Stress-Dependent Magnetic Equivalent Circuit for Modeling Welding Effects in Electrical Steel Laminations

Abstract

Welding has a severe impact on the efficiency of electrical machines. The heat added during the welding process affects the microstructure of the material and causes residual stress. This results in local degradation of the magnetic permeability and facilitates additional iron losses in the machine core. With the purpose of modeling and simulating welding effects in electric machines, this paper proposes a stress-dependent magnetic equivalent circuit (MEC) model for welded non-grain-oriented electrical steel laminations. A modified iron loss model is proposed to accommodate these welding effects. Furthermore, the proposed MEC model is applied to a M270-35A stator core as a case study. It was demonstrated that the core losses increase by 25% when four welding joints are applied. With a limited number of magnetic measurements on a welded and unwelded core, the model can be fully parametrized. Finally, the model was successfully validated on a core with eight welding seams at 100 Hz. The proposed model can be integrated into the design of electric machines to consider the welding effects.

1. Introduction

Throughout the production process of electric machines, several techniques are applied to shape and stack the electrical steel laminations to make the stator and rotor core. After shaping and stacking the individual laminations, welding connects the laminations into a rigid stack and provides the torsion strength necessary for machine operation. An example of a welded stack of stator laminations is shown in Figure 1. Many researchers have studied the impact of welding on the core losses in electrical steel quantitatively, as reported in [1,2]. Others have studied the effects of welding on the microstructure of the material in and near the weld seam [3]. In [4], the residual stress near welding seams in electrical steel was measured and it was found that tensile stress appears within the weld seam while compressive stress appears in the heat-affected region near the weld seam. In [5], the relation between stress and the magnetic properties of electrical steel was reported. The variation of magnetic properties under stress can be linked to the impact on core losses of residual stress near welding seams. Additionally, in [6], it was noted that the welding process destroys the thin insulation layer between the lamination sheets, introducing short circuit paths, and resulting in increased classical losses. This effect is shown in the cross-section view in Figure 2.

It is clear from previous research that welding introduces residual mechanical stress in the area around the welding seam, which is known to change the local magnetic properties. Welding locally impairs the insulation layer between adjacent laminations, resulting in increased classical losses.

Recently, researchers have attempted to model the effects of welding in electrical steel in several ways. In [7], an analytical model is presented that shows the increase in classical losses when the weld bead radius increases. In [6], a 3D-FEM motor model was proposed where the yoke had uniform magnetic properties, which were identified using measurements on welded ring cores. In the [1] Epstein frame, measurements were used to parametrize a degradation model for magnetic material properties assuming homogenously damaged areas. It is considered that the model incorporates cutting effects adequately and allows incorporation of these effects in a finite-element method-based motor design software.

To the author’s knowledge, there are no reports of an analytical MEC model that accommodates local welding effects. The existing models apply either time-consuming finite element (FE) models or measurements with a standardized Epstein setup, which have different geometric properties than ring cores. Moreover, the existing models neglect the effects of residual stress or locally changing hysteretic material properties around the weld seam. A fast and accurate parametrized model for simulating welding effects on iron losses in electrical steel would be a valuable tool for studying the impact of each parameter. In this paper, a stress-dependent MEC model is presented that calculates the iron losses in welded stacks of electrical steel. The model is based on the known relations between mechanical stress and magnetic properties of non-grain oriented (NGO) electrical steel, with few geometrical and magnetic simplifications. Simultaneously, the model incorporates the effects of local eddy currents in the weld seam. Using Bertotti’s loss separation principle [8], a modified coarse-scale stress-dependent iron loss model is proposed. Finally, the model is applied to magnetic measurements on a welded stator core in the form of a case study.

2. Proposed Model

In this section, the specific aspects of the MEC-model will be presented. Further, the stress-dependency of the material model and loss model will be described in more detail. Afterward, in the Methodology section, it is explained how the model can be applied to calculate core losses in a welded stack of laminations.

Table 1. Symbols commonly used in the model.

