Stator Short Circuits Detection in PMSM by means of Higher Order Spectral Analysis (HOSA)

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1 Stator Short Crcuts Dtcton n PMSM by mans of Hghr Ordr Spctral Analyss (HOSA) J. Rosro 1, J. Ortga, J. Urrsty, J. Cárdnas, L. Romral 1 ABB, -mal: javr.rosro@s.abb.com Moton Control and Industral Applcatons Group, Tchncal Unvrsty of Catalona, C/ Colon 1 Tr Trrassa. Catalona. Span, -mal: romral@l.upc.du Abstract Ths papr prsnts and analyzd short crcut falurs for Prmannt Magnt Synchronous Motor (PMSM). Th study ncluds statd and dynamc condton for xprmntal and smulaton tst. Th stator currnt s analyzd by mans of hghr ordr spctral analyss (HOSA). HOSA tchnqus ar usd for stator currnt analyss undr stabl condtons s powr frquncy spctrum dnsty (PSD); and Multpl Sgnal Classfcaton (MUSIC), and bspctrum ar usd undr dynamcs condtons. Thrfor, t s possbl to mprov th accuracy and ffcncy of tchnqu. Exprmntal rsults valdat th analyss and dmonstrat for HOSA can b appld to dtct and dntfy short crcut falurs n synchronous machns. Indx Trms PMSM drv, Short Crcut, Hghr Ordr Spctral Analyss, HOSA, powr frquncy spctrum dnsty, PSD, Bspctrum, Multpl Sgnal Classfcaton, MUSIC. I. INTRODUCTION Synchronous-typ lctrc motor drvs ar now xtnsvly usd n many ndustral applcatons rqurng hgh systm rlablty, hgh ffcncy and xtndd hghspd opraton. In many of ths crtcal drv applcatons, t s rqurd th drv comprsng th motor and th convrtr, undr fault condtons, to oprat stably and mt bas drv nds for a prod of tm bfor th systm can b (slf) rpard [1]. Stator or armatur faults ar usually rlatd to nsulaton falur. In common parlanc, thy ar gnrally known as phas-to-ground or phas-to-phas faults. It s blvd that ths faults start as undtctd turn-to-turn faults that fnally grow and culmnat nto major ons []. Short crcuts ar th most mportant falurs n PMSM [3-5], not only to b th man rason of th motor damags, but also for th problms ths falurs can provok n hgh scurty applcatons. Ths applcatons rqur an arlr fault dtcton, n ordr to solat th falur and assur fault tolrant post control of th drv. Thr ar many tchnqus to dtct turn-to-turn faults, th majorty of thm basd on th analyss of stator voltags and currnts, axal flux and d-q currnt and voltag componnt. Tolyat [6, 7] and Pnman [8] hav dscrbd th opraton of nducton motor drvs undr th condtons of loss of on phas, brokn bars and shortd stator turns usng wndng functons. Howvr, th modls thy consdrd dd not nclud thr saturaton or control ntractons. From yars ago, th falur dtcton n a motor s studd by analyzng th stator currnt harmonc by mans of FFT [9]. But, FFT cannot b appld n no-statonary sgnals [1, 1]. Fortunatly, sgnal procssng thors provd svral algorthms for applcatons wth no-statonary sgnals [11]. Jont tm-frquncy analyss s a novl approach n th motor dagnoss applcatons. Succssful us of ths tchnqus rqurs undrstandng of thr rspctv proprts and lmtatons [1-14]. Th hghr ordr spctral analyss (HOSA) has bn consdrabl ntrst to rsarchrs n th sgnal procssng and ths ntrst has rcntly bn xtndd to th condton montorng [15, 16]. HOSA rqurs no a pror data for fault dtcton and quantfcaton. Th dsadvantags of HOSA s that has a hgh computatonal ovrhad, and thr complx ntrprtaton [16, 17]. Howvr, th modl ordr and th procssng tm can b rducd usng fltrng and frquncy dcmatng tchnqus as hav dscrbd n [18-]. Ths papr suggst that s possbl to study and dntfy short crcuts n