CALCULATION IN THE FIELD OF SEGMENTAL ROTOR MACHINES TAKING INTO ACCOUNT WINDING HARMONICS AND ROTOR AIRGAP IRREGULARITIES

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Cora Ferguson
5 years ago
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1 CLCULTION IN THE FIELD OF SEGENTL OTO CHINES TKING INTO CCOUNT WINDING HONICS ND OTO IGP IEGULITIES Y STCT The stator mmf over a segmet of the segmetal rotor reluctace machie is treated as a ifiite array of geerators feedig a commo busbar, ad the magetic potetial of the rotor segmet is obtaied as the potetial of the equivalet busbar. The rotor potetial for ay airgap profile is readily obtaied ad it is show how the method may be exteded to axially lamiated machies ad those of the flux barrier type. This approach is derived from cosiderig the flux due to a sigle stator coductor carryig curret.. INTODUCTION feature of the aalysis of segmetal rotor reluctace machies is the ecessity to determie the magetic potetial as zero, because the rotor, ulike that of the segmetal machie is oe itegral piece. The magetic potetial maifests itself i the reversal of flux i the air gap. I referece [] the quadrature axis potetial was evaluated by usig the fact that flux caot accumulate o the pole, ad otig that at the poit of reversal, the rotor potetial must equal the stator applied magetic potetial. The method was further exteded i referece [] to iclude a rotor with chaels cut over the cetral part of each segmet. The rotor potetial is calculated i this paper by cosiderig the stator mmf as a ifiite array of parallel geerators feedig a commo busbar (the rotor segmet) ad the iteral impedace of each geerator represets the reluctace of the airgap at the poit of actio of each of these geerators. The parallel geerator approach adopted here leads to the same basic expressio for rotor potetial as i ref. [] ad [3]. The determiatio of the rotor potetial is however exteded to iclude ay rotor air gap cofiguratio ad shows how harmoicas of stator mmf are take ito accout.. FLUX DUE TO SINGLE STTO CONDUCTO I fig. (a) is show a rolled-out segmetal-rotor machie, ad also show are flux paths of a sigle stator coductor which lies over oe of the rotor *L.. GU segmets. ost of the flux will follow the path show i the diagram. No serious error will result if all the flux is regarded as cofied to that path. This meas that all frigig flux usig paths like ad flux betwee adjacet segmet, i.e. usig paths such as CC ad DD are igored. The flux desity distributio due to curret i the sigle stator coductor will have the form show i fig. (b). If X ad Y represet the magetic reluctaces of the airgap betwee the sectio of segmet of width X ad the complemetary sectio of width Y respectively, the the amplitude ad of the rectagular wave of flux desity distributio are give by *r. L.. gu is a eader i Departmet of Electrical ad Electroic Egieerig of the Uiversity of Nigeria, Nsukka i i ( X Y )x ( X Y ) y iy gβ ix g 3. FLUX DUE TO SYETICLLY WOUND STTO WITH FULL PITCH COILS Cosider the same segmetal rotor machie which has a sigle stator slot per pole i a symmetrical arragemet. Let each slot machie cotai a coductor carryig curret i amps with directios as idicated i Fig. (a). The flux desity distributio over two pole pitches will be as show i Fig. (b). The values of ad are the same as obtaied for Fig.(b). If the airgap flux desity was calculated usig the product of mmf distributio. Fig. 3(c) ad the permeace distributio Fig. 3(d) of the air gap b betwee rotor segmet ad stator, the result will be as show i Fig. 3(a). The height of the graph of Fig. 3(a) is /g. This graph clearly differs from that of Fig. (b). I fact the algebraic differece betwee the two graphs is the graph of Fig. 4. The height of the graph i y x of Fig. 4 is g It represets a flux desity distributio actig i oppositio to the flux desity distributio of Fig. 3(a). It is as if the rotor segmets acquire potetials ad become sources of mmf directig flux ito the stator.

