Investigation of Force Generation in a Permanent Magnet Synchronous Machine

Transcription

1 Investigation of Foce Geneation in a Pemanent Magnet Synchonous Machine W. Zhu, Student Membe, IEEE, S. Pekaek, Membe, IEEE, B. Fahimi, Senio Membe, IEEE and B. Deken, Student Membe, IEEE Abstact Taditional analysis of pemanent magnet synchonous machines has focused upon establishing a elationship between the quadatue (q and diect (d axis stato cuent (o voltage and the electomagnetic foce ceated to establish otation (toque. In this pape, an altenative analysis of electomagnetic foce poduction is consideed. Specifically, the influences of q- and d-axis stato cuent on both the adial and tangential components of the aigap flux densities ae fist evaluated. Using a Maxwell Stess Tenso appoach, the fields ae then used to evaluate both the adial and tangential component of foce density ceated in the aigap of the machine. Fom this pespective seveal inteesting obsevations ae made. Fist, it is shown that the d-axis cuent has zeo influence on the aveage tangential foce (toque, as pedicted using taditional analysis, but it has a significant influence on the aveage adial component of foce. Second, it is shown that the q-axis cuent contibutes to both the aveage adial and aveage tangential components of foce. Inteestingly, it is also shown that unde standad opeating conditions, the aveage adial foce fa exceeds that of the aveage tangential component of foce. Theefoe, one can conclude that the magnetic fields established ceate a significant component of foce in a diection that cannot poduce toque. Index Tems Pemanent Magnet Synchonous Moto, Maxwell Stess Tenso, Foce Density, Toque Geneation. I INTRODUCTION Ove the past half-centuy, advances in the design of electic machiney have been elatively modest when compaed to quantum leaps made in the design of contolled switches (semiconductos. Although advances in electic machiney have been elatively modest, the contol of machines has changed damatically due to inceased computing powe and the powe handling capability of high powe semiconducto devices. Fo each class of electic machiney, contols now exist to achieve a vaiety of desied esponses (high-bandwidth cuent/toque tansduction, minimal toque ipple fo a desied aveage toque, maximum toque/amp, maximum efficiency, etc. New mateials, manufactuing techniques, compute technologies, and semiconducto devices all pomise to ensue futhe pogess is made along existing dive system design guidelines. The motivation fo this eseach has stated with a question being asked by seveal eseaches [,] ae thee paths fo new machine design/excitation stategies that could lead to significant changes in how electical/mechanical enegy convesion is achieved? To begin to addess this question, an analysis of an existing suface-mount PM machine is pefomed. Howeve, in contast to a taditional macoscopic appoach in which the electomagnetic toque is the focus of the enegy convesion pocess, in this eseach an altenative view of foce geneation in the PM machine is consideed. Specifically, the adial and tangential components of magnetic flux density in the aigap ae evaluated and used to calculate the electomagnetic foces (adial and tangential components. This so-called micoscopic view of the foce geneation is used to view foce poduction fom a diffeent pespective and helps to futhe undestand how magnetic fields lead to foce poduction. Based upon this analysis, seveal obsevations have been made. Fist, it is shown that the d-axis cuent has zeo influence on the aveage tangential foce (toque, as pedicted using taditional analysis (in the absence of satuation. Howeve, it has a significant influence on the aveage adial component of foce. Specifically, it is shown that the elationship between d- axis cuent and aveage adial foce is a quadatic function. Second, it is shown that the elationship between q-axis cuent and aveage tangential foce is linea (as pedicted using taditional analysis, while the elationship between the q-axis cuent and aveage adial foce is also a quadatic function although a diffeent function than fo the d-axis. Inteestingly, it is also shown that unde standad opeating conditions, the adial foce fa exceeds that of the tangential component of foce. Theefoe, one can conclude that the magnetic fields established ceate a significant component of foce in a diection that does not poduce motion. In ou opinion, the main contibutions of this eseach ae two-fold. Fist, the analysis povides the community with a moe complete undestanding of how typical excitation adjusts the fields in the aigap and influences the oveall foce pofile. Second, the esults point to a geneal diection fo futhe eseach in the aea. Specifically, the fact that the majoity of foce poduced does not lead to motion aises a question as to whethe altenative geometies/excitation schemes can be developed to yield a moe poductive foce pofile fo electic machines. Pio to poceeding with the analysis, it is impotant to highlight the woks of eseaches whose contibutions ae closely elated. Specifically, seveal eseaches have used analytical techniques to aive at the magnetic field distibution in the aigap of electic machines [-]. Among the ealy effots was Hague [], who povided analytical solutions fo the magnetic fields in the aigap of a machine due to cuent imbedded in ion o located inside the aigap of a machine.

