Effect of Change in Pole Shape Design on Harmonic Contents of PM Synchronous Motor Air Gap Flux Density Waveform

Transcription

1 Fourth LACCEI Internatonal Latn Amercan and Carbbean Conference for Engneerng and Technology (LACCET 2006) Breang Fronters and Barrers n Engneerng: Educaton, Research and Practce 2-23 June 2006, Mayagüez, Puerto Rco Effect of Change n Pole Shape Desgn on Harmonc Contents of PM Synchronous Motor Ar Gap Flux Densty Waveform Osama Mohammed, PhD Professor, Florda Internatonal Unversty, Mam, FL, USA S. Ganu, PhD student Vstng Researcher, Florda Internatonal Unversty, Mam, FL, USA N. Abed, PhD Student PhD student, Florda Internatonal Unversty, Mam, FL, USA Abstract Ths paper utlzes the wavelet pacet transform to study the effects of change n the shape of the magnetc pole on the harmonc behavor of ar gap flux densty waveform. A surface mounted PM motor s used as an example. The orgnal desgn contans 6 PM and 36 stator slots. The rotor and the stator wndng are redesgned to have 4, 8, and 2. The ar gap flux densty waveform s obtaned from the fnte element solutons. The results have been compared wth dfferent pole structure desgn. It has been found that there s not much dfference n the harmonc content due to change n PM pole structure on the ar gap flux densty waveform. Keywords PM synchronous motor, wavelet pacet transform, wndng change. Introducton Conventonal analytcal desgn and analyss methods of AC machnes follow the assumpton of havng snusodal flux waveform n the ar gap. An ncrease n the number of can be utlzed for ncreasng the machne power densty or for satsfyng the lower speed demands at some applcaton felds. Ths reduces the number of slots per pole per phase whch fnally leads to the ncrease n the harmoncs of ar gap flux waveform [Mohammed et al., 2004]. Increase n the harmoncs can cause a varety of undesrable effects n the energy system. For example, harmoncs can cause sgnal nterference, over voltages, and crcut breaer falure, as well as equpment overheatng, malfuncton, and falure. Harmoncs can cause excessve heatng n system components, resultng n shortened lfe or falure. Rotor heatng and pulsatng output torque caused by harmoncs can result n excessve motor heatng and neffcency. In other words, the harmonc ncrease wll affect both the machne performance and the machne control system performance. Wavelet transform proves to have some advantages when compared to the classcal FFT (Fast Fourer Transform)-based algorthms. Unle FFT, the perodcty of the sgnal s not requred and erroneous results are avoded usng wavelet pacets. The wavelet pacet method solves ths problem by decomposng the measurement sgnal nto a famly of tme fnte sgnals through a flter-ban resultng n

2 mult-scale wavelet decomposton. Harmoncs of the fundamental frequency, sub-harmoncs and nterharmoncs are consdered, dsregardng the length of ther occurrence n tme. Ths paper s organzed as follows. Frst the remodelng of the motors havng dfferent pole based on the orgnal machne s gven. It ncludes the rearrangng of the wndng n the slots and the PM on the rotor ron surface. FE computaton s performed on these dfferent desgns to obtan the feld dstrbuton and the flux densty waveform n the ar gap. Then the wavelet and wavelet pacet theory are brefly ntroduced. Ther mplementaton on the ar gap flux densty waveform s presented next. 2. Desgn of Dfferent Pole Numbers The orgnal surface mounted PM synchronous motor has 6 on the rotor ron and 36 slots n the stator ron. One pole of the motor s shown n fgure. The wndng s double layer and excted wth 3 phase supply. Slots per pole per phase (spp) are 2 (36/3*6). The magnets are radally magnetzed. Ther operatng pont s B r =.08 Tesla and µ r =.07. The angular magnet pole wdth s 50 degrees mechancal. Nonlnear BH curves are used for both stator and rotor ron. In order to form dfferent number of, the stator wndng needs to be rearranged so as to form same number of on stator as that of rotor. Wndng arrangement for dfferent number of s shown n fgure 2. Fgure : One pole of pm synchronous motor Fgure 2: Wndng arrangement n dfferent pole desgns up to 8 slots n all smulated machnes

