Switched reluctance motor core loss estimation with a new method based on static finite elements

Abstract

Core loss estimation in switched reluctance motor is a complex task, due to non-linear phenomena and non-sinusoidal flux density waveforms. Several methods have been developed for estimating it (e.g. empirical, and physical-mathematic models), each one with merits and limitations. This paper proposes a new method for core losses estimation based on Finite Element Method Magnetics software. The main idea is using the machine phase-current harmonics as input for estimating core losses. In addition, a comparative study is carried out, where the proposed approach is faced up to a different one, based on Fourier decomposition of the flux density waveforms in the machine sections. In order to systematically analyze and compare the applied estimation cores loss techniques, a case study of a three-phase 6/4 SRM for different simulation scenarios is introduced. The outcomes of both methods are discussed and compared, where core loss convergence is found for limited speed and load ranges.

Keywords:

• Switched reluctance motor
• core loss
• time-harmonic
• eddy-current
• hysteresis
• finite element method magnetics

REVIEWING EDITOR:

• Ai Qingsong
• Wuhan University of Technology
• China

SUBJECTS:

• Technology
• engineering and technology
• electrical and electronic engineering
• computer and software engineering
• power engineering
• power and energy

1. Introduction

Switched reluctance motors (SRM’s) are simple, low cost, rugged and fault tolerant. They can reach very high speeds, having a wide constant power zone, with high efficiency (Ahn & Lukman, 2018). They have been recognized as an alternative to permanent magnet (PM) machines and induction motors for several applications, like household devices, electric mobility, and industrial use (Bostanci et al., 2017; Lukman & Ahn, 2021). Losses are mainly concentrated in the stator since there are no windings or PM in the rotor (Han et al., 2021). Therefore, it is easy to cool them, making it suitable for high-temperature and harsh operating environments (Bostanci et al., 2017). Nonetheless, some drawbacks still need to be tackled, where high torque pulsation, vibration and noise are the most relevant ones (Han et al., 2021; Lukman & Ahn, 2021). In addition, SRM modeling is challenging, due to its highly non-linear operation conditions.

In order to improve electrical machine’s efficiency, accurate losses estimation is fundamental, having also impact on its design, operation, and control (Wang et al., 2016). Most often, lamination manufacturers provide core loss data under sinusoidal excitations in a limited frequency and flux density range. However, such data is not adequate for predicting losses in electrical machines with non-sinusoidal flux waveforms, as in SRM. This requires loss information at high flux density and frequency, particularly for high-speed operation where they become higher (Ibrahim & Pillay, 2013). Core loss estimation in SRM poses several challenges due to non-sinusoidal flux density waveforms in different stator and rotor core sections (some highly saturated), like unipolar alternating and dc-biased bipolar waveforms. They depend on several factors, as machine design (geometry, stator and rotor poles, yoke geometry, number of phases), operating conditions (conduction angle and mechanical speed have an impact on magnetic saturation and lamination skin effect) and control methods (Torrent et al., 2011). Moreover, due to skin effect, the lamination flux density distribution is not uniform.

Over the years, several approaches were proposed for calculating SRM core losses: Steinmetz-based empirical models, magnetic circuit and equivalent circuit models, physical-mathematic models (e.g. Preisach and Jiles-Atherton) and numerical finite element analysis (FEA) based approaches (Krings et al., 2012; Yan et al., 2020). The latter ones can give better estimation, since they can address to core loss physics, aiming to describe its non-linear mechanism under distorted flux (Parsapour et al., 2015). However, they are complex and computationally heavy. Empirical models are the preferred ones for engineers, due to its simplicity and faster processing. Parameter estimation is based on curve-fitting methods, validated by manufacturer iron sheets data, experimental results or through finite element modeling (FEM) (Ibrahim & Pillay, 2013; Parsapour et al., 2015). Accuracy is much sensitive to parameter values, which are valid only for specific conditions (e.g. flux density and frequency ranges). Moreover, the manufacturing process of the machine has also a deep impact on core losses, which is very difficult to address in the parameter estimation process (El-Kharahi et al., 2015; Krings et al., 2012) and modelling in general.

Core losses are usually addressed by three different approaches (in time and frequency domain or both): empirical equations, loss separation components and hysteresis models (Melo & Araújo, 2020; Rodriguez-Sotelo et al., 2022). For SRM, a popular approach is to take Fourier flux density decomposition in core sections with different waveforms. Core losses are then calculated in frequency domain, usually with a Steinmetz-based formula (Chen et al., 2021; Krishnan, 2001; Mthombeni & Pillay, 2005; Yan et al., 2018).

The main goal of this work is to propose a new approach for estimating static core losses in SRMs based on Finite Element Method Magnetics (FEMM) software. It takes advantage of FEMM modeling features, where lamination skin effect, hysteresis and saturation are considered for calculating lamination flux density non-uniform distribution. Naturally, FEMM limitation to sinusoidal assumption was the main challenge for modelling this machine. For a three-phase 6/4 SRM several operation modes were simulated, and the motor core losses are estimated with the proposed method. For comparison purpose, a second method is also employed for calculating core losses following a similar procedure as in (Krishnan, 2001), which is similar to the one addressed in the previous paragraph. The outcomes of both methods are analyzed and discussed, where some convergence is found for limited speed and load range. It must be stressed that this is a simulation study, based only on static methods. Nevertheless, this work seeks to shed some light over the existing challenges for core loss estimation based on an open-source finite element analysis tool. By comparing its outcome with a well-established static core loss estimation method, we intend to show the proposed approach may have some potential for SRM core loss estimation.

The paper is organized as follows: Section 2 presents an overview of the fundamental of this work: the proposed approach and a comparison study for SRM core loss estimation. The SRM model design is addressed in Section 3 while in Section 4 is discussed in detail the two methods for calculating core losses, particularly the proposed one. Simulation results and discussion are presented in Section 5 and the final conclusions are stated in Section 6.

2. Proposed method and switched reluctance motor core loss study

This section begins with a short overview on FEMM, followed by the fundamental idea of the proposed method. A discussion about the integration of the two methods for the comparing study purpose is also addressed.

2.1. Overview on finite element method magnetics

Finite Element Method Magnetics (FEMM) is an open-source software for solving low frequency electromagnetic problems on 2-D planar and axisymmetric domains (Meeker, 2009). It is user friendly, accurate, with low computational burden. These are the main reasons for FEMM wide acceptance in both research and education fields (Crozier & Mueller, 2016).

