SEPARATING LOAD TORQUE OSCILLATION AND ROTOR FAULTS IN STATOR CURRENT BASED-INDUCTION MOTOR CONDITION MONITORING

Transcription

1 SEPARATING LOAD TORQUE OSCILLATION AND ROTOR FAULTS IN STATOR CURRENT BASED-INDUCTION MOTOR CONDITION MONITORING A Dirtation Prntd to Th Acadmic Faculty By Long Wu In Partial Fulfillmnt Of th Rquirmnt for th Dgr Doctor of Philoophy in Elctrical Enginring Gorgia Intitut of Tchnology May 2007

2 SEPARATING LOAD TORQUE OSCILLATION AND ROTOR FAULTS IN STATOR CURRENT BASED-INDUCTION MOTOR CONDITION MONITORING Approvd by: Dr. Thoma G. Habtlr, Advior School of Elctrical and Computr Enginring Gorgia Intitut of Tchnology Dr. Jnnifr E. Michal School of Elctrical and Computr Enginring Gorgia Intitut of Tchnology Dr. Ronald G. Harly School of Elctrical and Computr Enginring Gorgia Intitut of Tchnology Dr. J. Rhtt Mayor School of Mchanical Enginring Gorgia Intitut of Tchnology Dr. Dpakraj M. Divan School of Elctrical and Computr Enginring Gorgia Intitut of Tchnology Dat Approvd: Dcmbr 8, 2006

3 Ddicatd to my parnt Mr. Zongwi Wu and Mr. Guangrong Zhai and my wif Mr. Ruilin Tian for thir lov and upport

4 ACKNOWLEDGEMENTS Fw goal ar vr achivd without th hlp of many othr popl. During my Ph.D. tudy at Gorgia Tch, I hav bn vry fortunat to rciv trmndou guidanc, ncouragmnt and upport from my mntor, collagu, frind and family. Dr. Thoma G. Habtlr ha bn a wi and trutd advior throughout th ntir proc. Hi confidnc in my capabiliti ha givn m immn opportuniti to timulat my rarch potntial and improv my profional communication kill. I gratly apprciat hi continuou guidanc and upport. I am alo gratful to Dr. Ronald G. Harly for hi patint guidanc, contant ncouragmnt and invaluabl uggtion for my rarch work. I hav bnfitd ignificantly from hi knowldg and xprinc. I would alo lik to thank Dr. Dpakraj M. Divan, Dr. Jnnifr E. Michal and Dr. J. Rhtt Mayor for taking tim to rad my thi and rv a my dirtation committ mmbr. I wih to xpr my gratitud to th Graingr Cntr for Elctric Machinry and Elctromchanic (CEME) at th Univrity of Illinoi and th Powrix Tchnologi, LLC for thir gnrou financial upport for th pat thr yar. I wa fortunat to hav many xcptional pr and collagu in my rarch lab. Among thm, I would lik to giv my incr thank to Dr. Xianghui Huang, Dr. Zhi Gao, Dr. Bin Lu, Wi Zhou, Young-kook L, Aritidi Zacha, Yi Yang, Wi Qiao and Dr. Sang-Bin L for thir hlpful dicuion and collaboration. I alo wih to thank Dr. Salman Mohaghghi, Dr. Satih Rajagopalan, Dr. Joy Mazumdar, Afroz Imam, iv

5 Harjt Johal, Jyoti Satry and othr fllow graduat tudnt in th lctric powr group for thir frindhip and company throughout my tudi at Gorgia Tch. Mot of all, I am dply indbtd to my parnt for bing an trnal ourc of upport, ncouragmnt and motivation throughout my lif. My dar wif and bt frind, Ruilin Tian, ha bn with m for vry ingl tp during thi long journy. Without thir gnrou lov, upport and undrtanding, vrything I hav accomplihd would not hav bn poibl. v

6 TABLE OF CONTENTS ACKNOWLEDGEMENTS... iv LIST OF TABLES... x LIST OF FIGURES... xi SUMMARY... xvi CHAPTER 1 INTRODUCTION Ovrviw Rotor Eccntriciti in Induction Motor Brokn Rotor Bar in Induction Motor Varying Mchanical Load Poition-Varying Load Torqu Ocillation Singl Frquncy Ocillating Load Torqu Priodic Load Torqu Dip Random Load Torqu Dip Objctiv of th Rarch Dirtation Outlin... 9 CHAPTER 2 SUMMARY OF PREVIOUS WORK ON SEPARATING LOAD EFFECTS FROM ROTOR FAULT DETECTION Ovrviw Modl Rfrnc Etimation Currnt Park Vctor Approach Vinna Monitoring Mthod vi

7 2.5 Othr Tchniqu for Eliminating Load Effct Mchanical Load Monitoring by Unuprvid NN Approach Load Diagnotic Utilizing Intantanou Powr Spctrum Load Fault Dtction by Tim-Frquncy Analyi Chaptr Summary CHAPTER 3 MAGNETIC FIELD ANALYSIS FOR ECCENTRIC MOTORS Ovrviw Purly Rotor Static Eccntricity Modifid Stator and Rotor MMF Motor Inductanc Rformulation Motor Currnt Charactritic Purly Dynamic Rotor Eccntricity Modifid Stator and Rotor MMF Motor Inductanc Rformulation Motor Currnt Charactritic Mixd Rotor Eccntricity Modifid Stator and Rotor MMF Motor Inductanc Rformulation Motor Currnt Charactritic Motor Currnt Charactritic for a Load Ocillation Chaptr Summary CHAPTER 4 EXTRACTING NEGATIVE SEQUENCE HARMONIC INFORMATION Ovrviw vii

8 4.2 Traditional Symmtrical Componnt Dcompoition Rotating Rfrnc Fram Tranformation Mthod Nw FFT Tchniqu to Sparat Squnc Information Application for Pha-Pha Maurmnt Chaptr Summary CHAPTER 5 DEVELOPMENT AND SIMULATION OF NEW ROTOR ECCENTRICITY FAULT INDICATORS Ovrviw A Nw Eccntricity Indicator in Main-Fd Machin Matlab Simulation for a Main-Fd Induction Motor Induction Motor Dynamic Modl Purly Static Eccntricity in a Main-Fd IM Purly Dynamic Eccntricity in a Main-Fd IM Mixd Rotor Eccntricity in a Main-Fd IM Poition-Dpndnt Load Ocillation in a Main-Fd IM A Nw Eccntricity Indicator for Driv-Connctd Machin Matlab Simulation for a Driv-Connctd Induction Motor Mixd Rotor Eccntricity in a Driv-Connctd IM Poition-Dpndnt Load Ocillation in a Main-Fd IM Simulation Rult Bad on Finit Elmnt Modl D Finit Elmnt Modl Mixd Rotor Eccntricity for a Main-Fd Machin Poition-Dpndnt Load Ocillation for a Main-Fd Machin FEA Simulation for a Driv-Connctd Machin viii

9 5.7 Chaptr Summary CHAPTER 6 SEPARATING ROTOR ECCENTRICITY INDUCED MOTOR ASYMMETRY FROM OTHER SOURCES Ovrviw Inhrnt Motor Aymmtry Stator Intr-Turn Fault Zro-Squnc Information Zro-Squnc Information for a Stator Intr-Turn Fault Zro-Squnc Information for Rotor Eccntricity Comparion of Zro-Squnc Information Chaptr Summary CHAPTER 7 EXPERIMENTAL SETUP AND IMPLEMENTATION OF LOAD TORQUE OSCILLATION Ovrviw - Baic Exprimntal Stup Motor Eccntricity Implmntation Motor Dynamomtr Sytm Clod-Loop Driv Configuration Voltag and Currnt Maurmnt Implmntation of Load Torqu Ocillation Purly Load Ocillation Implmntation Principl IGBT Configuration Chaptr Summary CHAPTER 8 RELIABLE ROTOR ECCENTRICTY DETECTION SCHEME AND EXPERIMENTAL RESULTS Ovrviw ix

10 8.2 Exprimntal Rult for Main-Fd Condition Exprimntal Rult for Clod-Loop Driv-Connctd Condition Rliabl Rotor Eccntricity Dtction Schm Chaptr Summary CHAPTER 9 CONCLUSIONS, CONTRIBUTIOINS, AND RECOMMENDATIONS FOR FUTURE WORK Concluion Contribution Rcommndation for Futur Work APPENDIX A MOTOR PARAMETERS USED FOR SIMULATION APPENDIX B MOTOR AND DRIVE PARAMETERS USED FOR EXPERIMENTS APPENDIX C INTRODUCTIOIN TO SHARK DSP BOARD USED FOR CREATING LOAD OSCILLATIOIN REFERENCES VITA x

11 LIST OF TABLES Tabl 7.1 Mchanim to implmnt a quai inuoidal load ocillation Tabl 8.1 Normalizd ngativ qunc componnt at f, ( f + frm) and ( f f ) in upply currnt rm Tabl A.1 Motor paramtr ud for imulation Tabl B.1 Namplat information for th xprimntal motor Tabl B.2 Impdanc information for th xprimntal motor Tabl B.3 Main paramtr for th Alln-Bradly driv and ncodr xi

12 LIST OF FIGURES Figur 2.1 Block diagram of th modl rfrnc timation Figur 2.2 Vinna Monitoring Mthod voltag modl (a) and currnt modl (b) Figur 2.3 Flowchart of th balin calculation during th initiation tag Figur 2.4 Gnral tructur of th ANN ud for ccntricity dtction undr diffrnt load condition Figur 3.1 Static air-gap ccntricity Figur 3.2 An lmntary 2-pol induction machin with tatic ccntricity, howing a clod magntic loop abcda Figur 4.1 Poitiv and ngativ qunc harmonic pac vctor in (a) tationary rfrnc fram and (b) rotating rfrnc fram Figur 4.2 Procdur to obtain pr pha qunc componnt from pha-pha maurmnt Figur 4.3 Rlationhip btwn qunc componnt from pr-pha and pha-pha quantiti Figur 5.1 Simulink chmatic uing rformulatd motor inductanc for main-fd machin Figur 5.2 Simulatd main-fd i a pctrum with ξ = 0.5, α = 0, lip = Figur 5.3 Simulatd main-fd i pctrum with ξ = 0.5, α = 0, lip = f q Figur 5.4 Simulatd main-fd I pctrum with ξ = 0.5, α = 0, lip = Figur 5.5 Normalizd ngativ qunc fundamntal componnt w.r.t. th load lvl whr ξ = 0.5, α = Figur 5.6 Normalizd ngativ qunc fundamntal componnt w.r.t. th dgr of tatic ccntricity whr lip = , α = xii

13 Figur 5.7 Pha angl diffrnc btwn poitiv and ngativ qunc componnt w.r.t. th motor fault poition whr ξ = Figur 5.8 Simulatd main-fd i a pctrum with ξ d = 0.5, β = 0, lip = Figur 5.9 Simulatd main-fd I pctrum with ξ d = 0.5, β = 0, lip = Figur 5.10 Simulatd main-fd i a pctrum with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip = Figur 5.11 Simulatd main-fd I pctrum with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip = Figur 5.12 Simulatd main-fd motor haft pd profil with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip = Figur 5.13 Simplifid Simulink chmatic to imulat main-fd rotor ccntriciti 74 Figur 5.14 Simulatd main-fd i a pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip = Figur 5.15 Simulatd main-fd I pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip = Figur 5.16 Simulatd main-fd motor haft pd profil with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip = Figur 5.17 Simulatd main-fd i a pctrum with 100% load torqu ocillation Figur 5.18 Simulatd main-fd I pctrum with 100% load torqu ocillation Figur 5.19 Simulatd main-fd motor haft pd profil for a halthy motor with 100% load torqu ocillation Figur 5.20 A typical clod-loop induction motor driv ytm Figur 5.21 Simulink chmatic uing rformulatd motor inductanc for clod-loop driv-connctd machin xiii

14 Figur 5.22 Simulatd driv-connctd i a pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = Figur 5.23 Simulatd driv-connctd v a pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = Figur 5.24 Simulatd driv-connctd I pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = Figur 5.25 Simulatd driv-connctd V pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = Figur 5.26 Simulatd driv-connctd i a pctrum with 100% load torqu ocillation Figur 5.27 Simulatd driv-connctd v a pctrum with 100% load torqu ocillation Figur 5.28 Simulatd driv-connctd I pctrum with 100% load torqu ocillation Figur 5.29 Simulatd driv-connctd V pctrum with 100% load torqu ocillation Figur 5.30 Maxwll 2D modl for th induction motor Figur 5.31 FEA modl imulatd main-fd i a pctrum with 30% tatic ccntricity and 30% dynamic ccntricity Figur 5.32 FEA modl imulatd main-fd I pctrum with 30% tatic ccntricity and 30% dynamic ccntricity Figur 5.33 FEA modl imulatd main-fd i a pctrum with 50% load torqu ocillation Figur 5.34 FEA modl imulatd main-fd I pctrum with 50% load torqu ocillation Figur 5.35 Maxwll 2D olvr tup intrfac Figur 7.1 Illutration of xprimntal tup with load ocillation control circuit xiv

15 Figur 7.2 Stator winding configuration for 230 Volt opration Figur 7.3 Schmatic to control load torqu ocillation Figur 7.4 Quai inuoidal load torqu ocillation profil Figur 7.5 IGBT board configuration Figur 7.6 Connction btwn th DSP board and IGBT board Figur 8.1 Main-fd ccntric motor I pctrum at 1776 rpm with contant load Figur 8.2 Main-fd halthy motor I pctrum at 1776 rpm with contant load Figur 8.3 Main-fd halthy motor I pctrum at avrag 1776 rpm with load ocillation Figur 8.4 Normalizd ngativ qunc harmonic currnt at ( f f ) + for an ccntric motor with contant load and a halthy motor with load ocillation Figur 8.5 Normalizd ngativ qunc harmonic currnt at ( f f ) rm for an ccntric motor with contant load and a halthy motor with load ocillation Figur 8.6 FFT pctrum of tator currnt pac vctor for th ccntric motor Figur 8.7 FFT pctrum of tator voltag pac vctor for th ccntric motor Figur 8.8 FFT pctrum of tator currnt pac vctor for th halthy motor Figur 8.9 FFT pctrum of tator voltag pac vctor for th halthy motor Figur 8.10 Normalizd fundamntal ngativ qunc tator currnt v. rfrnc pd Figur 8.11 Normalizd fundamntal ngativ qunc tator voltag v. rfrnc pd Figur 8.12 Normalizd total fundamntal ngativ qunc componnt v. rfrnc pd Figur 8.13 Propod complt onlin rotor ccntricity fault dtction chm for induction motor rm xv

16 Figur C.1 ADSP EZ-KIT Lit board layout xvi

17 SUMMARY Stator currnt pctral analyi tchniqu ar uually ud to dtct rotor fault in induction machin. Magntic fild anomali in th air-gap du to th rotor fault rult in charactritic id-band harmonic componnt in th tator currnt pctrum, which can b maurd a rotor fault ignatur. A poition-varying load torqu ocillation at multipl of th rotational pd, howvr, ha xactly th am ffct. Stator currnt harmonic du to a load torqu ocillation oftn obcur and vn ovrwhlm rotor ccntricity fault dtction inc th magnitud of load ocillation inducd harmonic i uually much largr. Although prviou rarch ha uggtd om mthod to diffrntiat btwn th two ffct, mot of thm rly havily on th accurat timation of motor paramtr. Th objctiv of thi rarch i to dvlop a far mor practical and computationally fficint mthod to dtct rotor fault ffctivly in th prnc of a load torqu ocillation. A ignificant advantag of th propod chm i that it do not nd any knowldg of motor paramtr. Th normalizd ngativ qunc information inducd by a mixd rotor ccntricity in th tator currnt or tator voltag pac vctor pctra, rv a a rliabl rotor fault indicator to liminat load ocillation ffct. Dtaild airgap magntic fild analyi for an ccntric motor i prformd and vral machin inductanc matric a wll a thir drivativ ar rformulatd accordingly. Carful obrvation of th inductanc matric provid a fundamntal undrtanding of motor opration charactritic undr a fault condition. Simulation xvii

18 rult bad on both induction motor dynamic modl and Maxwll 2D Finit Elmnt Modl dmontrat clarly th xitnc of th prdictd rotor fault indicator. Extniv xprimntal rult alo validat th ffctivn and faibility of th propod dtction chm. xviii

19 1 CHAPTER 1 INTRODUCTION 1.1 Ovrviw Th induction motor (IM) ha bn th hor powr of indutry for many yar. In an indutrializd nation, thy can typically conum 40%-50% of all th gnratd capacity of that country. Effctiv onlin condition monitoring of IM i critical to improving th productivity, rliability and afty to avoid unxpctd downtim and xpniv rpair cot. Thrfor, th diagnoing of th halth condition of IM i rciving mor and mor attntion from indutry in th pat dcad inc it can dtct an incipint fault at an arly tag. In gnral, noninvaiv dtction mthod, uch a th motor currnt ignatur analyi (MCSA) tchniqu [1]-[3], for condition monitoring i prfrrd ovr othr mthod inc it do not diturb th normal opration of IM. In addition, th only rquird dtction variabl ar th trminal upply currnt, which ar radily availabl in th motor control cntr (MCC). Th mot common two typ of rotor fault ar rotor ccntriciti and brokn rotor bar. Both of thm introduc om fault charactritic harmonic in th tator currnt which can b monitord to dtct rotor fault. Manwhil, om mchanical load condition hav imilar ffct on th tator currnt pctrum, which may cau ambiguity ithr in th condition monitoring of IM or in th mchanical fault dtction [5]-[21]. 1