th wndngs of th PMSM dtrmnd by mans of HOSA as PSD, MUSIC and bspctrum n th whol opraton rang. Th motor was xprmntal tstd and smulaton wth a short crcut n 4, 8 and 1 turns for spd btwn 6 rpm and 6 rpm. Exprmntal rsults wr compard wth thos obtand from a halthy machn, and conclusons ar don. II. HIGHER ORDER SPECTRAL ANALYSIS (HOSA) Thy ar manly basd on statstcal procdur and thy comput th powr frquncy spctrum dnsty (PSD). Ths mthods ar classfd n non-paramtrc, paramtrc and subspac tchnqus [16, 19, ]. Th non-paramtrc mthods hav th sam problms compard to th classcal FFT concrnng th frquncy rsoluton. Du to a pror knowldg, th paramtrc mthods hav mprovd prformancs (Burg, Yul-Walkr, covaranc and modfd covaranc). Thn, th paramtrc mthods do not valdat th tradtonal assumpton that th data outsd th wndow ar qual to zro and thr prformancs ar affctd by th sgnal-to-nos rato. Th subspac mthods ar known as hgh rsoluton mthods as MUSIC, bspctrum or trspctrum whch dtct frquncs wth low sgnal-to-nos rato [19] /9/$5. 9 IEEE 964

Radrs ar drctd to [15, 16] for a rgorous tratmnt of HOSA fundamntals and subsqunt drvatons, howvr, gvn a dscrt tm sgnal x(n), th dscrt Fourr transform (DFT) of s dfnd to b: X j π n ( f ) = x( n) (1)

2 Radrs ar drctd to [15, 16] for a rgorous tratmnt of HOSA fundamntals and subsqunt drvatons, howvr, gvn a dscrt tm sgnal x(n), th dscrt Fourr transform (DFT) of s dfnd to b: X j π n ( f ) = x( n) (1) n= Th wll-known scond-ordr masur, th powr spctral dnsty (PSD) P(f) of x(n) can b dfnd n trms of X(f) as: [ X ( f ) X *( )] P ( f ) = E f () Whr E[ ] s th statstcal xpctaton, or avrag, and X*(f) s th complx conjugat of X(f). Th PSD s a lnar transform and s a functon of th frquncy ndx f. Extndng ths dfntons to thrd-ordr masurs gvs rs to th bspctrum dfnd n (3). B ( f1, f) = E[ X ( f1) X ( f) X *( f1 + f) ] (3) From (3) t may b sn that, unlk th PSD, th bspctrum ar functons of mor than on frquncy ndx and, furthr, t may also b sn that thy ar complx quantts,.., thy contan both magntud and phas nformaton about th orgnal tm srs [16, 17]. A) Multpl Sgnal Classfcaton (MUSIC) Multpl Sgnal Classfcaton (MUSIC) mthod assocats th nos to th mnmum gnvalu of th nos varanc allowng low sgnal-to-nos frquncy componnts stmaton and th frquncy stmaton s basd on th orthogonalty of th sgnal vctors to th nos gnvctors [18, 19]. Th subspac mthods assum that data x[n] can b rprsntd by m complx snusods n a nos [n] [15, 18]: x m jπ fn jφ [] n = A + [] n A = A =1 (4) Whr n=, 1,, N-1, N s th numbr of sampl data, f, A and φ ar th frquncy, th magntud and th random phas of th complx snusod rspctvly, [n] s a squnc wth a zro man wht nos. Th mthod known as Psarnko harmonc dcomposton (PHD) shows that th frquncs of th snusods can b obtand from th gnvctor corrspondng to th smallst gnvalu of th autocorrlaton matrx. Whn [n] s wth zro man wht nos and a varanc σ, th autocorrlaton matrx R of x[n] can b rprsntd by: m R = A = 1 H ( f ) ( f ) +σ I f [ ] T jπf j4πf jπ ( N 1) ( f ) = 1... Th xponnt H s th Hrmtan transpos and I s th dntty matrx. It s sn that th autocorrlaton matrx R s composd of th sum of sgnal and nos autocorrlaton matrcs: R = Rs +σ I (7) Psarnko obsrvd that th nos varanc corrsponds wth th smallst gnvalu of R and that th frquncs of (5) (6) th complx snusods can b stmatd by usng th orthognalty of th sgnal and nos subspacs, that s: H ( f ) V ( m + 1 ) =, = 1,..., m (8) Wth V(m+1) th nos gnvctor n a matrx R