Non-sinusoidal time analysis and motion are not supported by FEMM since it is based on mechanical static configurations (Lehikoinen et al., 2018). Speed impact on the modeled system can be addressed through a simulation sequence of static configurations, where transients are disregarded. Hence, for electrical machines modeling these limitations must be considered. In spite this, there are several examples where FEMM has been used in electrical machines modelling and design (Velazquez Velazquez, 2016), but not so frequently for core loss estimation. By defining the proper boundary conditions, FEMM solves time-harmonic problems in phasor domain. In subsection 4.2 is provided a description of the methodology applied in FEMM for estimating core losses.

2.2. Conceptualization of the proposed method

The first step was designing the FEMM’s SRM model. In addition, some Matlab^® scripts were developed for:

• Implementing the proposed core loss estimation method;

• Decoupling the core loss components in each mesh element, since FEMM’s output gives the total losses.

As stated before, FEMM is limited to harmonic time-dependent magnetic problems which is a significant limitation for modeling SRM core loss, since flux density waveforms are not sinusoidal. Taking advantage of current as FEMM’s natural input source, this can be overcome by considering the SRM phase-current harmonic decomposition. In addition, since current pulses are easy to be estimated or measured, this can be seen as an advantage when implementing this approach, compared to other methodologies. Figure 1 shows the fundamental structure of the proposed method for SRM core loss estimation.

Figure 1. Concept of the proposed method for static SRM core loss estimation based on FEMM.

It should be noted the way our method deals with the SRM rotor static configuration is not explicitly stated in Figure 1. This is addressed in subsection 4.3.

2.3. Switched reluctance motor core loss estimation – interaction of the two methods

In order to take a preliminary insight on the proposed FEMM-based method effectiveness, a comparison study to a well-established approach was carried out (denoted as Method 1), which is supported by an SRM Equivalent Circuit Model. From Figure 2, it can be seen the two SRM models are closed coupled, but it should be noted the developed method (Method 2) can be implemented independently, as shown in Figure 1.

Figure 2. Core loss estimation with Methods 1 and 2, based on two SRM models and Fast Fourier Transform (FFT).

The former is a much simpler SRM model than the latter, since core saturation is the only non-linear phenomenon being addressed, but its computation burden is much lighter. In order to deal with saturation in the Equivalent Circuit Model, the SRM static magnetization (Ψ(i,θ)) and torque (T(i,θ)) characteristics are stored in two look-up tables, which are fundamental components of the equivalent circuit model, as shown in Figure 2. Furthermore, these characteristics were generated with FEMM model which has to be considered first. As for the equivalent circuit model, it was developed in Matlab^®/Simulink^®, where periodic time flux density waveforms in different SRM sections are generated.

Both Methods are in frequency domain, based on two loss decouple (hysteresis and eddy current). Nonetheless, loss calculation is performed quite differently. The first one is an empirical approach, based on a curve fitting approach, which requires loss data from lamination manufacturers. The second has a solid theoretical background with a relatively simple mathematical formulation but limited to sinusoidal applications. Both methods have merits and limitations which are addressed in the paper (subsections 4.1 and 4.3 describe Methods 1 and 2, respectively). In addition, the way they are applied for this particular SRM analysis is carefully discussed.

For this work, current waveforms generated by the SRM equivalent circuit were also decoupled in their harmonic content, and the most relevant ones were taken as the input of the FE model (Figure 2). Therefore, while for the empirical method core losses were estimated based on flux density harmonics, in the second method they were calculated addressing current harmonics.

3. Switched reluctance motor models

In this section, the SRM models addressed in Figure 2 are presented in detail and their merits and limitations are discussed.

3.1. Finite element model

The first step was to build a three-phase 6/4 SRM model with FEMM, where the machine in (Soares & Costa Branco, 2001) was taken as reference. The parameters of the designed SRM are listed in Tables 1–3.

Table 1. Electrical and mechanical data.

Parameter	Symbol	Value
Rated power	Pn	750 W
Rated speed	nn	1800 rpm
Stator poles	Ns	6
Rotor poles	Nr	4
Stator pole arc	βs	30°
Rotor pole arc	βr	30°
Turns/phase	N	185
Wire conductivity	σW	58 MS/m
Wire strand diameter	dW	1.024 mm
Supply voltage	VDC	150 V
Phase resistance	Rs	1.30 Ω
Maximum inductance	Lmax	58.33 mH
Minimum inductance	Lmin	4.33 mH
Inertia	J	0.0013 kg·m^2
Viscous friction coeff.	Kf	0.02 Nm·s·rad^-1

Table 2. Geometric data.

Parameter	Symbol	Value
SRM radius	RSRM	85 mm
Shaft radius	Rsh	10 mm
Rotor outside radius	Rro	42.5 mm
Rotor inside radius	Rri	29 mm
Stator outside radius	Rso	75 mm
Stator inside radius	Rsi	42.9 mm
Height of rotor pole	hrp	13.5 mm
Height of stator pole	hsp	32.1 mm
Air-gap length	lg	0.4 mm
Stack length	Lst	50 mm

Table 3. Core lamination data (M45G29).

Parameter	Symbol	Value
Thickness	d	0.36 mm
Conductivity	σlam	2.9 MS/m
Stacking factor	-	0.96
Mag. permeability (rel.)	µr	4689

Figure 3 shows a sectional view of the SRM, where Table 2 parameters are also represented.

Figure 3. Geometric parameters of the SRM designed with FEMM.

The SRM geometry is as simple as possible since the focus is to compare two different approaches for core loss estimation. Figures 4a,b and 5 show, respectively, the characteristics of magnetization, static torque, and self-inductance, for a single-phase of the designed SRM. Magnetic saturation is addressed, but not hysteresis phenomenon.

Figure 4. SRM characteristics: (a) magnetization [ψ(i,θ)]; (b) static torque [T(i,θ)].

Figure 5. SRM phase self-inductance characteristic [L(i,θ)].