2 Moe ecently, analytical solutions fo the fields in the aigap of PM machines have been povided in [4-]. Although these woks ae elated, diffeences exist in the methods used to account fo stato slots, the co-odinate system applied (i.e. pola/ectangula, and the solution method. Although vey useful fo undestanding field behavio and in design, none have exploed the link between q- and d-axis cuent excitation and the tangential component of the field ceated by the stato windings. Most ecently, [] has used analytical techniques to exploe the foce pofile unde q- and d- excitation. The pimay goal of thei effot was to conside the effect of using d-axis field weakening on foce ipple. Two key assumptions in thei analysis was that the stato windings poduce a pue sinusoidally distibuted field in the aigap, and that this field contains a negligible tangential component - i.e. only a adial flux density exists fom stato to oto. Heein, to be moe geneal, the fields ceated by the stato windings ae not assumed to be a pue sinusoid and in fact though FEA ae confimed to contain hamonics. Moeove, it is shown that in geneal one cannot assume that the stato windings poduce a unidiectional field. In fact the vecto natue of the fields ceated by the stato windings is citical to the poduction of both the tangential and adial components of foce. C' B A A' N B' C S S φm C B' Figue. A coss-sectional view of the PM machine and coodinate system. N A' A B C' II MACHINE MODEL φsm q-axis θm as-axis The PM machine studied in this eseach is shown in Fig.. It is a Hp, pm, -phase, 4-pole, -slot, suface-mounted pemanent magnet machine that is used in home appliance applications. In Fig., the phase-a winding magnetic axis and the q-axis of the oto ae shown, espectively. The mechanical oto position ( m is defined as the angle between the two axes. Fo the pupose of analysis that is used in Section V, the angle sm is defined as the position on the stato elative to the midpoint of the phase-a stato slot shown and the angle m is defined as the position on the oto elative to the midpoint of the pemanent magnet shown. Although not shown on the diagam, the sepaation between angles m and sm is the same as the angle between the q- and as-axis ( m. Heein, so-called electical angles s,, and ae defined by multiplying the mechanical angles by the numbe of pole pais. The elationship = ( s exists between electical angles: The following assumptions have been made fo the analysis povided: The stato teeth and pemanent magnets ae igid; no defomation due to adial and tangential foce is expeienced by these components. The stato windings ae concentated and ae wound at a full-pitch. The pemanent magnets ae paallel-magnetized. The pemanent magnets ae not demagnetized by the flux intoduced by the phase cuents. The flux density in the z-axis is assumed zeo (no end effects. Hysteesis and Eddy cuents ae neglected. III MACROSCOPIC VIEW OF FORCE PRODUCTION Pio to investigating foce poduction unde a so-called micoscopic view, it is useful to conside the analysis of the machine fom a moe taditional pespective. In taditional lumped-paamete-based machine analysis, a closed-fom expession fo the electomagnetic toque is established using an enegy balance appoach. This yields the expession [] P Wc Te = ( Whee P is the numbe of magnetic poles and W c is the coenegy of the coupling field that can be epesented in a fom Wc = iasaspm ibsbspm ics cspm Wcpm ( ( In (, W ( epesents enegy that is due to the pemanent cpm magnets being attacted to stato ion and the tems asm, bsm, epesent the influence of the flux of the pemanent csm magnets on the espective stato windings. Fo initial analysis, the flux linkages ae assumed of the fom: asm = mag sin ( (4 bsm = mag sin ( (5 csm = mag sin ( (6 Substituting ( into ( yields