Symbol	Description	Unit
Rweld	Radius of the welding seam	mm
Rdeg	Radius of the degraded zone	mm
σ1	Stress in stress-free zone (always assumed 0 MPa)	MPa
σ2	Residual stress in degraded zone	MPa
σ3	Residual stress in welded zone	MPa
σeq	Equivalent stress calculated in each reluctance by Equation (1)	MPa
Ch	Core loss coefficient for hysteresis loss
Ccl	Core loss coefficient for classical loss
Cex	Core loss coefficient for excess loss
Kµ	Stress-dependent correction factor for magnetic permeability
Kh	Stress-dependent correction factor for hysteresis loss component
Kex	Stress-dependent correction factor for excess loss component
wtot	Average yoke width	mm
ltot	Average yoke circumference	mm
d	Lamination thickness	mm
Bp	Peak flux density	T
µr	Relative permeability
lr	Length of reluctance	mm
Ar	Cross section of reluctance	mm2
f	Excitation frequency	Hz
ρ	Electrical resistivity	Ωm

contains the most important recurrent symbols in the model and

Table 2. Model assumptions.

1	The weld seam area and its surrounding heat affected area are simplified as a rectangular area with dimensions specified in Figure 2.
2	The curvature on the outer edge of the ring core is simplified to a straight line.
3	A uniform isotropic stress state is assumed in each reluctance of the considered MEC, but stress anisotropy with respect to magnetization direction is considered.
4	The skin effect is neglected which is a valid assumption when considering sufficiently low frequencies.
5	The applied welding technique does not change the metallurgic composition of the weld seam. The electrical steel melts and bonds with adjacent laminations without any additional binding material during the welding process.

includes all the assumptions that were made.

2.1. MEC Model

In the model, the stator yoke of the machine is simplified as a ring core with a rectangular welded region and magnetically degraded region (Figure 2). Each welded section of the stator can be represented by a MEC, as shown in Figure 3. The weld seam geometry in this model was simplified using rectangular areas. The weld seam is located on the outer edge of the lamination. In the welding seam, the laminations are all electrically connected. In the region near the welding seam (region 2 in Figure 2), the heat from the welding process causes residual stress in the material, but the electrically insulating coating remains intact. For regions 2 and 3, a uniform stress state is assumed in the model. This stress state influences the magnetic permeability necessary for calculating the magnetic flux in each reluctance. In [9,10], measurement data shows the relation between the magnetic permeability and uni-axial (when magnetization and stress direction are parallel) stress state in NGO steels. It is assumed in this paper that these data are representative for each grade of NGO electrical steel. Based on these data stress-dependent correction factors for the relative permeability K_µ can be constructed as shown in Figure 4. However, these expressions are only valid when the direction of magnetization is parallel to the direction of stress, while it is known that the stress dependency of electrical steel is anisotropic. The influence of this stress anisotropy can be modeled using the equivalent stress approach. This approach was used by several researchers, such as in [11,12], and enables the calculation of magnetic properties under multi-axial stress when only uni-axial measurements are available. The equivalent stress model was adopted from [11] and is defined by:

$${\sigma }_{eq}=\frac{1}{K}ln\left(\frac{2\exp \left(K{\mathit{h}}^T\mathit{s}\mathit{h}\right)}{\exp \left(K{\mathit{t}}_{\mathit{1}}^T\mathit{s}{\mathit{t}}_{\mathit{1}}\right)+\exp \left(K{\mathit{t}}_{\mathit{2}}^T\mathit{s}{\mathit{t}}_{\mathit{2}}\right)}\right)$$

(1)

where K is a material parameter (=4 × 10⁻⁹ for silicon-iron), s is the deviatoric part of the applied stress tensor (s = σ − (1/3)tr(σ)I), see [11] for more details, and I is the identity tensor. h, t₁ and t₂ are the direction vectors parallel to the applied field, orthogonal to the applied field and orthogonal to the sheet plane, respectively. In regions 2 and 3 (Figure 3), the stress is assumed to be directed radially away from the weld seam. This assumption corresponds to the results found in [2,4]. The stress in these areas is directed either 0°, 45° or 90° with respect to the magnetic flux direction (double-ended arrows in Figure 3). For the reluctances where the residual stress is not parallel to the magnetization direction, the corresponding σ_eq is calculated and used to determine the correction factors for each reluctance. In the proposed model, each reluctance in the network is defined by:

$$R({\sigma }_{eq},B)=\frac{l_r}{{\mu }_0{K}_{\mu }\left({\sigma }_{eq}\right){\mu }_r\left(B\right)A_r}$$

(2)

where l_r and A_r are the length and magnetic cross-section of each reluctance. The relative magnetic permeability µ_r is the stress-free permeability. In Figure 3, region 1 is the stress-free zone where K_µ = 1. Regions 2 and 3 are not stress-free and K_µ is determined by first calculating σ_eq and afterward applying K_µ based on Figure 4.