20 1.2 Rotor Eccntriciti in Induction Motor Baically, thr ar two typ of ccntriciti: tatic ccntricity and dynamic ccntricity. In th ca of a tatic ccntricity, th minimum radial airgap lngth i fixd in pac. Th cntr of rotation i locatd at th cntr of rotor and i away from th cntr of th tator bor. Static ccntricity might com from impropr poitioning of tator cor and rotor cag, tator cor ovality, an xciv axial load, or worn baring. Howvr, in th ca of a dynamic ccntricity, th minimum radial airgap lngth i rotating with th rotor around th tator innr priphry, yt th cntr of rotation i till locatd at th cntr of tator. Dynamic ccntricity might com from a bnt rotor haft or oprating at om critical pd which cau mchanical ronanc. In practic, both typ of ccntriciti tnd to coxit bcau thr i alway an inhrnt lvl of tatic or dynamic ccntricity vn in a nwly commiiond motor du to th manufacturing and ambly imprfction. Thrfor, purly tatic or dynamic ccntricity do not xit in a ral machin. Rotor ccntriciti produc a radial magntic forc on th rotor haft, which i wll known a th unbalancd magntic pull (UMP). UMP act to pull th rotor vn furthr away from th tator bor cntr. Thi nforcd rotor diplacmnt cau xciv tr on th machin and gratly incra baring war. Unl dtctd arly, th ffct may nowball finally into a tator-to-rotor rub, lading to ignificant damag to th tator cor, tator winding, and rotor cag. Evntually, thi damag rult in a major brakdown of th machin and a cotly rpair. Du to th airgap prmanc modulation of th inducd tator and rotor MMF a wll a th intraction among th tatic ccntricity, dynamic ccntricity, tator and 2

21 rotor lotting ffct, aturation ffct, and vn tator and rotor pac and tim harmonic, rotor ccntriciti giv ri to th apparanc of om high frquncy charactritic id-band harmonic in th tator currnt pctrum, who frqunci ar locatd at [27], 1 fcc _ high = ( nr ± nd ) ( ) ± 2na ± nw f p (1.1) whr n = 1, 2, 3, i th rotor lotting ffct ordr numbr, R i th numbr of rotor bar, n d i th dynamic ccntricity ordr numbr ( n d = 0 corrpond to a pur tatic ccntricity), i th pr unit lip, p i th fundamntal pol pair numbr, n a i th aturation ffct ordr numbr, n w i th tator MMF tim harmonic ordr numbr and f i th fundamntal upply frquncy, which i 60Hz in th U.S. Th principal lot harmonic (PSH) ar alo givn in (1.1) by n d = 0, n a = 0 and n w = 1. Frquncy componnt givn by (1.1) aociatd with tripln pol pair numbr hould b idally abnt in th upply currnt of a balancd 3-pha IM [4]. In othr word, th ccntricity rlatd high frquncy charactritic harmonic can only hav harmonic pol pair numbr of m p, whr m = 1, 5, 7, 11, 13,... Obviouly, th dominant charactritic frquncy corrpond to th fundamntal pol pair numbr p whn m = 1. If th motor can not b dimantld to find th xact numbr of rotor bar, dtcting rotor ccntriciti uing (1.1) might cau problm. In addition, it wa dmontratd in [4] that for om combination of p and R, th componnt givn by (1.1) may not b prnt in th tator currnt pctrum. Morovr, with th incraing u of invrtr-fd induction motor driv, th high frquncy harmonic bcom difficult to idntify du to th driv control loop filtring ffct [24]-[28]. 3

22 In th prnc of a mixd ccntricity, low frquncy id-band harmonic will alo appar in th tator currnt pctrum. Thy ar locatd at [24]-[26], 1 f = (1 ± k ) f p cc _ low (1.2) whr k i an intgr. Th dominant low frquncy charactritic fault componnt corrpond to th ca whr k = 1. Th xitnc of th harmonic i du to th intraction btwn both typ of ccntricity. Thortically paking, only whn both tatic ccntricity and dynamic ccntricity xit imultanouly, will harmonic givn by (1.2) b xpctd to appar. Sinc thr i alway om ridual lvl of tatic or dynamic ccntricity, (1.2) i commonly monitord to dtct rotor ccntriciti, irrpctiv of th combination of p and R. 1.3 Brokn Rotor Bar in Induction Motor For brokn bar typ of rotor fault, th corrponding id-band charactritic harmonic ar locatd at [1]-[3], f = (1± 2 k ) f brb (1.3) Again, th main fault indicator in th prnc of thi typ of rotor fault corrpond to th ca whr k = 1. Howvr, if i vry mall (at light load condition) or th FFT pctral rolution i too coar, dtcting brokn rotor bar uing (1.3) may po om difficulty du to th intrinic pctral lakag proprty of th FFT algorithm. 1.4 Varying Mchanical Load Stat of th art condition monitoring chm for IM rotor fault ar uually dvlopd undr th aumption of a contant load torqu condition. Abnormal mchanical load condition and varying load lvl, howvr, hav alo a ignificant 4

23 impact on th charactritic fault ignatur in th currnt pctrum. For xampl, th magnitud of th low frquncy id-band harmonic in (1.2) ud for dtcting rotor ccntriciti can vary rmarkably from no load to full load [12]. Som typ of mchanical fault uch a unbalancd load, angular and radial haft mialignmnt, tc., ntially crat a rotor ccntricity inid th motor. Thrfor, dtcting th mchanical fault can dirctly tak advantag of tho fault indicator xprd in (1.2). Othr typ of mchanical load anomali may alo introduc xactly th am id-band harmonic in th tator currnt pctrum whil thr i no impact on th rotor haft. Altrnativly paking, th mchanical load anomali do not produc any rotor fault inid th motor itlf. Hnc ambiguity may ari whn dtcting ithr rotor fault or mchanical load anomali by analyzing id-band harmonic in a ingl pha tator currnt pctrum. Th mchanical load anomali includ vral working mod. Thir xprion and corrponding ffct on th tator currnt pctrum ar brifly ummarizd blow Poition-Varying Load Torqu Ocillation A poition-varying load torqu ocillation ha th following xprion, T = T + η T co( θ ) load avg avg rm (1.4) whr T avg i th avrag load torqu, θ rm i th mchanical rotor angl with rpct to th tator, and th cofficint η dnot th load torqu ocillation lvl. Th valu of η can b a mor than 1.0 for om application uch a rciprocating compror [5]. Auming th mchanical ytm i linar, all th frquncy componnt in th load torqu will appar in th haft torqu, thu lading to a torqu rippl int m. In an idal 5

24 induction motor modl with purly inuoidal input voltag and ngligibl tator ritanc, th qd-axi tator flux linkag contain only fundamntal componnt. Th intantanou torqu in an arbitrary rfrnc fram i givn by, 3 Tm = ( ) p ( λdiq λqid ) 2 (1.5) Obviouly, any ocillation in th load torqu at a multipl of th rotational pd k f rm will induc corrponding id-band harmonic in th tator currnt pctrum at, 1 f = f ± k f = (1 ± k ) f p load rm (1.6) whr f rm i th rotor mchanical rotation frquncy. It i clar that, quation (1.2) and (1.6) rprnt th am tator currnt id-band harmonic. Sinc th load ocillationinducd harmonic uually hav a largr magnitud than tho rulting from rotor ccntriciti, it i vry difficult to diffrntiat th two ffct uing convntional ingl pha tator currnt pctral analyi tchniqu Singl Frquncy Ocillating Load Torqu Similar to th prviou ca, on can conidr an ocillating load torqu at a ingl frquncy f 0 uprimpod to a contant avrag torqu. Th load torqu can b xprd a, T = T + η T co(2 π f t) = T + T co(2 π f t) load avg avg 0 avg oc 0 (1.7) If both T oc and f 0 ar conidrably mall with rpct to T avg and f, rpctivly, th problm can b tratd uing a mall ignal approach. Analytical drivation [15] how that id-band harmonic at f ± f0 appar in th tator currnt pctrum and th 6

25 amplitud of th harmonic currnt modulation I i rlatd to th load torqu ocillation, T load, by, I Tm ξ Tload = = 2 Icoϕ Tavg ξ + (2 π fj) Tavg (1.8) whr 2Icoϕ i th fundamntal tator currnt corrponding to th avrag load torqu with coϕ bing powr factor, ξ i th ratio btwn th haft torqu rippl and rotor pd rippl with ξ = T () t ω () t, and J i th motor-load ytm inrtia. At vry low frqunci, if 2π fj 0 m rm ξ, th haft torqu rippl i almot qual to th load torqu rippl. A th frquncy incra, th ffct of load diturbanc on th haft torqu and tator currnt will dcra accordingly Priodic Load Torqu Dip A typical priodic load diturbanc i a dip in th torqu. In thi ca, th pctrum of th load torqu, auming a priod qual to 1 f 0, will contain a fundamntal componnt at frquncy f 0, and a qunc of high ordr harmonic at frqunci k f0, who magnitud obviouly dpnd on th duty ratio of th torqu dip. Th total ffct of thi typ of load anomaly can b conidrd a a uprpoition of a ri of ingl frquncy ocillating load. Thrfor, id-band harmonic locatd at f ± k f0 will appar in th tator currnt pctrum. In th ca that f 0 i vry clo to th doubl lip frquncy, i.., f 0 2 f, th dtction of brokn rotor bar uing fault indication in (1.3) may b confud by a priodic dip in th load torqu. 7

26 1.4.4 Random Load Torqu Dip In ca of a random dip in th load torqu, th liding window play an important rol in th pctral analyi. Diffrnt width of th lctd obrvation window aign diffrnt priod to th random dip ignal in th load torqu. Th claical tator currnt pctral analyi i not uitabl to dtct thi typ of load anomaly inc th location of fault ignatur in th tator currnt pctrum dpnd on th lctd width of th obrvation window. Tim-frquncy pctral approach uch a th Wignr Ditribution mthod ha to b mployd to dtct thi typ of load fault. 1.5 Objctiv of th Rarch A can b n from th background introduction prntd in prviou ction, both rotor fault and mchanical load anomali introduc imilar harmonic pattrn in th tator currnt pctrum, which may cau ambiguity in condition monitoring of IM. Th objctiv of thi rarch i to dvlop a impl, motor paramtr-indpndnt chm to ffctivly dtct rotor ccntricity fault in th prnc of a poition-varying load torqu ocillation. A mixd rotor ccntricity introduc unbalanc in th motor paramtr, which furthr lad to incrad ngativ qunc information in th thr pha upply currnt. Howvr, in th ca of a poition dpndnt load torqu ocillation, th motor itlf i till ymmtric. Idally only poitiv qunc information hould xit in th tator currnt. Thrfor, by dtcting th incrad motor aymmtry lvl from th motor trminal quantiti, th two ffct can b ffctivly paratd. 8

27 Th ultimat goal of thi rarch i to dvlop a rliabl, computationally fficint and load indpndnt algorithm for onlin rotor ccntricity dtction in both main-fd and driv-connctd induction motor. 1.6 Dirtation Outlin A comprhniv litratur rviw on th xiting tchniqu to parat load ffct from rotor fault dtction i givn in Chaptr 2. Thir rpctiv mrit and hortcoming ar ummarizd to familiariz th radr with th background framwork of thi rarch. In Chaptr 3, a dtaild magntic fild analyi in th airgap du to a purly tatic ccntricity, a purly dynamic ccntricity and a mixd rotor ccntricity i prntd. All kind of machin inductanc a wll a thir drivativ ar rformulatd accordingly. In ordr to accuratly xtract ngativ qunc harmonic information from thr pha currnt, a nw FFT tchniqu i propod in Chaptr 4. Thi mthod ha ignificant advantag ovr th traditional ymmtrical componnt dcompoition mthod for thi application. Bad on th fundamntal undrtanding of motor oprating charactritic undr a fault condition, a nw rotor ccntricity indicator i propod in Chaptr 5. Simulation rult from both a implifid Matlab Simulink modl and a tim-conuming Finit Elmnt Modl ar alo prntd in thi Chaptr. Conidring that th motor aymmtry may alo rult form a poibl tator intr-turn fault, zro-qunc information i utilizd to ditinguih th two cau. A dtaild mathmatical drivation on thi topic i givn in Chaptr 6. Chaptr 7 dcrib th baic xprimntal tup including ccntric and halthy motor, th dynamomtr, th clod-loop driv and th data acquiition ytm. Th 9

28 principl and implmntation mthod for th load torqu ocillation ar alo xplaind thrin. Exprimntal rult for a main-fd motor and a clod-loop driv-connctd motor, with man-mad ccntricity or with load torqu ocillation, ar all givn in Chaptr 8, in ordr to dmontrat th ffctivn of th propod nw rotor ccntricity indicator. Bad on th xprimntal rult, a rliabl, computational fficint and load indpndnt rotor ccntricity dtction chm i illutratd. Finally, Chaptr 9 conclud th dirtation and dcrib th contribution of thi rarch a wll a th rcommndd futur rmaining work. 10

29 2 CHAPTER 2 SUMMARY OF PREVIOUS WORK ON SEPARATING LOAD EFFECTS FROM ROTOR FAULT DETECTION 2.1 Ovrviw Both rotor ccntricity and brokn rotor bar fault may b confud with diffrnt typ of load anomali in th tator currnt pctrum. Although th two typ of rotor fault hav diffrnt natur, thy har th am baic rotor fault dtction chm. A ingl pha tator currnt i analyzd by FFT and corrponding charactritic fault ignatur ar locatd according to (1.2) or (1.3). In addition, th varying mchanical load lvl or mchanical ronanc point hav ignificant impact on th rotor fault dtction a wll. Th litratur rviw of thi chaptr will addr th iu inc thy all dal with th am topic of parating load ffct from rotor fault dtction. 2.2 Modl Rfrnc Etimation Th rarch work of [5]-[7] propo a chm to dtrmin th machin halth condition in th prnc of a poition-varying load torqu ocillation. Thi i accomplihd by comparing th actual tator currnt to a modl rfrnc valu which includ th load ffct. Th diffrnc btwn th two ignal provid a filtrd quantity which i indpndnt of load variation. Thi load indpndnt currnt information can b ud for continuou on-lin condition monitoring without concrn for th load ffct. 11

30 In th ynchronou rfrnc fram with th d-axi alignd with th rotor flux linkag, th d-axi voltag quation can b writtn a, v d = Ri ω λ + λ dt r r r r d d q d (2.1) v = d 0 = Ri + dt λ (2.2) r r r dr r dr dr Obviouly, quation (2.2) i indpndnt of th rotor angular pd ω r. Sinc th inductanc till kp contant in th prnc of a load ocillation, combining (2.2) with th d-axi flux linkag quation, λ = Li + L i r r r d d m dr λ = Li + Li r r r dr m d r dr giv th following xprion for th drivativ of th d-axi tator currnt, r r d r RLi r d Rλd L d r λd r i dt d = dt L L L 2 m r (2.3) (2.4) (2.5) xprd in trm of tator variabl only. Rarranging (2.1) allow th drivativ of th d-axi tator flux linkag to b writtn a, d dt λ = v Ri + ωλ (2.6) r r r r d d d q A hown in Figur 2.1, th drivativ quation (2.5) and (2.6) can b ud in th timator to dtrmin th prnt chang rat in th d-axi tator currnt for an idal machin undr arbitrary load condition. Onc d r id dt ha bn dtrmind in th timator from th maurd tator currnt and voltag, th d-axi tator currnt at th nxt ampling intant can b timatd by Eulr mthod, 12

31 v q v d i q i d λ q λ d λqr λdr r vq r v d r iq r i d r λq r λ d d dt λ r d d r id dt r id i ˆr d r i d i ˆr d ˆ d i = i + h* i dt r r r d d d h Figur 2.1 Block diagram of th modl rfrnc timation ˆ r r d r id ( N + 1) = id ( N) + h* id ( N) dt (2.7) whr h i th tim intrval btwn two ampling point. Th timatd d-axi tator currnt i ˆr d i thn ubtractd from th actually maurd quantity i r 13 d. Thi diffrnc i ignificant inc it provid a dtctabl ignal which i indpndnt of th load ffct whil till containing th tator currnt harmonic information introducd by th rotor fault ffct. Th major drawback of thi chm li in it havy dpndnc on th accurat timat of motor paramtr including R, R r, L m, L l and L lr. In gnral, it i vry difficult, if not impoibl, to achiv uch an accurat timat du to th tmpratur ri, magntic aturation and kin ffct. Sinc th timation chm aum th load torqu rmain contant during th ampling intrval, th ampling frquncy ha a dirct impact on th limination of load ffct. Manwhil, in ordr to produc a ufficintly high frquncy rolution in FFT analyi, th total ampling tim can not b too hort.

