Unbalanced magnetic pull in an 18-slot/8-pole dual three-phase machine under different modes of operation Open Access

Principal benefits of multiphase machines include power sharing among phases and fault-tolerant operation, which ensures continuous control over flux and torque even in the event of a fault. This capability not only enables uninterrupted operation (Yu et al., 2024) but also provides control flexibility to enhance torque production, minimize torque ripple and improve redundancy and safety (Rigatos et al., 2024). Additionally, multiphase machines offer higher rated power and reduced power losses, making them well-suited for a diverse range of applications (Levi, 2016).

Among the various multiphase configurations, the dual three-phase machine (DTM) with two neutral points is the most widely used (Zhu et al., 2021). This design capitalizes on conventional three-phase inverter technology, enabling the use of existing, well-established and cost-effective power conversion strategies. The dual three-phase topology also enhances drive versatility by enabling power sharing between winding sets. These sets can be connected to two independent three-phase power sources or loads simultaneously, facilitating efficient active and reactive power transfer across the windings (Subotić et al., 2019). Additionally, this capability supports synthetic loading, providing a simplified approach for determining machine losses (Drobnič et al., 2024; Zabaleta et al., 2019).

Unbalanced magnetic pull (UMP) in electrical machines can arise from three main factors: winding faults, rotor eccentricities and the diametrically asymmetric disposition of stator slots and phase windings (Dorrell et al., 2010). Rotor eccentricities, caused by manufacturing imperfections or operational wear, lead to UMP as the rotor is not perfectly centered within the stator (Wen et al., 2024). On the other hand, the asymmetric arrangement of stator slots and coils creates imbalanced magnetic fields even in the absence of rotor anomalies. This effect is particularly pronounced in PM machines with fractional-slot windings (Zhu et al., 2007). UMP negatively affects machine performance, increasing noise and vibrations while reducing bearing lifespan. Both analytical methods and finite element method (FEM) are commonly used to study these effects, quantify UMP and assess its impact on machine performance and durability.

However, alternative operating modes of the DTM can also induce UMP. In this paper, we analyze UMP characteristics during post-fault operation (PFO), where only one winding set is active, and synthetic loading operation (SLO), where the winding sets are energized in the opposite sense compared to normal operation (NO). Our focus is on the performance of an 18-slot/8-pole interior permanent magnet synchronous DTM under PFO and SLO conditions. We investigate the mechanisms leading to UMP and quantitatively assess it through a series of FEM simulations. The objective is to determine the UMP magnitude and evaluate its potential impact on the machine’s mechanical integrity.

The fractional-slot per pole winding configuration offers numerous advantages, including high efficiency, high torque density, low cogging torque and shorter end-windings. A typical slot/pole combination for three-phase machines occurs when the slot number and pole number differ by one. In such machines, the coil pitch is approximately equal to the pole pitch, ensuring a high winding factor for the working harmonic. The cogging torque is minimized because the least common multiple of the number of slots and poles is large, resulting in a high cogging torque frequency and a small amplitude (Patel et al., 2014).

However, a notable drawback of fractional-slot windings is the generation of undesirable MMF harmonics. These harmonics rotate asynchronously with respect to the rotor and contribute to additional core losses and eddy-current losses in the rotor magnets. Furthermore, fractional-slot machines may exhibit UMP due to the asymmetrical disposition of the slots and coils, leading to additional vibrations and acoustic noise (Zhu et al., 2007).

Figure 1 (left) shows a 9-slot/8-pole PMSM machine, which serves as a precursor to the 18-slot/8-pole machine used in this paper. In the 9-slot/8-pole machine, each phase consists of three adjacent coils connected in series, with the middle coil having opposite polarity to the other two. The windings of phases a, b and c are displaced by 120° electrical degrees, producing symmetrical and phase-shifted back electromotive forces (Zhu et al., 2007). However, the physical layout of the phase windings is asymmetrical, leading to UMP during on-load operation. The MMF spectral content is shown in Figure 2 (top). The 4^th harmonic, which corresponds to the rotor’s number of pole pairs, interacts with the 4 pole-pair rotor magnetic field to generate electromagnetic torque. This harmonic is consequently known as the working harmonic. In contrast, other harmonics do not contribute to useful energy conversion and are considered undesirable. The most prominent of these is the 5^th harmonic, which has an amplitude comparable to the working harmonic but propagates in the opposite direction relative to the rotor.