wth th dmnson (m+1) (m+1). MUSIC approach has bttr prformancs than Psarnko mthod bcaus of th ntroducton of avragng va th xtra nos gnvctors (k=m+1,m+,...,m). Th avragng rducs th statstcal fluctuatons prsnt n th Psarnko psudo-spctrum whch ar du to th rrors n stmatng th autocorrlaton matrx. Thn, th MUSIC psudo-spctrum of spac vctor s dfnd as: MUSIC 1 P ( f ) = M (9) H f V k= m+ 1 ( ) k Whr (f ) H V k =, k=m+1,...,m. It has th largst paks at th prncpal componnts and ths frquncs ar n th bandwdth [, F s / ] for a sgnal sampld wth F s samplng frquncy. Th MUSIC mthod taks a long computaton tm to fnd mor frquncs by ncrasng th ordr of th frquncy sgnal dmnson. To solv ths problm, an algorthm basd on zoomng n a spcfc frquncy rang s proposd wth MUSIC n ordr to mprov th dagnoss prformancs to xtract frquncs [18, 19]. Th Zoom-MUSIC algorthm s a tchnqu to mak th MUSIC fastr and mor accurat n xtractng frquncs n a spcfd bandwdth [18-]. III. WORKBENCH OF SIMULATION AND EXPERIMENTAL Th motors undr analyss hav bn a PMSM of 6 rpm nomnal spd,.3 Nm nomnal torqu, and 3 pols par [1]. Smulatons and xprmnts hav bn carrd out for motors wth 4, 8 and 1 short turns of stator phas wndng; th motors wr drvn at nomnal, mdum (45 and 3 rpm) and low (15 and 6 rpm) spd. Both, th smulatons and th xprmnts wr carrd out undr statonary and no-statonary condtons. Fg. 1. Schmatc of th fnt lmnt analyss (FEA) for a PMSM Fnt lmnt analyss (FEA) s proposd for th smulaton of lctrcal machns wth short crcut fault. FEA rvals tslf lk an accurat an asy mthod to dtrmn th ntracton btwn non-lnar ffcts. PMSM n a halthy stat and wth short crcut turns ar smulatd at dffrnt /9/$5. 9 IEEE 965

3 workng ponts usng a two-dmnsonal (-D) fnt-lmnt analyss softwar (FLUX D) [1]. Th PMSM workbnch for FEA analyss s shown n Fg. 1. Th Hghr Ordr Spctral Analyss Toolbox wth Matlab s usd to comput th HOSA, PSD, MUSIC and Bspctrum [15]. IV. SIMULATION AND EXPERIMENTAL RESULTS Som xprmnts hav bn carrd out for dffrnt motors wth shortd turns. Th analyss s mad for stabl condton of spd and constant torqu. Th rsults of machn wth short crcut ar compard wth thos obtand from halthy machns. A. Stabl Stat (spd and constant torqu) Th spctral analyss of stator currnt s mad usng Powr Spctral Dnsty (PSD) or Multpl Sgnal Classfcaton (MUSIC). Ths mthods allow to dntfy th most rprsntatv frquncs fltrng th Gaussan nos and gvng prorty th frquncs of hghr valu, that n ths cas ar th man frquncy of stator and falur frquncy. Ths way, th falur dtcton s assurd and th rsoluton for ntrst frquncs s mprovd. Fg. and Fg. 3 show MUSIC of stator currnt for 15 and 6 rpm rspctvly. For 6 rpm, th man harmonc 3 Hz and falur harmonc at 9 Hz for PMSM wth short crcut ar obsrvd, th othrs harmoncs lk 1 st, 5 th and 7 th of machns ar covrng by th swpng of spctrum and t can not sn. Th last harmoncs normally ar prsnt whn t s analyzd by mans of classc MCSA, thrfor, MUSIC has fltrd ths harmoncs and has cntrd th analyss n th man harmonc of stator and 9 th harmonc. Th xprmntal rsults ar show n Fg. 4 and Fg. 5 for MUSIC at 15 and 6 rpm rspctvly. MUSIC shows a hgh valu for 9 th harmonc for PMSM wth short crcut. For 6 rpm, MUSIC cntrs th analyss n th man harmonc of stator (3 Hz) and th falur harmonc (9 Hz) for PMSM wth short crcut. Howvr, for halthy machns th 9 th harmonc s fltrd. In ths cas of halthy machn th MUSIC analyss taks anothr frquncy from gratr ampltud lk 15 th and 1 st harmonc. Thos harmoncs