3.2. Equivalent circuit based model

A three-phase 6/4 SRM nonlinear simulation model was developed in Matlab^®/Simulink^®, as shown in Figure 6, based on the following parametric equations (Krishnan, 2001): (1) $$V_{\mathit{\text{phase}}}=R_s·i+\frac{d{\mathrm{\Psi }}_{\mathit{\text{phase}}}\left(i,\theta \right)}{\mathit{\text{dt}}}$$(1) (2) $${\mathrm{\Psi }}_{\mathit{\text{phase}}}(i,\mathrm{\theta })=f_h\left(i,\theta \right)$$(2) (3) $$T(i,\mathrm{\theta })=\sum _1^3\frac{\partial W_c^j(i,\theta )}{\partial \theta }$$(3) (4) $${\mathrm{T}(\mathrm{i},\mathrm{\theta })‐\mathrm{T}}_{\mathrm{\text{load}}}=\mathrm{J}\frac{\mathrm{d}{\mathrm{\omega }}_{\mathrm{r}}}{\mathrm{\text{dt}}}+{\mathrm{K}}_{\mathrm{f}}{\mathrm{\omega }}_{\mathrm{r}}$$(4)

Figure 6. SRM equivalent circuit simulation model.

Where V_phase, R_s, i, ψ_phase, θ, T, W^j_c, T_load, J, ω_r and K_f are, respectively, the stator phase voltage, resistance, current, flux linkage, rotor position, motor torque, co-energy in phase j, load torque, system inertia, rotor mechanical angular speed and viscous friction coefficient. Finally, f_h (i,θ) represents the magnetizing characteristics. As previously stated, both (2(2) $${\mathrm{\Psi }}_{\mathit{\text{phase}}}(i,\mathrm{\theta })=f_h\left(i,\theta \right)$$(2) ) and (3(3) $$T(i,\mathrm{\theta })=\sum _1^3\frac{\partial W_c^j(i,\theta )}{\partial \theta }$$(3) ) are modeled with two look-up tables. They were generated with FEMM and are represented in Figures 4 and 5. Since mutual magnetic inductances are neglected, the SRM model has three independent phase blocks.

The power converter dynamics and losses are disregarded here since this work is focused on SRM core loss. Three voltage levels are available – V_phase ∈{-150,0,150} V) –, like in the H-bridge asymmetric converter, and voltage PWM control and hysteresis current control can be implemented. No speed control was performed, there is only one inner loop for current regulation in hysteresis control. The flux linkage (ψ) is estimated from the inverter output voltage. For this reason, it was chosen as a state variable, together with θ. Therefore, for each simulation step, one has i = f(ψ,θ).

4. Core loss estimation methods

4.1 Method 1 (with Matlab^®/Simulink^® model)

The calculation is based on a two-step procedure: in the first one, the model in subsection 3.2 is used for predicting flux density waveforms in different parts of the SRM. In the second step, Fourier analysis is applied for harmonic characterization of the waveforms. For each flux density harmonic, core losses density (P_core) was calculated with the following improved Steinmetz expression (valid for sinusoidal waveforms), based on two-loss decoupling components (static hysteresis (P_h) and classic eddy-current (P_e) losses): (5) $$P_{\mathit{\text{core}}}{=P}_h+P_e=k_h{\mathit{\text{fB}}}_{\mathrm{\text{max}}}^{a+b\bullet B_{\mathrm{\text{max}}}}+k_e{\left(f{B}_{\mathrm{\text{max}}}\right)}^2$$(5)

Here, $k_h,k_e{,f}_h$ and $B_{\mathrm{\text{max}}}$ are, respectively, the hysteresis and eddy-current factors, the harmonic frequency, and its peak flux density. As for a, b they are fitting parameters (in addition to $k_h$ and $k_e$), regarding the lamination loss data (see Table 4). They were analytically calculated in the f < 150 Hz range. For the other ranges, small arrangements in $k_h,$ $k_e,$ and b were sufficient, as depicted in Figure 7. For each SRM section, flux density waveforms were first estimated with the described model (‘Flux Waveforms’ block in Figure 6). Fourier decomposition is then applied, and core losses are calculated as a sum of SRM section’s harmonic losses. In order to promote an even comparison with Method 2, the excess-loss term was suppressed. Nonetheless, Table 4 parameters allow a good match between the lamination manufacturer data (Proto Lam, 2022) and (5), as shown in Figure 7. This seems to be related to the p_h expression, which is also reported in (Bouchnaif et al., 2020). Equation (5)(5) $$P_{\mathit{\text{core}}}{=P}_h+P_e=k_h{\mathit{\text{fB}}}_{\mathrm{\text{max}}}^{a+b\bullet B_{\mathrm{\text{max}}}}+k_e{\left(f{B}_{\mathrm{\text{max}}}\right)}^2$$(5) was already applied for SRM core loss estimation (Akiror et al., 2012; Ganji et al., 2010), where some modifications were implemented, in order to address hysteresis minor loops and non-sinusoidal flux waveforms. This is not considered here. Nonetheless, the aim of the present work is to compare two static harmonic different methodologies for core loss estimation, where the focus is to identify eventual convergence results. They are based on harmonic decomposition, where Fourier analysis is common to both methods, with significantly differences. Naturally, both approaches have merits and drawbacks. Overall, (5(5) $$P_{\mathit{\text{core}}}{=P}_h+P_e=k_h{\mathit{\text{fB}}}_{\mathrm{\text{max}}}^{a+b\bullet B_{\mathrm{\text{max}}}}+k_e{\left(f{B}_{\mathrm{\text{max}}}\right)}^2$$(5) ) also suffers from considering uniform flux distribution in laminations. In addition, it is based on the superposition principle where, for a highly non-linear system as in SRM, must be carefully applied.

Figure 7. Core losses curve-fitting (lamination M45 G29): (a) f ≤ 200 Hz; (b) f > 200 Hz.

Table 4. Curve-fitting parameters (M45 G29).

Kh	a	b	Ke	f range
5.1 × 10^−3	1	0.71031	7.864 × 10^−5	f < 150 Hz
5.1 × 10^−3	1	0.71031	5.864 × 10^−5	f ∈ [150–600] Hz
3.1 × 10^−3	1	0.51031	4.864 × 10^−5	f > 600 Hz

4.1.1. Core loss calculation

In order to compute the core losses with this approach, the flux densities in different sections of the machine were first derived based on a stator pole flux waveform, as in (Krishnan, 2001). It should be noted they are similar in all stator poles, but are phase-shifted by the stroke angle (θ_S), given by: (6) $${\theta }_S=2\pi \left(\frac{1}{N_r}-\frac{1}{N_s}\right)=30°$$(6)

For a three-phase 6/4 SRM, Figure 8a,b represent, respectively, a stator phase flux linkage average path and the machine sections with different flux waveforms (the arrows represent the mean path contribution of each phase). There are 3 sections for the stator: poles; yoke 1 and yoke 2. For this particular SRM, yoke 1 flux waveform is given by the sum of the three-phase poles waveforms, while flux waveform in yoke 2 results from adding two-phase poles waveforms and subtracting the third one. It should be noted that the waveforms in two contiguous yoke 2 sections are similar but not identical, since they are phase shifted. Nonetheless, they have the same harmonic magnitude content. As for the rotor, there are 2 sections with different flux waveforms: poles and yoke. The total SRM core losses (P_SRM) are given as: (7) $$P_{\mathit{\text{SRM}}}=\sum _{k=1}^n{P}_{\mathit{\text{core}}\_k}·W_k$$(7)

Figure 8. 3 Phase 6/4 SRM: (a) Phase flux linkage mean path (dashed lines); (b) Core sections with specific flux waveforms.

