magnétiques par des courants de surface. Application aux noyaux ferrites 2D

Transcript

1 Modélisation PEE : epésentation des matéiaux magnétiques pa des couants de suface. Application aux noyaux feites D Hai ui Ngoc To cite this vesion: Hai ui Ngoc. Modélisation PEE : epésentation des matéiaux magnétiques pa des couants de suface. Application aux noyaux feites D. Enegie électique. Univesité de Genoble,. Fançais. <tel v> HAL Id: tel Submitted on 3 Oct (v), last evised Jan (v) HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.

2 ADEE ADAEFFE FADA AAAD D AFE EAADEAAD DDAE ED A EDE FFA AAAAE A AADEDED FE AEDFD AAA DE AAA

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4 Lun án này con dành tng b m thân yêu! À mes paents!

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6 Remeciements AEDEAAEEAA DDA ADAAAADADA EAEEE AAAEDDDAEAA EDADAADEDEDA DEDEAADDA AAAAA AEAEAEDADD DAAEDDADADEDEADE AAAADADED AAADE DEDAAAADDEEAAAAA DAADADDADDAEDDA DEAAADA AAAEADEDEDAE ADFDDEDADAEDEDADA ADDDA ADEDAA AAADEDFA DEDD ADAAA AEDDADAADAEDDA AAFAAADA D ADDAD DAED Je tiens à emecie bien chaleueusement mes amis vietnamiens : ám n anh Hùng, nh mt ngi anh tai ca em, Aã luôn giúp A ch ddn cho em te nhfng ngày Au em At chân lên nc Pháp. ám n anh Tng, anh c, anh Hiu béo, anh Tun bc Aã ch cho em nhfng kinh nghim v cuc sng cng nh tong hc tp và nghiên cu, cám n các anh Aã chia s vi em nhfng nm tháng Ay ý ngha ti Genoble này. m n anh ch em tong gia Aình in ti Genoble cng nh tên toàn nc Pháp vì s Aoàn kt và gn bó, vì nhfng tình cm nng m chúng ta Aã dành cho nhau, cám n anh ch em Aã luôn Ang viên và giúp A mình tong thi gian qua.

7 Mes denies mots iont à ma famille, qui m a continuellement soutenu : on cm n b m Aã luôn bên cnh, tin tng ng h và Ang viên con, cám n b m Aã dành cho con nhfng th tt Ap nht và to Aiu kin cho con Ac hc An ngày hôm nay. háu cám n bà Aã tn to dy bo và chm sóc cháu tong sut nhfng nm tháng cháu hc Ai hc. Không có bà chc chn cháu không có Ac ngày hôm nay, cháu cám n bà nhiu lm! ám n dì Aã luôn thng yêu và Anh hng cho cháu. ám n em yêu Aã An bên anh, cho anh nhfng giây phút ngt ngào và hnh phúc. ám n em Aã luôn bên cnh, ng h và chia s vi anh nhfng nm tháng vea qua! Meci encoe à tous!!! onne lectue! Genoble, Mai

8 AD F FEAFEE FEEFEA FAAEAF E DAEDE AAEDE E DAEDE EAEDDA AAEDE DEDDA DDADE EDAAEADA DA DADA FADAA DAEDAED ADAEADADDDDA D FADADDDDADDD ED

9 AD EAAADAAEAAEA EAEDA D DA DADDA FADAA DDAEDA FFFF DDADA DDDA DDDAD FFFF F E A A FEAFAEFE E FF AAEAAEA ADAA DE EDDE FAAADDAA FAAADEAADA AADAEA FF F DEAEAEAEEAEDED DEAEAEDED DEDEDD

10 AD AAAAADAAEA AAEEAAA ADAA ED DDAD ADADDA DDAED F FDFFEEA AFE FFF AAA AEEDDAAEADA AEEDDADED A DEADAA ADADAD D D A D A ADADAD DEADA EADADAD D AAD AAD EADADAD D AAD

11 AD AAD EADADA DEAD F DADDAEDDADDA AEAAA AEDAA EEAEA EDD EEAAA EEAEA EEA F AEDADAEDEAED AAAEDDEAAED ADAEDEAEDDEEA A F F AF A FFE AFFAFEFAF E F AAEDA ADAEAEAAEDADEA AADAE

12 AD DEEAEDAA DEDEAEAED ADAED ADAEADAEDAEDAAE AADADEE AADAAED AADADAAED AAD DDAE DEAEEDDADE EDE AAD DDAEDDAEAAAD ADDDDAED ADAEADDDDADEAA DA F EAFEF EFFFA EE EE A A EFA AE F FFEAAEFEFEEFA EEAAEFAAF FFEAAEEFA EEAAEAFF EE A A EFA AE F FFEAAEFAEFAFFEFEF

13 AD EE A A EFA AE F FFEAAEFEAEFAFEFEF EEE

14 F

15

16 EFADAFFD AADAEDAEADDDE EDAEDADAEAAAEDAADAE DADADEDEAAAADE EADDDDEDAEE DEADEADAEAADA DAADADAEAEADEDAAD EDEAEDDADADA EDAADADAEAADDAEE ADEDADADEAAA EDAEADAEAAAD DDAEEAEADDDAADAA DAEDEEAAADDAAADA EAA D ADAA E A E A AA EDEAEDDEA EADDAA DEADEDDAEADEAEDADADDAE EDAEDADADDAAAEAD ADDEEDADDAA DADADDA AAEAADADDADEAAED EADAEDDADAAEDDDD DDDAEEAADAEAAE AAAAEDDADAAEEAAA DAAADEADAEA DAEAD ADADDADDADA A E D DA D E D DDAFDDDEAAEDDE AADAEDEDAEDDADADDAEDA EAAAADDAD AAAAAAADDDA EEDAAEAADDDAA DAEAAEAEDAEDDE DAEAADEEDDA ADADADEAEDA AADDADEEAAAAE DDAEDAAAEDADDAD DADEDADADAAAE DDAAEDDDAAAAAEAE

