A magnetic circuit model with coupling effect for salient switched reluctance machines

Abstract

In this paper, a novel magnetic circuit network model is proposed for flux calculation on switched reluctance machines (SRM) of double saliency. Specifically, the airgap reluctance is modeled by a topology of paralleled magnetic paths that only depends on geometry and rotor position, while magnetic saturation on poles is modeled by a simple lookup approach. Such network is featured by allowing magnetic coupling effect from multiple phases. The flux linkage and torque generation considering the coupling can be calculated in a fast and accurate way without limit of power density or rotating speed.

Keywords

Switched reluctance machine magnetic equivalent circuit phase coupling magnetic saturation

1. Introduction

The magnetic equivalent circuit (MEC) model has been widely used in switched reluctance machines (SRM) to calculate flux linkage characteristics. In such a model, the machine is discretized into poles and yoke sectors that are represented respectively by a lumped magnetic reluctance element [1, 2, 3, 4]. However, the conventional MEC often results in less calculation accuracy, due to the double saliency structure that both stator and rotor have salient poles. As a result, the airgap path varies sharply with rotation [5, 6] and when power density increases, high local magnetic saturation occurs. That saturation distributes at pole tips and varies with rotor position and phase current level [7], serving as part of the magnetic reluctance.

The flux linkage calculation is required of high accuracy upon which further calculation such as torque and loss can be based. To account for such double saliency that leads to high nonlinearity of magnetic circuit, improvements were made. In [8], the airgap reluctance is modeled by a fitting method of polynomial function. However, the polynomial coefficients cannot be accurately obtained without finite element method. In [9] curve fitting is used instead by an exponent function, with coefficients that respectively represent flux linkage when the machine starts saturation, degree of saturation severity and inductance increment under high excitation current. However, higher order harmonics lead to inaccuracy by adding ripple to the predicted instantaneous torque. In [10], the magnetic circuit is calculated according to Ampere’s Law without empirical equations, and the calculation is divided whether the pole pair has alignment. However, calculation accuracy of flux linkage needs improvement by comparing with finite element (FE) method. Such numerical gap with FE is small, but would lead to deviation for further analysis. In [11], the flux tubes are refined according to flux path and calculation shows the desired accuracy. However, the model subjects to complexity when SRM prototype varies. To account for magnetic saturation distribution, variable sizes of magnetic reluctances as a function of rotor position and phase current excitation is applied in MEC [12, 13]. However, the circuit model is complicated in managing the magnetic reluctances. Alternatively direct measurements are used as part of the MEC simulation [14, 15, 16].

Further, magnetic coupling of phases occurs close to the full aligned rotor position where a couple of phase is excited simultaneously. Taking such a point can improve calculation accuracy [17, 18] in flux and torque control. However, most MEC models do not take into account the coupling and only a few are involved [19, 20]. In [19], the coupling effect is taken by adding equivalent airgap reluctance between phases. The elements values can be predetermined by FE analysis or measurement. However, the calculation accuracy needs further discussion if taking a closer comparison with FE, especially close to the aligned rotor position.

This paper proposes a FE assisted MEC network model for salient SRMs with phase coupling. The MEC part is for the rotor position dependent airgap reluctance calculation while the FE part is for magnetic saturation estimation. This paper is structured as follows: In Section 2, the proposed network model is introduced. Specifically, modeling of airgap reluctance and magnetic saturation are described respectively. In Section 3, calculation accuracy in terms of flux linkage is discussed, followed by torque application considering the magnetic coupling.

2. The MEC-FE model

In SRM operation, each phase is fluxed and defluxed, which is divided by rotor pole pitch. In general, the time to deflux the machine should equal to flux, and it is inappropriate to extend dwelling angle, which will result in reverse torque by residual flux. With speed increasing, the rate of phase current per rotor angular shift is reduced and therefore phase advancing control is applied to flux in advance as well as to reduce residual flux after alignment. At higher speed, phase current will continue, leading to phases being fluxed all the time during commutation. Table 1 shows overview of the studied 3-phase 12/8 SRM for traction application.

Table 1
Geometry overview of the studied SRM

Stator bore outer radius (mm)	67.50	Stator pole width (deg)	15.5
Stator bore inner radius (mm)	34.85	Rotor pole width (deg)	17.5
Rotor bore outer radius (mm)	33.95	Airgap length (mm)	0.9
Rotor bore inner radius (mm)	27	Stack length (mm)	125
Shaft radius (mm)	12	Stator yoke thickness (mm)	12

Accordingly Fig. 1 shows flux paths at typical operation conditions. In Fig. 1a, one phase at unaligned rotor position is excited, the magnetic circuit is unlikely to get saturated and airgap reluctance is considered as the main factor. In Fig. 1b, a couple of pole pairs, including the one of defluxing close to the full aligned position while the other pair of fluxing just after the unaligned position, are energized simultaneously. The multi-phase excitation may lead to not only magnetic saturation on poles, but also magnetic coupling between phases. Therefore, phases are related and the degree of dependence is deemed as a function of rotor speed and phase current.

