Impact of Slot/Pole Combination on Inter-Turn Short-Circuit Current in Fault-Tolerant Permanent Magnet Machines

This paper investigates the influence of the slot/pole (S/P) combination on inter-turn short-circuit (SC) current in fault-tolerant permanent magnet (FT-PM) machines. A 2-D sub-domain field computational model with multi-objective genetic algorithm is used for the design and performance prediction of the considered FT-PM machines. The electromagnetic losses of machines, including iron, magnet, and winding losses are systematically computed using analytical tools. During the postprocessing stage, a 1-D analysis is employed for turn-turn fault analysis. The method calculates self- and mutual inductances of both the faulty and healthy turns under an SC fault condition with respect to the fault locations, and thus SC fault current, considering its location. Eight FT-PM machines with different S/P combinations are analyzed. Both the performance of the machine during normal operation and induced currents during a turn-turn SC fault are investigated. To evaluate the thermal impact of each S/P combination under an inter-turn fault condition, a thermal analysis is performed using finite element computation. It is shown that low-rotor-pole-number machines have a better fault tolerance capability, while high-rotor-pole-number machines are lighter and provide higher efficiency. Results show that the influence of the S/P selection on inter-turn fault SC current needs to be considered during the design process to balance the efficiency and power density against fault-tolerant criteria of the application at hand.

2) overrating of the phase inductance, which limits the 41 phase short-circuit (SC) current to a safe value in the 42 case of winding short-circuit fault; where 102 e 1 electro motive force in the healthy turns; e 2 electro motive force in the shorted turns; I 1 phase current induced in the shorted turns; I s SC fault current; L h self-inductance of the healthy turns; L s self-inductance of the shorted turns; L m mutual inductance between the healthy and the shorted turns; R h resistance of the healthy turns; R s resistance of the shorted turns. 103 Hence, the steady-state SC fault current (I s ), after the 104 machine has been shorted via the converter terminals, can be 105 estimated using the following equation: where ω e is the angular electrical pulsation. From (3), it can 110 be seen that I s is related to three major parameters, which 111 are resistances R s and R h , inductances L h , L s and L m , and 112 operational frequencies. 113 For clarity, the terms in (3) can be substituted as follows: where ϕ represents the non-load flux linkage per turn. Dividing 122 the nominator and denominator of (6) by ω e 2 yields AQ:2 As ω e is significantly greater than b, c, and R h , (7) can be 125 simplified to For the considered single turn-turn fault condition, N s = 1; 128 therefore, the second term of (8) can be neglected   Fig. 2 represents the process involved in the optimization 170 of the electrical machine design and both the performance 171 and turn-turn SC fault analysis of the optimized design. 172 The optimization process starts with the initially selected 173 S/P combinations in Section III and the fixed outer diame-174 ter (OD) of 120 mm, which is limited by the envelope of 175 the target application. Other design variables such as split 176 ratio (SR), aspect ratio (AR), tooth-width-to-slot ratio (TR), 177 slot-opening (So), tooth-tip height (h t ), magnet span (α m ), 178 magnet height (h m ), the number of turns per slot (N t ), and 179 phase current (I p ) are set as variable parameters. The design 180 process is limited by the following three design constraints.  2) Phase winding inductances are overrated to have 1 pu 183 inductance in order to limit the phase SC current equiv-184 alent to rated phase current of the design. 3) DC link voltage limit of the converter is fixed to ±135 V. 186 The key design optimization target is to produce highly 187 efficient and high-mass-density PM machines while satisfying 188 the above-mentioned constraints and application requirements 189 given in Table I. A multi-objective GA is adopted for the 190 optimization process, in which a 2-D electromagnetic model 191 is used during the design process, while to investigate the 192 turn-turn SC fault current, the 1-D SC fault model is used. 193 It is worth noting that by adopting an analytical model for the 194 design and analysis, the computation time is greatly reduced 195 while maintaining a high level of accuracy. Finite element (FE) 196 is therefore not considered here. The adopted analytical model 197 and the GA technique for the design and analysis are discussed 198 in detail in the following sections.  1) The machine has a radial geometry as shown in Fig. 3. 210 2) The stator and rotor cores have an infinite permeability 211 and zero conductivity. 3) The magnets are magnetized in the radial direction and 213 their relative recoil permeability is unity (μ r = 1). 214 4) The current density (J c ) over the slot area is uniformly 215 distributed.