17 EFADAFFD DEDAAEDDAADAE EAEDAEAAAEADA AADAEDADAADEEEDAD DAEEAEEDAADAAEA DEDDDAEE AEDADDAADDA ADADADA EEEDED DDAEADAAEDDEAADA ADAEEADAEDE ADADDAEAADED EDDAEAADAD EAAEDDEEDADDAAADE DDDDDDADAEDAEAADDAE DADADEDAEEADDAA DAEDEADDAEADADAADAE D ADADDA DDAEDDAEDE DEAAAEEE EAAAADEDADDAADAED AEAEADADA DDAADADAAAEA EDEDDEAD AAADADAEDE D EA DA E DEA E AEDEAADADAA DADADAEAAA DDDED DDEAAEDEDD AAAEA AADADEDADAE DDEDAADEDD ADDAEEDDADADA DADEAAADDEDDA DADAEDDADEADAD AAAAADDAADAEDEAED

18 EFADAFFD ADADAAAEAA EDDAAEAAEDE EDDA EADAAAEADAED DAEDDEDDAAEAADA ADADAAAADA AEA

19 EFADAFFD

20

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22 ADFFAAFDAFDDA FEAFE DDAEDDAEDED EAAAEEEEAAAADE DADDAAAEEEDEDADADA AD DA A DAA A D EAADAADDAEEDADAEDDAE ADADADADAEAAAEE AADAAEAEAAE EAAFEADAEED EEEADAAE A EADEAEDEDAEDADDA EEAEAAA DDADADDDAEAEDD AEDEDDADEDADAAEAADA AEADADDEEAADDAEE AADDDAEDDDAAADA DEDEDEAADAE DEEDDADADADAAD DAAEADADEAEDAA A DADADAAEAE AAAADEEAEED AAEDAEAEADAEAA EDADDDAAEEAE ADADEDAAEA DADAEAADAEDDDADD ADDAEDDDAAEDEEA AEEDAADDEEADDA AEDEEDADAAEDDEAD DEDAADDAEADAEEAAAA DDADAEAE EAAA EDAEDDADAEAA EDEDDAAEDEDEEDDA

23 ADFFAAFDAFDDA DADDAEAAEDEAEDEDD EAAAEAADEA EEDEEAAADDDAAED EDEAAEAEADDEDEDAD EA AEDADADAAFADAED AEADEAEDDEEDADD DAEDAEAAA ot H = J ote = t div = dive = EDADEADEDAAD J = σ E = H ADAEADAEAAAE AAEADDAAD = ot A diva = A E = gadu t ADDAADAAEA = J A DDADDAADA DDAEDD U =

24 ADFFAAFDAFDDA DDDAEADAEA A( ) = 4π espace J ( ) dv EDDAAED AAAEDEEAEDAD DDAEEAADEAADAADDE DAEDADEAAE AAEDAAADA EDADDEDAEAAAE EDEAAEAEEADAA DDEDAA D DADAEDDADAED DADAEDDDD AEDDADAEEADAAEDA AEADDADAADAA EDA [ V ] = [ Z ][. I ] [ Z ] = R j. ω. i M ij DEEDDAEDEADAEA AEAEDEDADEAAADAEAED DAAADDAEAAA AEDDEEDAEDDA EAAEAAEAEDADAAAEDD DFAEDDAADEADDDA

25 ADFFAAFDAFDDA DDDADDDD EAEDDA M ij = AS. dl j I S i A S ADS j j I DED A A A E DA A A ADDAAAEDDAD AAEAADAEAEEDA EAAADEEDADA DA DADAAEDADAAA AEDADEDAEAAE AEAAD DADEDADEDD EEDDAEDD AEDADEDDDEAED ADDDAEDADD EADA A A D AADA ADEDE DEDDDDADDADD DAEDAAADDEDAEA AAEADAEEAAAADDADADAA

26 ADFFAAFDAFDDA EDEDEDAAEA EADAEDA AADAEDEAEEE EDADDADDADEDAADAA DDEADADAAAAD EDADDAADADA EDADE EDDEDDDEA EDEAEEEDADDA EAAAAAAEDADDADEA DADAADAAEAAA EAADDAA DAADAADAADADAE DAEAADDAADADDA AEA = H FAAA AEAEDDAAAAD EAEDEDEDED A ( ) = 4π espace J ( ) dv DEAADAADEDAAEA DA ot ( H gad H ) = ( oth H gad ) = J DDDAEAEA A = J H gad DA ADEEAADA DDDEDEDDADEAD

27 ADFFAAFDAFDDA H gad A ( ) = dv 4π espace DA J libe DAADEADAE DDAJ lié EADEEED DAADAAADEAADAAEDD AEDADDA M = χ H = ( ) H ADAEDAEAAEDE DAAEAAAAADAADADDDDAEDDEAA EEDEAAEDADE DEEADAEAA J = ( ) J lié libe DAEAEEEDEEA ADAEA DADADA H F D AAEAADDDEDAD DADDAD EDAAEDDAA EAAE DADEADDDD EADAEDEDAEDEDADAD DEAEEDDDA A ) = 4π H n ε εds = 4π ( S S ( ) H n ds DAAAEDED K A A K A ( ) = ds 4π S K = ( ) H n

28 ADFFAAFDAFDDA DADDED KDAA DDA J lié EEDDDDAED AEDEEADDAE DDFFDA D DEDDAAE H I = σ J lié ds = ( ) oth ds = ( ) H dl σ DDAD J SAEDDAADDD DE F AADDDDEA DED K = J S ε dldddadddaee F A A dσ = n dlε DEAED F DE I = ( ) J s ( n dlε ) = K( n dl) DD KDAAA F D AD D I = ( ) ( H n)( n dl) = ( ) H dl