3 Dependng on the number of, spp wll change. As the number of goes on ncreasng, each phase wll occupy less number of slots. The angular magnetc pole wdth s ept constant for all the pole varatons. The ar gap between permanent magnets gets adusted accordngly. The dmensons of the stator yoe and stator teeth n all the topologes are ept unchanged. All radal and axal dmensons for motor are same n all the cases. Obvously, ron part n the motor havng hgher number of wll be operatng at hgher flux densty level and goes towards saturaton. Saturaton s one of the mportant factors causng harmonc ncrease n the ar gap flux densty waveforms along wth slottng effects. 3. Fnte Element Analyss (FEA) Followng formulaton s used for solvng the transent magnetc soluton. σ A + ( A) V Hc t ν = σ + () Where, σ s electrcal conductvty, A s magnetc vector potental,ν s reluctvty, V s electrcal scalar potental, H c s magnetc coercvty. Nonlnear transent FE analyss was performed for the feld calculaton usng commercal software of Flux2d. The transent analyss conssts of electromagnetc feld formulaton and electrc crcut formulaton. Each formulaton provdes dfferent matrces. These are coupled and solved smultaneously at every tme soluton. All feld, crcut and moton equatons are descrtzed n tme doman. A converged soluton s obtaned at each tme step. Movng ar gap feature s utlzed for the transent operaton. The ar gap s dvded nto 3 layers, one belongng to stator, one to the rotor and mddle one whch actually causes the rotor to move wthout changng the mesh. Ratng of the motor s 2 hp, 3 phase, 200 rpm, 6 pole, 60 Hz, 36 slots, 2nch deep, half slot current 28 Amp-turns. Half slot current s mantaned constant throughout all pole arrangements. Consderng motor perodctes, only quarter geometry s modeled for 4, 8, 2 whle for 6 one sxth geometry s modeled. For all the smulaton cases, dfferent velocty s assgned to the rotor to adust the angular frequency n the stator wndng to be equal to 60 Hz. Speed s equal to 800, 200, 900 and 600 rpm n 4, 6, 8, 2 respectvely. The rotor s rotated through every one degree wth dfferent tme step dependng upon the speed of the rotor to complete 360 degrees mechancal cycle. Flux densty values are taen at the last nstant ponts were used n capturng the waveform n order to extract the frequences up to 32 bands from wavelet analyss. Fgure 3 shows feld lnes whle fgure 4 shows flux densty waveform n ar gap for dfferent pole desgns. Fgure 3: Feld lnes for dfferent pole numbers Fgure 4: Flux densty waveforms n ar gap

4 4. Wavelet Theory 4. Introducton As n fourer transform(ft), the wavelet transform(wt) conssts of decomposng a gven functon nto a set of buldng blocs. However, as opposed to the fourer transformaton n whch the buldng blocs are the well-nown complex exponentals, the wavelet transform uses the dlated and translated verson of a mother wavelet whch has convenent propertes accordng to tme/frequency localzaton. Unle FT whch gves a global presentaton of the sgnal, WT provdes a local representaton (n both tme and frequency) of a sgnal; therefore t s sutable for analyzng a sgnal where tme/frequency resoluton s needed. 4.2 Multresoluton analyss A multresoluton analyss of the followng propertes: 2 L (R) s defned as a sequence of closed subspaces V of 2 L (R), Z, wth V V + v( x) V v(2 x) V + v( x) V0 v( x + ) V U V s dense n L (R) and I V = {0} = = A scalng functon ϕ V o, wth a non-vanshng ntegral, exsts such that the collecton { ϕ{ x l) l Z}, s a Resz bass of V o A sgnal can be successvely approxmated by wavelets wth dfferent scales (multresoluton decomposton). Each step of the decomposton of sgnal corresponds to a certan resoluton. The decomposton process can be terated, wth successve approxmatons beng decomposed n turn, so that one sgnal s broen down nto many lower-resoluton components. Ths s called the wavelet decomposton tree. For n-level decomposton, there are n+ possble ways to decompose or encode the sgnal. The wavelet pacet transform (WPT) method s a generalzaton of wavelet decomposton. In wavelet pacet transform, the detals as well as the approxmatons can be splt. Ths yelds more than 2n- 2 dfferent ways to encode the sgnal. Wavelet pacet decomposton s depcted n fgure 4 (a). Let φ( t) and (t) be the scalng functon and the correspondng mother wavelet functon n the conventonal DWT and defne o ( t ) = φ( t ), and ( t) = ( t), For 2 N sampled waveform the wavelet transform coeffcents (WTCs) of a gven functon B(t) at the level and th pont can be evaluated by convolvng the sequence d ( ) wth a low pass flter (LPF), and then downsamplng by a factor of two. In the same manner, coeffcents (WTCs) of a gven functon B(t) at the (2+) th node can be evaluated by convolvng the sequence d ( ) wth a Hgh pass flter (HPF), and then downsamplng by a factor of two. Number of bands at th level s 2. Mathematcally, the (WTCs) can be wrtten as: d ( ) = B( t) ( t) dt = h( n) d (2 n) 2 2, n d ( ) = B( t) ( t) dt = g( n) d (2 n) 2+ 2+, n (2)