3 P aspm bspm cspm Wcpm ( Te = ias ibs ics (7 Wcpm ( whee epesents the toque due to the pemanent magnets being attacted to the stato (cogging toque. When the equations of the PM machine ae expessed in tems of physical (abc vaiables, the stato flux linkages ae functions of oto position (due to (4-(6 and thus the lumped-paamete model is time-vaying. To eliminate time-vaying components a tansfomation of vaiables is applied in which the dynamics of the machine ae epesented in the oto fame of efeence []. The esulting dynamic model of the machine expessed in the oto efeence fame is of the fom vqs = i s qs ds pqs (8 vds = i s ds qs pds (9 qs = Li ss qs ( ds = Li ss ds m ( P Te = miqs Tcog ( whee s epesents the stato esistance, the oto angula velocity, vqs and vds the tansfomed stato voltages, iqs and ids the tansfomed stato cuents, qs and ds the tansfomed stato flux linkages, and Tcog the cogging toque. Equations (8-( epesent the standad dynamic equations that ae used to analyze machine pefomance. Thei deivation is descibed in texts on PM machines and the esponse pedicted using the model is known to be accuate, povided that the paametes ae accuate. One item of inteest is that fom ( it can be obseved that the electomechanical toque geneated by the machine is only associated with q-axis cuent, the d-axis cuent has no contibution to aveage toque. Based upon this obsevation, the d-axis cuent is typically set to zeo o a minimum value to achieve a maximum toque pe ampee atio and minimum coppe loss. Thee ae instances whee the d-axis cuent is intoduced to have the effect of weakening the field ceated by the magnet to facilitate opeating at highe machine speeds. Although the standad equations offe the basic pinciple to design the contol of PMSMs, it is obseved that the infomation povided is limited fom the sense that thee is little infomation on the oveall foce poduction. Specifically, it is known that within the aigap thee ae both adial and tangential components of foce. To date, thee has been vey little focus on how the q- and d-axis components of cuent contibute to adial components of foce, with the exception of [], wheein the focus was on foce ipple and assumed that the fields ceated by the stato winding ae unidiectional (i.e. adial fom stato to oto. Thus an altenative micoscopic investigation on toque and foce chaacteistics is descibed in the following section to addess both of these issues. IV MICROSCOPIC VIEW OF FORCE PRODUCTION An altenative to deiving the electomagnetic toque using an enegy balance appoach is to deive the components of foce fom the magnetic field using a Maxwell Stess Tenso method [4]. Specifically, within the aigap of the machine the local adial and tangential components of foce density can be expessed [4] ft = BBt ( µ ( f = B B (4 µ t whee ft is the tangential component of foce density (N/m^, is the adial component of foce density (N/m^, B is the f adial component of the magnetic flux density and B t is the tangential component of magnetic flux density. µ is the pemeability of fee space. Equations ( and (4 povide the basis fo a micoscopic investigation of the foce poduction within the machine. As a fist step in viewing the foce unde the light of (- (4 it is useful to conside the magnetic flux densities in the aigap of the machine subject to altenative stato excitation. To facilitate this study, the machine was modeled using a finite element appoach wheein the adial and tangential components of flux density wee evaluated using a FEA model ceated using the commecial package Maxwell [5]. A mesh with 94 tiangles was used in the calculations. To minimize eo, the contou of integation was established in the middle of the aigap [6]. Hence the distibution of the flux density (and foce density components was calculated in the middle of the aigap. A convegence test was conducted to ensue accuacy. Specifically, the numbe of tiangles was inceased to oughly and the diffeence between the foces calculated was within.5%. The tangential and adial flux density distibution along the aigap fo a machine in which the stato is de-enegized and the oto is located at = is shown in Fig.. In Fig., the hoizontal axis epesents the position of an obseve inside the aigap of the machine. Fom the wavefoms it can be seen that the pemanent magnet ceates a significant adial component of flux density (B pm. Thee ae changes in the adial component aound the egions diectly below stato slots. In contast, the tangential distibution of the flux density (B tpm has elatively