Where P_{core_k} is the core loss density in section k (calculated with (5(5) $$P_{\mathit{\text{core}}}{=P}_h+P_e=k_h{\mathit{\text{fB}}}_{\mathrm{\text{max}}}^{a+b\bullet B_{\mathrm{\text{max}}}}+k_e{\left(f{B}_{\mathrm{\text{max}}}\right)}^2$$(5) ) for the most relevant flux density harmonics), W_k is the iron section weight and n is the number of sections (6 stator pole, 2 yoke 1, 4 yoke 2, 4 rotor pole and 4 rotor yoke).

Figures 9 and 10 show the flux waveforms and flux density harmonics at the SRM sections, for [T = 3.8 Nm; n = 1715 rpm], which is almost coincident to the designed SRM rated operation. As depicted in Figure 11, for T_av = 1.4 Nm (see Table 6), core loss distribution in the SRM sections occur mainly on stator, in particular at yoke 2. As for the rotor, they are mostly concentrated at the yoke. This is not surprising, after taking a look at the spectrum components in Figures 9 and 10. For the other torque ranges, loss distributions are similar.

Figure 9. Stator SRM core sections: (a) Flux waveforms; (b) Flux density harmonics.

Figure 10. Rotor SRM core sections: (a) Flux waveforms; (b) Flux density harmonics.

Figure 11. 6/4 SRM core loss distribution (T_av = 1.4 Nm): (a) Stator sections; (b) Rotor sections.

4.2. Core loss estimation in FEMM

The following sections provide a detailed description over the core loss method applied in FEMM, addressing its formulation, merits, and limitations.

4.2.1. Lamination eddy-current loss formulation (linear materials under time-harmonic excitation)

For general conductors and lamination, displacement current can be neglected even at high frequencies, since the involved areas and electric fields are small. Therefore, eddy-current can be considered as a quasi-stationary phenomenon. Laminations applied in electrical machines core have a very small width, compared to their length and height. Eddy-currents tend to flow near the lamination surface, where the penetration layer depends on the material conductivity, magnetic permeability, and excitation frequency (Stoll, 1974). Hence, this phenomenon can be suitably modeled as a one-dimensional problem (Lammeraner & Štafl, 1966; Stoll, 1974). In fact, this depends on the penetration depth compared to the lamination length and height, which is much smaller than these two measurements. An abstract semi-infinite lamination is represented in Figure 12, where only its thickness (2b) is finite.

Figure 12. Semi-infinite lamination: eddy-current one-dimensional modeling.

The magnetic field is represented parallel to the lamination, which corresponds to the main flux. However, there are also magnetic leakage fields whose direction is mainly normal to the laminations. This may cause significant eddy current loops flowing parallel to the laminations (Hollaus, 2001). It should be noted that this is not addressed in Figure 12. Moreover, this one-dimensional approach means that the lamination edge is not represented, i.e. eddy-currents have a single direction. Since current flows in x-axis, $\overrightarrow{\mathrm{H}}$ has only one component in the z-axis direction and $\left|\overrightarrow{\mathrm{H}}\right|$ is only ‘y’ dependent. It should be pointed that this is also acceptable when lamination length or height is much higher than its thickness (Lammeraner & Štafl, 1966). Taking the following Maxwell’s equations: (8) $$\mathrm{▿}\mathrm{\times }\overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}=-\frac{\partial \overrightarrow{B}}{\partial \overrightarrow{H}}\mathrm{·}\frac{\partial \overrightarrow{H}}{\partial t}$$(8) (9) $$\mathrm{▿}\mathrm{\times }\overrightarrow{H}=\overrightarrow{J}$$(9)

For a linear and isotropic material, one has $\mu =\frac{\delta \overrightarrow{B}}{\delta \overrightarrow{H}}.$ From vector Ohm’s law, (8) can be written as: (10) $$\mathrm{▿}\mathrm{\times }\overrightarrow{J}=-\sigma \mu \mathrm{·}\frac{\partial \overrightarrow{H}}{\partial t}$$(10)

Where σ is the electric conductivity of the lamination steel. For Figure 12, one has: (11) $$\frac{\partial J_x}{\partial y}=\sigma \mu \mathrm{·}\frac{\partial H_z}{\partial t}$$(11)

And (9) reduces to: (12) $$\frac{\partial H_z}{\partial y}=J_x$$(12)

Finally, combining (11(11) $$\frac{\partial J_x}{\partial y}=\sigma \mu \mathrm{·}\frac{\partial H_z}{\partial t}$$(11) ) and (12(12) $$\frac{\partial H_z}{\partial y}=J_x$$(12) ) yields: (13) $$\frac{{{\partial }^{2}H}_z}{{\partial y}^2}=\sigma \mu \mathrm{·}\frac{\partial H_z}{\partial t}$$(13)

• Sinusoidal-time variation

In phasor domain, (13(13) $$\frac{{{\partial }^{2}H}_z}{{\partial y}^2}=\sigma \mu \mathrm{·}\frac{\partial H_z}{\partial t}$$(13) ) is given by (subscripts are now suppressed, since this is a one-dimensional equation): (14) $$\frac{{\partial }^2\underset{¯}{H}}{{\partial y}^2}=\mathit{\text{jωσμ}}\mathrm{·}\underset{¯}{H}={\alpha }^2\mathrm{·}\underset{¯}{H}$$(14)

Where α = (1 + j)/δ, and δ is the skin-depth penetration factor, given by: (15) $$\delta =\sqrt{\frac{2}{\omega \sigma \mu }}$$(15)

