Magnetic Flux Density Measurement in Permanent Magnet Synchronous Machines

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1 Magnetic Flux Density Measurement in Permanent Magnet Synchronous Machines A Saadoun, Y Amirat o cite is version: A Saadoun, Y Amirat. Magnetic Flux Density Measurement in Permanent Magnet Synchronous Machines. 5 international conference on Sciences and echniques of Automatic control & computer engineering - SA 204, Dec 204, Hammamet, unisia. <hal > HAL Id: hal Submitted on 4 Mar 205 HAL is a multi-disciplinary open access archive for e deposit and dissemination of scientific research documents, wheer ey are published or not. he documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Magnetic Flux Density Measurement in Permanent Magnet Synchronous Machines A. Saadoun and Y. Amirat, Member, IEEE Abstract his paper describes two measurement techniques of e air gap flux density in a small permanent magnet synchronous machine. he first meod is based of e harmonic analysis of e induced voltage in a search coil placed on e inner periphery of e stator, while e second relies on e analysis of e induced voltage in e stator windings. Accurate results can be achieved since e measurement procedure relies only on well defined electrical quantities regardless of e dimensions of e machine. A furer appreciation of e accuracy of e proposed measurement procedures can be confirmed by e reconstruction of e stator induced voltages. Alough each meod has its own merit in terms of field sensing, ey are basically identical; noneeless ey enable a cross comparison of e measured results. Keywords Poisson s equation for magnetic field problems; finite element modeling; induced emf I. INRODUCİON he computation of e air gap flux density in electric machines is certainly e most important aspect of e design procedure. Despite e long experience accumulated by several generations of electric machines designers, e task is still a state of art []. It has become even more challenging as far as permanent machines are concerned because e air gap field distribution is difficult to predict by e classic meods [2]. o overcome e problem, finite element software packages are presently used to solve directly e governing magnetic field equation. However, e final assessment of e finite element solution can be only made on e basis of e experimental results. he present work has been carried out, initially, in e prospect to validate e finite element model of e machine, but during e course of e experiments, not only great discrepancies between e computed and measured results have been found but also unexpected field harmonics caused by mechanical asymmetry between two adjacent pair pole pitches have been observed. Since in permanent magnet machines, e air gap is made comparatively shorter in order to optimise e dimensions of e magnet, it has turned out at e calculation of e air gap flux density is very sensitive to e dimensions of e machine, particularly on e location of e mean air gap line [3,4]. o alleviate e impact of e machine dimensions on e calculation of e radial component of e air gap field, two meods have been used to check e consistency of e measurements. he first meod relies on e harmonic analysis of e induced emf in a search coil whereas e second is based on e analysis of e induced emf in e stator windings. However, e calculation of e air gap flux density harmonics still depends on e dimension of e machine; erefore anoer formulation is needed to get rid off ese parameters from e measurement procedures. II. FEM MODELLING OF HE PMSM he Poisson s equation governing e magnetic field problem A A ( ν y ) + ( νx ) = JΣ x x y y has been solved over a domain solution restricted to one pole pitch, using e freeware FEM package FEMM v4. he two dimensional finite element model of e permanent machine is shown in figure. Figure 2 represents e magnetic air gap Flux Density Distribution (FDD) along e mid-air gap line. Figure : Flux line contours from e FEMM package fem solution over a pole pitch. ()

3 fundamental of e air gap field. Furermore, e average flux per pole is 2 π Φ = τlb= τlb (6) Substituting (4) and (5) in (3) gives E π = Φ f (7) 2 c Figure 2: Computed magnetic flux density distribution along e mid-air line. III. MEASUREMEN PROCEDURE he key point of e measurement procedure is to derive a relationship between e induced voltage and e radial component of e flux density distribution in which e dimensions of e machine are not expressed explicitly. he induced voltage wiin a particular conductor of e winding is derived from e flux-cutting rule as From e last relation, it can be seen at e flux per pole is no longer affected by e main dimensions of e machine. It can be found easily and accurately from a single measurement of e induced voltage if e rotational speed of e machine is kept constant. he same relation holds for e high order harmonic expect for e change in e signal frequency E π = Φ k 2 ck 2 f = τ k lb mk f (8) e () t B( θ ) lv c = (2) where B( θ ) represents e air gap FDD along e mean air gap line. Provided at e peripheral speed of e rotor v is constant e induced emf ec () t signal, shown in figure 2, represents also e waveform of e air gap field which can be described by a Fourier series N B( θ) = Bmksin( kθ + ϕk ) (3) k = Figure 3: Measured air gap magnetic flux density he r.m.s value of e induced signal due to e fundamental is c= c = m = 0 0 (4) E e dt lv B dt lvb his relation is manipulated by expressing e peripheral speed v = πdn= 2τ f (5) in terms of e pole pitch τ and e frequency of e induced emf voltage due to e rms value B of e Figure 4: Measured air gap magnetic flux density obtained from digitizing signal of figure 3.