29 ADFFAAFDAFDDA DAEDDAAED DDEDDAEDDD DAAADEDADDAAAD AEAAADAADADADAAAEDE ADAAAAADA D DE E E E E EAAA AD E DAD DAADDEEAAAADADEAAA DADDAADDEDAEDA DDADEAADAEAD DADDAADDDAEDDEAAA ADAEAAAEAEEAAA DADADAAAEDADDAEEA EAADDDDAAEAAAEAEDAD DAEDEAAED EDEDEAADAEADED ADADEAA EDADEAAEDDE DEDAADAAAEDEAAAD AAEADADAAEAEAAAEAEADA t = t

30 ADFFAAFDAFDDA AEDEAAAADAEAAEAEAAA AEDADDAADADAEDEDEDD EDD Ht = Ht DEAAADEAAAEADADAAADA EDEEAADDDEAED ADADAADEDAEDDDAADEAAA AAAADADADAAEDDAA ADDEDADDEADA DA K Ht = K Ht DAEAAEADADAE DAAEDADDA K = Ht ADADAAEDAEEAA EDDEEDD DAAADDDAEAD = ( ) K Hft Hst AEDDADAEDAA DAEAEEADAEDAAADA DAAD EDDADADADA DDDEAEDADED A Hft i = Ki Hst i i = j Hst U. K ij j Hft K U K U K δ A = = D E D ij i i ij j ij ij j j j δ E

31 ADFFAAFDAFDDA AAADADAEDDEEAE EEAEDAADDDAAAED EEDADAA V ij = δ ij U [ Hft ] [ V ][. K] ij = AADEDEAEDAAAEDD DAAEDDA DAAED DEDAEAAEAA DDEAEDDAADAEEEAAAD DEADDA EADADAAEADDADAD EEAADAEAEAAAADAA AADAEDAEADAEAE DDDA ADDAEAEDDAEEEDAED EAADAEDAEADEDD DDEAAAAAAAAA AEDDDAADA ADAEAADEAADAAAAEAED DDDAEDEEDDAAEE AEDADADDAEDAD

32 ADFFAAFDAFDDA AADAEAEAADAD EA DEDDA A EDDA A DEDDADA ED A EAEDED DEAEAEA DEAEAEDEDD FFEFFE DDAAEEAA D AA DA A E AD E DA DDEDEDEDDAE DADAEADEDE DADDADEDED EAEAEDEDAA D AED AAEDAA F ADDDE AADDDEAEDA f, φ fddd

33 ADFFAAFDAFDDA AA DD ADAAEAEEAE AEADADEDADAADDDAAE AAEDA [ cos( φ φ )] Af (, φ) = I f ln f f f 4π DDDAEAEAADDEADAED EAAAADADADADAEEAEAADD AAAAEDA [ cos( φ φ )] Af f f Hft(, φ) = = I f 4π cos( φ φ ) EDDEAAAEAE π R N DEDAE f f f π A ( i, φ ) = R, i i N D E AAAAA FAF

34 ADFFAAFDAFDDA FA DEAEDDAEEDDEAE DADEDD πr K j N AADDAADDADADAD EDDEAEA πr K j N D AADDADADADD. A R, φ ja R, φi Hst πr [ R R cos( φ φ )] i j i, j = K j = 4π N R R R. R cos( φi φ j ) K j N ADADADD DADAEDAEADEADDE DDDADDAAE ADDDAEADA AD DDAEEADAEADDDAED = N U i, j = δ V i, j i, j U i, j AEEADDEEAEDDA

35 ADFFAAFDAFDDA EAEDAAEDD DEDAADADDD AD DF AAEDAEEADED AAEADAAAEDA AEAEEEAAAA AAA AAA F

36 ADFFAAFDAFDDA DEAADEAE DADDADDAAEADEDDAE DAEDEDDDED DDAAEDDDE ADAADDAEAED AA D A DDA E EDDDAEDD EAADA DEAAEAADDDD E AA E EEA A EDEAAA EDA DAAE Ai AE AA A F AADA AA DA A ADE EAA DDA At DDDDDEAA AEDDAAEDDAD AEDAAADAEAAAAA EDAADAAADAAEDAA = t A(, φ) = 4π I f t ln D f f f cos( φ φ ) f A E AEDAADEAAAE EAADAAAAAA EADAAA ( =, φ ) DD R f f D AADAAEA =

37 ADFFAAFDAFDDA A A f A(, φ) = I ln I ln f f 4π D f f cos( φ φ f ) 4π E D cos( φ φ f ) E DDEADADADDD DAADDEDAEDEDEDDEDD DEADADAADADAEDAAE EA K(φ ) K( φ ) = H( R, φ) H( R, φ) φ φ AADADAEEAEAEAEADAEAADD DDAA EAE A Hφ = φ = EEADEAAAEDEAEDAA K(φ ) I f cos( ) R f φ φ f K( φ ) = π R R cos( φ φ ) f f f AAADADDADDDDD DAEAD = π φ Itot K( φ) d( ) =,366. A ADADAADDDAEEAA ADAEAADAAE EDDDAEEADEAAE DAAEDAAAADAADEDADA DAADAEDEA

38 ADFFAAFDAFDDA AAAAAAAAA D AA A F A D DAEDDEAEDDAA ADDEAEDADDADEAEDD EEAEDEAEADADAEEDED D DADAEAAAAEAEA EDDEEAEDEADD EEAEAEADAAEADE ADADAEEDEAD ( R, φ) φ ( R, φ) φ K( φ) = DEA D F AD EAEADAEEDEDD AEADA

39 ADFFAAFDAFDDA A DADEAEDD EAEEDEDAA AEDAA EDAA ADDDF AA AEAE DA 6 K PEE K FLUXD K analytique AED / 3/ DE DEAEDEEEAEDDADA DAEDDDAD A AAEDADDDADA EAEEDDDADAD DDDA

40 ADFFAAFDAFDDA Eeu elative ente PEE et analytique 3 DA / 3/ DE DDADAEEDE A EDAEDEEDED DADDAEDAEDEDDDDA ADEDEDEDDADDDAAEAA DADADDDDAEDADD EAADAEAAADAADAEAA DDDDAEDA DDDAEDDDEDEEAE