Which is a measure of the flux penetration in the lamination. Equation (14)(14) $$\frac{{\partial }^2\underset{¯}{H}}{{\partial y}^2}=\mathit{\text{jωσμ}}\mathrm{·}\underset{¯}{H}={\alpha }^2\mathrm{·}\underset{¯}{H}$$(14) has a well-known solution: fixing $\underset{¯}{H}$ at the lamination surfaces as $\underset{¯}{H}$_(y=b) = $\underset{¯}{H}$_s1 and $\underset{¯}{H}$_(y=-b) = $\underset{¯}{H}$_s2, for a uniform applied field (i.e. $\underset{¯}{H}$_s1 = $\underset{¯}{H}$_s2 = H_s), $\underset{¯}{H}$ is given as: (16) $$\underset{¯}{H}=H_s\frac{\mathrm{\text{cosh}}\left(\alpha y\right)}{\mathrm{\text{cosh}}\left(\alpha b\right)}$$(16)

Whereas $\underset{¯}{J}$ is expressed as: (17) $$\underset{¯}{J}=\alpha H_s\frac{\mathrm{\text{sinh}}\left(\alpha y\right)}{\mathrm{\text{cosh}}\left(\alpha b\right)}$$(17)

The following expression can then be derived: (18) $${\left|\underset{¯}{J}\right|}^2=2{\left(\frac{H_s}{\delta }\right)}^2\frac{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2y}{\delta }\right)-\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2y}{\delta }\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}$$(18)

For this one-dimensional problem, eddy-current loss per unit surface of a lamination is given as: (19) $$P_{\mathit{\text{eddy}}\_s}=\frac{1}{2\mathrm{\sigma }}{\int }_{-b}^b{\left|\underset{¯}{J}\right|}^2\mathrm{·}\mathit{\text{dy}}$$(19)

Combining (18(18) $${\left|\underset{¯}{J}\right|}^2=2{\left(\frac{H_s}{\delta }\right)}^2\frac{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2y}{\delta }\right)-\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2y}{\delta }\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}$$(18) ) and (19(19) $$P_{\mathit{\text{eddy}}\_s}=\frac{1}{2\mathrm{\sigma }}{\int }_{-b}^b{\left|\underset{¯}{J}\right|}^2\mathrm{·}\mathit{\text{dy}}$$(19) ) yields: (20) $$P_{\mathit{\text{edd}}{y}_s}=\frac{H_s^2}{\text{σδ}}\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)-\mathrm{\text{sin}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}$$(20)

Most engineering approaches for iron loss estimation (both classical sinusoidal and non-sinusoidal methods) do not account for the complex interactions between lamination’s magnetic hysteresis and eddy currents (Steentjes et al., 2017). Therefore, an accurate prediction of iron loss and magnetization behavior of non-oriented soft magnetic steel lamination, under arbitrary excitation waveforms, must rely on coupled hysteresis and eddy current models (Du & Robertson, 2015; Steentjes et al., 2017). It should be pointed that in (5) both loss components are independently treated. Instead, FEMM deals with hysteresis and eddy current in a coupled way, even if it is limited to sinusoidal scenarios (this is addressed in subsection 4.2.3).

4.2.2. Hysteresis effect

For a linear magnetic material (i.e. without saturation effect) submitted to time-sinusoidal excitation, hysteresis can be represented by an elliptical symmetrical B(H) loop (Cardelli, 2015; Stoll, 1974, Sen & Adkins, 1956) as in Figure 13. This approach was employed by O’Kelly in a set of relevant papers for rotating electrical machines analysis , and lamination flux penetration combining both hysteresis and eddy-current effects, without (O'Kelly, 1972a) and with (O'Kelly, 1972b) saturation. In addition, in (O'Kelly, 1969) linear steady-state equivalent circuits for hysteresis and induction motors are derived, taking hysteresis and eddy-current into account, whereas the former is addressed with an elliptic hysteresis loop.

Figure 13. Hysteresis symmetric loop and equivalent elliptical loop.

There is now a phase lag θ_h between B and H. In frequency domain, a complex hysteresis permeability (${\underset{¯}{\mu }}_h$) can be defined as: (21) $${\underset{¯}{\mu }}_h=\frac{\underset{¯}{B}}{\underset{¯}{H}}=\frac{{\mu }_h\underset{¯}{H}{e}^{-j{\theta }_h}}{\underset{¯}{H}}={\mu }_h{e}^{-j{\theta }_h},\mathit{\text{with}} {\mu }_h=\left|{\underset{¯}{\mu }}_h\right|$$(21)

The parameters of this idealized loop can be easily estimated from the measured (real) hysteresis loop: µ_h is the ratio between B_max and H_max, whereas θ_h is such that the areas of the elliptic loop and the measured loop are the same. It should be noted that for Figure 13 blue hysteresis loop, an applied sinusoidal H leads to a flux density waveform with harmonics. Even if only the fundamental component contributes to hysteresis loss, neglecting the flux density harmonics has an impact in characterizing flux penetration in the lamination. In addition, modelling hysteresis with complex permeability has the following limitations:

• Hysteresis due to pulsating and rotational magnetic fields are considered to have similar effects;

• Hysteresis loss is assumed to be frequency-dependent and proportional to the square of the flux density;

• Saturation is neglected;

• Only static hysteresis is considered.

4.2.3. Combined hysteresis and eddy-current impact in lamination losses

In order to couple static hysteresis with eddy-current effects in (14(14) $$\frac{{\partial }^2\underset{¯}{H}}{{\partial y}^2}=\mathit{\text{jωσμ}}\mathrm{·}\underset{¯}{H}={\alpha }^2\mathrm{·}\underset{¯}{H}$$(14) ), μ is replaced by (21(21) $${\underset{¯}{\mu }}_h=\frac{\underset{¯}{B}}{\underset{¯}{H}}=\frac{{\mu }_h\underset{¯}{H}{e}^{-j{\theta }_h}}{\underset{¯}{H}}={\mu }_h{e}^{-j{\theta }_h},\mathit{\text{with}} {\mu }_h=\left|{\underset{¯}{\mu }}_h\right|$$(21) ): (22) $$\frac{{{\partial }^2\underset{¯}{H}}_z}{{\partial y}^2}=\mathit{\text{jωσμ}}{e}^{-j{\theta }_h}\mathrm{·}{\underset{¯}{H}}_z=m^2\mathrm{·}{\underset{¯}{H}}_z$$(22)

Where: (23) $$\begin{array}{l}=\alpha e^{-j\frac{{\theta }_h}{2}}=p+\mathit{\text{jq}}=\\ =\frac{\sqrt{2}}{\delta }\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)+j\frac{\sqrt{2}}{\delta }\mathrm{\text{sin}}\hspace{0.17em}\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)\end{array}$$(23)