4 small amplitude and only occus at the location immediately aound the stato slots. The tangential flux density is attibuted to flux fom the magnet taveling to the ion wall of the slots. Though visual inspection, it can be seen that multiplication of B and Bt will yield a tangential component of foce density that has equal positive and negative components. Theefoe, integation of the tangential component of foce density will yield a value of (confimed numeically, meaning zeo aveage toque (as expected. In contast, at zeo stato cuent, the adial component of foce density will be nonzeo and is dominated by the adial component of the flux density. Flux density [T] angula position in aigap (φsm Figue. Tangential and adial flux density in aigap geneated by PMs. As a second study, the oveall tangential and adial flux density distibution fo a case in which cuent was applied in a manne i ds = and i qs = 4.6 A (ated, and the pemanent magnets ae maintained in the oto, is shown in Fig.. It is noted that the figue contains the distibution at a single oto position ( =. Simila cuves ae obtained at all positions of the oto. Flux density [T] B Bt B Bt angula position in aigap (φsm Figue. Radial and tangential flux densities along the aigap when i ds = and i qs = 4.6 A. Compaing the esults shown in Fig. with the flux densities of the de-enegized machine (Fig., it can be seen that the q-axis stato cuent influences both the adial and tangential components of flux density. As one might expect, this influence is most clea in the vicinity of stato slots. In contast to the de-enegized machine, in each pole-pitch the tangential component of flux density is positive ove the egion that the adial component is positive. It is negative ove the egion that the adial component of flux density is negative. Theefoe, the esult is that the tangential component of foce has a positive aveage value (i.e. aveage toque is poduced. It is also inteesting to conside that between stato slots, the adial components of flux density ae lage than the values obseved in the de-enegized machine. This leads to an incease in the aveage adial component of foce. The foce densities computed at a single oto position fo ds qs i = and i = 4.6 A ae shown in Fig. 4. Fo claity, the negative value of the tangential component of foce is plotted. Fom these plots it can be obseved that the foce density distibutions along the aigap ae fa fom unifom. This is due to the existence of slots and a spatial cuent distibution that is discontinuous. Consideing the two wavefoms of foce density, it is inteesting to note that the peak values of both the adial and tangential foce densities ae oughly the same. Howeve, the tangential component of foce is ceated in the egions nea the stato slots. In contast, the adial components ae distibuted ove an entie pole-pitch. Theefoe, upon integation it is clea that the adial component of foce will be geate than the tangential component. It is also inteesting to note that a elatively small pecentage of the aigap space actively paticipates in the toque geneation pocess at a given oto position. It would appea that thee is an oppotunity to impove toque density of a PMSM though design modifications (geometies, winding configuations, excitation that yield a nonzeo tangential foce density ove a wide egion simila to the adial foce density. Methods to seach fo such altenatives using numeical techniques ae being pusued in ongoing eseach. Foce density [N/(m*m] x 5 f -ft angula position in aigap (φsm Figue 4. Tangential and adial foce density distibution along the aigap at a single oto position ( =. 4