4 VI. HARMONIC ANALYSIS HE SAOR VOLAGES Figure 5: Measured amplitude spectrum of e air gap magnetic flux density using a search coil at no load IV. DESCRIPION OF HE MEASUREMEN PROCEDURE wo meods have been used to analyse e harmonic content of e air gap FDD; e first is based on e harmonic analysis of e induced emf in a search coil whereas e second relies on e analysis of e induced voltage in e stator windings. he experimental harmonic analysis is carried as follow. First, e measured signal, after level adjustment, is fed to a spectrum analyser, e frequency band is selected and a specific harmonic is singled out using a marker. he marker is a luminous spot which can be moved on e amplitude spectrum of e signal displayed on e screen of e spectrum analyser. he amplitude, expressed in db and e frequency of e selected harmonic are directly displayed on e screen. he measurement procedure is en repeated for each relevant harmonic. A great accuracy can be achieved by is meod since e measurement procedure depends only on e frequency at is incidentally checked in permanence by e spectrum analyser. he second meod, basically similar to e previous, relies on e harmonic analysis of e induced voltage signals in e stator winding whose waveforms are shown in figures 6 and 7. In is meod, e stator winding behaves in e same times as a multipoint sensing device distributed uniformly along e air gap giving a snapshot of e whole air gap field, a selective filter owing to e distribution factors associated wi each harmonics and e phase winding connection and finally as a summing device because of e series connection of e coil turns wiin a phase winding. Due to e particularities of is sensing meod, some harmonics actually present in e search coil signal will not be retrieved in e terminal voltage waveform as it can be seen from e comparison of pictures of figure 5 and figure 8. Figure 6: Actual line voltage waveform. V. HARMONIC ANALYSIS OF HE AIR GAP FLUX Figure 5 shows e harmonic spectrum of e flux density waveform shown in figure 4. It can be seen at e flux density waveforms are heavily polluted by e high order spatial harmonics and an unexpected 25 Hz fundamental and its multiples have been found in e amplitude spectrum, even ough e machine has been run for 50 Hz operation. hese harmonics are due to a slight asymmetry between e consecutive pair poles. heir amplitudes are relatively small, but ey still affect e accuracy of e measurements. In order to appreciate e accuracy of e previous measurement meod, anoer meod has been used to extract e same information from e induced line voltage in e stator windings Figure 7: Actual phase voltage waveform.