41 ADFFAAFDAFDDA Eeu elative ente PEE et FLUXD.% DA / 3/ DE A DADAEADDDAED DAAEAEADAADAE EAADADEAEDDD DDDA DDAAAEDDDA DDDAAADAAEEAD AA D D D AA E D EE E E D EAADAADAEDE ADDAEDDDDADA DAAAEDDDEAADA DEDAEEDEDEA AAEDAEAAAEADADADAA AADAAEDAEDEAA DADEDEDEDDA ADA

42 ADFFAAFDAFDDA ADEFDFEAE ADAAEDEEDEADAEA EAEDEDEDADDAADD DEAAEDADA DEEAADDAEAEAED DADDADADEDDADD DAEAAADDAEDA DEDEAAEDEADAED ADADDDDA D AEDEDADDAADADD ADAED W m = V Hdv ADAEDEDDD DADEDEDAEDEDDE ADAAEDDEAED DADADAEAEAAD div( A H ) = Hot( A) Aot ( H ) AA V div ( P) dv = Pds S EAE div A H ) dv = ( A H ) ds = H dv V ( AJdv S V V W m = H dv = AJdv ( A H ) ds V V S AAAEEEAEDADA EAEHDEDDAAAAADEDE

43 ADFFAAFDAFDDA DDAAAAEDDDDDAAAE AAADDEADAEEDDD EDDADEAAAAE AEDDADAEDAEDD AAAEEEAEAADE ADE ADADDDDADDA AEAA ADDADDAEDADDDA DADDDEAEDD DA DAAEDEAD π ( A H ) ds = l AHφ dφ S DEAE AEDAEDDE n { [ α cos( nφ) β sin( nφ) ]} = Aext(, φ) I α ln( ) n n π n= n { n [ α cos( nϕ) β sin( nϕ ]} I = Hext(, φ) φ α n n ) π n= DDDEEAEAA EAAA AAADEEF ADA AH φ = I π I π [ α ln( ) ] ADD l π I π [ α ln( ) ]. dϕ = l [ α ln( ) ] I I π π AADDADDADEDAE EADEAAAEAEEDAA ADADDAAAADDE DDAAADAAEDDADDE

44 ADFFAAFDAFDDA DAAAEDDDAD AE D AEDADEDEA AAADAAAADDD EDDADEDEDEA ADEAEDA AEAAADAEAAEAAAED D F DDD D F DEADEEA ADEDAED D DADAAEAA Af int = I f 4 R AAEA π D Af ext = I f AEA π 4 R A D E ADDEDD ADADDDED DEAAEAAADDADAEA EDAEADDDAEAFADA DADAEAEDAAAEAA DAEDADDEAADA A E Af int = ln I f 4π d d cos( φ) D R A E

45 ADFFAAFDAFDDA R Af ext = ln ln cos( ) I f 4 φ π d D d A E D d A E DDADADAE = Wm AJdv = A T J f ds AT J f ds V S S F F AAEA EADEA DAEAADA AAADDA AEAEDEA DDADADAEA EAEDEAEAD S = πr S = πr I f I f Wm = A T ds AT ds S S S DADEDADEADDDA DAEEAEEAEDEA DEF S A T ds = S S S S S ( Af int Af ext ) Af int ds = ( ) I f πd = I f S πr R 4π ds A D R E 8π EDE S S Af extds = πr 4π R π R ln D d A E ln D d d A cos( φ) I E f dφd DEAAAEEDEE EADAADAEA

46 ADFFAAFDAFDDA EAADDADAEE DDD = E A D = E A D ) cos( ) cos( ln n n d n n d d φ φ AD ln 4 f S I d R extds Af S E A D = π DADDDEAA DEADEAADADDAE EEEADDEAEA ( ) ( ) F F F F E A D F F F F E A D = ln 4 8 ln 4 8 f f f f f f m I d R I I I d R I I W π π π π ADAEEDD ( )( ) 4 4ln 8 f f f f f f f f m I I I I I I R R d I I W E A D = π π AAADEEDDDAEA DADAEDAEDAADAAAEEDEAEADA EAEEEAAEAAADAAE D D F ln 8 I R R d W m E A D = π

47 ADFFAAFDAFDDA D DA D E D E ADADDADDA EAEDA DDADEDAA EA E AA D D D DA D F DA DDED AEDAD DAE F D DAEDEDEDA DAEEDADD DA DEAAADAD DAAEDEAEDEAEDED AAEDDDAAEAEA D W m I f = S ( A A A ) p i S noy ds f A I S S A p f EA I S S ds A i EA D EA ds EAEADAD EA F EEDEDED DA f AA I S S A noy ds EAEADAD DDAEADEDD D. ADDADDADD A

48 ADFFAAFDAFDDA ( A A A ) ( A A A ) m = Wm Wm = p i noy p i noy W DEAEADAEAEA DDAAADAEEDEDAD DAD DED D F DAA DADEAADDEE DDDAEDEE EDADADDDDADD ADDEEDEED AEDAEDDFAEDAAE EDEDAAADDADE A n jnφ Aext (, φ) = c e DDDEAD DDD EADAEAEEDDA n R π R π n n A jnϕ jnϕ MoyA = c n e dϕ d = c n e dϕ d πr n= πr n= D E R π R F π n A jnφ n F jnφ MoyA = c = n e dφd cn e dφ d πr πr F D E F ADDEEDD EAEAFA R π π π R MoyA = c d c R c π R = = AAADDEADEAD AEAEDDDD DEEDAEDDADDAD AEAE I f DADEEDDE DA

