Inductance Calculations of a Dual Stator Machine

290 Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 p ISSN: 2635-3342; e ISSN: 2635-3350 Original Research Article Inductance Calculations of a Dual Stator Machine *Awah, C.C. Department of Electrical and Electronic Engineering, College of Engineering and Engineering Technology, Michael Okpara University of Agriculture, Umudike, PMB 7267, Umuahia, Nigeria. *[email protected] http://doi.org/10.5281/zenodo.18060919 ARTICLE INFORMATION ABSTRACT Article history: Received 25 Jun. 2025 Revised 22 Sep. 2025 Accepted 03 Oct. 2025 Available online 30 Dec. 2025 Due to the great impact of inductance on performance of an electric machine, its values and characteristics were studied in this investigation. Special consideration is made to the machine’s pole number effects. The considered number of poles (Nr) were: 10, 11, 13 and 14. Two-dimensional finite element analysis method was applied in the calculations. The considered inductance parameters were: self, mutual, direct-axis and quadrature-axis inductance. Resulting magnetic flux value of the compared machine types at rated current were 6.19 mWb, 11.23 mWb, 8.98 mWb and 5.20 mWb, respectively. High value of flux linkage would incidentally influence the resulting electromagnetic outputs of the machine, positively. The results revealed that the compared machine types have comparable selfinductance values; although, the 13-pole machine type has marginally higher self-inductance value than its compared equivalents. The 11-pole (Nr=11) and 13-pole (Nr=13) machine types exhibit high fault-tolerance potential owing to its relatively larger inductance ratio. Nevertheless, the Nr=13 machine category has the largest amount of axes inductance and this has high electromagnetic saturation implications. It is also revealed that the resulting magnetic flux magnitudes and its matching inductances of the investigated machine are dependent upon its pole number and on the magnitude of supplied current. © 2025 RJEES. All rights reserved. Keywords: Dual stator Finite element analysis Flux Inductance Machine 1. INTRODUCTION An electric machine’s winding inductance profiles would influence its fault withstand capacity and other output features, in addition to its electromagnetic saturation effects (Chen et al., 2005; Lee et al., 2015). Nevertheless, the torque density potential of a machine may be compromised at the expense of its enriched fault withstand ability (Chen et al., 2014); additionally, the speed range of a machine could be influenced positively, if it possesses a high amount of direct-axis inductance. However, the resulting inductance of a given machine is dependent upon many other factors such as: the machine’s geometric 291 C.C. Awah / Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 dimensions, air gap length, slot-pole combinations, amongst others (EL-Refaie et al., 2008). Owing to these winding inductance impacts, this current investigation predicts and compares the resulting quantities of the winding inductances with reference to different pole numbers. The dependence of winding inductance worth on its slot-pole combination is reconfirmed by (Ni et al., 2014). It is revealed by Dutta et al., (2012) that non-sinusoidal waveforms of winding inductances are mainly caused by its inherent magnetomotive force (MMF) harmonic contents; however, a machine configuration that has distributed winding arrangement is not affected by these MMF harmonics effect. The study also established that resulting winding inductance of a machine is a function of its rotating angular positions; besides, the relative difference between axis inductance of a machine is as a result of its mutual inductance harmonic components. It is noteworthy that the above-mentioned assertions are in agreement with the predicted values of this current investigation. An analytical approach is deployed in estimating the inductance of a permanent magnet machine with improved precision, as presented by (Chen et al., 2014); however, predictions using finite element analysis (FEA) approach usually exhibits higher accuracy than analytical methods. It is worth noting that FEA technique is implemented in this present analysis. Machine performances can be greatly affected by application of skewing method; particularly, its crosscoupling and saturation levels (Lazari et al., 2014). These impacts can be heightened by the presence of high armature reaction; thus, leading to significant alterations in the machines flux-linkages and corresponding output inductances (Jeong and Nam, 2015). These possible changes in the machine’s output features are validated by (Liu et al., 2016) with experimental tests. Meanwhile, the implemented rotor configuration of a machine is also critical in defining the resulting saturation level of the system (Fasil et al., 2016). Moreover, a permanent magnet machine