hav hgh ampltud n halthy machn by th constructv form: concntratd wndng, slot numbr by pol and phas (); n addton, n th cas of smulaton th slot nclnaton s not consdrd. Also, Not that n cas of halthy machn at 15 rpm, MUSIC taks th man harmonc and 1 th harmonc lk man valus and not 9 th harmonc. Ths s du to 9 th and 1 th harmoncs hav smlar valus. But n any cas, th falur condton to low frquncs s possbl to dntfy w=6 rpm Fg.. MUSIC of stator currnts, Smulaton rsults at 6 rpm w=15 rpm Fg. 3. MUSIC of stator currnts, Smulaton rsults at 15 rpm w=6 rpm Fg. 4. MUSIC of stator currnts, Exprmntal rsults at 6 rpm w=15 rpm Fg. 5. MUSIC of stator currnts, Exprmntal rsults at 15 rpm Smulaton rsults of PSD ar show n Fg. 6 and Fg. 7 for 6 rpm and 15 rpm rspctvly. Th dffrnc of ampltud s hgh n 9 th harmonc btwn halthy and faulty machns. Ths dffrnc s gratr than th MCSA classc mthod rportd bfor [4, 5]. PSD shown an uppr valu of /9/$5. 9 IEEE 966

4 3 db for PMSM wth short crcut and mantan ths dffrnc vn for low spd whr classc MCSA mthod prsnts troubl. Fg. 8 and Fg. 9 show PSD for 6 rpm and 15 rpm rspctvly. Thy hav bn takn from xprmntal rsults. Thy approv th smulaton and valdat th mthod for short crcut dtcton. Although th rsoluton s good around of falur frquncy, n th rst of spctrum sn t as much, rppl ampltud wth rang of ±5dB can b sn w=15 rpm Sort Crcut w=6 rpm Short Crcut Fg. 6. PSD of stator currnts, Smulaton rsults at 6 rpm w=15 rpm Sort Crcut Fg. 7. PSD of stator currnts, Smulaton rsults at 15 rpm w=6 rpm Fg. 9. PSD of stator currnts, Smulaton rsults at 15 rpm In rlaton at MCSA standard, th PSD and MUSIC mthods allow to mprov th rsoluton of short crcut harmonc; and thy lmnat th nos of sgnal ffcntly du to xtrnal sourcs or vn ntrnal sourcs of lss ntrst, ths ar: unbalancd tnson, ntwork harmoncs, sub harmoncs, modulaton of nvrtr, lak of solaton, vbratons, spd changs, rppl torqu, stator and rotor magntc loss, saturaton, tc. Th advantag of MUSIC and PSD mthods s that stmatd spctrum s vry clan and flat (wthout nos) n frquncy rang of hgh ntrst (outstandng). On th othr hand, HOSA allows stmatd spctrum for dffrnt mthods from thrd and fourth ordr. Thus, bttr rsoluton n th most notabl frquncs at a proportonal computatonal cost can b stablshd. Although th computatonal cost can b dcrasd f th analyss s cntrd n ntrst frquncs rang lk svral mthods rportd by Zoom-FFT and Zoom- MUSIC [18-]. Bspctrum looks for sum and multplcaton rlatonshps of two ntrst frquncs that n ths cas ar man frquncy of stator and falur frquncy. Fg. 1 and Fg. 11 show bspctrum for halthy and fault PMSM. Ths smulaton dats ar takn th whol stator currnt. For PMSM wth short crcut ar two ovals n th graph. Th ntrnal ovals corrspond to man frquncy (3 Hz) of stator currnt and xtrnal oval s around 9 Hz that s th falur frquncy Short Crcut Fg. 8. PSD of stator currnts, Exprmntal rsults at 6 rpm Fg. 1. Bspctrum of halthy PMSM, Smulaton rsults at 6 rpm /9/$5. 9 IEEE 967