Following similar steps to (16(16) $$\underset{¯}{H}=H_s\frac{\mathrm{\text{cosh}}\left(\alpha y\right)}{\mathrm{\text{cosh}}\left(\alpha b\right)}$$(16) )-(20(20) $$P_{\mathit{\text{edd}}{y}_s}=\frac{H_s^2}{\text{σδ}}\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)-\mathrm{\text{sin}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}$$(20) ), the lamination eddy-current loss per unit surface is given by: (24) $$P_{\mathit{\text{eddy}}\_s}=H_s^2\omega \mu b\frac{\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)}{2\mathit{\text{pb}}}-\frac{\mathrm{\text{sin}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}{2\mathit{\text{qb}}}}{\mathrm{\text{cosh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}$$(24)

The total lamination loss per unit surface can be derived from Poynting’s theorem as: (25) $$\begin{array}{l}{P}_{\mathit{\text{lam}}\_s}=\\ ={2H}_s^2\omega \mu b\frac{\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)}{2\mathit{\text{pb}}}\bullet {\mathrm{\text{cos}}}^2\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)-\frac{\mathrm{\text{sin}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}{2\mathit{\text{qb}}}\bullet {\mathrm{\text{sin}}}^2\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}\end{array}$$(25)

Whereas hysteresis loss per unit surface is given by: (26) $$\begin{array}{l}{P}_{\mathit{\text{hyst}}\_s}=\\ {=H}_s^2\omega \mu b\bullet \mathrm{\text{sin}}\left({\theta }_h\right)\frac{\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)}{2\mathit{\text{pb}}}+\frac{\mathrm{\text{sin}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}{2\mathit{\text{qb}}}}{\mathrm{\text{cosh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}\end{array}$$(26)

Obviously, P_{lam_s} = P_{hyst_s} + P_{eddy_s}.

The impact of lamination thickness on core losses is quite evident. Also, it is easy to see that if θ_h = 0, (24(24) $$P_{\mathit{\text{eddy}}\_s}=H_s^2\omega \mu b\frac{\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)}{2\mathit{\text{pb}}}-\frac{\mathrm{\text{sin}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}{2\mathit{\text{qb}}}}{\mathrm{\text{cosh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}$$(24) ) reduces to (20(20) $$P_{\mathit{\text{edd}}{y}_s}=\frac{H_s^2}{\text{σδ}}\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)-\mathrm{\text{sin}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}{\mathrm{\text{cosh}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{2b}{\delta }\right)}$$(20) ), since P_{eddy_s}=P_{lam_s}. On the other hand, for non-zero P_{lam_s} (i.e. with b, ω and μ not null), the condition {p,q} = 0 gives P_{hyst_s}= P_{lam_s}. From (23(23) $$\begin{array}{l}=\alpha e^{-j\frac{{\theta }_h}{2}}=p+\mathit{\text{jq}}=\\ =\frac{\sqrt{2}}{\delta }\mathrm{\text{cos}}\hspace{0.17em}\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)+j\frac{\sqrt{2}}{\delta }\mathrm{\text{sin}}\hspace{0.17em}\left(\frac{\pi }{4}-\frac{{\theta }_h}{2}\right)\end{array}$$(23) ) and (15(15) $$\delta =\sqrt{\frac{2}{\omega \sigma \mu }}$$(15) ) this stands for σ = 0. Naturally, P_{eddy_s} is null since eddy-currents cannot exist in such a material. Once again, it is stressed that magnetic hysteresis and eddy currents are somehow combined in these expressions, which is a significant difference from Method 1. Also, this highlights the physics criticism of treating hysteresis and eddy-current losses as independent phenomena (Cullity & Graham, 2011). Nonetheless, a deeper discussion on this very interesting topic is beyond the paper scope.

For the selected lamination in this study (Table 3), Figure 14 shows the impact of frequency and θ_h over the core loss components (more precisely, surface_loss/$H_s^2$), for a wide frequency range. There is a relatively narrow frequency range where p_h > p_ec, which slightly increases with θ_h. All losses increase with θ_h whereas the difference between p_ec and p_h decreases (i.e. the former and latter, respectively, decrease and increase with θ_h).

Figure 14. Influence of frequency and θ_h (°) on lamination surface losses.

4.2.4. Saturation effect

FEMM deals with magnetic saturation assuming that hysteresis loss is proportional to $B_{\mathrm{\text{max}}}^2,$ where hysteresis power loss density is given as: (27) $$P_h^{\mathit{\text{vol}}}={\frac{1}{2}\mu }_h{\omega H}_{\mathrm{\text{max}}}^2\mathrm{\text{sin}}\hspace{0.17em}\left({\theta }_h\right)=\frac{1}{2}{\frac{\omega }{{\mu }_h}B}_{\mathrm{\text{max}}}^2\bullet \mathrm{\text{sin}}\hspace{0.17em}\left({\theta }_h\right)$$(27)

It should be noted (27(27) $$P_h^{\mathit{\text{vol}}}={\frac{1}{2}\mu }_h{\omega H}_{\mathrm{\text{max}}}^2\mathrm{\text{sin}}\hspace{0.17em}\left({\theta }_h\right)=\frac{1}{2}{\frac{\omega }{{\mu }_h}B}_{\mathrm{\text{max}}}^2\bullet \mathrm{\text{sin}}\hspace{0.17em}\left({\theta }_h\right)$$(27) ) is derived from (26(26) $$\begin{array}{l}{P}_{\mathit{\text{hyst}}\_s}=\\ {=H}_s^2\omega \mu b\bullet \mathrm{\text{sin}}\left({\theta }_h\right)\frac{\frac{\mathrm{\text{sinh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)}{2\mathit{\text{pb}}}+\frac{\mathrm{\text{sin}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}{2\mathit{\text{qb}}}}{\mathrm{\text{cosh}}\hspace{0.17em}\left(2\mathit{\text{pb}}\right)+\mathrm{\text{cos}}\hspace{0.17em}\left(2\mathit{\text{qb}}\right)}\end{array}$$(26) ), for {p,q} = 0, where P_{lam_s}=P_{hyst_s}. For small θ_h, $P_h^{\mathit{\text{vol}}}$ can be approximated as: (28) $$P_h^{\mathit{\text{vol}}}\approx \frac{1}{2}{\omega B}_{\mathrm{\text{max}}}^2\bullet \frac{{\theta }_h}{{\mu }_h}$$(28)

Which means that $\frac{{\theta }_h}{{\mu }_h}$ must be constant. Therefore, for a magnetic material saturation level, with a specific μ_h, θ_h is given by: (29) $$\frac{{\theta }_h}{{\mu }_h}=\frac{{\theta }_{h\_\mathit{\text{max}}}}{{\mu }_{{}_{h\_\mathit{\text{max}}}}}\to {\theta }_h={\mu }_h·\frac{{\theta }_{h\_\mathit{\text{max}}}}{{\mu }_{{}_{h\_\mathit{\text{max}}}}}$$(29)

Where θ_{h_max} is the hysteresis angle at the maximum permeability (μ_{h_max}) point of the lamination material. Saturation level on each element has a direct impact over θ_h, meaning that hysteresis loop in FEMM is not simply modelled with an elliptical shape, in order to address the saturation effect.

