Software Methods and Tools for the Design and Optimization of Electromechanical Devices

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1 oftwae Methos an Toos fo the Desgn an Optmzaton of Eectomechanca Deces Paa 5 Lectue : The Numeca outon of the Fe Equatons mpe Fnte Eements Paa 5

2 Outne The fe equatons Repesentng sufaces One-Dmensona fnte eements Two-Dmensona fnte eements A physca ntepetaton of the mat Paa 5 The Fe Equatons Eectomagnetc system pefomance s escbe by Mawe s Equatons: Note that these equatons say nothng about eatonshps between B an H o E an J D H J t B E t D ρ B Paa 5 4

3 Mateas Mateas o athe the physca popetes ceate the ns between the fes.. The Consttute Reatonshps: B H J E D E Pemeabty Conuctty Pemttty Paa 5 5 Mateas Thus a hgh pemeabty w ceate a age oca magnetc fu fom a magnetc fe.. Mateas ae use to conto the shape of magnetc an eectc fes Basc constants: Pemeabty of fee space Pemttty of fee space Henes / m Faas/ m Paa 5 6

4 Matea Pemeabty M9 Magnetzaton Cue.5 B (T) H (A/m) Paa 5 7 Low Fequency Magnetcs () Conse a stuaton n whch f=,.e. statcs: t Mawe s Equatons euce to: H J E D ρ B Paa 5 8 4

5 Low Fequency Magnetcs () Ang n the consttute eatonshp: B H B J In two-mensons, B has two components the ecto es n the -y pane. Howee, snce A substtuton of the fom B A Paa 5 9 B s aowe Low Fequency Magnetcs () Ths esuts n the cu-cu equaton A J Whch euces to A J Ths s Posson s Equaton. If J=, t s Lapace Note that A has ony one component: A z, n two mensons Haf the computng compae to B Paa 5 5

6 The Fe as a uface () Posson s equaton poes a eatonshp between: The Fe Vaabe, A The geometc poston, an y The matea popety, µ The eectc cuent ensty, J Paa 5 o we hae: A = A(,y) µ = µ(,y) J = J(,y) The Fe as a uface () A can be consee as a suface oe the geometc space Paa 5 6

7 The Fe as a uface () The ffeent coous epesent the heght of the functon aboe the pane.. Paa 5 The Fe as a uface (4) The output pot s a esponse to the nput ata: The geomety The ectatons Goene by Posson s Equaton: A J Paa 5 4 7

8 The Fe as a uface (5) Note ths s a physca athe than a mathematca ew o the pobem s: How o we fn the suface? tep : How can sufaces be moee? Conse a one-mensona pobem the suface s a cue e.g. the otage stbuton aong a bue ppene.. Paa 5 5 A One-Dmensona Pobem () Conse a bue ppene beng use to cay sgnas E.g. a measuement of the ntegty of a gas ppene. Pobem: fn the otage stbuton (ef: este an Fea, Fnte Eements fo Eectca Engnees) Bue Ppes + g + Equaent Ccut of shot ength,, of ppe. Paa 5 6 8

9 A One-Dmensona Pobem () Ths pobem has an anaytca souton.. But we w not use that.. In the fnte eement appoach, the tansmsson ne equatons w not be soe ecty.. (tue of most f.e. appomatons) Instea use the physca pncpe that the otage aong the ne w ajust tsef to mnmze the powe oss. Paa 5 7 A One-Dmensona Pobem () The steps to soe ths pobem ae: Epess the powe, W, ost n the ne n tems of the otage stbuton, (): W W () ube the ente tansmsson ne nto K fnte sectons = eements Paa 5 8 9

10 A One-Dmensona Pobem (4) Appomate the otage () usng a sepaate set of appomatng epessons n each eement: ( ) M f ( ) f ae some conenent nown functons (chosen n aance), ae unnown coeffcents (an thee ae M of them) Epess the powe n each eement n tems of f () an the M unetemne coeffcents, Paa 5 9 A One-Dmensona Pobem (5) Because f ae chosen n aance, they ae nown. The powe s thus a functon of the unnowns,.. W W (,,..., M Intouce constants on the MK coeffcents to ensue the otage s contnuous fom eement to eement. The ensembe of eements w possess some N egees of feeom, N MK. Fnay mnmze the powe by ayng each coeffcent n tun subject to the constant that the otage aong the ne must ay n a contnuous fashon.. ) Paa 5