216
5) The end-effects are neglected and thus the mag-217 netic vector potential has only one component along 218

222
The magnetostatic partial differential equations governing 223 in the behavior of the machine in the different sub-domains 224 can be derived from Maxwell's equations. 225 These equations are formulated in terms of vector potential 226 as in where A represents the magnetic vector potential and its 229 subscript is related to the associated sub-domains. μ 0 is the 230 permeability of air, J c is the current density, and M r is the 231 magnetization radial component.
Hence, the phase self-inductance and voltage can be 258 represented as a function of flux as described in where N ph is the number of turns per phase, K f is the fill 262 factor, and ω is the rotor angular speed.
where A j is magnetic vector potential in the j th slot, σ is 277 the conductivity, and r c1 , r c2 , σ c1 , and σ c2 are the radial and Both hysteresis and eddy current losses associated with 283 the stator iron are estimated using the well-known Steinmetz 284 equations, where the losses generated due to localized satura-285 tion phenomena are neglected. As given in Fig. 4, the stator 286 iron is divided into three parts. The flux density in each part 287 is evaluated considering the average flux density in the air-288 gap domain. Finally, the iron losses are estimated using the 289 evaluated flux density together with the material properties 290 from its associated data sheet. It is worth highlighting here 291 that the flux density harmonic effects in localized point and 292 time harmonics associated with pulsewidth modulation (PWM) 293 are not accounted for. 294 Since the total electromagnetic losses (P t ) are known, the 295 efficiency (η) can be obtained from

C. Optimization Process of the Design 298
The design process is carried out using an optimization 299 routine based on a non-dominated sorting genetic algorithm, 300 where the above-mentioned 2-D electromagnetic computa-301 tional methodology is integrated to evaluate the perfor-302 mance [20]. The goal of the GA is to maximize the efficiency 303 and minimize the mass of the machine. As previously 304 mentioned, the optimization envelope was constrained by 305 the no-load air-gap flux density (B airgap ), phase self-306 inductance (L p ), and converter voltage limit. The per-unit base 307 inductance L pu is set as follows: where PM is flux linkage due to the permanent magnets and 310 I p is the rated phase current of the machine. Thus, the SC fault 311 current during a fault will be limited to its nominal value.  The design parameters of the selected machines for different 323 S/P combinations are summarized in Table II.   Fig. 6. Comparison of the individual losses across the studied machines (ac + dc represents ac and dc copper losses, including the end winding losses; Iron and Magnets represents eddy current and hysteresis losses in the stator iron and magnets, respectively).

340
In this section, results from the investigation of the effect 341 of S/P combination on inter-turn SC current in FT-PM are 342 presented. This section is divided into three subsections, where 343 the outcomes of the individual analyses are explained. Losses 344 and SC fault current were analyzed for each S/P combination 345 and thermal analysis was performed for the selected 346 S/P variants. In addition, a method that minimizes the SC fault 347 current is proposed.

349
The loss breakdown for each of the machines studied is 350 shown in Fig. 6. While the ac and the dc winding losses 351 are a major part of the total losses in all cases, the low slot 352 number machines show high winding losses. The increase in 353 the winding losses is mainly due to the bigger end windings' 354 length of the machines with a low slot number. The high-355 pole-number machines have high iron losses due to the higher 356 electrical frequency necessary for their operation. Also it is 357 worth noting that the 12/14 machine has higher iron losses 358 than the 24/16 and 24/20 machines. The stator iron loses are 359 dictated not only by the fundamental frequency of the phase 360 current, but also by the mass of the machine's stator core. 361 As is shown in Table II, the mass of the 12/14 machine's 362 stator core is bigger than the mass of both 24/16 and 363 24/20 machines' stator core and so are the iron losses of the 364 12/14 machine.

365
From Figs. 6 and 7, it can be seen that the 6/4 machine 366 proved to have the highest losses and thus lowest efficiency. 367 This is mainly due to high winding losses and magnet eddy 368 current losses. If the segmentation is adopted for the machine, 369 the magnet eddy current losses can be reduced. Although this 370 would be possible, the resultant efficiency will depend on the 371 number of segments adopted in the design.