49 ADFFAAFDAFDDA DD W noy = I f AK ( x f, y f ) EDAEEE ADDED EAEADDAEA DAEEDDAAE ADAAADAEAEDDD ADAEAEDEAD ADDD ( φ ) ( ) f, f f, φ f φ f = φ f = φ f D DAAAEDDED A D DAEEAADDAAEAAAD EDEDDDADDA DEEADDAEAAEDDAEEAEDD EAADDDEDAAEDDED DDEAEADAA EDEAEDDDADAEDAEDA

50 ADFFAAFDAFDDA EDDADDAED AEE DDADEDAD EEDDAADAE DAADADEDADEDAEDD ADAADADAADDA AEADDEDADAEDAEDD AAEEAEDAEDDD EDAEDDEAADA DEAAEAAEDDDAAAE EEAAEDDAADADAEA AEDDEAEDDAE DAE A ADADAADDDAEEAA ADAEAADAAEED EED DDAEEA DEAAEDAAEDAAAADAAD EAADADADAADAED EA AAAAAAAA AA D

51 ADFFAAFDAFDDA DDAEDAADEAAAD DADDAADAEDAA DAAE DAE DEAEDEAEF AEDDDDEDDEDAE DADAED AEDD EEAA Wm L = I DDAAEDEDDDE DDDDADAD DDA F A A AAEEDEEDD DADAEAEDE AEDEAAEDE AF EDEDDAEAAEDE DAAAAEADDEEADDA

52 ADFFAAFDAFDDA DAAEADDAEDA EDEDDADADA DA A D DAD DA F A D A D E D F A D DADDA F D AADDDDDEADDAEAAA EAEDAEEADEADED DDAEAEDDADAEDDAE DDEEDDEADAAEAD DDEEAEDEDEDADDAE DEDAEEDDDDADDE DDDDAADADAE

53 ADFFAAFDAFDDA AADAEDDEEADDAE DDDADEADEDE EAADAAADAEAADAA EDDDDAAEDDA AAAA AAAAAAA AAAAAA DAA AA AEEDEDDAAEEAAEDAA.DDADAEDD. DEDDAADDDF AAA AAEDD AAAEDD F DAD DA DAAADADAEED ADAADA

54 ADFFAAFDAFDDA AA AA DADEDDAAEDDEA AAAEDEAAAA AEAADDAEDDADDA EADDEDAA DAAD D DEAAAAE DADDAEDAA DDEAA DDEDAADEEAE EAAEA F AAAE DADDA D AAE DADDA DAADAAEEAADADDA EEDAADDADDA EAAEADEDED EE

55 ADFFAAFDAFDDA DADEDDDEDDAD DADDADAAEDDA DEDDAADA DDD F AADADADED DEAAEDADDAEAAAA EDEAAAAADDAED DADDA AA AA D

56 ADFFAAFDAFDDA DAADEEDDDADAA EEAADADDAEEE DDDEDAAEAADAA DDAEA F AAAE DADDA D AAE DADDA DDADEDDDEE ADDDDA AADADADED DEAAEDADDAEAAAA DAADADAAEDDADE ADEEAE EAED

57 ADFFAAFDAFDDA A A AA AA D DAADEEDDDADAA EEAADADDAEEE DDEAADDDAA DDAEA F AAAE DADDA D AAE DADDA

58 ADFFAAFDAFDDA EAFEF DDADDEA ADEDADDAAEDEED AADDADDEAAD EDADDAADDEEAA AEDEEDAEDADDA AEEDDAE E AEDDAAAEEE DAD D ADA E E E DDA ADAAEDDAA EEADEDEE AAEDADDDEDA EEAAADDEDEAEDD ADA AADDADA EEAAADEAADADEADAE EAAAAEADEAEDADDEADDAA EEEAADDADD

59 ADFFAAFDAFDDA

60

61

62 ADFDDFAFFFAD FEAFE EDADDEDEEDA AADAEAEDEDDD DAADAADDDE DADAAEDDDADAAEADAD AA DADEDAED D DA E A E AD D A AAA EAAA A DDA A ADDDDAEDAEDDEDEADAD AA FEAEEAFAAEFAF DADEAAEEAAADA ADEAAEDAAAAAD EAEDAEEED EDEAADD AEDAEAAAEDEDAEDAA AEAADEDEEAAEEAAAEA DEDDAEEDDEAA E yoz

63 ADFDDFAFFFAD DAEEAAADDDD EDAAEEDDADDD DAADAEDAADADDAEDAD EADEDDEAAEEDE. AD DDEEDEAADEAEDE AADDDDE AAAEDEAD EAAEAEADDAAADDAD AAEDDADAAD D DDADDAEADAEE DADEDEAEA DAEEAADDA AAA F AE DD

64 ADFDDFAFFFAD DAAADDAEADD EDEDAEDD DADAAEDADAA DEAADAEADEAAAAAEDDE A EA 3 3 omposante x x_pee x_fluxd EA. 5 3 y_pee y_fluxd omposante y AEA AEA D DADAEDDAD AEDAAAEDAED DAEAEEAEDEAEDAEADA A DEAEDAAEEAEAEA AADADDDDDADDADAD EADDDDADD AEDDEAADAED AEDDADDAEEDDA DAAEDADA DE FDEADDADDDE DAAADADEADDE DEADAADAEAEADAEE EDDEDEDAEAEDEAAAAD DEADAEEAEADEAEDAAA

65 ADFDDFAFFFAD AED K_PEE tacée su les contous intéieu et extéieu K_int K_ext AED K_FLUXD tacée su les contous intéieu et extéieu K_int K_ext E E DAAAEEDDDADAEAD DAEAAAADEDD DDADEAAEDEAD DAEDAD DAEAADEDDE ADAEAAEAEEED DDADDEEDADADAEAE ADA DAEAEAEDAED AADEDAADADAED DAEAAAAEDEDADAEDAADDED DAADADDAAEDDADAE EDDDAADD DDDAED EAADA DAADED ADDDDAAEDEADD AEDAEDDDADADADDEAADA AAEAADAADADA DDAAADADADD