5 (a) (b) Fgure 5(a): Wavelet Pacet decomposton wth successve flterng and down samplng. (b) Three level wavelet Pacet decomposton Where, t, ( t) = ( ) = h( n),2 n ( t) 2 2 n t, ( t) = ( ) = g( n),2 n ( t) 2 2 n They are the wavelet bass functon =0,,.,N = 0,,, 2 Snce the wavelet bass s orthogonal, the followng propertes should be fulflled: P, ( t) dt = 0 p=nteger p=q ( t) ( t) = 0 p q P q,, = 2 ( φ, ( t)) φ ( t) ( t) = 0 p 0 p,, (3) A three level wavelet pacet decomposton tree s shown n fgure 4b. Each tree represents the sgnal decomposton as depcted n fgure 4a. 5. Wavelet representaton of RMS and total harmonc dstorton In the wavelet theory, any waveform can be expressed n terms of t s weghted sums of wavelet bass functon, hence B(t) can be expressed as:

6 N J N J N J N J ,, φ,, = 0 = 0 = 0 = 0 = 0 = = 0 (4) B( t) = d ( ) ( t) + d ( ) ( t) = d ( ) ( t) + d ( ) ( t) Where d 0 ( ) s the scalng functon coeffcents. RMS of a waveform wth perod T can be expressed n terms of wavelet at a certan level as follows: N J N J N J N J = φ, +, φ, +, = 0 = = 0 = 0 = = 0 B( t) dt [ d ( ) ( t) d ( ) ( t)] dt = [ d ( ) ( t)] dt [ d ( ) ( t)] dt] N J d ( ) d ( ) φ, ( t), ( t) = = 0 N J N J N J = [ ( d ( )) [ φ, ( t)] dt + ( d ( )) [, ( t)] dt +2 d ( ) d ( ) φ, ( t), ( t) dt = 0 = = 0 = = 0 (5) Usng the wavelet orthogonalty property, (5) becomes: = 2 2 N 2 2 dt d = 0 = 0 B(t) ( ( )) (6) The RMS of the flux waveform can now found by dvded by the wave perod T and tae the root square of (6). B B t B n d B (7) N N T rms = ( ) = ( ) ( ( )) ( ) 0 N = N = T 2 n= 0 2 n= 0 = 0 n= 0 B d N 2 2 = ( ( )) N (8) 2 = 0 Where, B s the RMS value of frequency band at node. The Total Harmonc band dstorton (THBD) s defned by the rato of the RMS value of the harmonc bands at >0 (.e. excludng the lowest band) to the RMS value of the dstorted waveform. The THBD for the flux densty waveform s gven by the followng relaton: THBD ( ) 2 2 = B (9) Brms = 6. Results and dscusson Fgure 6: THBD change wth number of change