5 A plot of the aveage component of adial and tangential components of foce/length of the machine (N/m as a function of q-axis cuent is shown in Fig. 5. Compaing values of adial and tangential components of foce, it is clea that that adial component is much geate than the tangential component. It is also obseved that the tangential component of foce is linealy elated to q-axis cuent (as pedicted fom the maco-scopic model. Howeve, it can be seen that an incease in q-axis cuent also leads to an incease in the adial component of foce. As will be shown in Section V, the elationship between adial foce and q-axis cuent is a quadatic function. Aveage value of adial and tangential foce [N/m] tangential foce adial foce q-axis cuent [A] Figue 5. Aveage tangential and adial foce with inceasing q-axis cuent ( i ds = A. The flux density distibutions along the aigap fo a case in which the d-axis phase cuent is set to i ds = 4. A ( i qs = 4.6 A ae shown in Fig. 6. Compaing Fig. 6 to Fig., it can be obseved that the intoduction of the d-axis cuent leads to a change in both the adial and tangential components of flux density. If one compaes values of the adial component of flux density ove a pole-pitch, thee is a geneal incease in value. In contast, the tangential component does not see a geneal incease in magnitude ove a pole pitch. Rathe, as shown by the tangential components that ae encicled in the figue, thee ae places whee the tangential component of flux changes sign elative to thei value when i ds = A. Theefoe, in contast to the case in which i ds = A, the multiplication of B and Bt leads to egions in which the tangential foce density distibution is negative. The adial and tangential components of foce density ae shown in Fig. 7. Flux density [T] B Bt angula position in aigap (φsm Figue 6. Radial and tangential flux densities along the aigap when i ds = 4. A and ( i qs = 4.6 A. Foce density [N/(m*m] x angula position in aigap (φsm Figue 7. Tangential and adial foce density distibution along the aigap at a single oto position i ds = 4. A ( i qs = 4.6 A. Shown in Fig. 8 ae the flux density distibutions along the aigap fo a case in which the d-axis phase cuent is set to i = 4. A ( i = 4.6 A. The adial and tangential components ds qs of foce density ae shown in Fig. 9. Compaing the adial component of flux density ove a pole-pitch to those shown in Fig. and Fig. 6, thee is a geneal decease in value. In contast, the magnitude of the tangential component inceases slightly ove a pole pitch. Simila to the case in which d-axis cuent is positive, thee ae locations whee the tangential component of flux density changes sign elative to its value when i ds = A. Multiplication of B and Bt leads to egions in which the tangential foce density distibution is negative. Compaing adial foce density, it is seen that the decease in adial flux density leads to a geneal decease compaed to the cases in which i = A and i = 4.6 A. ds ds f -ft 5

6 Flux Density [T] Angula Position in the Ai Gap, [degees] sm Figue 8. Radial and tangential flux densities along the aigap when i ds = 4. A and ( i qs = 4.6 A. Foce Density [Nm ].5.5 x Angula Position in the Ai Gap, sm [degees] Figue 9. Tangential and adial foce density distibution along the aigap at a single oto position i ds = 4. A ( i qs = 4.6 A. Unde both positive and negative d-axis cuent, changes in adial and tangential flux densities ae such that the aveage value of the tangential component of foce is not elated to d-axis cuent (as pedicted by the macoscopic model. The aveage value of the adial and tangential components of the foce/length ae plotted as a function of d-axis cuent ( i qs = A in Fig.. It can be seen in Fig. is that the adial component of foce changes significantly with change in d-axis cuent. B B t f f t Aveage value of adial and tangential foce [N/m] 5 5 tangential foce adial foce d-axis cuent [A] Figue. Aveage tangential and adial foce unde diffeent d-axis cuent excitation (q-axis set to. V ANALYSIS OF OBSERVATIONS In ode to povide some insight into the esults obseved in the pevious section, the contibutions of the aigap flux densities is exploed analytically. Fo a PM machine, both B and B t ae