγ, mW	Pαβ, mW	Time, s
AVA	168	110	58	2,840
AτA*	165	108	57	8,220
AτA	165	108	57	2,250

Source: Authors’ own work