1 Introduction

Eddy current losses can make a significant part of total losses in power transformers, generators or motors. This has resulted in continuous application of eddy current loss calculation in nonlinear magnetic steel in various industries of electromagnetic devices.

Calculation of losses in magnetic steel with constant permeability can give good results if problem is restrained only to the linear part of B–H curve [1]. Linear modeling was, however, at one point replaced by various nonlinear models. This followed the understanding of importance of saturation of magnetic materials in loss calculation [1–3]. Nonlinear attempts were carried out by employing rectangular B–H curve assumption. One-dimensional (1-D) calculations were made for a sinusoidal plane wave penetrating into semi-infinite magnetic plate. Further significant modeling improvements were investigated by including analysis of hysteresis effects. Barth [4] used harmonic analysis for field and hysteresis representation, while Gieras [5] gave a solution for differential diffusion equation assuming variable complex magnetic permeability. All authors pointed out the difference between power loss calculation with forced sinusoidal varying flux and magnetic field. Further on, Emanuel [6] extended the research to forced non-sinusoidal flux and magnetic field.

With the development of high-speed computers, numerical methods such as finite element method (FEM) enabled calculation of eddy current losses in complex three-dimensional (3-D) geometries [7]. Unfortunately, in its early years, commercial FEM programs did not include solving nonlinear problems in calculations on large 3-D models in time-harmonic domain. Methods for nonlinear calculation [1–6] were still confined to smaller 1-D or 2-D computations. Nowadays, FEM solvers have started to include nonlinear modeling even for 3-D geometries in time-harmonic domain [8–11]. Some works suggest inclusion of higher harmonics in calculation by the so-called harmonic balance method [12]. Application of aforementioned approaches for solving complex 3-D problems can result with long time-consuming computations. In some cases, this is not suitable for utilization in usual industry design. Some producers of FEM software packages recently introduced quasi-nonlinear operating point look-up procedure in sinusoidal time variation to solution of such problems [13] in time-harmonic domain. They claim that although such procedure is not permissible from a strict mathematical point of view, it provides a good agreement with measurements. Such approaches of nonlinear modeling in time-harmonic domain have to be checked on simple configurations [14]. Investigations have shown quantitative importance of nonlinearity inclusion in loss calculation.

The purpose of this paper is to compare results of quasi-nonlinear time-harmonic approach with results of transient time-domain “step-by-step” computations, and with measurements. Due to its possibilities in modeling complicated 3-D geometries, commercial FEM package MagNet has been chosen for this research [13]. Final goal is to give evaluation of proposed nonlinear time-harmonic approach for calculation of stray losses in configuration such as transformer tanks.

2 Solving strategies

Transient solver can make solutions of nonlinear problems with higher harmonics included in electromagnetic wave shape. This enables us to analyze distortions of sinusoidal waves caused by magnetic steel saturation and its effect on eddy current losses. Transient solvers can be used for calculation on simple models, such as for 1-D or 2-D models, because 3-D models can turn out to be computational expensive (time and memory consuming).

Time-harmonic solver is more often used for calculations on 3-D models. Although time-harmonic solver assumes that the materials are linear, it is still possible to model non-linear materials using a quasi non-linear model. The assumption is that B and H are both in phase. The magnitude of the field is used to determine the value of the permeability (based on the usual B–H relationship). This is a non-linear process, and a simple update scheme is used, the element permeability is updated based on the B–H curve, and the equation resolved, until a converged solution is obtained [13]. It should be noted that this whole procedure is not permissible from a strict mathematical point of view. However, time-harmonic solver is significantly faster and easier to use on complex geometries such as transformers. Therefore, question that remains open is: how well do calculation results agree with measurements?

To get correct field distribution inside magnetic steel, mesh has to be created within depth of penetration of magnetic steel. Depth of penetration of electromagnetic wave is:

$$\begin{aligned} \delta =\sqrt{\frac{2}{\omega \mu \sigma }} \end{aligned}$$
(1)

where \(\omega \) is the angular frequency, \(\mu \) is magnetic permeability and \(\sigma \) electrical conductivity of magnetic steel. For excitations used in these models, depth of penetration varies from 1 to 3 mm. Therefore, the dimensions of the elements used for meshing of the plate should not exceed around 0.5 mm.