Wang (Wang, 2015) proposed a general redesign method for fractional-slots machines to achieve the complete elimination of all odd (or even) space harmonics, while retaining the key advantages of fractional-slot winding schemes.

For a 9-slot/8-pole machine, the process involves doubling the number of slots to 18 and dividing the stator windings into two three-phase sets. The coil span is increased to two slot-pitches to maintain the same mechanical angle of 40° as in the original 9-slot/8-pole machine. The coils associated with each winding set are placed in alternate stator slots along the circumference (Figure 1, middle).

The second three-phase winding set is shifted in space by 180° mechanical degrees or 9 slot-pitches resulting in an offset of 720° electrical degrees (Figure 1, bottom). If individual phase windings are connected in the same sense, the 4^th (working) harmonics produced by both winding sets are in phase and their effect adds up.

The 180° shift between the winding sets has an opposite effect on odd harmonics (v = 1, 3, 5…). The phase shift between the odd harmonics produced by the first three-phase winding set and those produced by the second three-phase winding set is v·180°. These harmonics have equal magnitudes but opposite directions, leading to their cancelation.

Figure 2 (bottom) shows the MMF spectrum of the redesigned 18-slot/8-pole machine. We can see that all odd harmonics are eliminated, while the even harmonics are doubled in magnitude. Besides the working harmonic, the 2^nd, 14^th and 22^nd are also amplified due to the offset between the winding sets. However, the 14th and 22nd harmonics are not problematic as their relatively long wavelengths lead to an attenuated magnetic field. Similarly, the 2^nd harmonic has a limited impact due to its relatively low amplitude.

The redesign method produced two three-phase winding sets, hereafter referred to as abc and xyz. If the winding sets remain independent (i.e. without interconnection), each can be fed separately, creating a DTM (Figure 3). Data and parameters of the machine are given in  Appendix (Table A1). In normal operating mode, with equal currents in both winding sets, the DTM inherits all the advantages of an 18-slot/8-pole three-phase machine, namely the high winding factor, near sinusoidal EMF, and the cancelation of odd space harmonics. Within the scope of this 18-slot/8-pole machine, the dual three-phase configuration offers other offset possibilities between the winding sets which leads to similar favorable properties in NO (Patel et al., 2014).

The dual three-phase configuration provides additional degrees of freedom. First, in addition to NO, where both winding sets are energized, the machine can operate in PFO, which results in a derated torque output with only one active winding set. Second, SLO, a method for determining electrical machine losses, can be applied by energizing the winding sets in opposite directions (Zabaleta et al., 2019).

Figure 4 shows the MMF spectra for different operating modes of a DTM. In NO, the abc and xyz currents are supportive, resulting in an MMF spectrum that contains only even harmonics (Figure 4, top). The 4^th working harmonic, which interacts with the main rotor field to produce electromagnetic torque, is the most significant – as in the case of the original three-phase machine. In PFO, the xyz currents are zero, meaning only the abc winding set is active. Unlike NO, the MMF spectrum of PFO (Figure 4, middle) also contains odd harmonics, with the 5^th harmonic being particularly prominent (Wang and Wu, 2024). A similar observation applies to SLO, where only odd harmonics appear in the MMF spectrum (Figure 4, bottom).

The harmonic content of the MMFs reveals that both PFO and SLO introduce odd harmonics, which can be problematic. As the main rotor field harmonic is the 4^th, its interaction with the armature’s odd harmonic spectrum leads to undesirable effects, such as UMP. In NO, the radial magnetic force produced by the abc winding set is counteracted by the xyz winding set, effectively canceling out UMP. However, in PFO and SLO, this compensation does not occur, resulting in the presence of UMP. The objective of this paper is to evaluate UMP across these three operating modes.