32 Thrfor, th computational cot and mmory rquirmnt ar rlativly high for thi chm. 2.3 Currnt Park Vctor Approach Cruz and Cardoo uggt uing th ynchronou rfrnc fram Park Vctor Approach to dtct rotor fault [8]-[9]. It i claimd that, by uing thi approach, ffct of rotor fault can b diffrntiatd from tim-varying load. Simulation rult prntd in thi rarch work dmontrat that, for a tim-varying mchanical load, th currnt Park Vctor rprntation in th ynchronou rfrnc fram i ntially a traight lin, which i tangntial to th claical circl diagram of th induction motor. Howvr, th occurrnc of brokn rotor bar fault i charactrizd by th apparanc of an lliptic pattrn, who major axi lngth i dirctly rlatd to th vrity dgr of rotor fault. Onit tt rult giv a vagu indication to idntify rotor fault from timvarying load. In ummary, although thi mthod can provid om clu to diffrntiat rotor fault from load ffct, th confidnc lvl of thi judgmnt i rlativly low, thu prvnt it from rving a a rliabl rotor fault indicator in th prnc of a load torqu ocillation. 2.4 Vinna Monitoring Mthod Th Vinna Monitoring Mthod (VMM) i bad on th comparion of th calculatd torqu valu from a voltag modl and a currnt modl [10]-[11], a hown in Figur 2.2. For an idally ymmtric machin, torqu valu from th two modl hould b qual. Rotor fault, howvr, lad to two diffrnt torqu valu. Thrfor, brokn bar typ of rotor fault can b dtctd by aing th rippl componnt in th torqu diffrnc at th doubl lip frquncy. 14

33 i r λ _ V r i xr τ r r λr_ I v 1 τ r (a) (b) Figur 2.2 Vinna Monitoring Mthod voltag modl (a) and currnt modl (b) In th voltag modl, th tator flux linkag pac vctor can b calculatd by, d v ri dt λ = _ V (2.8) in th tationary rfrnc fram. Th currnt modl i dcribd in th rotor rfrnc fram a, d 1 x i dt λ + λ = r r r r r_ I r_ I τr τr (2.9) Tranformation of th tator currnt pac vctor btwn th tationary and rotor rfrnc fram i dfind by, i = i ω r j rt (2.10) Th voltag modl and currnt modl rlatd torqu valu can b calculatd a follow, t = Im( conj( i ) λ ) V _ V (2.11) t = Im( conj( i ) λ ) I r r r_ I (2.12) A brokn bar rotor fault lad to a torqu diffrnc from two modl. In ordr to liminat th dpndnc on th load torqu lvl, it i dirabl to divid th torqu diffrnc by an timatd load torqu and obtain a normalizd torqu diffrnc ignal, 15

34 V I t t t = (2.13) t load Th amplitud of thi normalizd quantity i indpndnt of load lvl. Th cond harmonic componnt in (2.13) will rvr a th rotor fault indicator. Th primary advantag of VMM mthod i it indpndnc on th varying load torqu lvl and inrtia although thi chm i till not rliabl for light load condition. In gnral, ituation with load torqu valu blow a crtain thrhold (around 40% of th full load) hould b xcludd for diagnotic uing VMM. Th ffct of timvarying load torqu ocillation on rotor fault dtction in VMM ar not clar yt. Similar to th modl rfrnc timation mthod, th VMM alo uffr ignificantly from th inaccurat timat of motor paramtr including tator ritanc r, rotor ractanc x r and rotor tim contantτ r. 2.5 Othr Tchniqu for Eliminating Load Effct In rcnt publication, om othr tchniqu ar alo propod to liminat varying load torqu lvl and mchanical ronanc ffct on th tator currnt bad-condition monitoring of IM. Obaid [12]-[13] tak into account th iu by claifying th load lvl into vral bin and calculating an avrag thrhold valu for ach load bin. To avoid miing mchanical ronanc point within any load bin, th final balin i obtaind by wping th load lvl throughout ach bin and thn calculating an avrag balin for diffrnt load lvl within th bin. All final balin for diffrnt load bin ar computd during th initialization tag of th algorithm and combind togthr to rv a th fault thrhold with a 50% tolranc margin. Whn th thrhold valu i continuouly compard with actually maurd fault ignal, th halth condition of IM 16

35 can b dtctd undr any load lvl. Th flow chart of thi balin initialization procdur i hown in Figur 2.3. Huang [14] ralizd th imilar ida for th rotor ccntricity dtction in a clodloop driv-connctd induction motor by uing a uprvid multi-layr prcptron fd forward nural ntwork (MLP-FF-NN). In an AC driv, th mchanical pd can vary widly. Thi rult in a variation in th ccntricity-rlatd charactritic fault frquncy, load lvl and magnitud of th fault harmonic. Th rlationhip btwn th harmonic magnitud and th load lvl (rotor pd) i non-monotonic in gnral du to th mchanical ronanc ffct. Sinc it i vry difficult or vn impoibl to formulat thi rlationhip in a trictly analytical mannr, an artificial nural ntwork (ANN) can b ud to larn th complx rlationhip with only a finit numbr of oprating point, and thn timating rmaining fault ignatur amplitud for othr oprating condition. Th actual motor pd ω m and corrponding fundamntal driv upply frquncy f 1 ar ud a input to th ANN to dtrmin th motor lip and thu th oprating point. Th output layr of th ANN giv th ytm output, namly th ccntricity rlatd total harmonic amplitud v fcc + i a hown in Figur 2.4. Aftr th fcc training procdur i complt, th ANN i rady to dtct rotor ccntricity fault undr any oprating condition. In both tchniqu, data mut b collctd for a halthy motor at vral oprating point during th initialization or training tag. For om application, howvr, it may not b poibl to nforc thi larning procdur. 17

36 Dfin load bin Incra load lvl by 5% No St load to th firt lvl in bin Acquir tator currnt Load lvl i lat in Bin? Y Comput and tor avragd balin for load bin Comput FFT of currnt Stor RMS a balin for load lvl Entr accptabl margin a prcntag of balin Extract th componnt f ± f rm Calculat RMS valu at f ± f rm Calculat and tor thrhold for dfind load bin Initiation complt Figur 2.3 Flowchart of th balin calculation during th initiation tag maurd v fcc +i fcc ω m f 1 timatd v fcc + i fcc Updating wight training dtction Figur 2.4 Gnral tructur of th ANN ud for ccntricity dtction undr diffrnt load condition 18

37 2.6 Mchanical Load Monitoring by Unuprvid NN Approach A dicud in ction 1.4, diffrnt typ of mchanical load anomali alo introduc harmonic pattrn in th tator currnt pctrum. Although induction motor itlf can rv a a torqu nor to dtct load troubl, thi procdur may b miundrtood du to rotor fault ffct. Sall and Filipptti [15]-[18] uggt an unuprvid lf-organization mapping (SOM) nural ntwork to ditinguih among th ignatur introducd by diffrnt load fault and tho introducd by othr machin troubl uch a rotor fault. Thi tchniqu utiliz a lctd t of currnt pctral componnt a input to th ytm to prform fault claification. Th data t contructd from computr imulation rult ar ud to train th nural ntwork offlin. Thrfor, thi mthod may not b uitabl for onlin condition monitoring du to th rlativly high computational cot. In addition, for ach nw typ of motor, a t of nw imulation rult ha to b producd to facilitat th training proc. Exprimntal rult dmontrat a good capability to ditinguih among th priodic torqu dip, ingl frquncy ocillating torqu and halthy mod. Howvr, th ditinction btwn th ocillating torqu and brokn bar fault, a wll a that btwn th priodic dip and random dip ar not xplicit. Although it i poibl to incra th dimnion of th NN iz to achiv bttr prformanc, th total computation cot incra accordingly. 2.7 Load Diagnotic Utilizing Intantanou Powr Spctrum Auming an idal thr pha upply and a halthy tator winding, th intantanou powr i dfind a [19]-[20], p() t = v () t i () t (2.14) LN L 19

38 whr vln ( t ) i th lin-to-nutral voltag and il ( t ) i th trminal pha currnt. For an idal machin running at a contant load, th rfrnc intantanou powr i givn by, p () t = V co(2 π f t) I co(2 π f t ϕ) 0 m m Vm Im = [ co(2 π (2 f ) t ϕ) + co( ϕ)] 2 (2.15) whr V m and I m dnot th voltag and currnt amplitud, rpctivly, and ϕ i th load angl. It i clar to th intantanou powr pctrum includ both DC componnt and fundamntal componnt at 2 f. Auming a mchanical fault cau a inuoidal modulation of th tator currnt amplitud and th load angl ϕ do not chang ignificantly, th modulatd tator currnt can b xprd a, i () t = I co(2 π f t ϕ)[1 + Mco(2 π f t)] L m MIm = Imco(2 π ft ϕ) + co[(2 π( f f0) t ϕ)] 2 MIm + co[(2 π( f + f0) t ϕ)] 2 0 (2.16) whr M i th modulation indx and f 0 i th modulation frquncy. Clarly, two id-band harmonic appar in th tator currnt pctrum. Combing quation (2.14) and (2.16) giv th modulatd intantanou powr a, MVmI m p= p0() t + co( ϕ) co(2 π f0t) 2 MVmI m MVmI m + co[(2 π (2 f f0) t ϕ)] + co[(2 π(2 f + f0) t ϕ)] 4 4 (2.17) Excpt for th fundamntal and two id-band harmonic at (2 f f0) and (2 f + f0), thr xit an additional componnt locatd at th modulation frquncy f 0 in (2.17). Th lat componnt provid xtra diagnotic information in load fault dtction. 20

39 Simulation rult in [19] dmontrat that thi tchniqu ha om advantag ovr othr tchniqu in dtcting mchanical load anomali. Howvr, it i till difficult to ditinguih btwn th load ocillation ffct and rotor aymmtry fault. On th contrary, thi approach rliv th FFT pctral lakag phnomnon in om xtnt inc th modulation frquncy componnt appar far from th fundamntal in th intantanou powr pctrum. 2.8 Load Fault Dtction by Tim-Frquncy Analyi Thi rarch work [21]-[22] introduc vral ignal procing mthod in tator currnt fault ignatur analyi. Thortical dvlopmnt uing MMF wav approach in thi rarch claim that a priodic load ocillation rult in a inuoidal pha modulation in a tator currnt componnt. Th modulatd tator currnt can b xprd in a gnral form by, I() t = i () t + i () t = I in( ω t+ ϕ ) + I in( ω t+ βco( ω t)) (2.18) t rt t rt c whr ω c i th load ocillation charactritic frquncy in rad/, th paramtr β i calld modulation indx and ϕ i th initial pha diffrnc btwn th rotor and tator MMF. Th trm i ( t ) rulting from th tator MMF i not modulatd. Howvr, th t trm i ( t ) which i cloly rlatd to th rotor MMF how a pha modulation du to rt th conidrd load torqu ocillation. Th valu of β i zro for a contant load condition. In th prnc of a load ocillation, th tator currnt ignal i no longr trictly tationary inc th frquncy of i ( t) i varying inuoidally with tim. rt Th claical powr pctrum dnity (PSD) mthod rprnt th baic ignal analyi tool for tationary ignal in frquncy domain. It giv a firt indication of a 21

40 poibl load ocillation rlatd mchanical fault by an incra of id-band harmonic at f ± fc. Sinc PSD may lad to an ambiguity btwn load fault and inhrnt rotor ccntricity, tim-frquncy pctrum analyi tool including intantanou frquncy (IF) timation and Pudo Wignr Ditribution (PWD) ar propod to achiv bttr dtction capability of pha modulation in a tranint, non-tationary ignal. Th ffct of rotor ccntricity fault on th tator currnt amplitud and pha modulation i only ad on a purly dynamic ccntricity modl, not a mixd ccntricity. Th xprimntal load torqu ocillation i implmntd by mounting an unbalanc dik on th haft [22], which ntially crat an ccntricity in th airgap in addition to th load ocillation. In othr word, both th rotor ccntricity and th load torqu ocillation, which i impod on th motor haft, yild mchanical fault inid th motor. Thrfor it i till qutionabl to diffrntiat rotor mchanical fault from normal indutrial load with poition-dpndnt torqu ocillating charactritic, uch a rciprocating compror. In addition, thi tchniqu nd a vry high ampling frquncy for tim-frquncy ignal analyi and lad to a rlativly havy computational burdn. 2.9 Chaptr Summary A comprhniv litratur rviw on th xiting tchniqu of parating load ffct from rotor fault dtction i ummarizd in thi chaptr. Thir rpctiv mrit and drawback ar analyzd a wll. Although it i an important indutrial concrn in onlin condition monitoring of induction motor, it wa flt that vry littl work ha bn don on th topic of ffctivly dtcting rotor ccntricity fault in th prnc of a load torqu ocillation for both main-fd and driv-connctd induction motor. Thi 22

41 tak hould b achivd without dpndnc on th accurat timat of motor paramtr. Thi rv a th motivation and objctiv of th currnt rarch work. 23

42 3 CHAPTER 3 MAGNETIC FIELD ANALYSIS FOR ECCENTRIC MOTORS 3.1 Ovrviw It i wll known that th ffct of a purly tatic ccntricity ar quantitativly imilar to tho caud by a tator aymmtry, and a purly dynamic ccntricity lad to ffct imilar to a rotor aymmtry [32]-[33]. In practic, both typ of ccntriciti tnd to coxit bcau thr i alway an inhrnt lvl of tatic or dynamic ccntricity vn in a nwly commiiond machin du to th manufacturing and ambly imprfctn. 3.2 Purly Rotor Static Eccntricity An lmntary 2-pol, 3-pha induction motor with inuoidal winding ditribution on both tator pha and quivalnt rotor pha i modld to valuat th ffct of a purly tatic ccntricity. For th purpo of clarity, and without lo of gnrality, pac and tim harmonic in th inducd MMF, tator and rotor lotting ffct, and magntic aturation ffct ar all nglctd in th modl. Undr th aumption, it i ay to undrtand that thr i only fundamntal componnt in upply currnt. For a purly tatic ccntricity, th firt ordr approximation of th invr air-gap lngth can b xprd a [24]-[26], 1 g a0 a1 ( φ ) = + co( φ α) (3.1) 24

43 whr φ i th mchanical angular maur around th tator innr priphry, a 0 i th invr of th avrag ffctiv airgap lngth, a 0 = 1 g 0, and a 1 i trmd a th tatic ccntricity prmanc factor, 2 ( δ ) 21 1 a1 = δ 1 δ 2 (3.2) whr δ rprnt th dgr of tatic ccntricity. That i, δ i ratio of th ditanc btwn th cntr of rotation and th cntr of th tator bor, to th nominal airgap lngth. Dtaild Fourir ri analyi ha hown that, at a 40% dgr of tatic ccntricity, th cond harmonic cofficint of th Fourir ri producd by th invrion of th airgap lngth i only 20% of th firt harmonic cofficint [23]. It alo how that th firt harmonic cofficint ha a magnitud of on a normalizd bai in thi ca. Thrfor, quation (3.1) giv a vry raonabl approximation of th invrion of th airgap lngth for a purly tatic ccntricity. Th angl α in (3.2) dnot th rlativ poition btwn th motor fault and tator magntic axi a hown in Figur Modifid Stator and Rotor MMF Auming a inuoidal winding ditribution, th 3-pha tator winding ditribution can b writtn a [37], N N N N ( φ ) = in ( φ ), -π φ 0 2 N 2π π 2π ( φ ) = in φ, - φ N 4π π 4π ( φ ) = in φ, φ a b c (3.3) 25

44 whr N i th total numbr of turn pr tator pha. Bad on (3.3), th turn function for 3-pha tator winding hav th following form, n n n N ( φ ) = co ( ) +1 2 φ N 2π ( φ ) = co φ a b N 4π ( φ ) = co φ c (3.4) For a halthy motor with contant airgap lngth, g 0, th MMF producd by a unit ingl pha tator winding currnt can b aily obtaind by ubtracting th avrag valu of th turn function (dc offt) from itlf. Howvr, in th ca of a rotor ccntricity, thi mthod do not work inc it do not atify th fundamntal Gau Law. Th following drivation for th modifid MMF in th prnc of a purly tatic ccntricity i carrid out in a imilar way to [29]-[31] whr it wa originally drivd for ynchronou machin. Conidring a clod magntic loop abcda in a 2-pol induction motor with purly tatic ccntricity and unit currnt xcitation a hown in Figur 3.2, on can obtain th following xprion from Ampr Law, H dl = J ds (3.5) abcda or MMF(0) + MMF( φ ) = n( φ ) (3.6) S whr th prmability in both tator and rotor iron cor i aumd to b infinit and thr i no MMF drop along th magntic path inid iron. In addition, all magntic fild quantiti ar aumd to hav only radial componnt and rmain contant along ach radial dirction. Th fundamntal Gau Law ay that, 26

45 b-axi Rr R Cr α a-axi C c-axi Figur 3.1 Static air-gap ccntricity. C i th cntr of tator and Cr i th cntr of rotor. R i th tator innr radiu and Rr i th rotor outr radiu. b-axi Rr R Cr C φ a d b c a-axi c-axi Figur 3.2 An lmntary 2-pol induction machin with tatic ccntricity, howing a clod magntic loop abcda. 27