Figure 1.

Flux paths overview at typical rotor positions, (a) by one single phase excitation, (b) by two phase excitations with coupling effect.

The proposed MEC-FE model is shown in Fig. 2. The stator and rotor part is divided into sectors that each is represented by a lumped magnetic reluctance element, including stator yoke (SY), stator tooth (ST), rotor tooth (RT), rotor yoke (RY). Due to magnetic saturation that predominantly resides at the exciting poles, elements on the stator and rotor teeth are marked with arrows, indicating variability to be determined by lookup approach. Each rotor slot is represented by an element LK for slot leakage.

Figure 2.

Overview of the proposed network model.

The airgap reluctance (AG) is defined as magnetic connection between a stator and rotor pole pair. The connection is valid from the aligned position till the position when the adjacent pole pair is aligned. That is for one pole pair, the minimum airgap reluctance occurs at its alignment while the maximum occurs at the position when the adjacent pair is aligned. Therefore in Fig. 2, each stator pole has a couple of connections to the previous and next rotor poles. Further for each connection, 3 paralleled branches are shown. The middle branch stands for the tip-to-tip path while both side branches stand for the in-profile paths. For each branch the equivalent reluctance is represented by a ring, and the arrow indicates variability.

Figure 3.

Flux path details of airgap connections, showing misalignment and partial overlap of a pole pair.

Figure 3 shows the airgap flux path details for the airgap reluctance rings, with typical positons of both misalignment and partial overlap. In either case, airgap paths are modeled by 3 branches B1-B3-B4 and each is represented by a lumped magnetic reluctance element. The reluctance value can be calculated according to the length and width of the regarded airgap path. When poles are not aligned, B1 refers to the shortest distance between the stator and rotor pole tips. The length of B1 is determined by such shortest distance while the width is by a fitting approach [21]. B3 refers to the side path through rotor slot. The length is determined by the straights both in airgap and rotor slot while the width is by comparing with slot leakage path B2 according to the magnetic circuit principle. B4 path can be seen symmetrical to B3. When poles are partially aligned, B1 refers to the overlap airgap zone and the path is determined by airgap length $\delta$ and overlap width CD. In analogy B3 is calculated, to which B4 is likewise symmetrical.

Besides airgap, magnetic saturation as another source of reluctance affects flux linkage curves. The saturation serves as a function of phase current excitation, as well as rotor position that depend on airgap path. It is assumed that the degree of saturation is quantifiable and this paper uses a lookup approach to identify the value in the form of the permeance drop $\lambda$ , the reverse of reluctance. Due to double saliency, the lookup is separated according to whether the pole pair has alignment. The drop is described as the reduced permeance value in the MEC network if both the ideally non-saturated and actually saturated electrical steels are respectively applied, provided that the operation conditions are the same.

To make the lookup for magnetic saturation dimensionless, which means adaptability to variation of machine geometry, how to model the rotor positon with minimum variables is described. When poles are not aligned, rotor position is represented by the length AB in Fig. 3 that the main part of flux lines is going through airgap. Here it is defined as distance $L_{ab}$ that integrates rotor angular shift $\theta$ and radial airgap length $\delta$ by the mentioned fitting approach in [21]. Then, permeance drop is defined as $\lambda$ ( $i_{ph}$ , $L_{ab}$ ), where $i_{ph}$ donates phase current excitation. Let $L_{\theta}=L_{ab}$ , $\lambda$ ( $i_{ph}$ , $L_{ab}$ ) $=$ $\lambda$ ( $i_{ph}$ , $L_{\theta}$ ) holds true. When poles have alignment, Fig. 3 shows a featured horizontal overlap CD as main flux paths. In this case, the overlap width $L_{cd}$ and airgap length $\delta$ are defined. According to permeance calculation $\Lambda_{m}=$ ( $\mu\cdot S$ )/ $L$ , where $L$ , $S$ are respectively length and width of magnetic path, we define the airgap length per unit overlap width $\delta$ / $L_{cd}$ , which in analogy integrates the radial airgap length $\delta$ , as well as the rotor angular shift $\theta$ in the form of the overlap length $L_{cd}$ . Let $L_{\theta}=L_{bc}$ , the permeance drop $\lambda$ ( $i_{ph}$ , $L_{bc}$ ) $=$ $\lambda$ ( $i_{ph}$ , $L_{\theta}$ ) holds true.