UMP refers to an asymmetrical force that acts on the rotor and occurs due to a non-symmetric distribution of the magnetic field in the airgap. To determine UMP, the following procedure is used. First, finite element (FE) analysis predicts the radial and tangential air-gap flux density components (B_r and B_t). Both quantities are functions of stator angle α and rotor position θ_mech. Then, these components are used to calculate the radial traveling force wave density (Zhu et al., 2007):

And tangential traveling force wave density:

Both traveling waves are then integrated along a surface with an airgap radius r_AG to determine the cumulative force on the rotor with an axial length of l_rot. The impact of an individual traveling wave can be conveniently expressed in Cartesian coordinates. The force components F_rx and F_ry due to radial traveling force wave are:

and similarly, for F_tx and F_tr due to tangential traveling force wave:

UMP is then calculated as the vectorial sum of the components:

From the equations (3) and (4) it is evident that the UMP arises only from the 1^st order (fundamental) harmonic of the force wave. The equations involve the integral of the force wave function, which is rich in harmonics (cf. Figures 9 and 10), multiplied by a fundamental trigonometric function over one mechanical period. Therefore, all product terms containing non-similar frequencies will be zero and only the fundamental component of this space wave will evaluate to net contribution.

The fundamental harmonic of the force wave is caused by the interaction between field harmonics differing by one. As a result, a machine with only even or only odd field harmonics does not produce UMP, as such fields are symmetric (Chao et al., 1996).

To estimate UMP, a series of time-stepped 2D transient FEM analyses were performed using the Motor-CAD EMag module. Radial B_r and tangential B_t flux density within the airgap were calculated for a) no-load and four current levels, b) three operating modes (NO, PFO and SLO) and c) 160 rotor positions θ_mech from 0° to 180° mechanical degrees. Current displacement angle was set to 90°.

Figure 5 shows the radial flux density for all three operating modes at t = 0 s and I_s = 100A. The radial force density distribution on the inner surface of the stator is the primary cause of electromagnetically induced noise and vibrations.

Harmonic analysis of the field solution (Figure 5, bottom) reveals that only even harmonics exist in NO, whereas in PFO and SLO odd harmonics also appear, as predicted by MMF profile (Figure 4). As expected, the 4^th harmonic is dominant in all three cases. The odd harmonics are caused by armature reaction and increase with rising current load. As already mentioned, the UMP originates from the first spatial order harmonic of the force wave, which in turn is generated by the interaction of field harmonics with spatial orders differing by one. In case of PFO and SLO, the UMP is thus primarily caused due to interaction of 4^th and 5^th field harmonics (Figure 5, bottom). A smaller contribution can be attributed to the interaction of the 12^th and 13^th harmonics, while SLO also exhibits an interaction due to the 13^th and 14^th harmonic pair. The tangential flux density has a similar spectrum, with a prominent 5^th harmonic in case of PFO and SLO, but with significantly smaller amplitudes (Figure 6).

The radial and tangential traveling force waves are calculated from the air-gap flux-density using equations (1)–(2). The space harmonics distribution of the force waves arises from the combined effect of the permanent magnet and armature reaction fields. Figure 7 shows the waveforms for radial (top) and tangential (bottom) components for all three operating modes at t = 0 s and I_s = 100A. The radial force exhibits a similar amplitude regardless of the operating mode, and these amplitudes are an order of magnitude greater than those of the tangential wave.

Figure 8 presents the results in polar coordinates, providing a clearer visualization of the spatial symmetry required to avoid UMP. The radial force component (top) and tangential force component (bottom) are drawn to scale. Once again, it is evident that the radial force component dominates over the tangential one. In NO (Figure 8, left), the force profile along the airgap exhibits half-wave symmetry, where the force at each point along the air gap is counteracted by an equal force at the diametrically opposite point. This balance is maintained in both the radial and tangential direction and prevents the emergence of UMP. In contrast, PFO and SLO (Figure 8 middle and right) lack this symmetry, leading to the occurrence of UMP.

Figure 9 illustrates the harmonic spectra of radial force wave. In NO, the dominant harmonic is the 8^th harmonic, with the 1^st harmonic, indicative of UMP, being absent. In PFO, the 8^th harmonic remains dominant; however, additional odd harmonics, particularly the 9^th, also appear. In SLO, the 9^th harmonic becomes dominant, while the 8^th harmonic is attenuated. The most important observation is that the 1^st harmonic is present both in PFO and SLO, indicating the presence of UMP. Although its amplitude is relatively small compared to other components, it is important to note that the forces from higher harmonics are spatially compensated and induce vibration modes, whereas the 1^st harmonic directly contributes to UMP.