46 whr B( φ ) S B( φ ) ds = 0 (3.7) rprnt th radial magntic fild flux dnity in th airgap and th urfac intgral can b takn in a clod cylindrical urfac jut locatd outid th rotor outr priphry. Bad on th aumption, quation (3.7) can b rwrittn a, or 2π 0 µ MMF φ g φ r l dφ = 0 1 ( ) ( ) 0 (3.8) 2π 1 MMF( φ) g ( φ) dφ = 0 (3.9) 0 whr µ 0 i th airgap prmability, r i th tator innr radiu, and l i th axial tack lngth of th machin. Multiplying both id of (3.6) by g 1 ( φ ) and intgrating from 0 to 2π, on obtain, 2π 2π 1 1 { MMF(0) + MMF( )} g ( ) d = n( ) g ( ) d 0 0 φ φ φ φ φ φ (3.10) Subtituting (3.9) into (3.10), MMF(0) = 2π 0 n φ g φ dφ 1 ( ) ( ) 2π 0 g 1 ( φ ) dφ (3.11) Finally, from (3.6) and (3.11), th modifid tator MMF can b xprd a, MMF( φ ) = n( φ ) 2π 0 n φ g φ dφ 1 ( ) ( ) 2π 0 g 1 ( φ ) dφ (3.12) Combining (3.1), (3.4) and (3.12) togthr, th modifid 3-pha tator MMF with a unit currnt xcitation and a purly tatic ccntricity can b calculatd a, 28

47 MMF MMF N ξ ( φ ) = co( φ ) co( α) 2 2 a N 2π ξ 2π ( φ ) = co φ co( α ) b N 4π ξ 4π MMFc ( φ ) = co φ co( α ) (3.13) whr ξ = a1 a0 i th tatic ccntricity cofficint. Similarly, th modifid quivalnt 3-pha rotor MMF undr th am condition can b calculatd a, MMF MMF N ξ ( φ, θ ) = co( φ ) co( α θr) 2 2 r ar r r r N 2π ξ 2π ( φ, θ ) = co φ co( α θr ) r br r r r Nr 4π ξ 4π MMFcr ( φr, θr) = co φr co( α θr ) (3.14) whr N r i th total numbr of turn pr quivalnt rotor pha, φ r i th mchanical angular maur around th rotor outr priphry, and θ r i th mchanical angular diplacmnt btwn th tator a-pha magntic axi and rotor A-pha magntic axi. Not that, in ordr to obtain (3.14), th wll known tranformation btwn φ and φ r, φ = φr + θr, ha to b applid into th calculation Motor Inductanc Rformulation Bad on th modifid tator and rotor MMF, all kind of machin inductanc including tator lf and mutual inductanc, rotor lf and mutual inductanc, and tator to rotor mutual inductanc ar calculatd uing th rgular doubl intgral mthod [34]. For intanc, 29

48 0 φ + π 1 Laa = Ll + Na ( φ ) µ 0 r l MMFa ( ξ) g ( ξ) dξ dφ π φ (3.15) All othr machin inductanc can b calculatd in a imilar way. Auming th quivalnt rotor pha ha th am numbr of turn a th tator pha, N = Nr, on can furthr implify th final inductanc matrix xprion uing th following prt ymbol, 2 N N Nr πµ 0 0 πµ 0 0 (3.16) Lm = rla = rla Thrfor, th tator lf inductanc matrix, tator to rotor mutual inductanc matrix and rotor lf inductanc matrix can b xprd a, Ll 1 1 [ L ] = 0 Ll 0 + Lm 1 (3.17) tatic L l π 4π co( α)co( α) co( α )co( α) co( α )co( α) ξ 2π 2π 2π 4π 2π L m co( α)co( α ) co( α )co( α ) co( α )co( α ) π 2π 4π 4π 4π co( α)co( α ) co( α )co( α ) co( α )co( α )

49 2π 4π co( θr) co( θr + ) co( θr + ) 3 3 T 4π 2π [ Lr ] = [ Lr ] = Lm co( θr + ) co( θr ) co( θr + ) (3.18) tatic tatic 3 3 2π 4π co( θr + ) co( θr + ) co( θr) 3 3 2π 4π co( α)co( α θr) co( α)co( α θr ) co( α)co( α θr ) ξ 2π 2π 2π 2π 4π Lm co( α )co( α θr ) co( α )co( α θr ) co( α )co( α θr ) π 4π 2π 4π 4π co( α )co( α θr) co( α )co( α θr ) co( α )co( α θr ) Llr 1 1 Lrr = 0 Llr 0 L m 1 tatic L lr [ ] (3.19) 2π 4π co( α θr)co( α θr) co( α θr )co( α θr) co( α θr )co( α θr) ξ 2π 2π 2π 4π 2π Lm co( α θr)co( α θr ) co( α θr )co( α θr ) co( α θr )co( α θr ) π 2π 4π 4π 4π co( α θr)co( α θr ) co( α θr )co( α θr ) co( α θr )co( α θr ) Th drivativ of th motor inductanc ar givn blow and will b ud in Chaptr 5 for dynamic imulation of th motor prformanc undr fault condition. [ ] d L dθ r tatic = (3.20) 31

50 2π 4π in( θr) in( θr + ) in( θr + ) 3 3 T d[ Lr ] d[ L ] tatic r tatic 4 2 L π π m in( θr ) in( θr ) in( θr ) (3.21) = = + + dθr dθ r 3 3 2π 4π in( θr + ) in( θr + ) in( θr) 3 3 2π 4π co( α)in( α θr) co( α)in( α θr ) co( α)in( α θr ) ξ 2π 2π 2π 2π 4π Lm co( α )in( α θr ) co( α )in( α θr ) co( α )in( α θr ) π 4π 2π 4π 4π co( α )in( α θr) co( α )in( α θr ) co( α )in( α θr ) [ ] 2 d L rr tatic dθ r 2π 4π in(2α 2 θr) in(2α 2 θr ) in(2α 2 θr ) 3 3 ξ 2π 4π = Lm in(2α 2 θr ) in(2α 2 θr ) in(2α 2 θr ) π 2π in(2α 2 θr ) in(2α 2 θr) in(2α 2 θr ) 3 3 (3.22) Motor Currnt Charactritic It i clar to that th firt row in (3.17), (3.18) and (3.19) ar jut th rgular inductanc matric for a halthy induction motor. Howvr, du to th xitnc of a purly tatic ccntricity, additional inductanc matric ar introducd into [ ] [ L ], and [ ] r tatic L. rr tatic L, Carful analyi of th inductanc matric provid om inight into th undrtanding of th motor oprating charactritic undr a fault condition. Obviouly, th diagonal lmnt of [ L ] ar no longr qual, vn though [ ] tatic tatic tatic L i till ymmtric and contant. It i th paramtr unbalanc in th inductanc matric that introduc ngativ qunc information into motor trminal quantiti for an ccntric ca. In addition, th dgr of motor aymmtry i cloly rlatd to th dgr of rotor 32

51 ccntricity. Thrfor, th highr th dgr of ccntricity; th largr th magnitud of th corrponding ngativ qunc componnt. Sinc th implifid induction motor modl conidr a inuoidal input, only th fundamntal componnt appar in th tator currnt pctrum. 3.3 Purly Dynamic Rotor Eccntricity For a purly dynamic ccntricity, th firt ordr approximation of th invr airgap lngth can b xprd a, 1 g r a0 a2 ( φ ) = + co( φ β) (3.23) r Modifid Stator and Rotor MMF Similar to th ca of a tatic ccntricity, th modifid 3-pha tator and rotor MMF for a purly dynamic ccntricity hav th following form, MMF MMF MMF N ξd = ( )[co( φ ) co( θ + β)] i 2 2 N 2π ξd 2π = ( )[co( φ ) co( θ + β )] i N 4π ξd 4π = ( )[co( φ ) co( θ + β )] i a r a b r b c r c (3.24) whr ξ d = a2 a0 i th dynamic ccntricity cofficint. MMF MMF MMF Nr ξd = ( )[co( φ ) co( β)] i 2 2 Nr 2π ξd 2π = ( )[co( φ ) co( β )] i Nr 4π ξd 4π = ( )[co( φ ) co( β )] i ar r ar br r br cr r cr (3.25) 33

52 3.3.2 Motor Inductanc Rformulation All machin inductanc a wll a thir drivativ for a purly dynamic ccntric motor can b xprd a follow, Ll 1 1 [ L ] = 0 Ll 0 + Lm 1 (3.26) dynamic L l π 4π co( θr + β)co( θr + β) co( θr + β )co( θr + β) co( θr + β )co( θr + β) ξd 2π 2π 2π 4π 2π Lm co( θr + β)co( θr + β ) co( θr + β )co( θr + β ) co( θr + β )co( θr + β ) π 2π 4π 4π 4π co( θr + β)co( θr + β ) co( θr + β )co( θr + β ) co( θr + β )co( θr + β ) π 4π co( θr) co( θr + ) co( θr + ) 3 3 T 4 2 [ Lr ] [ Lr ] L π π m co( θr ) co( θr ) co( θr ) (3.27) = = + + dynamic dynamic 3 3 2π 4π co( θr + ) co( θr + ) co( θr) 3 3 2π 4π co( θr + β)co( β) co( θr + β)co( β ) co( θr + β)co( β ) ξd 2π 2π 2π 2π 4π Lm co( θr + β )co( β) co( θr + β )co( β ) co( θr + β )co( β ) π 4π 2π 4π 4π co( θr + β )co( β) co( θr + β )co( β ) co( θr + β )co( β )

53 Llr 1 1 [ Lrr ] = 0 Llr 0 + Lm 1 (3.28) dynamic L lr π 4π co( β)co( β) co( β )co( β) co( β )co( β) ξd 2π 2π 2π 4π 2π L m co( β)co( β ) co( β )co( β ) co( β )co( β ) π 2π 4π 4π 4π co( β)co( β ) co( β )co( β ) co( β )co( β ) [ ] 2 d L dynamic d dθ r 2π 4π in(2θr + 2 β) in(2θr + 2 β ) in(2θr + 2 β ) 3 3 ξ 2π 4π = Lm in(2θr + 2 β ) in(2θr + 2 β ) in(2θr + 2 β) π 2π in(2θr + 2 β ) in(2θr + 2 β) in(2θr + 2 β ) 3 3 (3.29) 2π 4π in( θr) in( θr + ) in( θr + ) 3 3 T d[ Lr ] d[ L ] dynamic r dynamic 4 2 L π π m in( θr ) in( θr ) in( θr ) (3.30) = = + + dθr dθ r 3 3 2π 4π in( θr + ) in( θr + ) in( θr) 3 3 2π 4π in( θr + β)co( β) in( θr + β)co( β ) in( θr + β)co( β ) ξd 2π 2π 2π 2π 4π + Lm in( θr + β )co( β) in( θr + β )co( β ) in( θr + β )co( β ) π 4π 2π 4π 4π in( θr + β )co( β) in( θr + β )co( β ) in( θr + β )co( β ) [ ] d L rr dθ dynamic r = (3.31) 35

54 3.3.3 Motor Currnt Charactritic Again, th firt row in (3.26), (3.27) and (3.28) ar jut th rgular inductanc matric for a halthy induction motor. Howvr, du to th xitnc of a purly dynamic ccntricity, additional inductanc matric ar introducd into [ ] [ L ], and [ L ] r dynamic rr dynamic L, dynamic. A purly dynamic ccntricity do not introduc ngativ qunc information into tator currnt inc th cntr of rotation i till locatd at th cntr of th tator bor. Th airgap prmanc modulation by th dynamic ccntricity produc om idband harmonic into th tator currnt. 3.4 Mixd Rotor Eccntricity Th firt ordr approximation of th invr airgap lngth for a mixd rotor ccntricity can b xprd a [24]-[26], 1 g r a0 a1 a2 r ( φ, φ ) = + co( φ α) + co( φ β) (3.32) whr φ = φr + θr Modifid Stator and Rotor MMF Th modifid 3-pha tator and rotor MMF for a mixd rotor ccntricity ar baically th ummation of tho for a purly tatic ccntricity and a purly dynamic ccntricity, MMF MMF MMF N ξ ξd = ( )[co( φ ) co( α) co( θ + β)] i N 2π ξ 2π ξd 2π = ( )[co( φ ) co( α ) co( θ + β )] i N 4π ξ 4π ξd 4π = ( )[co( φ ) co( α ) co( θ + β )] i a r a b r b c r b (3.33) 36

55 MMF MMF MMF Nr ξ ξd = ( )[co( φ ) co( α θ ) co( β)] i Nr 2π ξ 2π ξd 2π = ( )[co( φ ) co( α θ ) co( β )] i Nr 4π ξ 4π ξd 4π = ( )[co( φ ) co( α θ ) co( β )] i ar r r ar br r r br cr r r cr (3.34) Motor Inductanc Rformulation Machin inductanc matric for a mixd rotor ccntricity ar vn mor complx inc thr i an intraction btwn th modifid MMF and airgap prmanc modulation for diffrnt typ of ccntriciti. In othr word, th modifid MMF componnt from a purly tatic ccntricity intract with th prmanc modulation componnt from a purly dynamic ccntricity, and vic vra. All inductanc matric and thir corrponding drivativ for a mixd rotor ccntricity, which ar litd in th following pag, hav a clo rlationhip with tho for a purly tatic or dynamic ccntricity. By tting ξ = 0 or ξ d = 0, machin inductanc for a mixd rotor ccntricity bcom tho for a purly tatic or dynamic ccntricity. 37

56 Ll 1 1 L = 0 Ll 0 L m 1 mixd L l [ ] (3.35) 2π 4π co( α)co( α) co( α )co( α) co( α )co( α) ξ 2π 2π 2π 4π 2π L m co( α)co( α ) co( α )co( α ) co( α )co( α ) π 2π 4π 4π 4π co( α)co( α ) co( α )co( α ) co( α )co( α ) π 4π co( θr + β)co( θr + β) co( θr + β )co( θr + β) co( θr + β )co( θr + β) ξd 2π 2π 2π 4π 2π Lm co( θr + β)co( θr + β ) co( θr + β )co( θr + β ) co( θr + β )co( θr + β ) π 2π 4π 4π 4π co( θr + β)co( θr + β ) co( θr + β )co( θr + β ) co( θr + β )co( θr + β ) π 4π co( α)co( θr + β) co( α )co( θr + β) co( α )co( θr + β) 3 3 ξ ξd 2π 2π 2π 4π 2π Lm co( α)co( θr + β ) co( α )co( θr + β ) co( α )co( θr + β ) π 2π 4π 4π 4π co( α)co( θr + β ) co( α )co( θr + β ) co( α )co( θr + β ) π 4π co( α)co( θr + β) co( α)co( θr + β ) co( α)co( θr + β ) 3 3 ξ ξd 2π 2π 2π 2π 4π Lm co( α )co( θr + β) co( α )co( θr + β ) co( α )co( θr + β ) π 4π 2π 4π 4π co( α )co( θr + β) co( α )co( θr + β ) co( α )co( θr + β )

57 2π 4π co( θr) co( θr + ) co( θr + ) 3 3 T 4π 2π Lr = Lr = Lm co( θr + ) co( θr ) co( θr + ) mixd mixd 3 3 2π 4π co( θr + ) co( θr + ) co( θr) 3 3 [ ] [ ] (3.36) 2π 4π co( α)co( α θr) co( α)co( α θr ) co( α)co( α θr ) ξ 2π 2π 2π 2π 4π Lm co( α )co( α θr ) co( α )co( α θr ) co( α )co( α θr ) π 4π 2π 4π 4π co( α )co( α θr) co( α )co( α θr ) co( α )co( α θr ) π 4π co( θr + β)co( β) co( θr + β)co( β ) co( θr + β)co( β ) ξd 2π 2π 2π 2π 4π Lm co( θr + β )co( β) co( θr + β )co( β ) co( θr + β )co( β ) π 4π 2π 4π 4π co( θr + β )co( β) co( θr + β )co( β ) co( θr + β )co( β ) π 4π co( α θr)co( θr + β) co( α θr )co( θr + β) co( α θr )co( θr + β) 3 3 ξ ξd 2π 2π 2π 4π 2π Lm co( α θr)co( θr + β ) co( α θr )co( θr + β ) co( α θr )co( θr + β ) π 2π 4π 4π 4π co( α θr)co( θr + β ) co( α θr )co( θr + β ) co( α θr )co( θr + β ) π 4π co( α)co( β) co( α)co( β ) co( α)co( β ) 3 3 ξ ξd 2π 2π 2π 2π 4π L m co( α )co( β) co( α )co( β ) co( α )co( β ) π 4π 2π 4π 4π co( α )co( β) co( α )co( β ) co( α )co( β )

58 Llr 1 1 Lrr = 0 Llr 0 L m 1 mixd L lr [ ] (3.37) 2π 4π co( α θr)co( α θr) co( α θr )co( α θr) co( α θr )co( α θr) ξ 2π 2π 2π 4π 2π Lm co( α θr)co( α θr ) co( α θr )co( α θr ) co( α θr )co( α θr ) π 2π 4π 4π 4π co( α θr)co( α θr ) co( α θr )co( α θr ) co( α θr )co( α θr ) π 4π co( β)co( β) co( β )co( β) co( β )co( β) ξd 2π 2π 2π 4π 2π L m co( β)co( β ) co( β )co( β ) co( β )co( β ) π 2π 4π 4π 4π co( β)co( β ) co( β )co( β ) co( β )co( β ) π 4π co( α θr)co( β) co( α θr )co( β) co( α θr )co( β) 3 3 ξ ξd 2π 2π 2π 4π 2π Lm co( α θr )co( β ) co( α θr )co( β ) co( α θr )co( β ) π 2π 4π 4π 4π co( α θr)co( β ) co( α θr )co( β ) co( α θr )co( β ) π 4π co( α θr)co( β) co( α θr)co( β ) co( α θr)co( β ) 3 3 ξ ξd 2π 2π 2π 2π 4π Lm co( α θr ) co( β) co( α θr )co( β ) co( α θr )co( β ) π 4π 2π 4π 4π co( α θr ) co( β) co( α θr )co( β ) co( α θr )co( β )