5 Flux per pole harmonics calculation is carried according to π E = 2 k Φ f N K 2 pk k c dk Examination of relation (8) shows at e bracketed part represents e r m s value of e induced emf in e search coil. Since Nc and K dk are well defined, e consistency of e experimental results may be assessed by evaluating e ratio (9) E pk = 2N c K (0) E dk ck he amplitude phase voltage and e corresponding flux density spectrum [, 4] are related to each oer by E = 2 2 τ lb f K () pk mk dk Figures 4 shows e flux density amplitude spectrum extracted from e line voltage signals depicted in figure 3. From e line voltage waveform, it is apparent at e inherent filtering property of stator windings is not fully effective since high order harmonics are still surimposed to e line voltage fundamental. signal for two main reasons. First, eir amplitude is negligible and eir associated distribution factors are only significant at much higher frequencies at which ey become equal to at of e 50 Hz fundamental frequency. he more pronounced harmonics, which persist in e air gap field, e phase and e line voltage, are e too ripples [4]. he frequency of e too ripples are related to e number of slots per phase per double pole pitch 2Q and eir order can be found from ν = 2Qk ±, k N (2) Bearing in mind at e number of slots per pole of e machine is 9, e first two slot harmonics are e 7 and 9. Despite eir high frequency, ey are not eliminated from e phase and line voltage waveform signal because eir distribution factors are equal to at of e fundamental. hese harmonics can be only processed selectively eier by chording e stator winding or skewing e stator slots, eir presence in e air gap field and induced line voltage waveforms is inevitable VII. SIGNAL RECONSRUCION o assess completely e accuracy of e measurement meods, a program has been developed to regenerate e stator voltages from e computed and measured air gap field distribution. For is purpose, a program has been devised to derive e stator winding voltages from e discrete harmonic analysis of e computed and measured magnetic flux density distribution shown respectively in figure 2 and 4. herefore, e accuracy of e measurement procedures exposed above can be readily assessed from e comparison e measured and computed phase and line voltage waveforms. his step provides only qualitative estimation. As far as signal waveform is concerned, it can be seen at e computed waveforms agree closely wi e measured ones. Figure 8: Measured amplitude spectrum from e line voltage at no load However, compared to e phase voltage shown in figure 7, e line voltage signal exhibits a great improvement in terms of high order harmonics elimination. Line voltage waveform improvement is due to 3rd order harmonic components removal by e wye connection of e stator windings. Besides eir own relatively small amplitudes, ey are furer attenuated because of eir low associated distribution factors. he 25 Hz sub harmonics, only present e search coil signal as shown in pictures of figure 5, are similarly removed from e line voltage Figure 9: Computed phase voltage waveform.

6 A final check of e accuracy of e measurement procedure has been made rough e comparison of e actual and simulated voltage in e stator winding. Alough, is measurement procedure has been used in connection wi a specific a device, it can be applied wiout any restriction to oer machines. LIS OF SYMBOLS Figure 0: Computed line voltage waveform. Finally, e accuracy of e finite element solution, in terms of harmonic content, is performed by comparing e measured and computed line voltages derived from eir respective magnetic flux density distribution. Figure is given to at purpose. B mk peak value of e k flux density harmonic peak value of e k flux harmonic Φ k E ck E pk r.m.s value of e k induced voltage r.m.s value of e k induced phase voltage FDD flux densitydistribution f fundamental frequency v pheripheral speed of e rotor l effective conductor leng τ pole pitch θ angular position N c number of conductors per phase Q number of slots per pole tee ripple harmonic order ν REFERENCES Figure : Comparison of e computed line voltage waveform. he difference between e measured and computed can be accounted from many factors such e magnetic history of e actual permanent machine under scrutiny, etc. Nevereless e last figure reveals a close agreement between e computed and measured waveforms. he experiment has been also carried out at full load. A close agreement has been also found between e measured amplitude spectrums using bo meods. VIII. CONCLUSION Magnetic flux density measurement is a vital aspect in e evaluation of e overall performance of a given design. In is respect, two measurement meods have been used to enable a cross comparison between e measured results and close agreement has been found. [] M.G. Say, Alternating current machines 5 edition, Pitman Publishing Ltd, London,982 [2]. orschanoff, Survey of numerical meod in field calculation, IEEE ransactions on Magnetics, Vol. MAG. 20, N 5, pp 92-97, Sept. 984 [3] J.K. SYKULSKI, Computational electromagnetics for design optimisation: e state of e art and conjectures for e future Bulletin of e Polish Academy of Sciencesechnical Sciences Vol. 57, No. 2, 2009 pp23-3 [4] K. B. Baltzis, he FEMM Package: A Simple, Fast, and Accurate Open Source Electromagnetic ool in Science and Engineering" Journal of Engineering Science and echnology Review (2008), pp [5] P. Zhou, W.N. Fu, D. Lin, S. Stanton, Z.J. Cendes, Numerical modeling of magnetic evices, IEEE rans. on Magn., Vol. 40, No. 4, pp , July [6] K..J. Binns,, C.P. Riley and.m. Wong, he efficient evaluation of torque and field gradient in permanent magnet wi small air gap, IEEE ransactions on Magnetics, Vol. MAG. 2, N 6, pp , Nov. 985 [7] M. Kostenko and, L. Piotroski, Electric machines: Alternating current machines, Vol.2, Mir Publishers, Moscow, 963