Figure 10 shows harmonic content of the tangential force wave, where significantly smaller amplitudes can be observed (note the 10x smaller scale). The DC component of the tangential force wave, present in both NO and PFO, generates torque. This component is notably absent in SLO. However, the presence of the 1^st harmonic in both PFO and SLO indicates that the tangential force wave also contributes to UMP, as described by equations (3) and (4).

In Figure 11, only the first spatial components of the radial (solid line) and tangential (dashed line) force waves are shown for PFO and SLO. It is evident that the 1^st harmonic of the radial force wave is approximately eight times larger than the corresponding harmonic of the tangential force wave in both operating modes.

The integral of the force waves along the circumference generates the radial F_r and tangential F_t components of UMP. Correct interpretation of the contributions from both waves is needed to accurately determine the spatial orientation of the corresponding UMP components (Zhu et al., 2011). As the same principles apply to both PFO and SLO, the analysis will focus on SLO for simplicity.

The angular direction of F_r always coincides with the peak of the radial force wave, as illustrated in Figure 12 (left). In contrast, the angular position of F_t is always 90° mechanical degrees ahead of the corresponding peak of the tangential force wave, as shown in Figure 12 (right). As indicated in Figure 11, the tangential force wave leads the radial force wave by 90° mechanical degrees in space, making F_t collinear with F_r. Additionally, the UMP components F_r and F_t are additive, and the resultant UMP, as shown in Figure 12, is the arithmetic sum of these components. However, since the radial force wave dominates, F_r is significantly larger than F_t.

Figures 13 and 14 depict the calculated UMP for 4 current levels (50A, 100A, 150A and 200A) across 160 rotor positions, ranging from 0° to 180° mechanical degrees. The amplitude range of UMP is similar for both PFO and SLO, remaining below 1.3 kN for 200A. However, as the current increases, the UMP ripple becomes more pronounced, particularly in the case of PFO.

Figure 15 illustrates the variation in the average UMP value with respect to stator current for both operating modes (solid lines). The average UMP value increases almost linearly in PFO, whereas in SLO, the relationship is more nonlinear. Since both winding sets are energized in SLO (unlike in PFO, where only the abc winding set carries current), higher saturation occurs, reducing the ratio of UMP to current. The shaded area represents the spread of minimum and maximum UMP values based on rotor position. As observed in Figures 13 and 14, the UMF ripple amplifies with increasing current and is more pronounced in PFO.

Based on the simulation results, an UMP of approximately 1.3 kN is expected at a maximum current of 200A. The DTM is equipped with bearings with a basic dynamic load rating C = 15 kN, where C represents the bearing load that reduces its lifetime to half after 1 million revolutions (SKF, 2011). Therefore, it is reasonable to assume that the PFO or SLO modes do not pose a significant risk of terminal damage to the bearing, especially since these operating modes are intended to be temporary.

For the 18-slot/8-pole DTM, PFO and SLO inevitably result in the emergence of UMP due to the presence of both odd and even harmonics in the field spectrum. This paper explores the specifics of UMP under these non-standard operating modes using FEA. We confirmed that UMP originates from the first spatial order harmonic of the force wave, which is generated by the interaction of field harmonics with spatial orders differing by one. This key finding enhances our understanding and ability to predict UMP in various operational scenarios. The analysis demonstrated that both PFO and SLO introduce significant UMP, with the radial force wave being the dominant component, far exceeding the tangential force wave in magnitude.

Although UMP introduces additional strain on the bearings, its magnitude remains an order of magnitude lower than the bearing load rating. Consequently, while UMP may reduce the bearing lifespan due to increased wear over time, it does not pose an immediate risk to the machine’s mechanical integrity. This indicates that the bearings can tolerate the extra load during the short durations typically associated with PFO and SLO operating modes.

The study is limited in scope, focusing on a single machine design to provide a detailed analysis of UMP. While the results offer valuable insights, they may not be directly applicable to other machine types without further investigation and exploration of design variations.

The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0258).

Table A1