59 [ ] d L dθ r mixd = (3.38) 2π 4π in(2θr + 2 β) in(2θr + 2 β ) in(2θr + 2 β ) ξd 2π 4π + Lm in(2θr + 2 β ) in(2θr + 2 β ) in(2θr + 2 β) π 2π in(2θr + 2 β ) in(2θr + 2 β) in(2θr + 2 β ) 3 3 2π 4π co( α)in( θr + β) co( α )in( θr + β) co( α )in( θr + β) 3 3 ξ ξd 2π 2π 2π 4π 2π + Lm co( α)in( θr + β ) co( α )in( θr + β ) co( α )in( θr + β ) π 2π 4π 4π 4π co( α)in( θr + β ) co( α )in( θr + β ) co( α )in( θr + β ) π 4π co( α)in( θr + β) co( α)in( θr + β ) co( α)in( θr + β ) 3 3 ξ ξd 2π 2π 2π 2π 4π + Lm co( α )in( θr + β) co( α )in( θr + β ) co( α )in( θr + β ) π 4π 2π 4π 4π co( α )in( θr + β) co( α )in( θr + β ) co( α )in( θr + β )

60 2π 4π in( θr) in( θr + ) in( θr + ) 3 3 T r mixd 4π 2π = = Lm in( θr + ) in( θr ) in( θr + ) dθ r 3 3 2π 4π in( θr + ) in( θr + ) in( θr) 3 3 [ ] d[ L ] d L r dθ r mixd (3.39) 2π 4π co( α)in( α θr) co( α)in( α θr ) co( α)in( α θr ) ξ 2π 2π 2π 2π 4π Lm co( α )in( α θr ) co( α )in( α θr ) co( α )in( α θr ) π 4π 2π 4π 4π co( α )in( α θr) co( α )in( α θr ) co( α )in( α θr ) π 4π in( θr + β)co( β) in( θr + β)co( β ) in( θr + β)co( β ) ξd 2π 2π 2π 2π 4π + Lm in( θr + β )co( β) in( θr + β )co( β ) in( θr + β )co( β ) π 4π 2π 4π 4π in( θr + β )co( β) in( θr + β )co( β ) in( θr + β )co( β ) π 4π in( α β 2 θr) in( α β 2 θr ) in( α β 2 θr ) 3 3 ξ ξd 4π 2π Lm in( α β 2 θr ) in( α β 2 θr ) in( α β 2 θr ) π 4π in( α β 2 θr ) in( α β 2 θr ) in( α β 2 θr)

61 [ ] d L rr dθ r mixd = (3.40) 2π 4π in(2α 2 θr) in(2α 2 θr ) in(2α 2 θr ) ξ 2π 4π Lm in(2α 2 θr ) in(2α 2 θr ) in(2α 2 θr ) π 2π in(2α 2 θr ) in(2α 2 θr) in(2α 2 θr ) π 4π in( α θr)co( β) in( α θr )co( β) in( α θr )co( β) 3 3 ξ ξd 2π 2π 2π 4π 2π + Lm in( α θr )co( β ) in( α θr )co( β ) in( α θr )co( β ) π 2π 4π 4π 4π in( α θr)co( β ) in( α θr )co( β ) in( α θr )co( β ) π 4π in( α θr)co( β) in( α θr)co( β ) in( α θr)co( β ) 3 3 ξ ξd 2π 2π 2π 2π 4π + Lm in( α θr )co( β) in( α θr )co( β ) in( α θr )co( β ) π 4π 2π 4π 4π in( α θr )co( β) in( α θr )co( β ) in( α θr )co( β )

62 3.4.3 Motor Currnt Charactritic Mixd rotor ccntriciti alo introduc paramtr unbalanc in th motor, a hown in (3.35). Sinc th machin inductanc matric for a mixd rotor ccntricity inhrit th charactritic from both purly tatic and dynamic ccntriciti, ngativ qunc information at id-band harmonic, which i rfrrd to th ngativ qunc harmonic information, i xpctd to appar in upply currnt. Thi rotor ccntricity charactritic information rv a th nw, propod ccntricity fault indicator for a main-fd induction motor in th prnc of a load torqu ocillation. Th motor aymmtry inducd by a rotor ccntricity alo yild a ngativ qunc componnt at th fundamntal in th upply currnt. Howvr, thi ngativ qunc fundamntal information i dominatd by th invitabl unbalancd powr upply for a main-fd machin. Thrfor, only ngativ qunc harmonic information, which i xmpt from th unbalanc powr upply, hould b proprly monitord to dcoupl th load ocillating ffct from th rotor ccntricity dtction. 3.5 Motor Currnt Charactritic for a Load Ocillation Idally, a poition dpndnt load torqu ocillation produc only poitiv qunc information (fundamntal and idband harmonic) in th tator currnt, conidring th motor and haft ar not impactd by thi typ of load variation. Howvr, thr i alway an inhrnt lvl of machin unbalanc during th manufacturing and ambly tag. In addition, intrumntation imprfction, uch a th currnt and voltag tranducr caling ratio mimatch, alo affct th maurmnt accuracy. Hnc, th normalizd ngativ qunc harmonic information with rpct to it poitiv qunc countrpart 44

63 in a main-fd machin rv a th mot rliabl indicator to accuratly valuat th incrad motor aymmtry lvl, which i olly caud by a rotor ccntricity fault. 3.6 Chaptr Summary In thi chaptr, a dtaild airgap magntic fild analyi for a purly tatic ccntricity, a purly dynamic ccntricity and a mixd rotor ccntricity i prformd to driv th modifid thr pha tator and rotor MMF for ach ca. Aftr that, all machin inductanc a wll a thir drivativ ar rformulatd by applying th claical doubl-intgration mthod. It i important to point out hr, although th modifid inductanc matric ar drivd for a 2-pol modl, thy ar wll uitd for a multi-pol motor by including a multiplir of machin pol-pair numbr. Th ccntricity rformulatd impdanc matric ar furthr compard with tho caud by a tator intr-turn fault in Chaptr 6. Finally, motor currnt charactritic corrponding to ach typ of rotor ccntricity a wll a a poition dpndnt load torqu ocillation ar dicud to provid an inight on ditinguihing load torqu ocillation from rotor ccntricity fault. 45

64 4 CHAPTER 4 EXTRACTING NEGATIVE SEQUENCE HARMONIC INFORMATION 4.1 Ovrviw In thi Chaptr, a nw FFT tchniqu i dvlopd to ffctivly xtract ngativ qunc harmonic information from th thr pha currnt. Th propod mthod ovrcom th hortcoming in th traditional ymmtrical componnt dcompoition mthod, which can only parat qunc componnt at th fundamntal or it multipl. It alo ha vral advantag ovr th rfrnc fram tranformation mthod to ffctivly parat harmonic qunc componnt. Thi nw tchniqu i important to achiv rliabl rotor ccntricity fault dtction undr arbitrary load condition. In addition, it alo a th xtraction of fundamntal pr-pha qunc componnt from pha-pha voltag maurmnt in many onlin condition monitoring application. 4.2 Traditional Symmtrical Componnt Dcompoition Symmtrical componnt ar mot commonly ud for analyi of thr-pha lctrical powr ytm [35]. If th pha quantiti ar xprd in phaor notation uing complx numbr, a vctor can b formd for th thr pha quantiti. For xampl, a vctor for thr pha currnt could b writtn a, I abc I a I I c = b (4.1) and th thr ymmtrical componnt phaor arrangd into a vctor a, 46

65 I ym I p I I z = n (4.2) whr th ubcript p, q, and z rfr rpctivly to th poitiv, ngativ, and zro qunc componnt. A pha rotation oprator a i dfind to rotat a phaor vctor forward by 120 dgr or 2 3 π radian, 2 j120 j 3 a= = π (4.3) A matrix A can b dfind uing thi oprator to tranform th pha vctor into ymmtrical componnt, 2 I p 1 a a I a Iym = I n = A I abc = a a I b 3 3 I z I c (4.4) Th traditional ymmtrical componnt dcompoition mthod may only parat poitiv and ngativ qunc componnt at fundamntal or it multipl, not at charactritic harmonic frqunci, inc thr pha phaor can b accuratly contructd from maurmnt only at th fundamntal or it multipl. For an invrtrfd motor with varying xcitation frquncy, thi mthod may not work vn at th fundamntal if th FFT pctrum rolution i not ufficintly high. Thrfor, thi chm i not uitabl to xtract ngativ qunc harmonic information for th purpo of onlin condition monitoring. 47

66 4.3 Rotating Rfrnc Fram Tranformation Mthod It i alo poibl to ditinguih ngativ qunc charactritic harmonic from thir poitiv qunc countrpart by prforming DQ tranformation in a nwly lctd rotating rfrnc fram [56]-[57]. In gnral, th mot dominant charactritic harmonic rulting from a rotor ccntricity or load torqu ocillation corrpond to th ca k = 1 in (1.2) or (1.6). Thrfor, on only nd to conidr th largt two harmonic in th tator currnt pctrum, L H f = f f and f = f + f (4.5) cc rm cc rm whr uprcript L dnot th lowr harmonic and uprcript H indicat th highr on. Thr ar both poitiv and ngativ qunc componnt xiting at L f cc a wll a H f cc in thr pha currnt. On can jut focu on a ingl charactritic harmonic,.g. f H cc. Th thr pha currnt harmonic at thi ingl frquncy can thn b xprd a, i = I co( ω t+ ϕ ) + I co( ω t+ ϕ ) H H a 1 cc 1 2 cc 2 i = I co( ω t+ ϕ 120 ) + I co( ω t+ ϕ ) H H b 1 cc 1 2 cc 2 i = I co( ω t+ ϕ 240 ) + I co( ω t+ ϕ ) H H c 1 cc 1 2 cc 2 (4.6) whr I 1, I 2, ϕ 1 and ϕ 2 rprnt th magnitud and initial pha angl for th poitiv H and ngativ qunc harmonic componnt at ω cc, rpctivly. Equation (4.6) can b alo writtn a, i = I co( ω t+ ϕ ) + I co( ω t+ ϕ ) H H a 1 cc 1 2 cc 3 i = I co( ω t+ ϕ 120 ) + I co( ω t+ ϕ 120 ) H H b 1 cc 1 2 cc 3 i = I co( ω t+ ϕ 240 ) + I co( ω t+ ϕ 240 ) H H c 1 cc 1 2 cc 3 (4.7) 48

67 whr ϕ3 = ϕ2. It i intrting to obrv that, th ngativ qunc componnt at H frquncy ω cc i actually quivalnt to th poitiv qunc componnt at th countr frquncy H ω cc. Hnc, th trm of ngativ qunc and countr frquncy can b ud intrchangably. Not that, imilar to th concpt of poitiv/ngativ qunc componnt, th notion of countr frquncy hr i only maningful to thr pha quantiti. On can furthr tranform th thr pha charactritic harmonic at f H cc into dqaxi componnt in a nwly lctd rotating rfrnc fram at ω = 2π f, ia f i 2 q co( θ) co( θ 120 ) co( θ 240 ) ( ) i f = b id 3 in( θ) in( θ 120 ) in( θ 240 ) i c (4.8) whr θ = ω t i th tranformation angl. Subtituting (4.6) into (4.8) and applying propr trigonomtric rlation [34] to implify th final quation, on obtain th dq-axi componnt a follow, f H H i q I1co[( ω ωcc ) t ϕ1] + I2co[( ω+ ωcc ) t+ ϕ2] f = H H id I1in[( ω ωcc) t ϕ1] + I2in[( ω+ ωcc) t+ ϕ2] (4.9) It i clar to that, in th nw rotating rfrnc fram, th original poitiv qunc harmonic componnt at H H ω cc i hiftd to ω ω cc and th corrponding ngativ H qunc harmonic componnt i hiftd to ω + ω cc. For th purpo of illutrating qunc componnt dcompoition in th nw rfrnc fram, pac vctor corrponding to th poitiv and ngativ qunc harmonic componnt ar hown in Figur 4.1, in th tationary and rotating rfrnc fram, rpctivly. 49

68 and Idally, poitiv/ngativ qunc harmonic magnitud hould b qual in th i f q f i d pctra. Howvr, du to th limitd FFT rolution and maurmnt noi, thr i alway a diffrnc in th pctral analyi of i f q and f i d. Thrfor, it bcom difficult to dtrmin th xact magnitud of th harmonic qunc componnt uing thi dq-tranformation tchniqu. In addition, th hiftd harmonic qunc componnt in th nw rfrnc fram might ovrlap with th qunc componnt from othr non-charactritic harmonic. H For xampl, if th thr pha currnt happn to contain harmonic at f + 2 f, which ha nothing to do with a rotor ccntricity or load ocillation, thn th poitiv qunc componnt at thi non-charactritic harmonic would intrfr with th ngativ qunc cc componnt at H f cc in th nw rfrnc fram rotating at f. Sinc th inhrnt noncharactritic harmonic vary from motor to motor, it i difficult to pcify a uniform rotating rfrnc fram to prform th dq-tranformation and accuratly xtract poitiv and ngativ qunc charactritic harmonic. 50

69 (a) (b) Figur 4.1 Poitiv and ngativ qunc harmonic pac vctor in (a) tationary rfrnc fram and (b) rotating rfrnc fram 4.4 Nw FFT Tchniqu to Sparat Squnc Information Th convntional pctral analyi prform an FFT on a ral numbr qunc,.g. a ingl pha tator currnt with a ampling frquncy f and total ampling tim T c, rulting in N = T f ampl in total. Again, on can conidr only a ingl frquncy charactritic harmonic in pha-a currnt, i = I co( ω t+ ϕ ) + I co( ω t+ ϕ ) (4.10) a 1 cc 1 2 cc 2 or i = I co(2 π f t+ θ ) (4.11) a cc whr th poitiv and ngativ qunc componnt at f cc ar combind togthr in (4.11) with [ co( ϕ ) co( ϕ )] [ in( ϕ ) in( ϕ )] I = I + I + I + I (4.12) 51

70 θ I in( ϕ ) + I in( ϕ ) = tan I1co( ϕ1) + I2co( ϕ2) (4.13) Equation (4.11) can b furthr dcompod into two frquncy componnt uing Eulr formula, a I I i 2 2 j2 π( fcc ) t jθ j2 π( fcc ) t j( θ ) a = ( ) + ( ) (4.14) Thrfor, FFT analyi of a ral numbr qunc, ia[ n ], plit a ingl frquncy quantity into two part, f cc and ( f cc ), in th pctrum. Each part har half of th total magnitud at that frquncy and tak oppoit pha angl. Obviouly, ( f cc ) or f f componnt in i [ n ] pctrum do not provid any additional information. cc a In ordr to ffctivly parat poitiv and ngativ qunc harmonic information, thr pha harmonic can b contructd into a harmonic pac vctor in th tationary rfrnc fram, I i i i 3 2 ( j0 j120 j240 cc = a + b + c ) (4.15) Similar to (4.14), ach of i a, i and b i c can b dcompod into four part whil conidring poitiv and ngativ qunc componnt for a ingl pha paratly, I I ia = ( ) + ( ) 2 2 I I + ( ) + ( ) j2 π( fcc ) t jϕ1 1 j2 π( fcc ) t j( ϕ1) 2 j2 π( fcc ) t jϕ2 2 j2 π( fcc ) t j( ϕ2) (4.16) I I ib = ( ) + ( ) 2 2 I I + ( ) + ( ) j2 π( fcc ) t j( ϕ1 120 ) 1 j2 π( fcc ) t j( ϕ ) 2 j2 π( fcc ) t j( ϕ ) 2 j2 π( fcc ) t j( ϕ2 120 ) (4.17) 52

71 I I ic = ( ) + ( ) 2 2 I I + ( ) + ( ) j2 π( fcc ) t j( ϕ1 240 ) 1 j2 π( fcc ) t j( ϕ ) 2 j2 π( fcc ) t j( ϕ ) 2 j2 π( fcc ) t j( ϕ2 240 ) (4.18) Subtituting (4.16)-(4.18) into (4.15), aftr a fw tp of mathmatical manipulation, on rach a vry impl xprion for th tator currnt harmonic pac vctor, I = I + I cc j 2 π( f cc ) t j ( ϕ1) j 2 π( f cc ) t j( ϕ2) 1 2 (4.19) FFT analyi of th complx numbr qunc, I [ n], plit th poitiv qunc harmonic magnitud, I 1, at th poitiv frquncy, f cc, and th ngativ qunc harmonic magnitud, I 2, at th countr frquncy, ( f cc ) or f fcc. Not that, th valu of f cc can b ithr poitiv or ngativ dpnding on th lction of k in (1.2) or (1.6). In ca th valu of f cc i a ngativ numbr in (4.19), thn I 1 appar at f cc or cc f + f whil I 2 appar at ( ). Thrfor, ngativ qunc harmonic cc f cc information i alway rflctd at th countr frquncy componnt in I [ n] pctrum. cc Although th abov mathmatical drivation dal only with a ingl frquncy charactritic harmonic, th propod FFT tchniqu can b aily xtndd to all frquncy componnt by contructing a tator currnt pac vctor, which contain all frquncy information, from thr pha maurmnt, I = I + I + I 3 2 ( j0 j120 j240 a b c ) By prforming FFT to th complx numbr qunc, I [ n] (4.20), poitiv and ngativ qunc componnt can b convnintly paratd at all availabl frquncy indx in th pctrum. In gnral, frquncy rolution in FFT dpnd on th total ampling 53

