Analysis of the Magnetomotive Force of a Three-Phase Winding with Concentrated Coils and Different Symmetry Features

Transcription

1 Analyss of the Magnetomotve Force of a Three-Phase Wndng wth Concentrated Cols and Dfferent Symmetry Features Deter Gerlng Unversty of Federal Defense Munch, Neubberg, 85579, Germany Emal: Deter.Gerlng@unbw.de Abstract- Permanent magnet machnes wth wndngs concentrated around each tooth gan more and more mportance. In ths paper, the Magnetomotve Force of a three-phase wndng wth concentrated cols wll be analysed. As an eample, a machne wth a two-col zone wdth wth opposng flu s regarded. It wll be shown that there are two dfferent wndng alternatves wth four dfferent rotor pole numbers that can generate a constant torque. All these alternatve realzatons are characterzed as fractonal-slot wndngs. Ths analyss may serve as a bass for the machne desgn concernng fundamental torque, as well as for acoustc nose calculaton because a major part of the harmonc content of the ar-gap feld s dentfed. I. INTRODUCTION Especally for Permanent Magnet (PM) motors the wndng confguraton wth concentrated cols around each tooth gets more and more nto the focus. Most often, such motors are desgned wth a sngle-col zone wdth [1-2], only sometmes more than one col per zone are regarded [3-4]. The specal nterest n such motors manly comes from the fact that smple and therefore very cost-effectve producton possbltes arse usng such a wndng topology. In addton, the end wndng s very small resultng n less volume, weght, costs, and losses. A typcal applcaton of a PM motor wth concentrated cols s descrbed n [5]. On the other hand, there are some techncal drawbacks aganst a sophstcated dstrbuted wndng, especally the hgh harmonc content n the Magnetomotve Force (MMF). Therefore, the MMF-dstrbuton wll be analysed n ths paper for the eample of a three-phase wndng wth concentrated cols and two-col zone wdth wth opposng flu. II. WINDING TOPOLOGY The followng fgure 1 llustrates the wndng topology under nvestgaton. As every tooth contans a wndng, ths s usually referred to as two-layer wndng (aganst a sngle-layer wndng, when just every other tooth carres a col). The nvestgatons descrbed n ths paper wll be conducted regardng the followng assumptons: - All cols wll have the same number of turns. - All teeth have equal wdth and they are equally spaced. - The three-phase ( m = 3) currents are symmetrc and purely harmonc: = ˆIsn( ωt) A ˆ 2π = Isn t B ω. (1) 3 ˆ 4π = Isn ωt C 3 - As the wndng harmoncs shall be nvestgated, the slottng effect wll be neglected. Therefore, the wndng dstrbuton can be appromated by an electrc loadng on a smooth stator surface, concentratng the current of a sngle slot nto the md-pont of the slot openng. - The wndng dstrbuton shown n fgure 1 for each phase s defned beng postve for the followng analyss,.e. the complete wndng dstrbuton shown n fgure 1 wll be denoted wth +A +B +C. - As an eample, a stator wth Z = 24 teeth wll be consdered. phase A phase B phase C Fg. 1. Analysed wndng topology. III. TORQUE GENERATION The torque of electrcal machnes can be calculated from the electrc loadng dstrbuton (or MMF-dstrbuton) and the flu densty dstrbuton. For PM-machnes, the rotor flu densty dstrbuton and the stator MMF-dstrbuton are essental. Generally, both dstrbutons are non-snusodal and they contan a fundamental wave and an nfnte number of harmonc waves. Torque s generated, f the ordnal number of a MMF-wave and the ordnal number of a flu densty wave concde. We wll see n the followng, that (under the eemplary assumptons made above) there can be a symmetry on a quarter