2.2. Coarse-Scale Stress Dependent Loss Coefficients

In the previous subsections, it is described how the MEC accounts for the residual stresses in and near the welding seam by assigning a stress-dependent magnetic permeability to each reluctance. Now, the MEC model can be solved for the flux density in each reluctance. However, the iron losses occurring in electrical steels also depend on stress [13]. In developing a stress-dependent iron loss model, a frequently used approach is to separate the core losses into hysteresis losses P_hyst, classical (eddy current) losses P_cl and excess losses P_exc and to investigate the relation to stress for each component:

$$P_{tot}=P_{hyst}+P_{cl}+P_{exc}$$

(3)

When the flux density waveform is sinusoidal, the total core loss depends on the peak magnetic flux density B_p and frequency f following:

$$P_{tot}=C_h{B}_p^{2}f+C_{cl}{B}_p^2{f}^2+C_{ex}{B}_p^{1.5}{f}^{1.5}$$

(4)

where C_h, C_cl and C_ex are grade-dependent coefficients. In [10,13,14,15], the stress-dependency of each loss coefficient was analyzed in more detail. These authors indicated the stress dependency of the hysteresis and excess loss coefficients clearly. Moreover, there appears to be a consensus concerning the independency of the classical loss to stress. In [14], the individual loss components are described as:

$$P_{hyst}=W_h\left(B_p\right)f$$

(5)

$$P_{cl}=\frac{B_p^2{f}^2{d}^2{\pi }^2}{\rho 6}$$

(6)

$$P_{exc}=8\sqrt{\left(1/\rho \right)G{A}_r{V}_0\left(B_p\right)}{\left(B_{p}f\right)}^{1.5}$$

(7)

where W_h is the hysteresis energy loss, ρ is the electrical resistivity, G = 0.1357, and d and S are the lamination thickness and cross-section, respectively. It can be assumed that the parameter V₀ together with hysteresis energy loss W_h could describe the effect of the applied external mechanical stress on the microstructure and on the magnetic properties of NGO electrical steels, where V₀ describes the influence on losses of various microstructural features such as (among others) grain size, crystallographic texture and residual stresses [14,16]. The models and data published by these authors enable the construction of a coarse-scale stress-dependent iron loss model. The stress-dependent hysteresis loss can be modeled using a correction factor K_hyst as illustrated in Figure 5. Similarly, the stress dependency of the excess loss can be found in the V₀ parameter, which can be modeled using a correction factor K_exc shown in Figure 6. However, B-dependency of V₀ in Equation (7) is neglected in this model, which results in inaccuracies when the material goes into saturation. Therefore, in the case study, the model is validated on magnetic flux values up to 1.3 T to avoid modeling errors. Applying these correction factors to the iron loss model results in:

$$P_{hyst}\left({\sigma }_{eq}\right)=K_{hyst}\left({\sigma }_{eq}\right)C_h{B}_p^{2}f$$

(8)

$$P_{exc}\left({\sigma }_{eq}\right)=\sqrt{K_{exc}\left({\sigma }_{eq}\right)}{C}_{ex}{\left(B_{p}f\right)}^{1.5}$$

(9)

3. Local Classical Loss in Weld Seam

In the previous section, it was mentioned that the classical losses remain unaffected by the stress state. However, as the welding seam forms a connecting bar of electrical steel at the outer edge of the stack, the local eddy currents in this bar can be calculated using the classical loss equation but with thickness d equal to the weld seam radius R_weld (as illustrated in Figure 2). This way, the influence of the weld seam radius on the classical losses in the weld seam (region 3 in Figure 2) is included in the model:

$$P_{cl,weld}=\frac{B_p^2{f}^2{R}_{weld}^2{\pi }^2}{\rho 6}$$

(10)

3.1. Methodology

In this section, it is described in three steps how the model can be applied in any specific case of welded electrical steel. In Figure 7, a flowchart shows how the methodology can be applied step by step.