Another way to restrain number of mesh elements of 3D models when using time-harmonic solvers is to employ a surface impedance (SI) boundary condition on the surface of magnetic steel. In this way, magnetic field is only calculated on steel surface. It is automatically assumed that the field and the losses decay exponentially inside the volume. This simplification leads to a simple formulation in losses per unit surface area \(P_\mathrm{S}\) \((\hbox {W/m}^{2})\) of the plate [15]:

$$\begin{aligned} P_\mathrm{S} =\sqrt{\frac{\omega \mu }{2\sigma }}H_\mathrm{rms}^2 \end{aligned}$$
(2)

where \(H_\mathrm{rms}\) is root-mean-square value of magnetic field at the surface of the steel plate. Expression (2) is only valid if thickness of the plate is many times larger than depth of penetration (1). It simplifies penetration in magnetic material using only one relative magnetic permeability value in whole depth of magnetic field penetration. This assumption has definite influence on calculation results. One of the purposes of the paper is to see the influence of this simplification. Within this research, surface impedance boundary condition is applied for all calculations made with time-harmonic solver.

3 Simple model

Evaluation of eddy current calculation methods is made on simple models for experimental loss evaluation in magnetic steel (shown in Fig. 1). Model consists of a coil and a magnetic steel ring. All loss computations and measurements are done at frequency 50 Hz using a 3-D FEM solver. As saturation effects of magnetic steel have to be considered, calculation has to be nonlinear. Nonlinearity due to saturation of magnetic steel is modeled as a single-valued nonlinear B–H curve. Investigation of hysteresis effects was not part of research included in this paper. However, similar contributions on this topic [16] suggest using magnetic induction vs loss curves developed from separate measurement of hysteresis loop areas.

Fig. 1
figure 1

a Experimental ring and b cross-section of the model in MagNet

Full size image

Coil can be excited by a sinusoidal voltage or current source. This distinction is important if material is nonlinear. Voltage source will cause total flux in steel to be sinusoidal, while magnetic field on steel surface will be distorted. In case of excitation by a sinusoidal current source, magnetic field on steel surface will be sinusoidal, while total flux in steel will have distortions. Difference between two sinusoidal excitation types can be simply evaluated when making a transient calculation. In case of time-harmonic solvers with nonlinear permeability in steel plate considered, magnetic field H and total flux \(\varPhi \) will be sinusoidal at every point of the model. Using time-harmonic solver, high time and memory consumption may be avoided and the problem can be restrained to computation that includes only the 1st harmonic. In this case, resulting fields and losses will be equal, regardless of whether the coil is excited with voltage or current source.

4 Calculation results and comparison with measurements

Calculation results are compared with measurements made on two magnetic steel rings. Dimensions and parameters of the rings are shown in Table 1. Additionally, DC magnetization curve was measured for both experimental rings and used as a B–H curve in the numerical model. Measured data are shown in Fig 2.

Table 1 Ring parameters and dimensions
Full size table
Fig. 2
figure 2

Measured B–H curves of experimental rings

Full size image

Coils wound on the rings were excited with sinusoidal voltage source. In Figs. 3 and 4, calculated and measured losses of the two experimental rings are compared with peak flux \(\varPhi _\mathrm{peak}\) on the x axis. Flux is expressed in webers per circumference of ring cross section \(2a + 2b\) (Fig. 1).

Fig. 3
figure 3

Calculated and measured losses on steel ring A

Full size image
Fig. 4
figure 4

Calculated and measured losses on steel ring B

Full size image
Fig. 5
figure 5

Comparison of calculated losses in case of different excitations: compared by a peak flux and b peak magnetic field

Full size image

Losses calculated with transient solver are lower than measurements up to 20 %. These highest deviations were noticed at lower excitation (flux) values where losses are lower than \(1000\hbox { W/m}^{2}\). Further on, results of time-harmonic calculation are lower at most 15 % from results obtained by the transient solver. These highest differences were evaluated at higher excitation values where losses are higher than \(1000\hbox { W/m}^{2}\). It can be concluded that both types of calculation do not differ more than 20 % from measurements. However, both types of calculations do not include hysteresis losses. It should be emphasized that contributions to loss estimation in magnetic materials [16] have shown that hysteresis loss can constitute up to 30 % of total loss in magnetic steel. This value falls to 10 % at higher saturation levels of magnetic material.

5 Sinusoidal current source vs sinusoidal voltage source

Transient “step-by-step” solver can be used to evaluate influence of different excitations: sinusoidal voltage or sinusoidal current source. Because this is a time “step-by-step” procedure, it is important to reach steady state where a certain DC component (if developed) will be damped. To avoid development of a DC component, sinusoidal functions of the sources had been assigned with such phase value that no flux linkage was present at instant \(t = 0\). For example, in case of voltage source, voltage had peak value at instant \(t=0\) (cosine function). Simulation lasted 12 periods (240 ms), and steady-state was reached.