11 A One-Dmensona Pobem (6) W,,..., N Ths mnmzaton etemnes the coeffcents an thus pouces an appomate epesson fo the otage aong the ne.. Questons: How many eements? What sot of appomatng functons? How ae constants ntouce? What type of mnmzaton technque? The answes efne whoe fames of fnte eement methos.. Paa 5 A One-Dmensona Pobem (7) Bue Ppes Epct epesson fo powe: Conse the secton,. Powe enteng s: W n + g + Equaent Ccut of shot ength,, of ppe. Powe eang s: ( )( ) W out Negectng secon oe tems, the powe ost n s: An g, W Paa 5

12 A One-Dmensona Pobem (8) Thus the powe oss pe unt ength s: W g ( ) An the tota oss fo the whoe ne s: W L g Now use pecewse-staght appomatons fo the otage stbuton aong the ne...e. assume the otage aes neay aong each of the K segments (eements).. Note that the otage nees to be contnuous fo ffeentaton but the sope oes not hae to be Paa 5 A One-Dmensona Pobem (8) Descbe the otage n a segment by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Whee () s the eement, s the ght en an the eft fe aue Eement Paa 5 4

13 A One-Dmensona Pobem (9) ) ( ) ( Epess ths, fo an eement, as: whee an o the tota powe ost n the ne s: g W W W K ) ( ) ( whee Paa 5 5 A One-Dmensona Pobem () g W ) ( ) ( ) ( ) ( Puttng t a togethe: Assume that an g (the esstance an conuctance) ae constant n an eement. The powe ost n an eement can be epesse as: T T g W j j j j an j tae on both aues, an Paa 5 6

14 A One-Dmensona Pobem () Constuctng the matces: L ( ) ( ) Defne a nomaze oca coonate: o Now ( ) L Paa 5 7 A One-Dmensona Pobem () Conse an T: But W gt j j T j j L Inepenent of the eement ength Paa 5 8 4

15 5 A One-Dmensona Pobem () T g W L g L W T T () )() ( 6 ) ( Paa 5 9 A One-Dmensona Pobem (4) 6 T o An the powe oss n a snge eement s: 6 L g L W Paa 5

16 A One-Dmensona Pobem (5) Now we nee to oo at the whoe ne ' ' ' 4' 5' ' The sjont segments nee to be connecte togethe an then the en otages set up to be equa whee they jon to poe contnuty.. Thus = 6 s = 4 con Paa 5 A One-Dmensona Pobem (6) The sconnecte otages ae eate to the 6 connecte ones though: s ' ' ' 4' 5' ' con ' ' ' 4' 5' ' Vs CV con Paa 5 6

17 7 A One-Dmensona Pobem (7) s s T s s T s V M V V L g L L g L L g L V W T... T T Wtng the tota powe as the sum of the eement powes: an C M C M CV M C V W s T con s T T con Paa 5 A One-Dmensona Pobem (8) If the esstance an conuctance pe unt ength ae the same fo a eements an a eements ae of ength L e, the fna connecte matces fo the 5 eement moe ae: L C C e T Paa 5 4

18 A One-Dmensona Pobem (9) An, fo the T mat: gc T T L eg C Paa 5 5 A One-Dmensona Pobem () The tota powe oss s: W V W V T con T con T T C C gc TC V MV con The ast step s to mnmze the powe oss. A the otages ae fee to ay ecept one the one at the souce en of the ppe W,,,..., N con Paa 5 6 8

19 A One-Dmensona Pobem () Dffeentatng ges: M M.. '' '' M M '' ''..... M M 5'5' '5' M M 5'' '' ' '.. 5' ' Howee, s actuay fe t s not an unnown Paa 5 7 A One-Dmensona Pobem () Mong the nown aues to the ght han se ges: M M.. '' '' M M '' ''..... M 5'5' ' M ' M.... 5' M '' ' '' ' 5'' ' Ths equaton epesents eacty as many smutaneous agebac equatons as thee ae fee noes Paa 5 8 9

20 Appomaton Eos What etemnes the eo n moeng the souton by fnte eements? It epens how many eements ae use It epens on the shape of the cue beng moee It epens on the functons use n an eement Paa 5 9 ummaze De the pobem nto sma peces (eements). Assume the fe behao oe the eement can be moee usng a nea combnaton of smpe functons pecfy the functons Detemne the coeffcents of each functon by mnmzng an epesson fo a physca quantty (e.g. enegy) Paa 5 4