4.3. Method 2 (with FEMM model)

For each simulated SRM operation, the following steps were carried out:

• Phase current from the SRM equivalent circuit model is decoupled into its main harmonic components, as in Figure 15;

• For a 6/4 SRM, phase current period corresponds to a rotor shift of 90° (mechanic). Core losses were calculated for k_max = 45 rotor positions since a step angle Δθ = 2° was found to be an adequate resolution (Figure 16 shows two rotor positions for a specific current harmonic). For each harmonic ‘h’, core losses are calculated with FEMM for the 45 rotor positions (p_h,p is the core loss value for the harmonic h at the rotor position θ_p). Then, the average core loss for all positions (p_{core_h}) is taken as the current harmonic contribution for the total losses;

• The procedure repeats for all current harmonics (h = 1 to h_max in Figure 17). For each harmonic, p_{core_h} is stored in vector M. The total core loss is the sum of the individual harmonic contributions (i.e. all the elements in M). Figure 17 depicts the flowchart of the proposed method for SRM core losses estimation, supported with FEMM. A most significant issue for this method is choosing θ_{h_max}. According to (O'Kelly, 1972a), typical data for non-grain oriented sheet steel (low-loss) is θ_h = 30° for B below the knee of the B-H curve. At the saturated region θ_h decreases. In addition (Stoll, 1974), states that for reasonably high-flux densities, one has θ_h < 20°. Even if ‘reasonably high-flux densities’ is not quantified, both references seem to agree. For this study θ_{h_max} was fixed as 20°.In order to simplify the procedure, Method 2 is based on a single-phase analysis, i.e. the overlapping currents in magnetizing and demagnetizing phases are not considered. In other words, it is always assumed that at the initial instant of a phase magnetization all the others are demagnetized. Compared to Method 1, this is a limitation of the proposed approach. A comparative synthesis of both methods is shown in Table 5, highlighting their merits and limitations.

Figure 17. Flowchart of the proposed method (Method 2).

Figure 15. SRM phase-current: (a) waveform; (b) harmonic content.

Figure 16. B distribution for two rotor positions (I_peak = 2.5 A, f = 182 Hz): (a) unaligned; (b) aligned.

Table 5. Features of Methods 1 and 2.

	Method 1	Method 2
Formulation	Based on Eq. Circuit Model (saturation). Empirical core loss estimation, with B uniform distributions: pcore = ph + pe.	Based on static 2D FEM (hysteresis, lamination skin effect and saturation). Analytical core loss estimation: pcore = ph + pe.
Input	B’s harmonic decomposition.	Phase current harmonic decomposition.
Merits	Relatively simple SRM model. Low time consuming Method 1 is a well-known approach	Based on a popular open-source FEM tool. Suitable for experimental data (currents). Flux estimation takes several phenomena.
Limitations	Individual harmonic impact is only valid for linear systems. Needs B’s waveforms Core loss calculation with sinusoidal data. Flux estimation takes only saturation.	Individual harmonic impact is only valid for linear systems. Minor loops and rotating losses are not considered. Time consuming
Computation burden	Low	High

5. Simulation results

This section addresses the simulation scenarios, results, and discussion. Different SRM operation was considered, where hysteresis phase-current control was implemented. As shown in Table 6, they are divided into three load-ranges, each one addressing an average torque (T_av). In order to clarify the discussion, in the following Method 1 and Method 2 are denoted as ‘B_harm’ and ‘FEMM’, respectively.

Table 6. SRM simulated operation modes.

	T [Nm]	n [rpm]
Tav = 1.4 Nm	1.5	688
	1.5	1487
	1.3	2621
	1.3	3625
Tav = 2.9 Nm	3.0	651
	3.8	1715
	2.6	2451
	2.2	3028
Tav = 6.2 Nm	6.1	688
	6.5	1360
	6.0	1885

5.1. Analysis and discussion of results

Figures 18 and 19 show, respectively, the SRM core loss and the FEMM/B_harm loss ratio (i.e. Method 2/Method 1 ratio) for the three load ranges in Table 6. For T_av = 1.4 Nm & 2.9 Nm, both methods give close results around 3000–3300 rpm and 700–1100 rpm, respectively.

Figure 18. SRM core loss estimation: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

Figure 19. FEMM/B_harm ratio: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

The core loss ratio in Figure 19 tends to increase with speed (for each load range), as well with the load (for the same speed). This puts in evidence the influence of both speed and saturation. As for T = 6.2 Nm, the ratio FEMM/B_harm is always higher than 1, which highlights the trend for higher difference as load increases, in core loss estimated by the two methods. Speed and load impact is more pronounced in FEMM: for low speed and load it gives lower core loss than B_harm method, but for higher speed and/or load FEMM’s core losses are much higher. This seems to reflect the non-uniform B distribution, which is more pronounced in high-speed range, where its value decreases inside the lamination (in B_harm method, flux distribution is uniform). On the other hand, as load increases the saturation effect becomes more significant; however, this phenomenon is addressed differently by the two methods.

Figures 20 and 21 show different behaviors for p_hyst and p_ec, in both B_harm and FEMM methods, which is more pronounced as speed increases. For T = 1.5 Nm and n < 3500 rpm, p_hyst is significantly higher for B_harm. At this speed range, p_hyst is dominant which explains the higher total losses compared to FEMM (Figure 18). For n > 3500 rpm, FEMM gives substantially higher p_ec, which is now the main loss component. This agrees to FEMM’s higher total losses in Figure 18. As load increases FEMM’s losses tend to overcome B_harm values. It can be seen that for T_av = 2.9 Nm and n < 1500 rpm, p_hyst are relatively close in both methods, while p_ec’s are much similar. For T = 6.3 Nm FEMM’s loss components are higher in all speed range. It should be pointed that these particular values depend on the chosen θ_{h_max}.