that has low quadrature-axis inductance over a given electric load condition would consistently have high overload potential (Qu and Zhu, 2021). Ordinarily, armature reaction tends to increase the torque pulsation and demagnetization vulnerability of a machine, in addition to reduction of its overload capability (Qi et al., 2023); these are all drawbacks, and it is usually higher or more conspicuous in machines that have consequent pole configuration. It is reconfirmed that the overload potential of machine as well as its longer speed coverage is a direct consequence of the machine’s direct-axis inductance value (Guo et al., 2022). Essentially, axes inductance significantly affects the field-weakening and output torque performance of an electric machine (Xu and Wang, 2023). Nevertheless, the individual values of a machine’s axes inductance can be manipulated through the use of flux-barriers (Kashif and Singh, 2022) or mechanical flux adjusters (Zhao and Liu, 2023), in order to suit specific applications. Moreover, the control mechanism of an electric machine performs better at a sufficient inductance magnitude, as noted by Ebadi et al. (2019). In general, the impact of winding inductances of a dual stator machine is analyzed and compared in this study amongst varying pole numbers, with a view to having a quantitative understanding of these parameters’ implications on a machine's electric outputs and performances. This study will find suitable applications in fault-tolerant and reliability uses. 2. METHODOLOGY Inductance profiles of a dual stator (DS) permanent magnet (PM) machine are presented in this study using the finite element analysis method. The analyzed machine has magnets on both its two stator sectors, for enhanced output performance; plus, a rotor which is sandwiched between these stators amidst a dual air gap arrangement. The cores and the magnets are made of steel and rare-earth materials, respectively. The windings are made of copper materials. Note that the windings are supplied with balanced three-phase alternating currents. The overall machine size is 90 mm. The simulation was conducted at an ambient temperature of about 20 °C, in one electric cycle. The predicted inductances are calculated on both load and no-load conditions with preference to a varying number of poles. The considered varying pole numbers (Nr) include: 10, 11, 13 and 14 poles, which are designated as: Nr=10, Nr=11, Nr=13 and Nr=14, respectively. The investigated machine model is displayed in Figure 1. The mathematical expression of the predicted selfinductance (Laa) and mutual inductance (Mab) is given in Equation (1) and Equation (2), respectively. The 292 C.C. Awah / Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 predicted flux linkage of a flux-switching PM machine to which the analyzed machine belongs to is provided in Equation (3), as highlighted in (Awah, 2022). Figure 1: Machine model (Awah and Nnabuenyi, 2023) a nlla aa I L  − = (1) a nllb ab I M  − = (2) Where: ψla and ψlb are the Phase A and Phase B excited flux-linkages, ψnl is the flux linkage on no-load. Ia is the applied Phase A current magnitude (Awah et al., 2024a). a s i sprfpml P D RFkNB   = (3) Where: N is number of turns, Bm is maximum flux density, kfp is the flux linkage ratio between the fundamntal and its peak value, Fr is the ratio of exciting flux to total flux, Rsp is the ratio of stator tooth thickness to pole pitch, Ps is the stator teeth number, Di is the stator diameter and la is the machine’s stack length (Li et al., 2016). 3. RESULTS AND DISCUSSION Figure 2 shows the self and mutual inductance outlines of the analyzed machine types over different rotor positions. It is observed that the compared machine categories have comparable self-inductance values. Therefore, the varying machine types have almost equal ability to sustain short circuit fault, owing to their nearly equal self-inductance values (Bianchi et al., 2006). The calculated inductance values are listed in Table 1. The absolute values of mutual inductance presented in Table 1 show that the Nr=10 and Nr=14 machine categories have relatively lower mutual inductance values; this property is a desirable quality for improved fault-tolerance ability (Awah et al., 2016). Again, the presumed high fault-tolerance proficiency Permanent magnets Inner stator Rotor piece Outer stator 293 C.C. Awah / Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 of the 10-pole and 14-pole machine types is reconfirmed by its low absolute ratio of mutual-inductance to self-inductance. The calculated negative mutual-inductance values of Table 1 imply that the assumed current directions of the 11-pole and 13-pole machine types are actually reversed. (a) (b) Figure 2: Inductances (a) Self-inductance and (b) mutual-inductance Table 1: Inductance values Items Values Pole number (Nr) 10 11 13 14 Self-inductance (Laa), mH 