21 ummaze Jon a the eements togethe to coe the compete pobem oman. Connect equaent ponts Mnmze the goba quantty oe the esutng set of nea equatons to etemne the potenta aues eeywhee. Paa 5 4 Two-Dmensona Pobems Not too many ea magnetc eces can be moee n one-menson Two-mensons,.e. an y o an θ, s the mnmum fo many eces. o, how can the metho be etene to -D? Paa 5 4

22 Two-Dmensona Fnte Eements () Fst.. What eement shape shou be use fo twomensona pobems? In theoy, any poygona shape cou be use If we eten the -D appoach, then use a nea shape functon,.e. nea n an y.. U a by c Ths s a fst oe shape functon.. Note U s an appomate souton A s the tue souton A=U f the moe s accuate Paa 5 4 Two-Dmensona Fnte Eements () The functon has unnowns: a,b an c o, we nee to be abe to efne equatons whee U s efne at thee ponts n space (,y ), (,y ), (,y ) to compute them.. o the basc shape has to hae thee etces o noes,.e. a tange. Ths s a smpe eement Paa 5 44

23 Two-Dmensona Fnte Eements () A smpe s the smpest geometc stuctue n any patcua menson.. In -D the smpe s a ne In -D the smpe s a tange In -D the smpe s a tetaheon Paa 5 45 Two-Dmensona Fnte Eements (4) Thee etces - noes Usuay numbee n an ant-cocwse ecton. Lnea shape functon: U a by c Paa 5 46

24 4 Two-Dmensona Fnte Eements (5) y U Geometc Eement Potenta aues U U U c b a y y y U U U U U U y y y y U Paa 5 47 Two-Dmensona Fnte Eements (6) y y y y y A y U U ) ( ) ( ) ( ), ( Rewtng: α s nown as a poston functon Obtane by a cycc otaton of the subscpts They ae ntepoatoy functons,.e. j y j y j j j j ), ( ), ( Paa 5 48

25 Two-Dmensona Fnte Eements (7) The enegy n an eement can now be efne: Gng: W ( e) U W ( e) j U U U U U j j Paa 5 49 Two-Dmensona Fnte Eements (8) Wtng ( e) j j The enegy n the eement can be epesse by: W U ( e ) T ( e) One tem n the mat can be eauate (the othes ae obtane by cycca pemutaton of nces) ( ) ( y y)( y y) ( )( 4A e U ) Paa 5 5 5

26 Two-Dmensona Fnte Eements (9) The enegy n the system s gen by the sum of the eement eneges: 6 5 W e W ( e) 4 Paa 5 5 Two-Dmensona Fnte Eements () As n -D, the eements must be jone to mpose contnuty A Connecton mat can be ceate an the noes enumbee Paa 5 5 6

27 Two-Dmensona Fnte Eements () Fo the two eements, the combne mat s: As befoe, the souton of the fe pobem eques the mnmzaton of stoe enegy W U Paa 5 5 Two-Dmensona Fnte Eements () The souton to ths pobem s ta,.e. U=, f thee ae no pescbe bounay aues If some bounay potentas ae fe, then the U ecto can be e nto U f (fee) an U p (pescbe). If the fee noes ae numbee fst: W U [ U ] f U U T T ff fp f U U f p pf pp p Paa

28 Two-Dmensona Fnte Eements () Ths esuts n: ff ff U f U f fp U p U ong fo U f esuts n a souton to the pobem fp p.e. U s appomatey equa to A Paa 5 55 Eo Estmaton Conse a two-mensona pobem goene by Lapace s equaton: u= u It s en by the bounay aues: Ethe u=u spec on conucto sufaces u= u O on symmety panes. n The stoe enegy s: W ( u) u u D. Lowthe

29 Eo Estmaton Assume that u(,y) s the tue souton to the pobem whe h(,y) s a suffcenty ffeentabe functon wth a aue of at eey bounay pont whee u has a specfe aue fom the bounay contons.. o (u+θh) whee θ s a scaa paamete s an appomate souton whch has the same pescbe bounay contons as u. The enegy of ths appomate souton s, then: W ( u h) W ( u) u. h.5 h. h D. Lowthe 5 57 Two-Dmensona Fnte Eements (4) How goo s the souton? o Whee ae the eos? Ths s a pecewse nea appomaton.. The best ft fo the numbe of eements The souton s EXACT at the pescbe bounay aues (the U p ). u W ( u h) W ( u) W ( h) h u h s n Paa