Now for both cases, the permeance drop $\lambda$ ( $i_{ph}$ , $L_{\theta}$ ) is deemed as a function of phase current $i_{ph}$ , as well as relative position of a pole pair $L_{\theta}$ that measures airgap reluctance variation. Further to make the lookup independent of a specific machine geometry, the permeance drop lookup function $\lambda$ ( $i_{ph}$ , $L_{\theta}$ ) is transformed. Let $\lambda^{\ast}=$ $\lambda L_{z}$ / $\mu_{0}$ , meaning the unit permeance drop in vacuum, where $L_{z}$ is the axial length of the machine and $\mu_{0}$ is the permeability of air. Now the lookup function is $\lambda^{\ast}$ ( $i_{ph}$ , $L_{\theta}$ ).

Figure 4.

The permeance drop lookup function obtained from FE, (a) non-overlapping; (b) at partial and full overlapping.

The permeance drop is modeled by a 3D lookup. Figure 4 shows the map at non-overlap and overlap rotor positions respectively. Note that the lookup applies to the machine with number of turns per pole $N_{tp}=$ 10, and if $N_{tp}$ differs, the phase current varies accordingly in reversely proportion, as long as the ampere-turn value keeps constant. In Fig. 4a, $\lambda$ ${}^{\ast}$ value rises up with the increased phase current $i_{ph}$ , as well as the decreased rotor position $L_{\theta}$ . When $L_{\theta}$ closes to 0 that poles are about to overlap, $\lambda^{\ast}$ increases sharply with $i_{ph}$ . This is because of the low $L_{\theta}$ value that is comparable to airgap length $\delta$ , making $\lambda^{\ast}$ variation sensitive. In Fig. 4b, $\lambda^{\ast}$ value rises up with both $i_{ph}$ and $L_{\theta}$ . The increased $L_{\theta}$ means more degree of overlap till full alignment, where the $\lambda^{\ast}$ value rises up with $i_{ph}$ in a saturated way, indicating strong magnetic saturation on poles.

Figure 5.

The flux linkage under selected rotor positions without magnetic coupling.

3. Results and discussions

Figure 6.

Comparison of flux linkage curves under selected rotor positions, with and without magnetic coupling between phases.

Figure 7.

Comparison of flux linkage variation due to the coupling effect, where different levels of current are applied.

The flux linkage curves of the studied machine by one single phase excitation at selected rotor positions are comparatively shown in Fig. 5, with a wide current range till 170 A. Note that 0 ${}^{\circ}$ is the unaligned rotor position while 22.5 ${}^{\circ}$ is the aligned position. The selected positions include the total unaligned 0 ${}^{\circ}$ , partial unaligned 5 ${}^{\circ}$ , just before the start of overlap 10 ${}^{\circ}$ , just after the start of overlap 14 ${}^{\circ}$ , wide overlap 17 ${}^{\circ}$ and total aligned 22.5 ${}^{\circ}$ . Each curve by the proposed network model in general agrees with FE method. At unaligned position 0 ${}^{\circ}$ where magnetic saturation unlikely happens, both methods agree, which proves high calculation accuracy of airgap reluctance. At the total aligned position 22.5 ${}^{\circ}$ , the airgap path calculation is simple, and the agreement indicates high lookup accuracy of magnetic saturation. As to 10 ${}^{\circ}$ and 14 ${}^{\circ}$ , there exists a numerical discrepancy between FE and the network model with increase of phase current. When close to the start of overlap, high local magnetic saturation occurs on tips of poles, making the lookup less accurate. This is because of the simplification by integrating rotor position $\theta$ and airgap length $\delta$ into one variable in the lookup table. When poles are far unaligned or widely aligned (5 ${}^{\circ}$ and 17 ${}^{\circ}$ ), such numerical discrepancy reduces, as magnetic saturation either doesn’t exist or is more evenly distributed on poles. In most cases, the proposed network model is sufficient for a wide power range.

Further Fig. 6 comparatively shows magnetic coupling effect at selected rotor positions. At unaligned 0 ${}^{\circ}$ and 10 ${}^{\circ}$ where only one phase is working, according to operation principle in Fig. 1a, both curves are exactly the same. However when poles are further approaching (at 14 ${}^{\circ}$ and aligned 22.5 ${}^{\circ}$ ), magnetic coupling by a couple of phase excitations is involved, as is shown in Fig.1b. In this case, both phases are loaded with current till 170 A. It is shown that the numerical gap of the curves at 14 ${}^{\circ}$ or aligned 22.5 ${}^{\circ}$ whether the coupling effect is taken is enlarged with increase of current. Also the gap is more enlarged at 22.5 ${}^{\circ}$ , indicating stronger coupling effect.