66 ADFDDFAFFFAD EA omposante x x_pee x_fluxd EA y_pee y_fluxd omposante y AEA AEA DADDDAADAD ADADDAADA ADDDDED ADAEEDDA DDDADDA AAAAA AAA DDEADAAAEDEAADAAEA DAAADDADDAADAD FDADEAADAADEDEAE AEDAEDD DADEDAEAADAAADAA E DADADEDDDD AAEDEEAADADDEDDAADED AAEAEAEAAADEAAAAA DAAAAAAEAAADDD DADEDAAEEDAD DDADADADDDEDAD DAEAAEEAAEAEA EDDDDAAADEADAE EDAAADEAEDEDADED DA

67 ADFDDFAFFFAD EA omposante x x_pee 88 x_fluxd x_pee 8 EA omposante y y_pee 88 y_fluxd y_pee AEA AEA AAAAAA AAAA DEAEDDDEADAEDA AAEAAEDADADEADDE EDDDDAADED DDAAEDADEAADDADE ADADEDEDDA AEDAEEDADDD EAEDDDDDAAD EDADEAEDDADADA AA DADADAEDADAAE DDDEDDAEDEDAA AADDEDDEAAAADAED DDEDADAEDAEDADA EDDEDADAAED AAADDAD DDEEDAA EDDDDDDD, 3, 7, 7 A Rn = ( n ) n n, 7 D E

68 ADFDDFAFFFAD EAAADADAAEAAAD EADAEAAEDAEADAEAA EADAAAAAEDAEDA EAAEDADEDDDEED DADADDAADEAAD DAEDDDDEDDDA EAEAADADAADDDE DDADDDDAE EEADDEDDDE ADADADDEEAEDA AED K_PEE tacée su les contous intéieu et extéieu K_int K_ext AED K_FLUXD tacée su les contous intéieu et extéieu K_int K_ext E E ADAEDEDDAADDE DEADEADEE AEADEEAEA ADDEDAFAEDDA DAADDEAAAADAD DADAEADAADDEDAD AAAEDDDEEDADE EDDEEDDEAAA

69 ADFDDFAFFFAD E AE E AE DF FEAF DADDADDAEADADA AEDADDEEDDADD AAAAEDDDAD EAEAEDADDAAAAADADADD AEDDDA DADAADEADEAD EEAEAEADEDDEDAE DADDAAEDA Milieu 3A R A R A D AR AR ep ep ep Milieu h ep ep ep AR AR D A A R 3A R ep Milieu 3 EDEEDDDEADDA AEDEDADADEAEA ADDEDEDDEAAAE

70 ADFDDFAFFFAD ADDDDDADEEAEAE ADD DADAADAE i x ( x, y) I y h = π ( ) x y h ( ) ( ) I m y h m ep y h m ep ( x, y) = t ( ) x y ( h m ep) x y h ( m ) ep D { } px 3 3 π m = ( ) ( ) I m y h m ep y h m ep ( x, y) = ( ) x { y h ( m ) ep } x y h ( m ) ep D { } px 3 3 π m = ( ) ( ) I m y h m ep y h m ep ( x, y) = ( ) x y ( h m ep) x y h ( m ) ep D { } nx 3 3 π m = I m y ( h m ep) y ( h m ep) ( x, y) = t ( ) D x y ( h m ep) x y ( h m ep) 3nx 3 3 π m = A E A E A E A E DADDDAAAEDAA EA EDDAEAAA j j i t = et = ==> = t ij ij ij ij j i j i AEAEADAA ADDDDEDDDAE AEADDEEAADAEDEDAD DEDAEDEDDDEDEA AEAAADAADDADF EEDDFDEDAAAEAEEA AADDDDAAAEA i x ( y) = K y h y h AEAEEEDEDDADAAEDEADD EDAEAAADEADAE DAADDDADDA ( ) ( ) K m y ( h m ep) y h m ep A px ( y) = t ( 3 ) 3 m = y ( h m ep) y h m ep D E

71 ADFDDFAFFFAD ( ) ( ) ( ) ( ) K m y h m ep y h m ep A px ( y) = 3 ( 3 ) m = y h m ep y h m ep D E ( ) ( ) K m y ( h m ep) y h m ep A nx ( y) = ( 3 ) 3 m = y ( h m ep) y h m ep D E ( ) ( ) ( ) ( ) K m y h m ep y h m ep 3nx ( y) = t3 ( 3 ) m = D y h m ep y h m ep A E AAEDEAAAEADDAA EADAAD F AEDDDDDADA EDAEDEDAAEED EDEDA F D K K t t t px = 3 3 = 3 3 m = m = m m ( ) ( ) ( ) K K t px = 3 3 = 3 3 m = m = m m ( ) ( ) ( ) K K t nx = 3 3 = 3 3 m = m = m m ( ) ( ) ( ) K K t t t m ( ) ( ) m 3nx = 3 3 = 3 ( 3 ) m = m = ADAADDAAAEDAAAA K t t K ; t K ; t K = = = ; = t t px px nx 3nx ADEDDAAAEE EAAEDDDDAAEADAD DAEDDDADDAADEAEE AAEA DF D DADDDAAEEDEEAAE AEADDAEAA AEADEAEDD DD E A E E D E D A DAAAEAEADAEAE EAADAEAD DEADAEADD DDDEEDDAAEEAEA

72 ADFDDFAFFFAD K t 3 = 3 K K 3 3t = 3 K DADEAEAADE EADDDEADEDAA DEAEDDAAAAEEAAD EDAEAEDEEAAA DF F DD EEDADADA AADADAEEDADDDDEADA EDDADAEADADDDEEE AAADAAA A A Hs = K K3 Hs A KA ==> = soit : U = Hs3 K3 Hs3 = K K D E 3 D E D E D E AAEA DDDEDDE DAAAEAAAADDDDE DEAAEDEAAEEDEAEEA V = U δ ij ij ij DAEDD EDEDEDEE AEA DF DDA F DDAAAEAD ADAEA FD DF. A A A A V = -U = - = D E D 3E D 3E D 3E DADDEDEADEAAD D AD Hft Hft 3 = K = K D A K D K 3E 3 A = E KA D K E