ceated by two souces: the stato windings and the pemanent magnets. Since the PM machine usually has a elatively lage effective aigap, fo the puposes of analysis, satuation is neglected. Theefoe, using supeposition, B = Bpm Bs (5 Bt = Btpm Bts (6 whee B pm, B tpm, B s and B ts denote the adial and tangential flux density ceated by the PM and stato windings, espectively. Substituting the field flux densities (5-(6 into (-(4 the tangential and adial foce density can be expessed as: ft = Bts Btpm Bpm Bs µ (7 f [( ( ] = B pm B s B ts B µ tpm (8 Using (7 and (8, the tangential and adial foce can be deived fom the design and excitation. Specifically, the oveall tangential and adial foce/length at a single oto position can be expessed as: P Ft = ft( Rd (9 P F = f( Rd ( and the electomagnetic toque expessed as Te = FRL t ef ( whee R is the adius of the contou upon which the Maxwell Stess Tenso is calculated and Lef is the effective stack length (meaning the physical stack length multiplied by a stacking facto. Fom (7, f t can be divided into fou pats: 6

7 (i BpmBtpm (ii Bs Bts (iii BpmBts (iv BsBtpm Tems (i and (ii poduce cogging and eluctance toque due to the existence of slots. Howeve, both will yield zeo aveage foce. A zeo aveage value of tem (i can be seen by consideing Fig.. Specifically, integation of the BpmBtpm shown in Fig. ove a pole pitch is zeo. The contibution of (ii on aveage tangential foce is due to the fact that B s is an odd function and B ts is an even function, which will become clea late in the analytical development. Theefoe, a conclusion is that only tems (iii and (iv can geneate an aveage toque. Fo the analysis pesented heein, the focus is on aveage values of foce and theefoe using (iii and (iv one can expess the foce/length due to these two tems as P Ft ( iii iv = [ Bpm Bts Bs Btpm ] Rds µ ( To conside the tems in ( it is useful to expand each of the flux densities in tems of a seies. Specifically, consideing Fig., one can see that Bpm can be expessed as a Fouie seies of the fom B = B cos( k ( pm pmk k = Similaly, it can be seen that Btpm can be expessed as a Fouie seies of the fom B = B sin( k (4 tpm tpmk k = Pio to poceeding with the flux densities poduced by the stato windings, it is inteesting to note that although Btpm is due to inteaction of the magnets with stato slots and theefoe one might expect the seies to only include components that ae multiples of the numbe of slots, it has a fundamental peiod of 6 (electical degees, i.e. Btpm. This is due to the behavio of the flux aound the tansition egion between magnets and can be seen in Fig.. To evaluate the effect of the stato windings, it is convenient to fist conside that the net field ceated by cuent in the stato windings is the sum of the fields ceated by the cuent in each stato slot. Specifically, if B sk and B tsk epesent the adial and tangential flux density ceated by cuent in k th slot, then the net flux densities ae expessed L B = B = B B B (5 s sk as bs cs k= L B = B = B B B (6 ts tsk tas tbs tcs k = whee L is the numbe of stato slots, B as, B bs, B cs, ae the adial components of flux density ceated by the phase windings. Bt as, Bt bs, and B tcs ae the tangential components of flux density ceated by the phase windings. The tangential and adial flux density distibution along the aigap ceated by the cuent in a single slot (obtained using the FE model is shown in Fig.. If both the stato and oto ion ae unsatuated, the local flux densities ceated by cuent in each slot ae popotional to the cuent. Thus we can define: Btsk ( s = islotb ( s (7 B ( = i b ( (8 sk s slot s whee b and b ae associated with the geomety of the machine. Fom the numeical esults it can be seen that fo the machine studied, b is an even function and b is an odd function (with espect to sm, espectively. This is consistent with the analytical deivations made on simplified machines. Specifically in [] solutions of the Poisson equation in cylindical coodinates ae obtained fo a machine assuming stato cuents embedded in stato ion and a machine assuming stato cuents in the aigap with a smooth stato inne suface. In both cases, the esults indicate a solution with even and odd symmety, espectively. Bsk [T] Btsk [T] Figue. Radial and tangential components of flux density intoduced by cuent in a single stato slot. Based upon the esult of cuent in a single stato slot, one can expess the tangential components of flux density contibuted by the individual phase windings in a fom Bt _ as ( s = iasb( s (9 Bt_ bs( s = ibsb( s / ( Bt _ cs ( s = icsb( s / ( φsm φsm 7