30 Two-Dmensona Fnte Eements (5) u W ( u h) W ( u) W ( h) h u h s n Eo wthn the aea Eo on the bounay On the bounay h s zeo wheee u s pescbe. ystem w ty to mae u/ n zeo n some aeage sense t w not be zeo at each pont Paa 5 59 Two-Dmensona Fnte Eements (6) Wthn the aea, the connecton between eements ensues that the aaton of u aong the common ege s the same n both eements. The aaton of u noma to the common ege s ffeent on ethe se of the eement.e. ths aaton s scontnuous (we not mpose C contnuty on the connecton) Ths scontnuty can be consee an ncaton of the oca eo. Paa 5 6

31 Two-Dmensona Fnte Eements (7) Paa 5 6 Two-Dmensona Fnte Eements (8) 5 6 Vaue of u aong ne though eements 4 Dstance aong ne Paa 5 6

32 Two-Dmensona Fnte Eements (9) How can the accuacy be mpoe? Moe eements the moe eements, the bette the suface shape can be matche.. Change the shape functons.. Use a hghe oe, e.g. a quaatc: U a by cy ey f Now we nee 6 ponts to fn the unnowns Paa 5 6 Two-Dmensona Fnte Eements () The mathematca eeopment foows the same pocess as befoe hape functons can be fst, secon, th, fouth, oe ystems can be but to ncue C contnuty.. Paa 5 64

33 Two-Dmensona Fnte Eements () What about Posson s equaton? Eectomagnets These systems hae souces whch ae not on the bounaes.. A J Co wth cuent ensty J Paa 5 65 Two-Dmensona Fnte Eements () The enegy eate functona s: F( u) u u uj It can be shown that ths w each a tue souton when mnmze, the same appoach as was taen wth Lapace can be use The mnmum of the functona occus when u=a, the souton of Posson s equaton. Paa 5 66

34 Two-Dmensona Fnte Eements () The fnte eement fomuaton pocees as befoe. The fst tem n the enegy s just the Lapace tem an can be epesente wth the same mat. Oe an eement, appomate the cuent ensty, J, n the same way as the potenta: J J (, y) Paa 5 67 Two-Dmensona Fnte Eements (4) The souce tem ntega s then: AJ Let T be efne as: T e j j A j J The eement nput enegy s then T ( e) AJ A T J j j Paa

35 Two-Dmensona Fnte Eements (5) Aowng fo the connecton of the eements, etc., the scetze equaton fo the tota enegy s: T T F( A) A A A TJ An mnmzng: F ff A f A Resuts n the fna system: TJ fp A p Paa 5 69 Two-Dmensona Fnte Eements (6) Incuson of matea popetes o fa the eeopment has assume that the mateas ae a a It aso assumes that the pemeabty s fe In a ea ece, the pemeabty, µ, aes wthn the geomety an s eay a functon of poston: (, y) In ths case, the pemeabty has to eman nse the cu opeato Paa 5 7 5

36 Two-Dmensona Fnte Eements (7) Thus the cu-cu equaton s: A J In ths case, the stoe enegy now has to tae nto account the enegy n the matea epesente by ntegatng up the B-H cue.. Paa 5 7 Two-Dmensona Fnte Eements (8) M9 Magnetzaton Cue Wong pont.5 Enegy B (T) H (A/m) Paa 5 7 6

37 Two-Dmensona Fnte Eements (9) The enegy n the matea s gen by: E B H. b The fnte eement fomuaton can now be ewoe substtutng fo B an H wth A an foowng the same pocess as befoe. Note that snce the pemeabty s a functon of A now, ths s a non-nea pobem. Paa 5 7 Two-Dmensona Fnte Eements () A typca tem n the mat s now gen by: ( ) ( y y)( y y) ( )( ) 4A e Ths s a nease eson.. To soe the pobem, t eques an teate pocess whee the µ aues fo each eement ae upate as a esut of each souton of the nea equatons. It s usua to assume that µ s constant wthn an eement. Paa

38 Intepetng the mat The tems n the mat hae the mensons of euctance.. In fact, ths cou be consee to escbe an equaent ccut epesentng the magnetc ece.. Paa 5 75 ong the Equatons The esutant fnte eement base set of equatons fo a magnetostatc pobem s VERY age an non-nea A typca pobem fo an eectca machne may nee 4 o 5 noes.. Two soe pocesses ae neee: A LINEAR equaton soe A NON-LINEAR pocess.. Paa

39 ong the Equatons Any nea equaton soe w wo.. Howee the system s: Vey spase Can be bay contone A ect (gaussan emnaton) soe has appomatey O(N ) compety N s the numbe of egees of feeom = fee noes. Paa