72 tim. In ordr to accuratly locat th charactritic harmonic in I [ n] pctrum, it i ncary to ampl thr pha quantiti for a ufficintly long priod. For a four-pol induction motor, th ngativ qunc componnt corrponding to th charactritic harmonic, f L = f f (poitiv valu), i locatd at f ( f f ), cc rm rm whil th poitiv qunc componnt corrponding to th charactritic harmonic f 3 f (ngativ valu) i locatd at f + ( f 3 f ). Whn th motor i running at rm rm light load condition, th two componnt li vry clo in th pctrum and may ovrlap with ach othr du to pctral lakag ffct. Thrfor, ngativ qunc H information locatd at f = ( f + f ) i a bttr rotor fault indicator to dtct cc rm ccntricity inducd motor aymmtry. Th propod nw FFT tchniqu ha ignificant advantag ovr th traditional mthod uing ymmtrical componnt dcompoition. Firt, ymmtrical componnt mthod rquir contruction of thr phaor at a ingl frquncy through rgular FFT analyi for ach pha. Du to th finit frquncy rolution in FFT, th phaor can b rliably tablihd only at fundamntal and it multipl. For a pcific charactritic harmonic f cc, th pha angl information calculatd from FFT might b far away from it tru valu, lading to an inaccurat phaor contruction and incrdibl ymmtrical componnt dcompoition. Scond, th propod tchniqu xtract poitiv and ngativ qunc information at all frquncy componnt imultanouly by xcuting th FFT on th pac vctor complx numbr qunc whil th ymmtrical componnt mthod prform th dcompoition only for a ingl frquncy quantity at ach tim. 54

73 4.5 Application for Pha-Pha Maurmnt In many onlin condition monitoring application, it oftn rquir xtracting fundamntal qunc componnt from motor currnt a wll a motor voltag. For tator intr-turn fault dtction, th impdanc matrix i calculatd from both currnt and voltag qunc componnt to valuat th motor aymmtry lvl [46]-[49]. In ordr to accomplih an accurat timation of motor fficincy or tator winding tmpratur [54], poitiv and ngativ qunc componnt in th trminal quantiti nd to b valuatd in th corrponding pr-pha quivalnt circuit modl paratly. Diffrnt from th thr pha currnt maurmnt, pha-nutral (auming Y- connctd tator winding) voltag maurmnt ar uually inaccibl. Although pha-ground maurmnt produc xactly th am poitiv and ngativ qunc componnt a pha-nutral voltag [46], thy ar inconvnint in many ca. (Not that th only diffrnc btwn pha-nutral and pha-ground maurmnt rid in thir zro qunc componnt.) In gnral, only pha-pha voltag maurmnt ar availabl for onlin motor condition monitoring. Th traditional procdur to obtain pr-pha qunc componnt from phapha maurmnt i hown in Figur 4.2. Firt, FFT analyi of ach pha-pha voltag yild a phaor at fundamntal. Aftr that, qunc componnt in pha-pha voltag ar obtaind by applying ymmtrical componnt dcompoition to all thr phaor. Finally, pr-pha qunc componnt can b drivd from corrponding pha-pha qunc componnt a follow, j( 30 ) 1 1 j( 30 ) Vpha = Van = Vab = V pha pha (4.21)

74 j j30 Vpha = Van = Vab = V pha pha (4.22) 3 3 whr uprcript 1 and 2 rprnt poitiv and ngativ qunc componnt rpctivly. Th rlationhip ar alo drawn in Figur 4.3. Th propod nw FFT tchniqu alo implifi th tranformation procdur in Figur 4.2 to obtain fundamntal pr-pha qunc componnt air and mor accuratly. Thi i don by applying th FFT dirctly to th contructd tator voltag pac vctor. Th dfinition of a tator voltag pac vctor i bad on pha-nutral voltag, V = V + V + V 3 2 ( j0 j120 j240 an bn cn ) (4.23) Figur 4.2 Procdur to obtain pr pha qunc componnt from pha-pha maurmnt 1 V pha-pha 2 V pha 1 V pha 2 V pha-pha Figur 4.3 Rlationhip btwn qunc componnt from pr-pha and pha-pha quantiti 56

75 Conidring th idntity xprion blow, 2 3 j0 j120 j240 0 = ( Vcn + Vcn + Vcn ) (4.24) th dird tator voltag pac vctor can b aily contructd from th two phapha voltag maurmnt by ubtracting (4.24) from (4.23) a follow, V = V + V 3 2 ( j0 j120 ac bc ) (4.25) Th FFT of th tator voltag pac vctor immdiatly giv th pr-pha poitiv qunc voltag at f and pr-pha ngativ qunc voltag at pctrum. f or f f in th For driv-connctd induction motor, th tator xcitation frquncy i varying with rpct to th rfrnc pd and load lvl. In thi ca, it bcom vn mor difficult to xtract fundamntal qunc componnt in tator currnt and voltag uing ymmtrical componnt dcompoition. Phaor at th fundamntal may not b accuratly contructd for a varying xcitation frquncy du to th limitd FFT rolution. Thrfor, th propod FFT tchniqu how grat advantag in driv application. 4.6 Chaptr Summary Thi Chaptr prnt an ffctiv and computationally fficint tratgy to xtract ngativ qunc harmonic information from tator currnt, which can b furthr applid in a rliabl diagnotic of rotor ccntricity fault in th prnc of a load torqu ocillation. By applying th FFT to th tator currnt or th tator voltag pac vctor, thi mthod can parat qunc componnt at all frqunci imultanouly. Thrfor, th propod tchniqu ha ignificant advantag ovr th traditional 57

76 ymmtrical componnt dcompoition mthod. In addition to rotor ccntricity dtction, thi tchniqu can b applid in many othr onlin condition monitoring ara uch a tator turn fault dtction, thrmal timation, and motor fficincy valuation. Thi mthod i particularly uful for driv application with a varying xcitation frquncy. 58

77 5 CHAPTER 5 DEVELOPMENT AND SIMULATION OF NEW ROTOR ECCENTRICITY FAULT INDICATORS 5.1 Ovrviw Th nw rotor ccntricity fault indicator in th prnc of a load torqu ocillation ar dvlopd for main-fd and driv-connctd induction motor, rpctivly, in thi chaptr. Eithr a implifid Matlab Simulink modl or a complx Finit Elmnt Analyi (FEA) can b applid to imulat induction motor dynamic prformanc. Th Matlab modl nglct th tim and pac harmonic in th inducd MMF. Magntic aturation and lotting ffct ar alo not conidrd. Thi modl pcifically focu on th airgap prmanc modulation caud by rotor ccntriciti. On th othr hand, FEA Modl includ all abov nglctd factor and produc much mor accurat rult. Howvr, it i xtrmly tim-conuming to prform th tranint analyi ncary to obtain a ufficintly high FFT pctral rolution. 5.2 A Nw Eccntricity Indicator in Main-Fd Machin A rotor poition-dpndnt load torqu ocillation at multipl of th rotational pd, uch a in rciprocating compror, introduc xactly th am harmonic componnt a a rotor ccntricity, in th ingl pha tator currnt pctrum. Th load ocillation-rlatd harmonic uually hav a much largr magnitud and ar likly to compltly mak ccntricity-inducd fault ignatur. In ordr to achiv rliabl 59

78 ccntricity fault dtction in th prnc of a load torqu ocillation, om nw ccntricity-only rlatd information ha to b dtctd. Magntic fild analyi how that a rotor ccntricity introduc paramtr unbalanc inid th motor. On th contrary, in th ca of a load torqu ocillation, th motor itlf i till balancd. Thrfor, th additional motor aymmtry, caud olly by th rotor fault, rprnt th nw ccntricity-only rlatd information which hould b xamind a wll, in onlin condition monitoring of induction motor. In a main-fd induction motor, motor aymmtry i compltly rflctd into th ngativ qunc componnt in tator currnt. Du to th invitabl unbalanc in upply voltag, th ngativ qunc information in fundamntal currnt i not a rliabl rotor fault indicator. Thu it i ncary to proprly monitor ngativ qunc charactritic harmonic information in thr pha currnt. Conidring th inhrnt motor unbalanc and intrumntation imprfction, it i dirabl to calculat th normalizd ngativ qunc componnt at charactritic harmonic with rpct to th corrponding poitiv qunc harmonic magnitud to accuratly valuat th incrad motor aymmtry lvl caud by rotor ccntriciti. Thrfor, th additional rotor fault indicator for a main-fd machin i I I * n_ cc δ = n_ cc= (5.1) I p_ cc whr I dnot th magnitud of th ngativ qunc charactritic harmonic n_ cc componnt, I dnot th corrponding poitiv qunc countrpart and th p_ cc uprcript rprnt th normalizd valu. To ffctivly liminat load ocillation 60

79 ffct, both th rgular charactritic id-band harmonic a wll a th propod additional motor aymmtry indicator hould b monitord imultanouly. 5.3 Matlab Simulation for a Main-Fd Induction Motor Induction Motor Dynamic Modl Airgap magntic fild analyi prntd in Chaptr 3 how that rotor ccntriciti introduc paramtr unbalanc in motor inductanc matric and ngativ qunc information i xpctd to appar in upply currnt. For th tim bing, thr i no confidnc about th voltag lvl at th tator (and th quivalnt rotor) winding nutral point. Conquntly, dynamic imulation of a faulty motor i prformd in th phyical ABC ytm intad of uing a qd-axi tranformation. All imulation quation ar concludd a follow, va v _ n r 0 0 ia λa d vb v _ n 0 r 0 i b λ = + b dt vc v _ n 0 0 r i c λ c (5.2) 0 v r_ n r 0 0 iar λar d 0 v = 0 r 0 i + λ r_ n r br br dt 0 v r_ n 0 0 r i cr λ cr (5.3) [ ] [ r ] T [ L ] [ L ] λabc L L iabc λ = abcr i r rr abcr (5.4) i + i + i = 0 a b c i + i + i = 0 ar br cr (5.5) 61

80 whr r and r ar th tator and rotor pr pha ritanc, v _ n and v r_ nar th unknown tator and quivalnt rotor nutral point voltag. All th inductanc matric in (5.4) hav bn rformulatd for a purly tatic ccntricity, a purly dynamic ccntricity and a mixd rotor ccntricity, rpctivly. Not that thr pha tator and rotor currnt ar forcd to um to zro in (5.5) during th dynamic imulation inc thr i no zro qunc componnt in ithr i abc or i abcr. Th mchanical quation of motion dpnd on th load charactritic. It i aumd that, for implicity, th load torqu conit only of an inrtial toqu and it xprion i xplicitly known. Thrfor, th mchanical quation i imply, dω T T J J dt dω dt rm r m load = q = q (5.6) for a 2-pol machin. Th dvlopd lctromagntic torqu, T m, can b obtaind from th magntic conrgy, W co, a follow [30], [36] T m W W co co = = θ rm θ I, I r r I, Ir (5.7) whr I and I r dnot tator and rotor currnt vctor, T I = ia ib ic (5.8) T I = i i i [ ] [ ] r ar br cr In a linar magntic ytm, th conrgy i qual to th tord magntic nrgy, 1 [ L ] [ L T T r ] I Wco = I I r T 2 (5.9) [ Lr ] [ Lrr ] I r Thrfor, in th 2-pol machin modl, T m ha th form of, 62

81 1 T T d [ L ] [ Lr ] I Tm = I I r T 2 (5.10) dθr [ Lr ] [ Lrr ] I r Not that in th ca of rotor ccntriciti, all inductanc matric hown in (5.4) and (5.10) vary a a function of θ r du to th non-uniform airgap prmanc modulation ffct. Matlab Simulink modl chmatic uing th rformulatd machin inductanc for a main-fd machin ar givn in Figur 5.1. Motor paramtr ud to imulat th Matlab Simulink modl ar litd in Appndix A for radr rfrnc. 63

82 Va [V_NS] [Ib] Vb [V_NS] Vc [V_NS] [Ia] [Ib] [Ic] [Iar] [Ibr] [Icr] [thta] Rb [Ia] [lamda_a] Ra [Ia] [Ia] [Ib] [lamda_b] Ia+Ib+Ic 1 [Ib] [lamda_a] [Ic] [lamda_c] 1 [lamda_b] -1/3 Mmory V_NS [V_NS] [lamda_ar] [lamda_br] [lamda_cr] [thta] MATLAB Function lamda--->i [Ic] [Iar] [Ibr] [Icr] Rc [Ic] [Iar] 1 [lamda_c] Rar [Iar] [Ibr] Iar+Ibr+Icr 1 0 [lamda_ar] [Icr] 60/(2*pi) Wmch rpm [V_NR] 1 [Ibr] Rbr 1 p/2 [thta] 0 1-1/3 Mmory1 [lamda_br] MATLAB Function Tm 1/Jq [V_NR] Rcr [Icr] TL_0+0.5*TL_0*co(u) 0 T_load 1 [lamda_cr] T [V_NR] Figur 5.1 Simulink chmatic uing rformulatd motor inductanc for main-fd machin Iabc_ Ia_ Iabc_r Ia_r [V_NR] V_NR 64

83 5.3.2 Purly Static Eccntricity in a Main-Fd IM A main-fd 3-pha, 2-pol induction motor i imulatd undr a contant load condition for a purly tatic ccntricity fault with ξ = 0.5, α = 0 and lip = It i clar from Figur 5.2 that only th fundamntal componnt xit in th ingl pha i a pctrum. By taking th rotating rfrnc fram tranformation mthod, fundamntal poitiv and ngativ qunc componnt ar paratd in th i f q pctrum at 50 Hz and 70 Hz, rpctivly, a hown in Figur 5.3, whr f = 10 Hz. Although imulation rult hr how that thi mthod work wll to manift th ngativ qunc componnt for a purly tatic ccntricity, th rotating frquncy of thi nw rfrnc fram ha to b lctd vry carfully to avoid undird ovrlapping with othr non-charactritic harmonic for a practical ccntric motor. In addition, ngativ qunc harmonic obtaind from thi tchniqu may hav diffrnt magnitud for d-axi and q-axi componnt, which lad to an uncrtainty in dtrmining th actual lvl of ngativ qunc charactritic harmonic. A bttr way to ditinguih poitiv and ngativ qunc componnt i to apply an FFT to th tator currnt pac vctor, I, o that th fundamntal qunc componnt ar paratd xplicitly in Figur 5.4 at ± 60 Hz. Du to it rlativ advantag ovr th rfrnc fram tranformation mthod, applying th FFT to th tator currnt or trminal voltag pac vctor will b ud latr on to xtract ngativ qunc information in th trminal quantiti. Rpatd imulation, including varying th contant load lvl, th tatic ccntricity cofficint, ξ, and th motor fault poition, α, lad to om inightful obrvation. Th poitiv qunc fundamntal currnt incra ignificantly from no 65

84 load to full load, but th ngativ qunc fundamntal componnt almot rmain unchangd whil kping a contant ξ = 0.5 qunc fundamntal componnt,. Thrfor, th normalizd ngativ * I n, with rpct to th poitiv qunc fundamntal countrpart, dcra a th load lvl incra (or rpm dcra) in Figur 5.5. In othr word, th valu of of tatic ccntricity. * I n i highly dpndnt on th load lvl vn for a fixd dgr On th othr hand, th ngativ qunc fundamntal componnt incra rmarkably whn th dgr of tatic ccntricity incra. Manwhil, th poitiv qunc componnt tay narly unchangd for a contant load lvl. Thu, th valu of * I n incra a th vrity dgr of tatic ccntricity incra if th load lvl i fixd, a hown in Figur 5.6. Variation of th motor fault poition or th angl α do not impact th magnitud of poitiv or ngativ qunc componnt. Howvr, th pha angl diffrnc btwn th fundamntal poitiv and ngativ qunc componnt in i f q, f i d or I pctrum vari a a function of th angl α, a hown in Figur 5.7, if both th load lvl and th dgr of tatic ccntricity ar fixd. Simulation rult alo clarly dmontrat that th voltag lvl at both tator and quivalnt rotor winding nutral point maintain zro in th prnc of a tatic ccntricity. Thi obrvation will b furthr thortically vrifid in Chaptr 6. Actually, zro qunc information act a an important ditinction btwn th rotor ccntricity and th tator intr turn fault although both of thm yild paramtr unbalanc inid th motor. 66

85 Magnitud (db) Frquncy (Hz) Figur 5.2 Simulatd main-fd i pctrum a with ξ = 0.5, α = 0, lip = Magnitud (db) Frquncy (Hz) f Figur 5.3 Simulatd main-fd i q pctrum with ξ = 0.5, α = 0, lip =

86 Magnitud (db) Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.4 Simulatd main-fd I pctrum with ξ = 0.5, α = 0, lip = Prcntag (%) Avrag Rotor Spd (rpm) Figur 5.5 Normalizd ngativ qunc fundamntal componnt w.r.t. th load lvl whr ξ = 0.5, α = 0 68

87 8 7 6 Prcntag (%) tatic ccntricity cofficint Figur 5.6 Normalizd ngativ qunc fundamntal componnt w.r.t. th dgr of tatic ccntricity whr lip = , α = Pha angl diffrnc (dgr) Motor fault poitioin xprd in alpha (dgr) Figur 5.7 Pha angl diffrnc btwn poitiv and ngativ qunc componnt w.r.t. th motor fault poition whr ξ =