Figure 20. B_harm and FEMM comparison of p_hyst: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

Figure 21. B_harm and FEMM comparison of p_ec: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

Figures 22 and 23 show p_hyst and p_ec distribution for, respectively, B_harm and FEMM methods. As expected, in low speed range p_hyst is higher than p_ec, while for higher speed the latter are dominant. This is particularly clear in FEMM for all load range. For T = 6.3 Nm, the speed for which p_hyst = p_ec has the lowest value, compared to T = 1.4 Nm and 2.9 Nm. Saturation effect may explain this, since for the highest load range it is expected to have lower θ_h in several mesh elements of the FE SRM model (see (29)). A simple look at Figure 14 reveals that as θ_h decreases, so does the frequency for which p_hyst = p_ec. Even if this is a rough analysis, it is somehow in agreement with Figure 20.

Figure 22. Estimated p_hyst and p_ec with Method 1: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

Figure 23. Estimated p_hyst and p_ec with Method 2: (a) T_av = 1.4 Nm; (b) T_av = 2.9 Nm; (c) T_av = 6.2 Nm.

The result comparison of this study was undertaken by considering merits and limitations of both methods – e.g. uniform (Method 1) and non-uniform (Method 2) flux density lamination distribution, uncoupled and coupled p_hyst and p_ec (respectively in Method 1 and Method 2) and section 4.2.2 last paragraph limitations (Method 2) – nonetheless, it should be pointed that FEMM takes saturation effect into account. In face of such different conditions, this comparative analysis is a non-trivial task. Core loss results for both methods are depicted in Table 7, including p_h and p_ec. In addition, the ratios of loss and its components are also included. It is interesting to note that for low load (T_av = 1.4 Nm), the p_ec ratio is higher than p_core ratio for all speeds. For T_av = 2.9 Nm and 6.2 Nm, this occurs only for the higher speeds (2451 rpm and 3028 rpm for T_av = 2.9 Nm and 1885 rpm for T_av = 6.2 Nm). Naturally, the opposite can be seen for p_h ratio.

Table 7. SRM estimated core losses (Methods 1 and 2).

		Method 1 (Bharm)			Method 2 (FEMM)			(FEMM/Bharm)
nr (rpm)	T (Nm)	ph (W)	pec (W)	pcore (W)	ph (W)	pec (W)	pcore (W)	ph ratio	pec ratio	pcore ratio
688	1.5	6.6	2.7	9.3	2.7	1.4	4.1	0.41	0.52	0.44
1487	1.5	10.9	8.1	19.0	5.2	4.9	10.1	0.48	0.60	0.53
2621	1.3	11.9	13.5	25.4	6.8	11.6	18.4	0.57	0.86	0.72
3625	1.3	14.1	20.2	34.3	9.7	32.5	42.2	0.69	1.61	1.23
651	3.0	7.9	4.5	12.4	6.7	3.1	9.8	0.85	0.69	0.79
1715	3.8	12.4	13.1	25.5	21.8	23.1	44.9	1.76	1.76	1.76
2451	2.6	13.6	17.6	31.3	20.4	33.4	53.8	1.50	1.89	1.72
3028	2.2	14.1	20.1	34.2	21.8	45.5	67.3	1.55	2.26	1.97
688	6.1	8.2	5.0	13.2	21.1	12.2	33.3	2.57	2.44	2.52
1360	6.5	12.5	14.4	26.9	33.4	36.3	69.7	2.67	2.52	2.59
1885	6.0	14.8	20.5	35.3	43.8	67.0	110.8	2.96	3.27	3.13

6. Conclusion

This paper presents a study where core losses of a three-phase 6/4 SRM are estimated and compared with two different approaches, based on sinusoidal waveforms. The first one takes Fourier analysis of flux density waveforms in different sections of the machine for estimating the total losses. The second approach is based on the popular FEMM software, where lamination core losses are calculated with a solid theoretical method, which is addressed in the paper. It should be stressed that our main goal was to introduce and discuss a new approach for SRM static core loss estimation, suitable for being used with FEMM. In addition, by comparing its outcome with a well-known static core loss estimation method, we intended to point its eventual potential for estimating SRM core loss. This is the main contribution of this paper, even if both approaches have limitations (particularly for non-sinusoidal electric machines analysis), as discussed throughout the paper. In spite this, several SRM operation scenarios were simulated, and comparison and discussion of core loss results were carried out. Some convergence was found for high load and speed, whereas for other load and speed ranges there are significant differences. It must be emphasized that none of the estimated core loss values can be taken as a granted reference, even for converging results in both methods. Even so, by proposing a new approach based on a well-known open-source software package, this work also intends to contribute for increasing the interest on the challenging non-sinusoidal core loss estimation topic, in particular for young researchers. Experimental tests and/or transient FEA are crucial for validation, which will be the following step of this work.

Acknowledgment

This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project LA/P/0063/2020. DOI 10.54499/LA/P/0063/2020 | https://doi.org/10.54499/LA/P/0063/2020.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data will be provided upon request.

Additional information

Notes on contributors

Pedro Sousa Melo

Pedro Sousa Melo received the Electrical Engineering and MSc degrees from the University of Porto, Porto, Portugal. Since 2001, he is with the Department of Electrical Engineering, School of Engineering, Polytechnic Institute of Porto. His main research interests include modeling of electrical machines, in particular switched reluctance machines, and magnetic materials for high-efficient machines.

Rui Esteves Araújo

Rui Esteves Araújo (M’99) received the Electrical Engineering diploma (5-year university degree) from University of Porto, Porto, Portugal, and the MSc and PhD degrees in Electrical and Computer Engineering from the University of Porto, Porto, Portugal, in 1987, 1992 and 2001, respectively. From 1987 to 1988, he was an Electrotechnical Engineer in Project Department of Adira - Metal Forming Solutions Company, Porto, Portugal, and from 1988 to 1989, he was Researcher with INESC, Porto, Portugal. Since 1989, he has been with the University of Porto, where he is currently an Associate Professor with the Department of Electrical and Computer Engineering, Faculty of Engineering, University of Porto. He is Senior Researcher in the INESC TEC, focusing on power electronics systems and its industrial applications to motion control, electric vehicles and renewable energies.