3.1.1. Model Construction

A parametrized MEC model should be constructed, as specified in Figure 3. In this figure, a MEC model is shown for a core with only one welding seam. R_weld, total length (l), total width (w) and thickness (d) should be set to the radius of the weld seam, the average yoke circumference, the average yoke width and lamination thickness, respectively. In the model, R_deg is a parameter that will be optimized in the next steps. Consequently, the dimensions of each reluctance in the model should be a function of R_deg. The reluctances are either located in the stress-free area, the degraded area or the welded area. The location of each reluctance will determine the permeability and core loss components of each reluctance. In the stress-free area, the reluctances have magnetic properties as defined by Equations (2), (5), (6) and (7). In the degraded zone, the reluctances have alternative magnetic properties which are defined by Equations (2), (6), (8) and (9). The reluctance which represents the welded zone is defined by the same equations as the degraded zone, except for the classical loss component, which is defined by Equation (10).

3.1.2. Magnetic Measurements

First, the single-valued BH-curve should be measured on an unwelded core. This curve defines the permeability of the reluctances in the stress-free zone. Secondly, the grade-dependent loss coefficients C_h, C_cl and C_ex need to be identified using core loss measurements at different frequencies and peak flux densities on an unwelded core. Any fitting algorithm can be used, as long as the accuracy of the fitted model is sufficient. Once these coefficients are identified, they will represent the loss coefficients of the reluctances in the stress-free zone. Finally, core loss measurements should be done on the welded stack at several frequencies and peak flux densities. These measurements will be used for fitting the MEC model parameters.

3.1.3. Model Parameter Fitting

With the parametrized MEC model and welded core measurements available, the model parameters σ₂, σ₃ and R_deg can be fitted using an optimization algorithm. The initial parameter values for σ₂ and σ₃ are set to 0 MPa. The initial value for R_deg is set to R_weld + 1 mm, because R_deg can never be smaller than or equal to R_weld. Then, the optimization is started using a constrained least squares algorithm. Once these parameters are optimized, the calculated core losses should correspond to the measured core losses on the welded core. Additionally, the model parameters give a coarse estimation of the stress near the weld seam and the size of the area which is damaged by the welding heat. The reluctances with their respective magnetic properties can now be copied into a full MEC machine model.

4. Case Study: Welded Stator Core Measurements

As a proof of concept, core losses were measured at different peak flux densities and excitation frequencies on a stack of M270-35A stator laminations before and after welding. The elastic limit of this material is 450 MPa and the inner and outer yoke diameter of the stator laminations was 147.6 mm and 193.9 mm, respectively. Consequently, the yoke section width is 23.15 mm. The laminations were laser cut and afterward stress relief annealed. This way, the cutting effect is reduced and a more accurate modeling of the welding effects is made possible. A stack of 15 laminations was welded on 4 equidistant locations around the edge of the yoke using tungsten inert gas (TIG) welding (compatible with assumption 5) with welding current 90 A at a welding speed of 4 mm/s. The weld seam radius R_weld was measured to be 3 mm. Global measurements of magnetic field strength H_s and magnetic flux density B_a were performed for a range of frequencies and peak flux densities similar as described in [17] using the setup displayed in Figure 8. The core losses are:

$$P_{core}=\frac{1}{T}\underset{0}{\overset{T}{{\displaystyle \int }}}{H}_s\frac{d{B}_a}{dt}dt$$

(11)

Core loss measurements were done on the unwelded and welded (with four seams) core over a range of frequencies (10, 50 and 200 Hz) and peak flux densities (0.1 T to 1.3 T in steps of 0.1 T). The measured losses in the unwelded stack were used to fit the parameters C_h, C_cl and C_exc of the undamaged material using Equation (4). The identification of these loss coefficients was made using the MATLAB surface fitting toolbox by plotting the measured and modeled core loss against the frequency and peak flux density.

Afterward, the unknown model parameters (R_deg, σ₂ and σ₃) are fitted using a combination of the MATLAB genetic algorithm and f_mincon toolbox. In this section, the measurements will be compared to the losses calculated by the fitted model and a validation of the model will be presented.