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3 NIJOTECH GU 5 Therefore, if the mmf method used i ormal machie aalysis is to apply to segmetal rotor machies, the effect of this rotor potetial must be take ito accout. The et flux desity will be the differece betwee what it would have bee if the rotor potetial were zero mius the flux desity produced by the differece betwee the potetial of the rotor ad that of the stator (assumed zero). 4. CLCULTION OF OTO SEGENT POTENTIL The polarity of the rotor potetial alterates betwee adjacet segmets. It follows that the zero potetial will coicide with the dotted lies as show i Fig.5. Let the effective reluctace of the flux paths betwee segmet ad the zero potetial be o, the Parallel- Geerator Theorem ca be used to determie the rotor potetial. The mmf actig at the airgap over the rotor i Fig. 5 is i, this acts dowwards over the sectio x of reluctace X ad dowwards over the sectio y of reluctace y. The potetial of the rotor is give by the equatio: I I Or X y I X y X y o The expressio is to F ( ) ( ) d o I X y equivalet Where F ( ) is the mmf distributio ad ( ) the permeace distributio expressed as Fourier series. The expressio X Y, the total permeace of the airgap over the segmet is also give by ( ) d T F ( ) T ( ) F( ) O ( ) d ( ) O d d 4.. FLUX DENSITY OF NUE OF COILS The flux desity distributio due to ay other group of symmetrical coductors ca be similarly obtaied. Provided there is o saturatio i the magetic circuit, the total flux desity distributio will be the sum of the separate flux desity distributios. The flux desity distributio of a represetative coil ca be expressed as F ( ) x f ( ) ( ) Where F ( ) x i the mmf distributio of the coil, is the rotor potetial due to that coil ad f ( ) is a fuctio i Fourier Series, that takes ito accout the area occupied by the rotor segmets ad also their polarity. The flux desity distributio due to all coils of the stator will be F ( ) x ( ) ( ) f ( ) F ( ) x ( ) ( ) f ( ) Where F( ) i the resultat mmf due to all coils expressed as a Fourier series F ( ) T O Which is the same as F( ) T O ( ) d ( ) d Where T is the total permeace

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5 NIJOTECH GU 7 fuctios whe the outer sphere is removed, () ad f ( ) are respectively the permeace ( ) d ad potetial fuctios whe the outer sleeve is removed ad the cetre chael that is left is regarded as havig ifiite reluctace, T is the total The total flux desity distributio may reluctace of the airgap betwee the outer sleeve therefore be writte ad the stator ad T is the total reluctace of the airgap betwee the ier sleeve ad the F( ) ( ) d stator. The calculatio of the flux desity F( ) f ( ) distributio i a machie with a axially lamiated rotor will follow much the same lie of T reasoig, oly that more sleeves are to be O cosidered [4]. The flux-barrier machie is oly a special case of the multilevel rotor machie 5. OTO WITH CENTL CHNNEL [5]. Usig the methods outlied above, the rotor potetial ad the flux desity distributio were calculated for a rotor with a cetral chael. The results are exactly as give i ref. []. 6. INTELEVED OTO CHINE This method ca be applied very readily to the aalysis of a iterleaved segmetal [3] rotor machie (Fig. 6). Let ad be the potetials of the outer ad ier rotor sleeves respectively. These ca be computed from the followig expressios F( ) ( ) d ( T T F( ) ( ) d F( ) ( ) d The flux desity i the stator airgap will be give by [ F( ) ) ( F( ) f ( )( )] f ( )] ( ) where ( ) ad ) ( ) / ( ) f are respectively the permeace ad potetial fuctios whe the ier sphere is removed, S(θ) ad f (θ) are respectively the performace ad potetial 7. CONCLUSION The method described above yields the same result for rotor potetial ad is equivalet to the other method i which the rotor potetial is calculated by determiig the poit of flux reversal. However it has the importat advatage that the sigle expressio that results, gives the flux desity distributio as a fuctio of the positio of the rotor relative to the stator mmf axis. This is importat whe cosiderig the field of reluctace frequecy chagers where it is desirable that the output emf be kow as a cotiuous fuctio of the rotor displacemet. Effects of stator harmoics ad their relatioship with harmoics of rotor permeace distributio for productio of sychroous torque at differet field sychroous speeds are easily established. This is useful for dealig with multispeed [6] ad charge pole [7] machies. EFEENCES [] LWENSON, P.J., GU, L.., Theory ad Performace of Polyphase reluctace machies. Proc.I.E.E. Vol.3, No.8, 964 pp [] LWENSON, P.J., GUPT, S.K., Developmets i the performace ad theory of

6 NIJOTECH GU 8 segmetal-rotor reluctace motors, Proc. I.E.E. Vol.4, No.5 967, pp [3] KISKO, J.K., Polyphase reactio Sychroous motor. Joural, merica Ist. Elect. Egieers 93 4 pp [4] CUICKSHNK,.J.O., ENZIES,.W., ad NDESON,.F., xially lamiated aisotropic rotors for reluctace motors. Proc. I.E.E. Vol.3, No. 966, pp [5] W.FONG ad HTSUI, J.S.C., New type of reluctace motor. Proc. I.E.E. Vol.7, No.3, 970 pp [6] LWENSON, P.J., GUPT, S.K., UTHY- VJU, S.. ultispeed performace of segmetal-rotor reluctace machies. Proc. I.E.E. Vol. 5 (5), pp [7] FONG, W., Chage-Speed reluctace motors. Proc. I.E.E. Vol.4, No.6, 967, pp

PROBABILITY AMPLITUDE AND INTERFERENCE

PROBABILITY AMPLITUDE AND INTERFERENCE PROILITY MPLITUDE ND INTERFERENCE I. Probability amplitude Suppose that particle is placed i the ifiite square well potetial. Let the state of the particle be give by ϕ ad let the system s eergy eigestates