8 whee B ( s is an even (zeo aveage peiodic function with espect to s. Specifically, s k s k= B ( = B cos( k ( Similaly, one can expess the adial components of flux density due to individual phase windings in a fom B _ as ( s = iasb ( s ( B _ bs ( s = ibs B( s / (4 B_ cs( s = icsb( s / (5 whee B ( s is an odd (zeo aveage peiodic function with espect to s, i.e. s k s k = B ( = B sin( k (6 Assuming phase cuent excitation of the fom: ias = is cos( i (7 ibs = is cos( i (8 ics = is cos( i (9 and substituting (7-(9, into (9-( and (-(5 and the net esult into (, one obtains the expession P Ft ( iiiiv = is cos( i B( s Bpm ( s µ B ( s Btpm ( s Rds P / is cos i B s Bpm( s µ / B s Btpm( s Rds P / is cos i B s µ / pm ( s B B s Btpm ( s Rds (4 Substituting (, (4, (, and (6 into (4, applying tigonometic identities, and using the elationship that i = i cos( (4 qs s i ds s i i = i sin( (4 one can obtain a geneal fom fo the tangential foce/length due to BpmBts and BtpmBs as P Ft ( iiiiv = Bpmk Bk Btpmk Bk iqsr Ft ( 8µ k k (4 whee Ft( epesents a component of tangential foce that is a function of oto position (i.e foce ipple. Heein the focus is on aveage tangential foce which is epesented by the fist tem on the ight hand side of the equal sign in (4. Consideing the expession, it is clea that the aveage tangential foce is only a function of the q-axis stato cuent. The deivation of adial foce follows a simila set of steps, although the inteaction of all components is a facto in the aveage adial foce. Plugging (5 and (6 into (8 and the esult into ( yields P F = Bpm( s B_ is 4µ i= a, b, c (44 Btpm ( s Bt _ is Rds i= a, b, c Substituting the Fouie seies epesentation of each of the components into (44 and using tigonomic identities to simplify, one obtains a esult fo the adial foce: P! 9 F = " ( Bk Bk ( iqs ids 4µ!# 4 kk,,9, etc $ ( Bpmk Btpmk ( BBpm B Btpm ids % R F ( k & (45 whee F( epesents a component of adial foce that is a function of oto position (foce ipple. Foce ipple is evaluated in [] fo the special case in which the stato windings yield only a pue sinusoidal adial flux density. A value of aveage nomal foce is also povided in [] based upon the sinusoidal-adial appoximation. Equation (45 will simplify to the esult povided in [] if the same assumptions ae applied although based upon the findings heein, these cannot be made in geneal. Heein the focus is on the aveage foce which is epesented by the fist thee tems on the ight hand side of the equal sign in (45. Fom (45 it is clea that the aveage adial foce is a quadatic function of both q- and d-axis cuents. Inteestingly, the q- and d-axis expessions take slightly diffeent foms. Specifically the thid tem on the ight hand side of the equal sign only includes i ds. This accounts fo the moe significant effect that the d-axis cuent has on the adial component of foce (seen in Fig. 8. Moeove, this same tem also shows how applying a negative d-axis cuent educes the adial component of foce. In contast, application of a q- axis cuent can only act to incease the adial component of foce. Fom (45 one can also obseve how the diffeent components of the flux densities play a ole in the poduction of aveage adial foce. As noted in Section II, the effects of satuation, eddycuents, and hysteesis have been neglected in the analysis of foce poduction. In satuation, the analysis stating with (5 is invalid, since one cannot assume lineaity. Initial eseach indicates that in satuation the odd/even symmety natue of the adial and tangential components of flux density emain the same. Howeve, since the flux density components ae nonlinea 8