372
As can be seen from Fig. 7, it is obvious that among 373 the considered machines, the 24/20 machine variant, which 374 delivers rated output with 95.7% efficiency, is the best design 375 choice in terms of performance.  Among other candidates, S/P combinations of the 12/8 and 396 12/10 machines have a similar SC behavior. It can also be seen 397 in S/P combinations of the 12/14 machine and 24/16 machine. 398 This is because of the associated electrical frequencies, which 399 are almost equal. Although these pairs of machines provide 400 almost identical results regarding SC, in terms of efficiency, 401 the 12/8 and 12/14 machines show increased efficiency.

403
In order to visualize the thermal behavior, the thermal 404 analysis was performed using the FE software and was carried 405 out in a coupled electromagnetic and thermal FE environment. 406 Two states, healthy and faulty, are studied. The healthy state 407 is simulated with a nominal phase current.

408
For the faulty state, to minimize the evaluation time, the 409 steady-state SC current obtained in the inter-turn SC fault 410 analysis is injected into the faulty turn. The remaining healthy 411 windings are separately excited using the nominal phase 412 current. In the analysis, thermal continuity between stator and 413 rotor is taken into account and the thermal boundaries (stator 414 outer surface temperature is fixed to 120°C) are kept the 415 same for all cases. The conductors' cross-sectional area and 416 insulation thickness are carefully selected considering slot fill 417 factor K f = 0.5. Results obtained for four cases are presented 418 in Fig. 10.

419
The SC analysis proved that the 6/4 machine is the most 420 tolerant to the inter-turn SC fault, and the difference in 421 the thermal distribution in the slot between the healthy and 422 fault conditions is almost negligible. As expected, high-pole-423 number variants 24/16 and 24/20 show a noticeable tempera-424 ture rise at the fault condition. Fig. 10(g) and (h) shows that 425 the 24/20 machine variant has critical hotspot due to the larger 426 fault current. It is worth highlighting here that although the 427 24/16 machine variant is subjected to less magnitude of worst 428 case SC current than the 18/12 variant, it has poor thermal 429 behavior. This is due to the windings resistance associated 430 with the 24/16 machine variant, which is higher than in the 431 18/12 variant, as evident from Fig. 6. to be the best compromise for such FT designs, since they have 438 higher efficiency and the SC current is almost twice the rated 439 value.

441
Although the 12/8 and 12/10 machine variants are the best 442 choice in terms of FT and efficiency, those machines have 443 almost twice the rated current when fault occurs close to the 444 slot-opening region. One way of minimizing the fault current 445 is to design the machine with a larger inductance, which can 446 be even higher than one per unit inductance. When possible, 447 this would result in a lower power factor and a significant 448 reduction in the achievable torque density.

449
Alternatively, the maximal SC current can be maintained 450 at twice the rated current by avoiding the placement of the 451 winding closer to slot-opening region. From Fig. 9, it is 452 obvious that using only 90% of the slot for the winding and 453 avoiding 10% closest to the slot-opening region replaces the 454 maximal SC fault current significantly. For the 12/8 and 12/10 455 machine variants, the SC current can be limited to under 2 pu, 456 if the 10% slot region is avoided. However, this will reduce the 457 slot fill factor, consequently increasing the dc losses. However, 458 it would be beneficial if the machine is operated at high speeds, 459 as the ac losses would be reduced [21].  It has been shown that the most critical inter-turn fault 468 location is near the slot-opening region and the magnitude of 469 the SC fault current can be significantly reduced by avoiding 470 winding placement near this region.

471
Furthermore, the inter-turn fault current magnitude depends 472 on the selection of the slot and pole numbers, which 473 influence the windings' parameters, namely, resistance and 474 self-inductance of both healthy and faulty turns and mutual 475 inductance between them.

476
Lower S/P combinations have better FT capability, while 477 high S/P combinations have improved efficiency. To balance 478 the efficiency and FT criteria of the application, the impact 479 of the S/P combination on inter-turn SC fault current must be 480 considered for the design process.    He was subsequently a Researcher with The University of Nottingham, 601 where he was involved in high-performance electrical drives and the 602 design and modeling of electromagnetic actuators for aerospace applications.