Using transient solver, calculated losses on experimental ring B are compared for cases with sinusoidal current and voltage source. Results are shown in Fig. 5. Losses are compared by two parameters: peak flux \(\varPhi _\mathrm{peak}\) and peak value of magnetic field at middle diameter of ring at steel surface calculated as \(\sqrt{2}H_\mathrm{rms}\). Figures show that calculated losses can be 40 % higher when changing type of excitation from current to voltage. Such differences have to be taken into account when making calculation that will be compared to measurements.

Origin of these differences is wave distortion caused by B–H curve nonlinearity in steel plate. Figure 6 shows these two cases compared. Total coil flux and magnetic field on steel surface are given for voltage and current excitation.

Fig. 6
figure 6

Flux and magnetic field for coil supplied by a sinusoidal current, b sinusoidal voltage

Full size image

6 Complex model

As already elaborated, time-harmonic solvers may easily be used for calculation of losses in complicated 3D geometries. Additionally, assigning surface impedance boundary condition on metal parts can substantially reduce time and memory consummation during FEM calculation. Such approach is used for loss calculation on a special stray flux model shown in Fig. 7.

Fig. 7
figure 7

Experimental model

Full size image
Fig. 8
figure 8

Model dimensions in millimeters

Full size image

Basic idea of the model is to simulate a winding stray field penetrating into tank wall of a transformer [17]. Model consists of three separate components:

  • winding with 200 turns of single strip conductor 7.0 \(\times \) 3.0 mm

  • core made of silicon steel sheets

  • magnetic steel plate 8 mm thick

Model dimensions are given in Fig. 8. Surface impedance boundary condition was used for loss calculation in the plate. Size of triangular elements used to discretize plate surfaces was set to 40 mm, while volume discretization of winding and core components was done with elements smaller than 50 mm.

Losses in the plate were equal to total losses measured by a power analyzer subtracted with winding and core losses. Winding stray losses were negligible because of chosen conductor dimension. For this reason, it was possible to evaluate losses in the winding with high precision. Here, it should be noted that core was oversized to keep values of iron losses inside the core low. In this way, introduction of errors during separation of core losses from total losses was minimized. However, to estimate losses in the core, plate was removed from the model and measured losses were subtracted with only winding losses. Obtained core losses were used in the following analysis of experiments with the plate. Such approach was verified with plates made from nonmagnetic steel and aluminium [17]. Calculation and measurement results did not differ more than 5 %. Nevertheless, stray losses in magnetic plate were dominant when winding was supplied by an ac source. This enabled quite precise measurement of losses in steel plate.

Within this paper, losses in the plate for different air gaps d between core and plate were studied and compared with quasi-nonlinear time-harmonic calculation. Model setup in laboratory is shown in Fig. 9.

Fig. 9
figure 9

Stray flux model in laboratory: a without plate, b with plate

Full size image

Winding was excited by a voltage source. Air gap between plate and core constitutes a dominant part of magnetic resistance of the circuit. Thus, even in saturated conditions, both current and voltage waveforms on winding terminals were sinusoidal. It can be stated that a configuration such as the complex model is different from the simple model presented in Sects. 2 and 3. Therefore, comparison of experimental and calculation results on complex model has to be made.

Comparison of measured and calculated losses in the plate is shown in Fig. 10. It can be stated that calculated losses are within ±7 % margin compared to measured losses. Quasi-nonlinear time-harmonic solver with surface impedance boundary condition set on plate surfaces gives quite accurate results for all three air gap values. Elaborated approach has shown to be useful for calculation of stray losses in 3D geometries such as large power transformers. It should be pointed out that such approach did not include hysteresis losses. For this reason, both experimental and theoretical research should be continued to improve modeling of similar practical configurations.

Fig. 10
figure 10

Comparison of measured and calculated losses in the plate

Full size image

7 Conclusions

Paper discusses two different calculation methods applicable in 3D FEM modeling: transient and time-harmonic solvers. Their application is evaluated on a nonlinear 2-D problem and compared with measurement results. Calculated losses with time-harmonic and transient solvers can be up to 20 % lower than measured losses. According to differences between transient and time-harmonic calculation, drawbacks of nonlinear time-harmonic simplification are visible at higher excitation values. However, it should be noted that hysteresis losses were neglected in all calculations.

It is pointed out that type of excitation can have certain influence on losses in magnetic steel. Calculated losses for voltage and current excitations can differ up to 40 %. When making quasi-nonlinear time-harmonic calculation, type of excitation does not have influence on loss values. This should be especially kept in mind in cases where sinusoidal flux or magnetic field could be distorted by nonlinearities in magnetic materials.

Measurement results of a special transformer stray flux experimental model are compared to calculation results and give good agreement. It can be stated that quasi-nonlinear time-harmonic solver can be found useful in industrial application for calculation of stray losses in power transformer tanks. However, further research should focus on improvement in nonlinearity and hysteresis inclusion in time-harmonic approach.