7 Table :Total harmonc dstorton for dfferent number of n case of orgnal rotor pole structure THD % RMS Band Hz Band 75-50Hz Band Hz Band Hz Band Hz Table 2:Total harmonc dstorton for dfferent number of n case of smooth rotor pole structure THD % RMS Band Hz Band 75-50Hz Band Hz Band Hz Band Hz Wavelet Pacet technque has been appled to the flux densty n the mddle of the ar gap obtaned from the fnte element model to extract the dfferent harmonc bands of the flux densty waveform. A fve level decomposton was selected to perform our study. Ths wll decompose the sgnal nto 32 frequency band ( 2 5 ). The frst eght are enough to evaluate the harmonc behavors. From the analyss t s clear that there s ncrease n harmoncs contents wthn the flux densty wth the number of ncrease. Fgure 6 shows the relaton between number of ncrease and the total harmonc band dstorton. There s a small ncrease n the THBD from 4 to 6 machne. On the other hand bgger ncrease s notced between 6 and 8, 6 and 2, and 8 and 2 respectvely. Table shows the analyss result of the flux waveform for the frst 4 frequency band for 4, 6, 8 and 2 of PM orgnal structure rotor respectvely. Table 2 shows smlar results for smooth rotor structure desgn motor. Detals and results about desgnng ths smooth rotor can be found n [Mohammed et al., 2004]. If we compare correspondng values for each pole, there s only mnmal change n the total harmonc band dstorton. Results show that changng the rotor structure from protrudng shape to smooth rotor does not change the harmonc contents apprecably n the ar gap flux densty waveform.

8 7. Concluson Fnte element smulaton presents an effectve method for studyng the nfluence of nternal structure changes on the harmonc content of the worng flux densty. A wavelet pacets technque was used to evaluate the harmonc behavor for dfferent number of. The property of WPT analyss shows an ablty to quantfy dfferent types of sgnals. It also shows hgh ablty of wavelets to extract the dfferent harmonc components dsregardng the length of ther occurrence n tme. The harmonc contents tend to ncrease wth number of ncrease but do not change wth the changes n the rotor structure desgn sgnfcantly. The results are very useful n the desgn and development of both the motor and drve s dagnoss systems. 8. References Angrsan L, Daponte P, D'Apuzzo M, and Testa A, A measurement method based on the wavelet transform for power qualty analyss, IEEE Transactons on Power Delvery, Vol.3, No. 4, Oct. 998 Ellott D, Handboo of dgtal sgnal processng: engneerng applcatons, Academc Press, INC, 987 Gall A. W, Analyss of electrcal transents n power systems va a novel wavelet recurson method, Purdue Unversty, 997 Goswam J. C., and Chen A.K, Fundamentals of wavelets, John Wley, 999 Hamd E, Kawasa Z, Yoshda H, and Do H, Wavelet analyss of voltage dsturbances for power qualty applcatons, IEE of Japan, Vol.22, No.2, February 2002 Kaser Gerald, A frendly gude to wavelets, Sprnger Verlag, 994 Mao P.L, and Aggarwal R.K, A novel approach to the classfcaton of the transent phenomena n power transformers usng combned wavelet transform and neural networ, IEEE Transactons on Power Delvery, Vol. 6, No. 4, Oct. 200 Madsett V. K., and Wllams D. B., Dgtal sgnal processng handboo, CRC Press, 999 Mohammed O.A., Abed N., Ganu S. and Lu S., Wavelet analyss of permanent magnet synchronous machne flux densty harmonc content wth dfferent pole number desgns, ACES 2004 Press W et al., Numercal recpes n fortran, Cambrdge Unversty Press, New Yor, 992 Wang Fan, On power qualty and protecton, Thess, Chalmers Unversty of Technology, 2000 Wornell G, Sgnal processng wth fractals: A wavelet-based Approach, Prentce Hall, 996 Yoon W and Devancy M, Power measurement usng the wavelet transform, IEEE Transactons on Instrumentaton and Measurements, Vol. 47, No.5, OCT 998