9 functions of stato cuent, the tangential foce cannot be expessed as a linea function of q-axis cuent. The effects of hysteesis and eddy cuents on the flux and foce densities may also be of inteest. Howeve, since machine models that include hysteesis and eddy cuent effects ae aely necessay to pedict the elationship between toque and stato cuent, these effects may not be significant (except possibly in high speed applications. VI CONCLUSION The magnetic flux density and foce density distibution along the aigap of a PM machine have been investigated analytically by leveaging a field solution obtained fom finite element analysis. The effect of both q- and d-axis phase cuent on the adial and tangential field and foce distibution has been analyzed. Based upon this analysis, seveal obsevations have been made. Fist, it is obseved that unde standad opeation the majoity of the foce poduced in the machine is in the adial diection, which cannot tanslate into oto otation. Second, it has been found that the q- and d-axis components of cuent both contibute to the aveage adial foce and that thei elationship can be expessed as a quadatic function. It is also veified that the aveage component of tangential foce (toque is linealy elated to q-axis stato cuent, while the d-axis cuent has no influence of aveage toque. Finally, though the analytical pocess an evaluation of how stato excitation influences flux densities to poduce toque and adial foce has been povided which helps to povide a so-called micoscopic view of foce poduction in the machine. Pemanent-Magnet Machines, IEEE Tansactions on Magnetics, Vol. 8, No., pp. 9-8, Jan.. [] Wang X., Li Q., Wang S., Li Q., Analytical Calculation of Ai-Gap Magnetic Field Distibution and Instantaneous Chaacteistics of Bushless DC Motos, IEEE Tansactions on Enegy Convesion, Vol. 8, No., pp. 44-4, Sept.. [] Jiao G. and Rahn C., Field Weakening fo Radial Foce Reduction in Bushless Pemanent Magnet DC Motos, IEEE Tansactions on Magnetics, Vol. 4, No. 5, Sept. 4. [] P. C. Kause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electic Machiney, IEEE Pess, 995. [4] Belahcen, A., Oveview of the Calculation Methods fo Foces in Magnetized Ion Coes of Electical Machines, Semina on Modeling and Simulation of Multi-technological Machine Systems, 9 Novembe 999, Espoo, Finland, pp [5] Maxewll D Field Simulato Manuals, Ansoft Copoation,. [6] L. Chang, A.R. Eastham, G.E. Dawson, Pemanent Magnet Synchonous Motos: Finite Element Toque Calculations, Industy Applications Society Annual Meeting 989, Vol., Octobe -5, 989, pp VII REFERENCES [] Goals of the Gainge Cente fo Electic Machiney and Electomechanics, UIUC, [] NASA Low Emissions Altenative Powe Poject CLEVELAND, OH [] B. Hague, The Pinciples of Electomagnetism Applied to Electical Machines, New Yok : Dove Publications, 96. [4] N. Boules, Two-Dimensional Field Analysis of Cylindical Machines with Pemanent Magnet Excitation, IEEE Tansactions on Industial Applications, Vol. IA-, pp , 984. [5] Q. Gu an d H. Gao, Aigap Field fo PM Electical Machines, Electical Machines and Powe Systems, Vol., pp , 985. [6] Boules, N., Pediction of No-Load Flux Density Distibution in PM Machines, IEEE Tansactions on Industial Applications, Vol. IA-, pp. 6-64, 985. [7] Q. Gu an d H. Gao, Effect of Slotting in PM Electical Machines, Electical Machines and Powe Systems, Vol., pp. 7-84, 985. [8] Zhu Z., Howe D., Bolte E., Ackeman B., Instantaneous Magnetic Field Distibution in Bushless Pemanent Magnet dc Motos, Pat : Open-Cicuit Field, IEEE Tansactions on Magnetics, Vol. 9, No., pp. 4-5 Jan.99. [9] Zhu Z., Howe D., Bolte E., Ackeman B., Instantaneous Magnetic Field Distibution in Bushless Pemanent Magnet dc Motos, Pat II: Amatue Reaction Field, IEEE Tansactions on Magnetics, Vol. 9, No., pp. 6-4, Jan.99. [] Zhu Z., Howe D., Chan C. Impoved Analytical Model fo Pedicting the Magnetic Field Distibution in Bushless 9