73 ADFDDFAFFFAD K D K 3 A = E 3 t D3t 3 A E K DDEADDDDE EAEADEAAEAA DDAADADADDDAD DE DF DADDD F ADDEDED ADAEAA FD AAAADAED ADAEDDEAAAD F D F AEAEDDDADAEADDF AADADAF EADADAF EAEAADAADDAAAEAA DEDEADAADAADAEA A DADAEDDAEADA AAEEEADEDDAAAEDAEA FD F EDAEAAADDEEAAD AAAADDADAAAA AAADAADDED DADEEAAEAAADADAAAAAED A AAAAEAAAEDD DEEDAEAADAEDE EDDADAAEDADDE DADDAAEDDE DEDADDDAAAEE DADAD AEEDAAAADA ADADDDEAEDDD DADADEAEDDADDADAD EDDADDDEAEDAADEA EADEADDAD EDDDADAEAAFAAEAAA EAADDDAEDAEAAD AAADDEDADAEDDDA DAEEDDADADAED DD

74 ADFDDFAFFFAD AAADEDAADAD AAAEEAADEDAEEAD FEEEDAEFAEFA DDDDAEA ADADDDDADDEDAA DDADDAAEDAA EDAEDAA DDAEDDEDA DDD A DDADEDADED ADEADAAEAAADDE ADAAAEADDDAAAE AEDEEEADDEAADAAA DADAAEEDDAAEADD EADADAAADDADEAAE DAEADAAAAEED DADDEDAEEADDEAD DDADDDADADADAE

75 ADFDDFAFFFAD DDDDAEAAADEADEAD DDEDADADDEAAEADEDA x y ( θ, ρ) EEADAEEEEA tx = ty = ρ ρ DEADADADADEEEAD A EEADADAADAEA yf If sinθ If Hfxi = If = = hfxi π πρ θ tx πρ ( x xf ) yf ( txi cos ) i xf cosθ If If Hfyi = If = = hfy π θ ty πρ πρ ( ) ( tyi sin ) xf y i i yf If DADDDA AEEDE πρ i DADADEADD ADADA,DADEADDEDA DADDAAAADEA Kx Hmx = Hmx = Kx Kx Hmx DAAAEADD A ED = Hmx = Kx AEADDADADAA D EADAEADAEEDD ADDEAD Hmxi = Hfxi Hsx i A EEADAEDADADAED EADFAA DDEEDAD AA : y y l Hsx = Ky l Hsx = A Ky avec : A = j j j i j j i, j j π xi y j π xi y j

76 ADFDDFAFFFAD AEADEDAA Kx Hfx A Ky = A Kx = Hfx A Ky D E ADDEADDDADAE AEADEEDDEAADA DEDDAD Ky A = Hfy A Kx D E DEDAAEDA DAAADEA A ij = π Kx = ( I A )..( Hfx. A. Hfy) Ky = ( I A )..( Hfy. A. Hfx) x DDAEEDAE EAADDAAEDEAD EDA EDEDAAAEADAAAE DEEAAAA ρ If E DADADEDDEEAD DADAEDDAAADDDAAEADED DADEADAADA AAEAAAEAADA EDDEADDEA DDADEDDA EDA DDDDEEAAAAEAD AEAAEDADEDA AAAAEDDAAEDAAAAAA D I h Kx = Hfx Ky = Hfy D K ( x) = = Hx π x h AEEAEDAEADAEDAEDA DD DDEDAEAADDDE DAEAAEDDADADDDD

77 ADFDDFAFFFAD DEDDADDDDAAAEA A EAAEADADEDDADA AEAAADEADDADAEDAD DDADEADDDAEEDEAEEDEAEDA DDAAEDDAAD AADADDADAAADAEEAADA DEEADDAAAEAEAEDD DEDAAAADAAADEDDA AEDAADDAEDAE AEEAAD DDDDDAADADAA DADEEAAEEAADEADD ADEAEDAAEADE DDADAAAEDDADD DAADEDEADEAAEDDA EDADDAEEDAA EEAAAA DEAADAEEDAEAAAEAE DAAADAEADEDEAAEDE EEADEDAAEEDEADEAD AADDEAAAA AAEDAAFD DDDADEAEDE

78 ADFDDFAFFFAD 4 3 AED Kx i Ky i L ADDDAA i D AEEADADEAADA DDAAEDDDADDE EADEEADAA DEEADDAADE EDAEADADA D. AEDADADAADEDDD DDA AAEAEDEAADA AEAEDDEEAAADD DAEDADDADE A DADAEDEADDADD AEADADEDADEADD AEAADADAAAEADAEAADA AADEDEAEDEADADAD ADAAEDDAEDDADDA

79 ADFDDFAFFFAD EAAE AA EAAE AA A EAAE AA y F A A D EAAE AA EAAE AA DADDA EAAE AA A EDADAAEE EAA EAD DA D D DD E DDAEDEEEADAAE DDDDAADAEDADD AED Densité de couant su le côté hoizontal Kx_FLUXD Kx_PEE AED Densité de couant su le côté vetical Ky_FLUXD Ky_PEE ADDDAA ADDDAA A DEEDAD EADEDDAEDADAADDAE AEAEDDDEDEEADD DADDEDEEEADEAAE