2 crcumference of the motor or on the half crcumference. The number of rotor pole pars on the regarded mnmum symmetry must concde wth the ordnal number of the man MMF-wave (relatve to ths symmetry) to generate a tme-ndependent torque. IV. WINDING ALTERNATIVES A. General Remarks The followng four alternatves are possble, f the wndng dstrbuton descrbed n chapters 1 and 2 s used: Case 1: +A +B +C A B C (mnmum symmetry: 12 slots, half machne crcumference) Case 2: +A B +C A +B C (mnmum symmetry: 12 slots, half machne crcumference) Case 3: +A +B +C +A +B +C (mnmum symmetry: 6 slots, quarter machne crcumference) Case 4: +A B +C +A B +C (mnmum symmetry: 6 slots, quarter machne crcumference) The change of two phases just changes the drecton of the MMF-waves (and therefore the rotatonal drecton of the rotor), but not the general MMF-dstrbuton. As ths rotatonal drecton s not of nterest here, ths wll not be regarded n the followng. The four alternatve cases mentoned above wll be nvestgated separately n the followng sectons. B. Case 1: +A +B +C A B C The followng fgure 2 shows the MMF-dstrbuton for the tme steps ω t = 0 and ω t = π /2 ; on the horzontal as the slots are numbered, on the vertcal as the relatve MMF-value Θ s gven (p.u. value). Fg. 3 shows the fourer components of these MMF-dstrbutons. The calculaton of the fourer components s done by usng the saltus functon method descrbed n [6]. On the horzontal as are gven the fundamental and harmonc wave numbers, on the vertcal as s shown the ampltude of each wave. It becomes obvous from these fgures that the ampltudes are not constant n tme, so that t s not possble to generate a tme-ndependent torque. Ths wndng dstrbuton s not useful. Θ Fg. 2. MMF-dstrbuton Θ for case 1 and ω t = 0 (left) and ω t = π / 2 (rght). Fg. 3. Fourer components of the MMF-dstrbuton Θ for case 1 and ω t = 0 (left) and ω t = π / 2 (rght).

3 C. Case 2: +A B +C A +B C The MMF-dstrbuton and ther fourer components for case 2 are shown n the followng fgures 4 and 5. It can be shown n general, that for ths case the ampltudes of the fundamental wave and the harmonc waves are constant (the ponts n tme shown n these fgures serve as a hnt). Therefore, wth ths wndng dstrbuton a constant torque can be generated. Fg. 4. MMF-dstrbuton Θ for case 2 and ω t = 0 (left) and ω t = π / 2 (rght). Fg. 5. Fourer components of the MMF-dstrbuton Θ for case 2 and ω t = 0 (left) and ω t = π / 2 (rght). Fg. 5 demonstrates that there are two domnant fourer components: ordnal number 5 and ordnal number 7. The respectve ampltudes are: A = and A = 0.509, 5 7 resultng n a rato of A /A = 1.4. Nevertheless, the 5 7 ampltude of the electrc loadng (whch s essental for the torque generaton) s the same for both harmoncs, as the MMF-dstrbuton s calculated from ntegratng the electrc loadng dstrbuton over one pole ptch. To get an overvew, whch harmonc should be used for torque generaton, the followng fgure of the fourer components of the electrc loadng s very useful (as just the p.u. data are of nterest, each MMF-harmonc s multpled wth the respectve number of pole-pars): Fg. 6. Fourer components of the electrc loadng dstrbuton A for case 2.

4 In general ths wndng dstrbuton can generate a tme-ndependent torque wth a 20-pole rotor (ordnal number 5 means 5 complete sne-waves or 10 half waves per half crcumference, resultng n a 20-pole confguraton) or a 28-pole rotor. Of course, even the other harmoncs can generate a tme-ndependent torque wth the approprate rotor pole number, but as the requred frequency s much hgher ths does not make sense because of hgher losses (eddy current losses, hysteress losses, and swtchng losses). D. Case 3: +A +B +C +A +B +C The MMF-dstrbuton and ther fourer components for case 3 are shown n the followng fgures 7 and 8: Fg. 7. MMF-dstrbuton Θ for case 3 and ω t = 0 (left) and ω t = π / 2 (rght). Fg. 8. Fourer components of the MMF-dstrbuton Θ for case 3 and ω t = 0 (left) and ω t = π / 2 (rght). Ths alternatve s the second possblty to generate tme-ndependent harmonc waves of the MMF-dstrbuton (the ponts n tme shown above serve as a hnt). Therefore, a constant torque can be produced. Lke n case 2 the fundamental component s not the hghest one. The same holds true for the fundamental components of the electrc loadngs (please refer to fgures 6 and 9). The hghest MMF-ampltude s reached fort the ordnal number 2 (.e. 2 pole pars for a quarter crcumference). Consequently, a 16-pole rotor has to be appled to use ths harmonc wave for torque producton. The fourer components of the electrc loadng of ths alternatve are shown n the followng fgure 9 (demonstratng that even a 32-pole rotor may be appled): Fg. 9. Fourer components of the electrc loadng dstrbuton A for case 3.