88 5.3.3 Purly Dynamic Eccntricity in a Main-Fd IM Singl pha tator currnt pctrum for a purly dynamic ccntricity with a contant load lvl in a main-fd machin i hown in Figur 5.8, whr ξ = 0.5, β = 0 and lip = Du to th dynamic airgap prmanc modulation in thi ca, idband harmonic locatd at (1 ± 2 k ) f alo appar in th tator currnt. In othr word, tator currnt harmonic at 61.5 Hz, 58.5 Hz, 63 Hz, 57 Hz, tc., xit for a dynamic ccntricity with lip = Th dynamic ccntricity inducd currnt harmonic ar imilar to tho caud by a brokn rotor bar fault inc both of thm introduc rotating aymmtry into th airgap magntic fild. FFT analyi of th tator currnt pac vctor, a hown in Figur 5.9, dmontrat clarly that no ngativ qunc componnt, nithr at fundamntal nor id-band harmonic, xit for a purly dynamic ccntricity. Again, th tator and rotor nutral point voltag ar confirmd to b zro in th imulation. d 70

89 Magnitud (db) Frquncy (Hz) Figur 5.8 Simulatd main-fd i pctrum a with ξ d = 0.5, β = 0, lip = Magnitud (db) Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.9 Simulatd main-fd I pctrum with ξ d = 0.5, β = 0, lip =

90 5.3.4 Mixd Rotor Eccntricity in a Main-Fd IM For a mixd rotor ccntricity, charactritic id-band harmonic at f ± k f rm can b dtctd in th ingl pha tator currnt pctrum, a hown in Figur 5.10, whr ξ = 0.5, ξ = 0.5, α =15, β = 0, and at a contant load lvl with lip = d Although harmonic at f k f rm can b idntifid by zooming in th currnt pctrum, thir magnitud ar not trutabl du to th FFT pctrum lakag ffct for thi 2-pol machin at a light load condition. Charactritic harmonic locatd at f + k frm rv a mor rliabl indicator for a mixd rotor ccntricity in thi ca. Th pctrum of th tator currnt pac vctor i hown again in Figur 5.11, whr th qunc componnt at th charactritic harmonic f + k f rm ar xplicitly paratd. Similar to a purly tatic or dynamic ccntricity, variation of th motor fault poition, i.., variation of th valu for angl α and/or β, do not chang th magnitud of qunc componnt for a mixd rotor ccntricity. Mixd rotor ccntricity alo introduc a haft pd ocillation a hown in Figur 5.12, whr th avrag pd i till kpt around 3555 rpm or lip = In othr word, th avrag haft pd for a halthy and an ccntric motor ar almot am whil maintaining th am load lvl. Sinc th tator and rotor nutral point alway rmain at zro voltag, imulation chmatic hown in Figur 5.1 can b implifid to tho hown in Figur 5.13 in ordr to acclrat th imulation pd for a main-fd machin. 72

91 f 20 Magnitud (db) f f f 2 f rm rm f f 3 f + f rm rm f + 2 f rm -30 f 4 f rm f + 3 f rm f 5 f Frquncy (Hz) rm Figur 5.10 Simulatd main-fd i pctrum a with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip = f 20 Magnitud (db) ( f + 2 f ) ( f + f ) rm rm f f + f rm f + 2 f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.11 Simulatd main-fd I pctrum with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip =

92 Shaft pd (rpm) Tim () Figur 5.12 Simulatd main-fd motor haft pd profil with ξ = 0.5, ξ d = 0.5, α =15, β = 0, lip = R [Ia] [lamda_a] [Ia] Va 1 [lamda_a] [lamda_b] [lamda_c] [Ib] Ia [Ib] Vb R 1 [lamda_b] [lamda_ar] [lamda_br] MATLAB Function lamda--->i [Ic] [Iar] [Ibr] R [Ic] [lamda_cr] [Icr] 1 [lamda_c] [thta] Vc Rr [Iar] 60/(2*pi) 0 1 [lamda_ar] [Ia] [Ib] Wmch 1 1 p/2 rpm [thta] [Ibr] Rr [Ic] 0 1 [lamda_br] [Iar] [Ibr] MATLAB Function Tm 1/Jq Rr [Icr] [Icr] T 0 1 [lamda_cr] [thta] TL_0+0.0*TL_0*co(u) T_load Figur 5.13 Simplifid Simulink chmatic to imulat main-fd rotor ccntriciti 74

93 Simulation rult hown in Figur 5.10 through Figur 5.12 corrpond to a mixd rotor ccntricity with ξ = 0.5 and ξ d = 0.5. Th purpo of applying th cofficint i to xplicitly how all th rotor ccntricity charactritic harmonic in th tator currnt pctrum and, thrfor, to furthr vrify th ffctivn of tho rformulatd machin inductanc matric prntd in Chaptr 3. Obviouly, th chon valu of ξ = 0.5 and ξ d = 0.5 ar far away from th practical ca. A mor raonabl tting for a mixd rotor ccntricity i to t ξ = 0.3 and ξ d = 0.2. Simulation rult corrponding to thi tting ar hown in Figur 5.14 through Figur In practic, only th two mot dominant charactritic harmonic f ± f rm hav a ufficintly larg magnitud to b ffctivly monitord in th tator currnt pctrum, whra ngativ qunc harmonic information at ( f + f ) i a mor rliabl indicator to dtct th inducd motor aymmtry. rm f 20 Magnitud (db) f f rm f + f rm Frquncy (Hz) Figur 5.14 Simulatd main-fd i pctrum a with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip =

94 40 30 f Magnitud (db) f f + f rm ( f + f ) rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.15 Simulatd main-fd I pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip = Shaft pd (rpm) Tim () Figur 5.16 Simulatd main-fd motor haft pd profil with ξ = 0.3, ξ d = 0.2, α =0, β = 0, lip =

95 5.3.5 Poition-Dpndnt Load Ocillation in a Main-Fd IM A 100% load torqu ocillation, T = T + T co( θ ), i implmntd in th load avg avg rm imulation to how th load ocillation-inducd id-band harmonic. Simulation rult hown in Figur 5.17 illutrat a much largr harmonic magnitud at f + f rm for a load torqu ocillation ca compard with that rulting from a rotor ccntricity in Figur Undr thi idal imulation condition, i.., purly inuoidal input voltag and prfctly ymmtric motor, thr i no ngativ qunc componnt in th tator currnt pac vctor pctrum, a hown in Figur A load torqu ocillation alo lad to a rlativly larg fluctuation in th imulatd haft pd profil of Figur Not that th avrag motor pd i till kpt at 3555 rpm. In practic, thi haft pd ocillation rulting from a load torqu ocillation i ignificantly diminihd by th mchanical damping ffct f Magnitud (db) f + f rm f f rm Frquncy (Hz) Figur 5.17 Simulatd main-fd i pctrum a with 100% load torqu ocillation 77

96 40 30 f Magnitud (db) f f rm f + f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.18 Simulatd main-fd I pctrum with 100% load torqu ocillation Shaft pd (rpm) Tim () Figur 5.19 Simulatd main-fd motor haft pd profil for a halthy motor with 100% load torqu ocillation 78

97 5.4 A Nw Eccntricity Indicator for Driv-Connctd Machin In principal, a clod-loop induction motor driv provid a currnt-rgulatd voltag ourc to th motor. It typically conit of an innr, high bandwidth, currnt-rgulating loop and an outr, low bandwidth, pd-rgulating loop, a hown in Figur Th currnt control loop i implmntd in a ynchronou rfrnc fram and PI controllr ar ud in both loop to achiv low tady tat rror. Du to th intrinic PI controllr low-pa filtring charactritic, part of th ccntricity rlatd charactritic harmonic information i tranfrrd from th tator currnt to th motor trminal voltag, dpnding on th control loop rgulation capabiliti. Thi complmntary rlationhip in th ditribution of rotor fault indicator rquir dtcting ccntricity rlatd harmonic in both tator currnt and tator voltag pctra imultanouly [41]- [43]. Similarly, a rotor poition dpndnt load torqu ocillation in a clod-loop, drivconnctd induction motor introduc xactly th am harmonic componnt in both tator currnt and trminal voltag pctra a wll. Th load ocillation rlatd harmonic uually hav much largr magnitud and ar likly to cau ambiguity in rotor ccntricity dtction. Thrfor, imilar to a main-fd machin, om nw ccntricity-only rlatd information ha to b xtractd to liminat load ocillation ffct in a driv-connctd induction motor. In a main-fd or opn-loop controlld induction motor, motor aymmtry i compltly rflctd into th tator currnt [44]. Spcifically, ngativ qunc harmonic information, intad of th ngativ qunc fundamntal information, hould b dtctd conidring th invitabl unbalanc in th main upply. Howvr, thi 79

98 chm can not b dirctly applid to a clod-loop driv application inc th ngativ qunc harmonic information i xtrmly mall in a driv-connctd induction motor du to th rlativly trong PI controllr rgulation capabiliti. On th othr hand, th driv-connctd motor trminal voltag com from th currnt rgulatd voltag ourc invrtr. Th inhrnt unbalanc in th invrtr output i vry mall, which i notably diffrnt from th main-fd machin ca. In addition, du to th control loop low-pa filtring charactritic, all ngativ qunc information would b compltly tranfrrd to th trminal voltag if th PI controllr rgulation capabiliti wr prfct. In practic, both tator currnt and trminal voltag contain ccntricity inducd motor aymmtry information. Thrfor, ngativ qunc fundamntal componnt in both tator currnt and trminal voltag nd to b monitord in a clod-loop driv. * ω m ω m PI * i q * i d i q PI PI * v q * v d PWM VSI i a i b i c i d θ ω m i q i d ω m θ Figur 5.20 A typical clod-loop induction motor driv ytm 80

99 Rotor ccntricity inducd motor aymmtry com from th paramtr unbalanc in th inductanc matric and i a function of th tator xcitation frquncy and th lip frquncy. Th ditribution ratio of th ngativ qunc information in th tator currnt and th trminal voltag i alo frquncy dpndnt. To accuratly valuat th rotor ccntricity-inducd motor aymmtry lvl, a ummation of th normalizd ngativ qunc fundamntal componnt with rpct to thir poitiv qunc fundamntal countrpart in th tator currnt and th trminal voltag i propod to b an additional rotor ccntricity fault indicator. That i, δ I V n n = In + Vn = + (5.11) I p Vp whr I n, I p, V n and V p dnot th magnitud of ngativ and poitiv qunc fundamntal componnt in tator currnt and trminal voltag, rpctivly. Sinc th bandwidth and clod-loop tranfr function charactritic of th rgulator ar normally unknown, it i not poibl to formulat a mor ophiticatd ummation of th normalizd voltag and currnt. For thi raon, a impl unwightd um i adoptd. For a clod-loop driv-connctd induction motor, th propod additional fault indicator, δ, combind with th traditional charactritic harmonic indicator in tator currnt and trminal voltag pctra, provid rliabl and nitiv dtction of rotor ccntricity fault vn in th prnc of a load torqu ocillation. 81

100 5.5 Matlab Simulation for a Driv-Connctd Induction Motor Th tator and quivalnt rotor nutral point hav bn confirmd from imulation rult to hav a zro voltag lvl for a main-fd ccntric motor. In othr word, th nutral point idally hav th am potntial lvl a th ground. Simulation chmatic for a clod-loop, driv-connctd induction motor i hown in Figur 5.21, whr th pd loop and currnt loop bandwidth ar t to b 25 rad/ and 2000 rad/, rpctivly. Corrponding PI controllr proportional gain, k p, and intgral again, k i, for ach loop ar calculatd accordingly [39]-[41]. Th mchanical rfrnc pd i lctd to b 3000 rpm, lading to f = 50 Hz, and th avrag load torqu i t to rm yild a driv xcitation frquncy of Hz. 82

101 ia ib m i_dq ic thta_da_t Tm_t Ia Ib Trminator thta_da_t Ic W_yn Wmch Iqd W_yn Etimatd Motor Modl ki i 1 I 3000/60*2*pi Wmch_rf ki kp Id_0 Id_rf 1 Id* Iq* kpi kii kpi 1 Vd* d q Vq* V_abc* thta_da_t qd --> abc 1 Idal SV_PWM m va vb ia ib ic vc Tm Wmch Thta Actual Motor Inrtia 1/Jq 1 Wmch 60/(2*pi) rpm Vqd TL_0+1.0*TL_0*co(u*2/p) T_load V Load Torqu Figur 5.21 Simulink chmatic uing rformulatd motor inductanc for clod-loop driv-connctd machin 83

102 5.5.1 Mixd Rotor Eccntricity in a Driv-Connctd IM Simulatd ingl pha tator currnt and trminal voltag pctra for a mixd rotor ccntricity ar hown in Figur 5.22 and Figur 5.23, whr ξ = 0.3, ξ d = 0.2, α =0, β = 0. Th corrponding pac vctor pctra ar hown in Figur 5.24 and Figur 5.25, rpctivly. It i clar from th imulation rult that for a clod-loop drivconnctd induction motor, th rotor ccntricity introduc dominant id-band harmonic in both th tator currnt and th trminal voltag. A for th ngativ qunc information, it mainly xit in th currnt and voltag fundamntal componnt. Manwhil, ngativ qunc harmonic information in a clod-loop driv i much mallr compard with that in a main-fd machin Poition-Dpndnt Load Ocillation in a Main-Fd IM Simulation rult for a 100% load torqu ocillation in a clod-loop driv ar hown in Figur 5.26 through Figur Idally, load ocillation yild only poitiv qunc information in tator currnt and trminal voltag. Thrfor, ngativ qunc fundamntal information can b monitord to liminat load ocillation ffct. 84

103 40 30 f Magnitud (db) f f rm f + f rm Frquncy (Hz) Figur 5.22 Simulatd driv-connctd i pctrum a with ξ = 0.3, ξ d = 0.2, α =0, β = f 40 Magnitud (db) f + f rm f f rm Frquncy (Hz) Figur 5.23 Simulatd driv-connctd v a pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = 0 85

104 40 30 f Magnitud (db) f f + f rm ( f + f ) rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.24 Simulatd driv-connctd I pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = f Magnitud (db) f f + f rm ( f + f ) rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.25 Simulatd driv-connctd V pctrum with ξ = 0.3, ξ d = 0.2, α =0, β = 0 86

105 40 30 f 20 Magnitud (db) f frm rm f + f Frquncy (Hz) Figur 5.26 Simulatd driv-connctd i pctrum a with 100% load torqu ocillation f Magnitud (db) f f rm f + f rm Frquncy (Hz) Figur 5.27 Simulatd driv-connctd v a pctrum with 100% load torqu ocillation 87

106 40 30 f 20 Magnitud (db) f f rm f + f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.28 Simulatd driv-connctd I pctrum with 100% load torqu ocillation f Magnitud (db) f f rm f + f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.29 Simulatd driv-connctd V pctrum with 100% load torqu ocillation 88

107 5.6 Simulation Rult Bad on Finit Elmnt Modl FEA modl-bad imulation rult ar mor convincing than tho obtaind from a Matlab modl inc FEA can incorporat magntic aturation, winding pac harmonic, a wll a lotting ffct in th imulation. Conidring th fact that FFT analyi rquir a fin frquncy rolution to proprly locat charactritic id-band harmonic, FEA imulation i xtrmly tim-conuming whn applying tranint analyi for a ufficintly long priod D Finit Elmnt Modl A 36-lot, 28-bar, 2-pol induction motor can b modld in Maxwll 2D a hown in Figur In ordr to imulat a purly tatic or dynamic ccntricity, th cntr of rotor can b movd away from th cntr of tator. In addition, th cntr of rotation can b alo diplacd from both th cntr of tator and th cntr of rotor to mulat a mixd rotor ccntricity fault. Figur 5.30 Maxwll 2D modl for th induction motor 89

108 5.6.2 Mixd Rotor Eccntricity for a Main-Fd Machin Auming D i th ditanc btwn th cntr of tator and th cntr of rotation, D d i th ditanc btwn th cntr of rotor and th cntr of rotation, and D i th nominal motor airgap lngth, thn th tatic ccntricity prcntag i dfind to b th ratio btwn th ratio btwn D and D. Similarly, th dynamic ccntricity prcntag i dfind to b D d and D. In addition, th cntr of tator, rotor and rotation li on a traight lin. A mixd rotor ccntricity with 30% tatic ccntricity and 30% dynamic ccntricity i implmntd in th FEA modl. Simulation rult, hown in Figur 5.31 and Figur 5.32, clarly dmontrat th xitnc of id-band charactritic harmonic a wll a ngativ qunc information in th tator currnt Poition-Dpndnt Load Ocillation for a Main-Fd Machin A 50% load torqu ocillation wa alo imulatd in th FEA modl and it lad to much largr id-band harmonic in th imulation rult hown in Figur 5.33 and Figur Again, only poitiv qunc information xit in th tator currnt. It i important to not that th valu of J q i critical to th magnitud of id-band harmonic rulting from a load torqu ocillation. Th mallr th valu of J q ud in th imulation, mor ignificant th motor haft pd ocillation and th largr th magnitud of inducd id-band harmonic. For xampl, imulation rult hown in Figur 5.33 and Figur 5.34 ar bad on a 50% load torqu ocillation and on tnth of th nominal valu of J q ( J q i only 0.01 kg m 2 for imulation rult hown in Figur 5.33 and Figur 5.34). It i clar that id-band harmonic hav much largr magnitud in Figur 5.33 and Figur 5.34 than tho in Figur 5.17 and Figur