5 f (khz) Fg. 11. Bspctrum of PMSM wth 1 short crcut turns, Smulaton rsults at 6 rpm Bspctrum allows vwd ntracton of harmoncs of machn and th sam as PSD and MUSIC cntrd th analyss n man frquncs assgnng thm bggr valu. Thus, Bspcturm mantans ntrst frquncy as th falur frquncy of short crcut for th whol opraton rang of machn. Smlar rsults for 6 rpm, 3 rpm and 45 rpm has bn obtand. Bspctrum can b usd for currnt analyss n dynamc condton of spd or torqu varaton. Ths vn would rnforc th ablty of mthod to show th ntracton of frquncs and magntud chang at th sam tm. B. Spd Chang (constant torqu) Th PMSM usually oprats to dffrnt spds and torqu condtons. Undr ths dynamc condtons, th short crcut dtcton can b carrd out by mans of analyss of tm frquncy of th stator currnt. As t could b sn n th prvous scton, th proposd mthod s abl to prform fault dtcton at a low spd, constant condtons. Nxt, t s dmonstratd that t s also usful for mdum and hgh spds undr varabl condtons. From Fg. 1 to Fg. 14 show stator currnt bspctrum for PMSM wth short crcut. Ths ar smulaton and xprmntal rsults for spd changs at 15 to 1 rpm. Thrfor, th short crcut frquncy n th spd chang s rnforcs wth ntracton wth othr harmoncs and mak mor vsbl th falur. Manly th ntracton wth man harmonc (75 Hz) and 5 th harmonc (15 Hz) can b sn. Although othrs frquncs ar prsnt n th graph ar lss notabl. f (khz) f1 (khz) Fg. 1. Bspctrum of stator currnts n PMSM wth 8 short crcut turns, Smulaton rsults for spd chang from 15 to 1 rpm f1 (khz) Fg. 13. Bspctrum of stator currnts n PMSM wth 1 short crcut turns, Smulaton rsults for spd chang from 15 to 1 rpm.4 f (khz) f1 (khz) Fg. 14. Bspctrum of stator currnts n PMSM wth 1 short crcut turns, Exprmntal rsults for spd chang from 15 to 1 rpm For spds of 6 rpm, 45 rpm, 3 rpm and 6 rpm smlar rsults ar obtand and thrfor, only th rsults for lss favorabl condton ar shown. Th proposd mthod dsplay xcllnt ablty to short crcut dtct n constant and dynamc condtons. Th analyss by mans of bspctrum allows composd a powrful tool for fault dtcton of short crcut n PMSM. Ths mthod can b appld for othrs falur typs as ccntrcty and dmagntzaton whr th harmoncs of ntrst ar 1 st and 5 th harmoncs. Th computatonal cost of HOSA tchnqus as PSD, MUSIC or bspctrum can b rducd sgnfcantly f t s appld only for ntrst frquncy rang as svral authors show for zoom-fft and Zoom-MUSIC [18-]. V. CONCLUSIONS Ths papr prsnts th short crcut dtcton n a PMSM by mans of Hghr Ordr Spctral Analyss (HOSA). PSD and MUSIC show fatur sgnatur dffrntat btwn halthy and faulty condtons, as wll as btwn dgrs of fault for statd stat. Th prvous mthods maxmz th dffrnc for short crcut harmonc and thr rsoluton s gratr n rlaton wth classc MCSA. Also, PSD and MUSIC can dtct short crcut for th whol spd rang. Ths mthods can b usd for prvntv mantnanc whr th tst n controlld and known condtons can b carrd out. Th computatonal cost of PSD, MUSIC or Bspctrum algorthm can b rducd wth zoom tchnqus that mak th /9/$5. 9 IEEE 968

6 mthod fastr and mor accurat n xtractng frquncs n a spcfd frquncy rang. VI. REFERENCES [1] J. Rosro, J. Cusdo, A. Garca, J. A. Ortga, and L. Romral, "Brokn barngs and ccntrcty fault dtcton for a prmannt magnt synchronous motor," n Th 3nd Annual Confrnc of th IEEE Industral Elctroncs Socty, IECON6, Pars - FRANCE, 6, pp [] J. A. Haylock, B. C. Mcrow, A. G. Jack, and D. J. Atknson, "Opraton of fault tolrant machns wth wndng falurs," IEEE Transactons on Enrgy Convrson, vol. 14, p. 149, [3] S. Nand, H. A. Tolyat, and X. L, "Condton Montorng and Fault Dagnoss of Elctrcal Motors - A Rvw," IEEE Transactons on Enrgy Convrson, vol., p. 719, 5. [4] J. Rosro, J. Cusdo, L. Romral, and J. A. Ortga, "Dtcton of Stator Short Crcuts n PMSM by man of jont Tm-Frquncy Analyss," n Th 6th IEEE Intrnatonal Symposum on Dagnostcs for Elctrc Machns, Powr Elctroncs and Drvs, IEEE SDEMPED, Cracow, Poland, 7, pp [5] J. Rosro, O. Almonacd, M. Amaya, and