5 Agan, even the hgher harmoncs can be used for torque producton, but because of the requred hgher frequency ths s not useful. As the vertcal as of fgure 6 and fgure 9 have the same values one can deduce that the torque of case 2 (fgure 6) s hgher. Although the frequency s hgher n case 2 because of the hgher number of pole-pars (20 or 28 aganst 16) most probably t s reasonable to make use of the hgher electrc loadng harmonc to realze the mamum torque. In addton, the fundamental wave (whch s not used for torque producton n both cases) s much smaller n case 2. E. Case 4: +A B +C +A B +C The MMF-dstrbuton and ther fourer components for case 4 are shown n the followng fgures 10 and 11: Fg. 10. MMF-dstrbuton Θ for case 4 and ω t = 0 (left) and ω t = π / 2 (rght). Fg. 11. Fourer components of the MMF-dstrbuton Θ for case 4 and ω t = 0 (left) and ω t = π / 2 (rght). Even ths alternatve shows tme-dependent harmonc waves of the MMF-dstrbuton, resultng n tme-dependent torque generaton. Agan, ths wndng dstrbuton s not useful. F. Summary It could be shown that there are two wndng alternatves resultng n a tme-ndependent MMF-dstrbuton. In general, both wndng alternatves can be used for generatng a constant torque. Case 2 (+A B +C A +B C) calls for a 20-poles rotor or a 28-poles rotor, case 3 (+A +B +C +A +B +C) calls for a 16-poles rotor or a 32-poles rotor. As the stator s realzed wth a 3-phase ( m = 3), 24-slot ( Z = 24) desgn, we get the followng wndng characterstcs: Case 2, 20-poles rotor ( 2 p = 20): q = = = p m 20 3 Case 2, 28-poles rotor ( 2 p = 28): q = = p m 28 3 Case 3, 16-poles rotor ( 2 p = 16): q = = = p m 16 3

6 Case 3, 32-poles rotor ( 2 p = 32): q = = = p m 32 3 All alternatves result n a fractonal-slot wndng. In case 2 and case 3 even a tme-ndependent torque generaton wth an ntegral slot wndng s possble, but then the fundamental wave wth qute a low ampltude of the electrc loadng has to be used, resultng n qute a low torque. V. CONCLUSION In ths paper, the Magnetomotve Force of a three-phase wndng wth concentrated cols and two-col zone wdth wth opposng flu has been analysed. It could be shown that there are two wndng alternatves resultng n a constant torque generaton. Havng a machne wth 24 stator slots, the frst wndng alternatve works wth a 20-poles or 28-poles rotor, the second wndng alternatve works wth a 16-poles or 32-poles rotor. All alternatves are featurng a fractonal-slot wndng. Analysng the electrc loadng harmoncs leads to the concluson that case 2 s advantageous concernng torque producton. To judge between the two alternatves of case 2 requres a detaled analyss of the respectve electromagnetc crcuts (e.g. usng FEM-calculatons). Such an analyss shows that n general the rotor wth 20 poles gves the mamum torque. REFERENCES [1] H.-D. Kolletschke: De Modulare Dauermagnetmaschne Aufbau und Egenschaften, Ph.D. dssertaton, Unversty of Federal Defense, Munch, Germany, 1987 (n German). [2] J. Fredrch: Bauformen und Betrebsverhalten Modularer Dauermagnetmaschnen, Ph.D. dssertaton, Unversty of Federal Defense, Munch, Germany, 1991 (n German). [3] H. Hofmann: Darstellung des Betrebsverhaltens drehzahlvarabler Dauermagnetmaschnen mt dem Kurzschlussstrom als Hauptparameter, Ph.D. dssertaton, Unversty of Federal Defense, Munch, Germany, 2005 (n German). [4] H. Polnder, M.J. Hoejmakers, M. Scuotto: Eddy-Current Losses n the Sold Back-Iron of PM Machnes for dfferent Concentrated Fractonal Ptch Wndngs, IEEE Internatonal Electrc Machnes and Drves Conference (IEMDC), May 3-5, 2007, Antalya, Turkey. [5] S. Abe, M. Murata: Development of IMA Motor for 2006 Cvc Hybrd, SAE Techncal Paper Seres [6] G. Koehler, A. Walther: Fourersche Analyse von Funktonen mt Sprüngen, Ecken und ähnlchen Besonderheten, Archv für Elektrotechnk, XXV. Band, 1931 (n German).