The magnetic coupling is further discussed. Figure 7 shows flux linkage variation under full rotor position range at selected levels of phase current $I_{ph\_1}$ and $I_{ph\_2}$ for a couple of phases. Phase 1 is the main phase for flux linkage calculation while Phase 2 is the auxiliary coupling phase, where 22.5 ${}^{\circ}$ is the aligned position for Phase 1. According to the operation principle, the coupling mainly occurs nearby 22.5 ${}^{\circ}$ where both phases are excited simultaneously. It is shown that 1) For each current level in Phase 1, the highest numerical discrepancy between the solid and dotted curves occurs near 22.5 ${}^{\circ}$ , the vertical axis to which the dotted curve with coupling is not symmetrical. This is because before 22.5 ${}^{\circ}$ , Phase 2 is has less degree of alignment and contributes less on the coupling flux while thereafter, it has more alignment that in turn strengthens the coupling flux. 2) Higher current excitation causes stronger coupling effect.

Table 2

Geometric variation (in %) from the studied SRM as M0

	$\beta_{s}$	$\beta_{r}$	$\delta$	$l_{sy}$
M0	100%	100%	100%	100%
M1			150%
M2			70%
M3		80%
M4	80%	80%
M5				75%

Figure 8.

The magnetic coupling effect due to variation of airgap length, where M1 and M2 are simulated, with M0 as reference; Note that 22.5 ${}^{\circ}$ is the aligned position and phase current 170 A is applied; Dots: with phase coupling; Lines: no phase coupling.

The model adaptability is verified by simulating typical SRMs of main geometric parameter variation. Table 2 shows the variation, including airgap length $\delta$ , stator pole width $\beta_{s}$ , rotor pole width $\beta_{r}$ and stator yoke thickness $l_{sy}$ , based on the studied machine as M0. M1 and M2 are respectively defined as the enlarged and reduced airgap length variation. M3 and M4 are the reduced pole widths. M5 is the reduced stator yoke thickness. In the following, each machine is simulated with and without magnetic coupling.

Figure 8 shows flux linkage due to variation of airgap length (M1 and M2). The values decrease with a higher airgap length, especially close to 22.5 ${}^{\circ}$ . Regarding magnetic coupling, the reduced gap length of M2 strengthens both the main and coupling flux, leading to higher discrepancy between the solid and dotted curves compared with M0 or M1.

Figure 9.

The magnetic coupling due to variation of pole widths where M3 and M4 are simulated, with M0 as reference; Note that 22.5 ${}^{\circ}$ is the aligned position and phase current 170A is applied; Dots: with phase coupling; Lines: no phase coupling.

Figure 10.

The magnetic coupling effect due to stator yoke widths where M5 is simulated, with M0 as reference; Note that 22.5 ${}^{\circ}$ is the aligned position and phase current 170A is applied; Dots: with phase coupling; Lines: no phase coupling

Figure 9 shows flux linkage due to variation of pole width. In M3, width of rotor pole is reduced by 20%, followed by M4 that the stator pole width is further reduced by 20%. It is found that the values at any rotor position are reduced in M3, and further sharply reduced in M4. Also, the peak width of the curves is narrowed with reduced pole width. However for each machine M0, M3 and M4, the numerical discrepancy due to the magnetic coupling remains similar. This is because nearby the aligned positon 22.5 ${}^{\circ}$ where the coupling takes principal effect, pole widths hardly link with degree of magnetic saturation.

Figure 10 shows flux variation due to stator yoke thickness. It is found that M5 values by reducing 25% yoke thickness, is reduced. This is due to the narrowed yoke that leads to magnetic saturation that weakens active flux. Regarding the magnetic coupling in M5, discrepancy happens in particular around 22.5 ${}^{\circ}$ , where the auxiliary coupling phase takes principal effect. This is because of the coupling phase that intensifies saturation in the narrowed stator yoke.

4. Conclusions

This paper proposes a FE assisted magnetic circuit model for SRMs, including phase coupling effect. The airgap reluctance is modeled by equivalent paths of simple geometric relations without empirical estimation. To account for local magnetic saturation, a fast FE lookup method is proposed, which can be applied to various machine geometries without power limit. The amount of computation is quite competitive compared with FE. Further this model can effectively simulate magnetic coupling effect between phases. It is shown significance the magnetic coupling affects flux and therefore the design can be better predicted under various control modes.

Footnotes

Acknowledgments

This work is supported by the Natural Science Foundation of China (51607180). The authors gratefully acknowledge Institute of Electrical Drives and Actuators (EAA), University of Bundeswehr Muenchen, Germany, for their support in this research.

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A magnetic circuit model with coupling effect for salient switched reluctance machines

Abstract

Keywords

1. Introduction

2. The MEC-FE model

Table 1 Geometry overview of the studied SRM

Footnotes

Acknowledgments

References

Table 1
Geometry overview of the studied SRM