0.27 0.27 0.29 0.26 Mutual inductance (Mab), mH 0.07 -0.11 -0.12 0.06 Absolute ratio aa L:Mab 0.26 0.41 0.41 0.23 Direct-axis inductance, mH 0.20 0.37 0.40 0.19 Quadrature-axis inductance, mH 0.21 0.39 0.41 0.20 Saliency ratio 1.05 1.05 1.03 1.05 Applied current, A 15 15 15 15 Speed, r/min 400 400 400 400 The 13-pole and 11-pole machine types have relatively higher axes inductance than its equivalent 10-pole and 14-pole categories, as depicted in Figure 3. It is worth noting that the higher the axes inductance values of a machine; then, the larger its electromagnetic output magnitudes (Hu et al. (2020), however, within nonsaturation limits of the machine. (a) (b) Figure 3: Axes inductance (a) Quadrature-axis inductance (b) Direct-axis inductance Figure 4 shows the winding inductance outlines of the compared machines at different axis currents. The results reveal that the axis currents as well as the adopted pole numbers will influence the resulting winding inductance values and shapes, of the machine. Noticeable inductance differences occur in the 11and 130 0.05 0.1 0.15 0.2 0.25 0.3 0.35 060 120 180 240 300 360 Self-inductance (mH) Angular position (deg) Nr=10 Nr=11 Nr=13 Nr=14 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0 60 120 180 240 300 360 Mutual-inductance (mH) Angular position (deg) Nr=10 Nr=11 Nr=13 Nr=14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 60 120 180 240 300 360 Quadrature-axis inductance (mH) Angular position (deg) Nr=10 Nr=11 Nr=13 Nr=14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 060 120 180 240 300 360 Direct-axis inductance (mH) Angular position (deg) Nr=10 Nr=11 Nr=13 Nr=14 294 C.C. Awah / Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 pole machine types at varying axis current settings compared to the corresponding 10and 14-pole counterparts. More so, it is obvious that the quadrature-axis inductance magnitudes of the analyzed machine types are higher than its direct-axis inductance values. These axes inductance values would have influence on the machine’s saturation propensity as well as its speed range (Oti and Awah, 2022). Hence, proper adjustment could be made through well established procedures, in order to realize a suitable value for a chosen application. Overall, the resulting axes inductance values and features of machines would be dependent upon its winding configuration and machine type (Awah et al., 2024b). (a) (b) (c) (d) Figure 4: Winding inductance (a) Nr=10 (b) Nr=11 (c) Nr=13 (d) Nr=14 The predicted PM flux at varying current settings is shown in Figure 5. The results show that the Nr=11 and Nr=14 machine typologies have the largest and smallest amount of PM flux, respectively. Also, the PM flux is virtually constant over the simulated electric loads, with a trivial reduction at higher current rating due to the electromagnetic saturation effect. Figure 5: Comparison of PM flux 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 10 12 14 16 18 Winding inductance (mH) Quadrature-axis current (A) Ld, id=-1A Lq, I=1A Ld, id=-3A Lq, I=3A Ld, id=-6A Lq, I=6A Ld, id=-9A Lq, I=9A Ld, id=-12A Lq, I=12A Ld, id=-15A Lq, I=15A 0.35 0.37 0.39 0.41 0.43 0.45 0 2 4 6 8 10 12 14 16 18 Winding inductance (mH) Quadrature-axis current (A) Ld, id=-1A Lq, I=1A Ld, id=-3A Lq, I=3A Ld, id=-6A Lq, I=6A Ld, id=-9A Lq, I=9A Ld, id=-12A Lq, I=12A Ld, id=-15A Lq, I=15A 0.35 0.37 0.39 0.41 0.43 0.45 0 2 4 6 8 10 12 14 16 18 Winding inductance (mH) Quadrature-axis current (A) Ld, id=-1A Lq, I=1A Ld, id=-3A Lq, I=3A Ld, id=-6A Lq, I=6A Ld, id=-9A Lq, I=9A Ld, id=-12A Lq, I=12A Ld, id=-15A Lq, I=15A 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 2 4 6 8 10 12 14 16 18 Winding inductance (mH) Quadrature-axis current (A) Ld, id=-1A Lq, I=1A Ld, id=-3A Lq, I=3A Ld, id=-6A Lq, I=6A Ld, id=-9A Lq, I=9A Ld, id=-12A Lq, I=12A Ld, id=-15A Lq, I=15A 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 2 4 6 8 10 12 14 16 Permanent magnet flux (Wb) Quadrature-axis current Nr=10 Nr=11 Nr=13 Nr=14 295 C.C. Awah / Nigerian Research Journal of Engineering and Environmental Sciences 10(2) 2025 pp. 290-296 4. CONCLUSION Inductance values and characteristics of a dual stator electric machine are investigated and quantified in this study. Finite element analysis is implemented in the result predictions. It is revealed that pole number and applied current values would influence the resulting electromagnetic magnitudes and features of a given electrical machine. Moreover, highest fault-tolerance ability is found in the machine types that have even number of poles i.e., the 10-pole and 14-pole machine categories, due to its low mutual to self-inductance ratio values relative to other compared machine types. The 13-pole machine is inferred to have high proneness to magnetic saturation due to its high possession of axis inductance value. 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