109 40 30 f 20 Magnitud (db) f f rm f + f rm Frquncy (Hz) Figur 5.31 FEA modl imulatd main-fd i pctrum a with 30% tatic ccntricity and 30% dynamic ccntricity f 20 Magnitud (db) ( f + f ) rm f f + f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.32 FEA modl imulatd main-fd I pctrum with 30% tatic ccntricity and 30% dynamic ccntricity 91

110 40 30 f f + f rm Magnitud (db) f f rm Frquncy (Hz) Figur 5.33 FEA modl imulatd main-fd i pctrum a with 50% load torqu ocillation f f + f rm Magnitud (db) f f rm Frquncy (Hz) Ngativ qunc Poitiv qunc Figur 5.34 FEA modl imulatd main-fd I pctrum with 50% load torqu ocillation 92

111 5.6.4 FEA Simulation for a Driv-Connctd Machin FEA imulation for a clod-loop driv-connctd induction motor can b implmntd by incorporating vctor control algorithm into FEA program. Th control algorithm can b writtn in C and compild into an xcutabl fil. In th Solvr Stup window a hown in Figur 5.35, thi xcutabl fil i pcifid to intract with th main FEA program aftr complting ach itration tp of th magntic fild olution [41]. Figur 5.35 Maxwll 2D olvr tup intrfac 93

112 5.7 Chaptr Summary In ordr to achiv an ffctiv dtction of rotor ccntriciti in th prnc of a load torqu ocillation, ccntricity-only rlatd information ha to b xtractd to liminat load ocillation ffct. Bad on th diffrnt motor opration charactritic, additional rotor ccntricity indicator ar dvlopd for a main-fd and a drivconnctd induction motor, rpctivly. Extniv imulation rult from both Matlab and FEA modl ar prntd in thi Chaptr to confirm th fficacy of th propod nw rotor fault ignatur. 94

113 6 CHAPTER 6 SEPARATING ROTOR ECCENTRICITY INDUCED MOTOR ASYMMETRY FROM OTHER SOURCES 6.1 Ovrviw Although rotor ccntricity inducd ngativ qunc information i a uful additional fault ignatur to liminat load ocillation ffct, ngativ qunc componnt in trminal quantiti might rult from othr ourc, uch a unbalancd powr upply, tator winding mimatch, magntic aturation inducd aymmtry, motor ambly dfct, intrumntation imprfction, tator intr-turn fault, tc., [52]-[53]. Thrfor, it i ncary to ffctivly diffrntiat ccntricity inducd motor aymmtry from othr factor. 6.2 Inhrnt Motor Aymmtry Similar to th tator intr-turn fault dtction, inhrnt motor aymmtry and intrumntation imprfctn hould b proprly valuatd throughout th motor oprating rang inc thy ar highly dpndnt on th working condition. In gnral, th ffct of motor non-idaliti ar rlativly air to b compnatd by ithr uing data lookup tabl mthod [45]-[46] or applying nural ntwork tchniqu [44] during th motor commiioning tag. 6.3 Stator Intr-Turn Fault A tator intr-turn fault lad to xactly th am ngativ qunc information a a rotor ccntricity, at both fundamntal and charactritic id-band harmonic. Thr ar 95

114 a fw othr tchniqu to a th tator winding inulation condition, uch a monitoring th qunc componnt impdanc matrix [48]-[49], or dtcting th voltag mimatch lvl [47]. Howvr, th mthod ntially till utiliz th tator turn fault inducd ngativ qunc information in motor trminal quantiti. In th xtrm ca whr both th load torqu ocillation and th tator turn fault occur imultanouly, vn th nwly propod additional rotor ccntricity indicator in (5.1) and (5.11) fail. Fortunatly, xcpt for xamining ngativ qunc information in tator currnt and voltag, an incipint tator turn fault can b alo dtctd by monitoring zroqunc componnt [50]-[53]. Th zro-qunc information rv a th lat rort to parat ccntricity inducd motor aymmtry from that caud by a tator turn fault. 6.4 Zro-Squnc Information In a Y-connctd induction motor, th zro-qunc voltag i dfind a, whr V a 1 V0 = ( Va + Vb + Vc ) (6.1) 3, V b and V b ar th lin-nutral voltag, which can b xprd in th following condnd form, whr V abc V = r [ E] i + [ L ] pi + p[ L ] i + [ L ] pi + p[ L ] i (6.2) abc abc abc abc r abcr r abcr, i abc and i abcr ar th tator voltag, tator currnt and rotor currnt vctor, rpctivly. Th matrix [ E ] i an idntity matrix for a halthy or ccntric motor, [ E] halthy = [ E] cc = (6.3) 96

115 Applying (6.2) into (6.1), a compact xprion for th zro-qunc voltag can b obtaind aftr a fw tp of mathmatical manipulation, r [ E] m, n in m= a, b, c 1 V = ( ) V = + [ L ] pi + p[ L ] i + [ L ] pi + p[ L ] i 0 n m, n n m, n n 3 n= abc,, n= abc,, m= abc,, m= abc,, r mnr, nr r mnr, nr m= abc,, m= abc,, (6.4) It i ay to how that th zro-qunc voltag for a halthy motor i idally zro conidring th fact that th ummation of thr pha currnt i zro Zro-Squnc Information for a Stator Intr-Turn Fault In th ca of a tator intr-turn fault, ach matrix in (6.2) ha th following form, 1 η 0 0 [ E] turn = (6.5) (1 η) (1 η) (1 η) [ L ] turn = Ll [ E] turn + L m (1 η) (1 η) (6.6) 2 4 (1 η) co( θr) (1 η) co( θr + π) (1 η) co( θr + π) [ Lr ] turn = L m co( θr + π) co( θr ) co( θr + π) co( θr + π) co( θr + π) co( θr) 3 3 (6.7) 97

116 0 0 0 [ ] = pl turn (6.8) 2 4 (1 η)in( θr) (1 η)in( θr + π) (1 η)in( θr + π) pl [ r ] turn = ωrl m in( θr + π) in( θr ) in( θr + π) in( θr + π) in( θr + π) in( θr) 3 3 (6.9) whr η dnot th fractional prcntag in th tator a-pha intr-turn fault. Subtituting (6.5)-(6.9) into (6.4) giv th zro-qunc voltag for a tator intr-turn fault, V 0 ηri co( ωt+ α) + ηωlliin( ωt+ α) 1 = ( η) ηωlmiin( ωt+ α) + ηωlmirin( ωt+ β ) 2 2 (6.10) whr I, I r, α and β ar th amplitud and th initial a-pha angl for th tator and quivalnt rotor currnt, rpctivly. It i clar to from (6.10), only fundamntal componnt at ω idally xit in th zro-qunc voltag for a tator intr-turn fault. Th magnitud of th zro-qunc information i dominatd by th contribution from th machin ingl pha magntizing inductanc, L m. In practic, for an invrtr-fd machin, high-frquncy componnt that ar intrinic to th PWM opration and tripln harmonic rulting from om typ of modulation mthod may alo xit in th zro-qunc voltag. Th harmonic componnt ar not rlatd to th tator intr-turn fault and can b aily low-pa filtrd in th maurmnt intrumntation. 98

117 6.4.2 Zro-Squnc Information for Rotor Eccntricity Similar calculation for th zro-qunc voltag can b applid to an ccntric motor by ubtituting (3.17)-(3.18) and (3.20)-(3.21), or (3.35)-(3.36) and (3.38)-(3.39), into (6.4) for a purly tatic ccntricity and a mixd rotor ccntricity, rpctivly. In both ca, rotor ccntriciti rult in a null zro-qunc voltag, which furthr confirm th concluion obtaind form imulation rult prntd in Chaptr Comparion of Zro-Squnc Information Thortical analyi conform to th phyical raoning vry wll. Rotor ccntricity fault only lad to a prmanc modulation in th airgap and th rulting motor paramtr unbalanc in inductanc matric i ymmtrically n by thr pha winding. Conquntly, for an ccntric motor, ummation of all column lmnt in inductanc matric (xcluding lakag inductanc componnt) a wll a thir drivativ i alway zro. On th contrary, a tator intr-turn fault yild diffrnt ffctiv numbr of turn for ach pha and th rulting motor paramtr unbalanc i not ymmtrical. Thrfor, zro-qunc information ari in thi ca. Although lin-nutral voltag may not b accibl in all ca, monitoring zroqunc information may till b incorporatd into rotor ccntricity diagnotic tratgi for om critical application. Actually, continuou maurmnt of zroqunc information do not incra th motor condition monitoring cot ignificantly [58]. With thi additional zro-qunc maurmnt, it i bnficial to improv th total diagnotic prformanc for both rotor ccntricity and tator intr-turn fault dtction. 99

118 6.5 Chaptr Summary Thi chaptr prnt a brif rviw of diffrnt factor that may produc ngativ qunc information in upply currnt and/or trminal voltag. Motor inhrnt nonidaliti and intrumntation mimatch can b dtrmind and tord during th commiioning tag, and ud latr on to proprly t up th fault ignatur thrhold valu in onlin motor condition monitoring. To ffctivly parat rotor ccntricity inducd motor aymmtry from that rulting from a tator intr-turn fault, upplmntary zro-qunc information ha to b xamind a wll. 100

119 7 CHAPTER 7 EXPERIMENTAL SETUP AND IMPLEMENTATION OF LOAD TORQUE OSCILLATION 7.1 Ovrviw - Baic Exprimntal Stup Th baic xprimntal tup i illutratd in Figur 7.1, whrin th xprimnt ar prformd for both main-fd and clod-loop driv-connctd induction motor. Motor trminal quantiti ar nd through currnt/voltag tranducr and fd into th National Intrumnt data acquiition (NI DAQ) dvic. Th dynamomtr of th motor conit of a DC gnrator and fiv paralll connctd ritor bank. By controlling th on/off tatu of th rhotat, an ocillating load condition i mulatd. A man-mad rotor ccntricity i intntionally cratd inid th motor and ttd undr a contant load. Th rult obtaind from an ccntric motor ar furthr compard with tho for a halthy motor undr both contant load and ocillating load condition Motor Eccntricity Implmntation To proprly crat a rotor ccntricity fault and avoid tator-to-rotor rub, th rotor outr urfac i firt machind to incra th airgap lngth from to A tatic ccntricity i cratd by nlarging th baring houing and placing 0.01 him on both id to offt th rotor. A dynamic ccntricity i cratd by machining th motor haft blow th baring and inrting a him on on id. For comparion purpo, anothr motor with th am pcification i alo machind around th rotor outr urfac to achiv xactly th am nlargd airgap lngth, whil no ccntricity fault i cratd for th cond motor. 101

120 Analog Dvic DSP board Shaft pd informatioin RS232 cabl Computr GPIB cabl NI DAQ dvic IGBT_0 IGBT_1 IGBT_2 IGBT_3 Controllabl Rhotat 0 Controllabl Rhotat 1 Controllabl Rhotat 2 Controllabl Rhotat 3 Contant Rhotat 0 Currnt/Voltag tranducr Incrmntal Encodr Induction Motor LovJoy Coupling DC gnrator Invrtr Output Main-fd DC gnrator To gnrator fild xcitation Shaft pd informatioin Alln Bradly 1336E Driv AC 230V Sourc DC 90V Sourc Figur 7.1 Illutration of xprimntal tup with load ocillation control circuit 102

121 Th motor ttd in th xprimnt i a thr pha gnral purpo induction motor, who namplat data and impdanc information ar litd in Appndix B. Th motor i only opratd with 230 Volt configuration, rulting in two iolatd nutral in tator winding. Sinc th airgap lngth i ignificantly incrad, th motor i opratd with a largr lip (rducd pd) and lowr output powr if th tator currnt i till limitd by th originally ratd valu Motor Dynamomtr Sytm A 10 hp dc gnrator i linkd to th motor haft through Lovjoy coupling. Svral ritor bank ar connctd in paralll to th dc machin armatur winding to control th motor load lvl. A ritor bank ha at mot ight 50 ohm ritor and ight 100 ohm ritor connctd in paralll. In gnral, for a contant load opration, on or two ritor bank ar ufficint to rach th ratd currnt condition with a dc fild xcitation of 90 Volt. To implmnt an ocillating load, fiv ritor bank ar connctd in paralll to th dc gnrator, whrin four of thm hav a rially connctd IGBT witch rpctivly, a hown in Figur 7.1. By controlling th on/off tatu of IGBT, diffrnt numbr of ritor bank ar plungd into th load ytm to imitat a quai inuoidal load torqu ocillation Clod-Loop Driv Configuration An Alln-Bradly driv 1336E driv i ud for th clod-loop, driv-connctd motor opration. Corrponding ncodr i an Alln-Bradly 845S incrmntal optical on. Main paramtr of th driv and th ncodr ar givn in Appndix B. 103

122 7.1.4 Voltag and Currnt Maurmnt For norl motor condition monitoring, only maurmnt of th upply currnt and th trminal voltag ar rquird. Thy can b nd via LEM LA-55P and LEM LV-25P tranducr, rpctivly. To analyz th charactritic id-band harmonic a wll a th qunc information in trminal quantiti, two pha currnt ( I a, I b ) and two pha-pha voltag ( V ab, V bc ) maurmnt ar ufficint. Howvr, to valuat zro-qunc information for a wy-connctd induction motor, thr phanutral voltag maurmnt ar mandatory. For th low voltag connction, thr ar two floating nutral in tator winding, a hown in Figur 7.2, whr both N 1 and N 2 ar tator nutral point. Idally, both nutral hould hav th am voltag lvl auming thr i no mimatch among pha winding. For a complt amnt of zro-qunc information, both point hould b dtctd. Figur 7.2 Stator winding configuration for 230 Volt opration 104

123 On-lin data acquiition u a 5 khz ampling rat and a total 5 cond ampling tim to acquir a 0.2 Hz frquncy rolution in th FFT pctral analyi. Thi maurmnt rquirmnt conum modrat mmory torag and provid ufficint accuracy for th practical implmntation. 7.2 Implmntation of Load Torqu Ocillation On way to crat a poition-dpndnt load torqu ocillation i to mount an unbalancd dik on th motor haft. Th unbalancd dik ha a tl bolt and nut placd in hol on a balancd mtal plat. Th hol ar drilld at diffrnt radial ditanc from th rotor haft. Th tangntial forc producd by th bolt and nut dpnd on thir total ma, th radial ditanc from th haft, and th haft pd [38]. Although thi xprimntal tup crat a mall amount of load torqu ocillation, it alo impo a ignificant cntrifugal forc on th motor haft. Thrfor, thi mthod ntially crat a rotor ccntricity (mainly dynamic ccntricity) inid th motor, which i not th concrnd ca of a purly load torqu ocillation in rciprocating compror Purly Load Ocillation Implmntation Principl Th chmatic to implmnt a purly load torqu ocillation i illutratd in Figur 7.3. In thi chm, ach of th four ritor bank (R 0, R 1, R 2 and R 3 ) i rially connctd to an IGBT to contitut a paralll branch of th dynamomtr. Th rmaining ritor bank (R 4 ) i dirctly connctd to th dynamomtr dc output. For th control purpo, th gat ignal for ach IGBT witch ar th output of an Analog Dvic DSP board (SHARK ADSP-2181). Dtaild information rgarding thi DSP board i givn in Appndix C. 105

124 Th main Viual DSP++ program on th computr produc a qunc of urdfind high and low voltag ignal (2 Volt and 0 Volt) at ach of th four I/O flag on th DSP board. Th DSP output ar furthr nt to th IGBT board to control th on/off tatu of th IGBT a wll a th paralll branch. Each of th ritor bank R 1 - R 4 ha only on ritor R conductd and th conumd powr would b V 2 dc R if thi paralll branch i on. Th ritor bank, R 4, ha m R ritor connctd in paralll and conum a total powr of 2 ( mvdc ) Rcontinuouly. Th witching qunc for ach paralll branch and th voltag lvl for th corrponding IGBT gat ignal ar givn in Tabl 7.1. Thi witching pattrn i rpatd to yild a quai inuoidal load ocillation a plottd in Figur 7.4. On advantag of th propod witching qunc i to kp th avrag load lvl unchangd bfor and during th load ocillation condition. Hnc, no ignificant haft pd variation occur aftr th load ocillation i nforcd. If th motor haft pd i fd back to th main Viual DSP++ program to ynchroniz th motor pd with th witching frquncy, a tim-varying or poition-varying load torqu ocillation i accomplihd with thi chm IGBT Configuration Th IGBT board configuration i hown in Figur 7.5. It conit of two IGBT modul and on ingl pha diod rctifir. Th IGBT chip ud in th xprimnt i IRGBC30U from Intrnational Rctifir, which ha a collctor-to-mittr blocking voltag of 600 Volt and a maximum continuou collctor currnt of 23 A at room tmpratur. Th rating ar ufficintly larg for th xprimntal invtigation of th load torqu ocillation prformd in thi rarch. Th IGBT modul i connctd to th DSP board through th BNC cabl a hown in Figur

125 Figur 7.3 Schmatic to control load torqu ocillation Figur 7.4 Quai inuoidal load torqu ocillation profil 107

126 Tabl 7.1 Mchanim to implmnt a quai inuoidal load ocillation 108