L. Romral, "Smulaton and Fault Dtcton of Short Crcut Wndng n a Prmannt Magnt Synchronous Machns (PMSM)," n ISEF 7 - XIII Intrnatonal Symposum on Elctromagntc Flds n Mchatroncs, Elctrcal and Elctronc Engnrng, Pragu, Czch Rpublc, 7. [6] H. A. Tolyat, S. P. Wakar, and T. A. Lpo, "Analyss and smulaton of fv-phas synchronous rluctanc machns ncludng thrd harmonc of argap MMF," IEEE Transactons on Industry Applcatons, vol. 34, pp , [7] H. A. Tolyat and T. A. Lpo, "Transnt analyss of cag nducton machns undr stator, rotor bar and nd rng faults," IEEE Transactons on Enrgy Convrson, vol. 1, pp , [8] G. M. Joksmovc and J. Pnman, "Th dtcton of ntr-turn short crcuts n th stator wndngs of opratng motors," IEEE Transactons on Industral Elctroncs, vol. 47, pp ,. [9] J. Rosro, L. Romral, J. Cusdo, A. Garca, and J. A. Ortga, "On th short-crcutng fault dtcton n a PMSM by mans of stator currnt transformatons," n Th 38th IEEE Powr Elctroncs Spcalsts Confrnc, PESC7, Orlando, Florda, USA, 7. [1] J. Rosro, J. Cusdo, A. Garca, J. A. Ortga, and L. Romral, "Study on th Prmannt Magnt Dmagntzaton Fault n Prmannt Magnt Synchronous Machns," n Th 3nd Annual Confrnc of th IEEE Industral Elctroncs Socty, IECON6, Pars - FRANCE, 6, pp [11] N. Arthur and J. Pnman, "Condton montorng wth non-lnar sgnal procssng," n IEE Colloquum on Non-Lnar Sgnal and Imag Procssng (Rf. No. 1998/84),, 1998, pp. 4/1-4/5. [1] R. Yan and R. X. Gao, "Hlbrt Huang Transform-Basd Vbraton Sgnal Analyss for Machn Halth Montorng," IEEE Transactons on Instrumntaton and Masurmnt,, vol. 55, pp. 3-39, 6. [13] J. Cusdo, J. A. Rosro, J. A. Ortga, A. Garca, and L. Romral, "Inducton Motor Fault Dtcton by usng Wavlt dcomposton on dq componnts," n IEEE Intrnatonal Symposum on Industral Elctroncs, ISIE6, Montréal, Québc, Canada, 6, pp , vol. 3. [14] Z. Yupng, "Hlbrt-Huang Transform and Margnal Spctrum for Dtcton of Barng Localzd Dfcts," n Th Sxth World Congrss on Intllgnt Control and Automaton, WCICA 6., Dalan, Chna, 6, pp [15] A. Swam, J. M. Mndl, and C. Nkas, Hghr Ordr Spctral Analyss Toolbox for us wth MATLAB: Untd Sgnals & Systms, Inc., [16] N. Arthur and J. Pnman, "Inducton machn condton montorng wth hghr ordr spctra," IEEE Transactons on Industral Elctroncs, vol. 47, pp ,. [17] A. C. McCormck and A. K. Nand, "Bspctral and trspctral faturs for machn condton dagnoss," IEE Procdngs Vson, Imag and Sgnal Procssng, vol. 146, pp. 9-34, [18] S. H. Ka, H. Hnao, and G. A. Capolno, "A Hgh-Rsoluton Frquncy Estmaton Mthod for Thr-Phas Inducton Machn Fault Dtcton," IEEE Transactons on Industral Elctroncs, vol. 54, pp , 7. [19] S. H. Ka, H. Hnao, and G. A. Capolno, "Zoom-MUSIC frquncy stmaton mthod for thr-phas nducton machn fault dtcton," n IECON 5. 31st Annual Confrnc of Industral Elctroncs Socty. IEEE, Nort Carolna, USA, 5, pp [] L. A. Prra, D. Frnands, D. S. Gazzana, F. B. Lbano, and S. Haffnr, "Applcaton of th Wlch, Burg and MUSIC Mthods to th Dtcton of Rotor Cag Faults of Inducton Motors," n Transmsson & Dstrbuton Confrnc and Exposton: Latn Amrca, 6. TDC '6. IEEE/PES, 6, pp [1] CEDRAT, CAD packag for lctromagntc and thrmal analyss usng fnt lmnts, Usr s gud. Flux D. Franc, 5, /9/$5. 9 IEEE 969

Analyzing Frequencies

Analyzing Frequencies Frquncy (# ndvduals) Frquncy (# ndvduals) /3/16 H o : No dffrnc n obsrvd sz frquncs and that prdctd by growth modl How would you analyz ths data? 15 Obsrvd Numbr 15 Expctd Numbr from growth modl 1 1 5