The identified core loss coefficients of undamaged material are C_h = 0.0126, C_cl = 1.195 × 10⁻⁵ and C_exc = 0.0011. After fitting the unknown model parameters, it was found that for a compressive stress state of σ₂ = 20 MPa in region 2 and R_deg =14 mm, the simulations correspond accurately with the measurements. The fitted stress state in the weld seam σ₃ was 330 MPa tensile stress; however, it must be noted that this factor appeared to play a minor role in the loss calculation due to the comparatively small size of Rweld in the total width of 23 mm. Figure 9 displays the measured and simulated specific iron losses for different frequencies and peak flux densities for the welded and unwelded case. This graph shows a good correspondence between the fitted model and the measurements. The deviation between the measured and simulated values for each case can partially be explained by the noise-sensitive nature of the measurement setup. Moreover, the relation between the V₀ and B_p was neglected in the model in order to simplify the stress dependency of V₀. This could lead to inaccuracies when the material is saturated. Therefore, measurements and fitting were conducted for B_p-values up to 1.3 T. The eddy current losses at 1 T and 50 Hz were 15.8 W/kg in the welded zone and approximately 0.21 W/kg in the degraded and undamaged zone. The total core losses at 1 T and 50 Hz were 16.3 W/kg in the welded zone, 1.84 W/kg (on average) in the stress-free zone and 2.14 W/kg (on average) in the degraded zone.

In Figure 10, the model (which was fitted with loss measurements at 10, 50 and 200 Hz on a 4-weld seam stack) was validated in case the number of weld seams doubled to 8 at 100 Hz. Here, measurements on a welded stack with eight weld seams are compared to model calculations when 8-weld sections are simulated. This shows that the model accurately predicts the effect of core losses for varying frequencies and varying amounts of weld seams.

5. Discussion

In Figure 9, the core losses show an exponential increase with increasing peak flux density and frequency, which is in correspondence with Equation (4). When the laminations are welded, this exponential trend remains but appears to have shifted upwards. The total core losses increase by about 25% at 50 Hz and 1 T when four welding seams are added. In

Table 3. Comparison of welded stator measurements with other examples found in literature.

Reference	Sample Geometry	Added Core Loss Due to Welding at Sinusoidal Flux Density of 50 Hz and Bp = 1 T
Case Study	Ring Core, 4 weld seams	25%
[1]	Epstein Strips, 3 weld seams	10%
[2]	Ring Core, 6 weld seams	42.5%
[6]	Ring Core, 8 weld seams	22%
[18]	Ring Core, 2 weld seams	10%

, this result is compared with other results found in the literature on similar welded cores. From this comparison, it is clear that the added core losses strongly depend on each specific case. Several specific parameters such as the steel grade, core geometry, welding speed and amount of welding seams all have an impact on the added losses.

The results obtained from the simulations demonstrate that the local eddy current losses in the welded zone are significantly higher than the other loss components. In this section, the effect of the weld radius R_weld on the total core losses is simulated and compared to the results presented in [7]. In this work, Wang and Zhang performed FE-simulations and analytical calculations on welded laminations where the total width was 5 mm and the sheet thickness was 0.27 mm. These results demonstrate that the eddy current losses increase exponentially when the weld radius increases from 0.8 mm to 1.3 mm at 50 Hz excitation frequency. With the optimized fitting parameters from the previous section kept constant, the effect of increasing the weld radius (from 3.7 mm to 6 mm, in order to keep the ratio of the weld radius over the total section width equal for both cases) on the total core losses is simulated and shown in Figure 11. This figure shows that the proposed stress-dependent MEC model adequately incorporates the effect of increasing core losses due to increasing weld radius, similar to the results in [7].

6. Conclusions

This paper presents a parametrized welding model that may be included in a complete MEC model of an electrical machine. The model accounts for local residual stress near the welded region and for local eddy currents in the weld seam. Although the model simplifies the welding geometry and magnetic sensitivity to stress and assumes uniform residual stress states in different sub-regions, it provides an interesting tool for studying the prominent effects of welding and allows for a qualitative study of the effect of different process parameters on the additional iron losses due to welding. When the model parameters are fitted correctly based on experimental observations on welded ring cores, it is possible to calculate additional core losses due to welding at varying amounts of welding seams, excitation frequencies and weld bead radii. The fitted parameters also give an estimation of the size of the degraded zone and the residual stress present in the area around the welded zone. The calculation time is low because the model does not require FE-simulations. In the case study, it was demonstrated that the core losses increase by approximately 25% when four welding joints are applied. The application of the model on cores of M270-35A grade steel has demonstrated that with a limited number of magnetic measurements on a welded and unwelded core, the model can be fully parametrized. The model was successfully validated on a core with eight welding seams at 100 Hz.