80 ADFDDFAFFFAD AADDDAAEAADAE DAEDEAEDEAEDADE DAAAADDE ADAADDAEDDE DDDE DEDAEADAEDAEAD EDEAEEDDAA ty Kx(. tx) Hfx( tx) Ky(. ty) dty ρ = ρ π tx ty tx Ky(. ty) Hfy( ty) ρ = π tx ty Kx( ρ. tx) dtx DEDDDDAD ty tx A Kx( ρ tx) Kx ( ρ tx) dtx dty =... π tx ty tx ty D π E ty A... = Hfx( tx) Hfy ( ty) dty D π tx ty E tx ty A Ky ( ty) Ky ( ρ ty) dty dtx =... π tx ty tx ty D π E tx A... = Hfy ( ty) Hfx ( tx) dtx D π tx ty E EDEADADDD DDA.If ADEEEAE πρ. If ADEAEADA Kx( ρ. tx) = kx( tx) EDAE πρ ADAA ty tx A kx ( tx) kx ( tx) dtx dty =... π tx ty π tx ty D E ty... hfx( tx) hfy ( ty) dty π tx ty

81 ADFDDFAFFFAD tx ty A ky ( ty) ky ( ty) dty dtx =... π tx ty D π tx ty E tx... hfy ( ty) hfx ( tx) dtx π tx ty A ρ DEEEAEDAA EEEDDEADDDDD AADD DAAD EEEEE AAEADDDEAADAAEEEDA DDADADE DAAADADEADADAA DDE ρ DDDEEEDE AAAAADEAEDAEAEADAAD EEAAEEADDAEEDEAE DDAADEEEADDAEE EADD ty Kx( ρ tx) = Ky ( ty) dty ρ π tx ty FA ty Kx tx Ky ty dty π tx ty ( ρ ) = ( ρ ) p ADAA Kx(ρtx) = tx AAA DAAAAADAADAEA E ty ty tx tx tya dty = tx d p tx ty ty tx ty p tx A A D E tx tx tx D E D E p ty p u π p du p tx u tx = = π A sin p D E

82 ADFDDFAFFFAD DEADAAAEEDDE AA EAAAA π π A p = sin p p tx π tx π = A sin p D E D E DAA EAEDEAAA / 3 5/ 3 EADAEAE Kx( ρtx) = tx Kx( ρtx) = tx ADAAEDAEEAA DAE AEE. A p = acsin = acsin π D E π D A E DAE AEDEEDD DDDADDAAD EDDDEDDDDDAEAD EAAA DADEDAADA AEDAADDAEEADD A ρdaadaaeee ED If xa Kx( x) = M πρ D ρe A acsin π D E AAAAAA EDAAEE ADEEAED If ρ t t t ( ) = p K t p DDDDADDAAD DDDAADDAAEADDEDA D DDAADADAEDDDA EAEDDAEEDEDADEA DADAEEDADDAAAADE DADEDDDDD

83 ADFDDFAFFFAD u ua S t k ( t) α ln k ( u) du = α.. u t t t D E A t t A t 4 t D E t D E A A S π A. t ln( t) ln t θ S π t t t D E D E D E EAFE DEDDADEDA EDADADDDEADA DAAEDAEDAEAAADEDA EEDDAAEADAAAEDA DDAADDADAEADA ADAEEDDAEDDADDA DDEEAEAAAD EDDDDADDDDADDD AADDAEDAAED DDDADADDADAA DDAEDADEEEDADAE DAEAADAEADE ADADAAAD D E A E DDA DA DA D A AEDDEDADADAE DADEEADDAD EDADEDAAEA AAAEDEAADDDAA DDADDAADADADADAA EAAADAAEDDADADAEDAAD EAEDADDDAADAAA EDDDDE EDDDEDAEAAA AAEADADAAE AEDAADAAADDADADA DEEAAEAADEADDDAAEDAAD DADAAAEAAADAAD EAAADDDDEAAEDD DEDAEAEAEDAEDDAD AEDADA

84 ADFDDFAFFFAD DADDAEDDAAED DAEEAEADDDAEA EDADAFAADAEDAAAEAEDD DEDDEDADA DADADADADEDDD EDDAEADAEDDEADDDADAA AEAADDADAEDEA DDDEAEDADA DDAADDDADADAEEDED DADADAAEDEDAD

85 ADFDDFAFFFAD

86 D

87

88 ADADAFAFDAF FEAFE EDADDDAEDA DADEDAADDADEE DAAAEAAADEDAD AEAEDDADAEAADEAED DADDEADDAEA DADDEDDDADDAEDADDA AAAEDAA. DDDDEADEDA DDAAEDAAAAEDADAEA EEAAADDADDAAEAE AADAAAAAEAAADEDDAD EDEEDDADAEDDEADA DADEDAADDADA DDDEDADADED AAADADAEAADDDAAE AEDADADD AAEADDDDDADD DADADADAEEDEDEDDEDA EDDAEDADE DEDEAAAADDEDADEDADA EEA. ADADDAEEDEAADAE DADAEAEE. DAADD EEEAAEDADDADAEAD EEAAEDDADDAEAA EA DFFE DAEAAAD DDADADDAEEAAEDDAD DAAAAEDAADA ADDD F EDDDAADDEEDEAAD DEAADEDDAADEAEDDD DEDDA

89 ADADAFAFDAF D E DDA D AA AAA DDDADADAAAEDDAEAD DDDAADDEDEAADAEA DDDAEDDAED DAEEDD D F DDDDADDA EAADADAAEDADDDEADAA AEAAAADEAAADA FD DEDADEEDADAAAADA AAAAAADADA AADADADADEA DADDAEAAEAEADD DAAADEEDDADADA EADAEAEDDAAAAADDAE EAADEAADADEAEAADA DDDDDAEADAE DADDAEDDD DEEDDAAADEDDE

90 ADADAFAFDAF DDADAAEDDDA ADDAADEEE DDADDA DD DAADAADEADEAAAD DDADAAEEEAAEE DDEEADDA DDEADAADDEAAAED DADEA DAAD D DADAADEEAAD AAEDAEDDDDAEDD DDAEDD ADDADDAAEDAAE D P H dl = P Htdl = N i= dli D Ti Ti A Htdl dl = i E N Ti i= Ti Htdl DEADAAEEDADE EAADADEAADDADAAAADEA DDDEDDDAEAAEE DDADAADFAD DAAE

91 ADADAFAFDAF D DADA D D D D E ADA DA DDADDDDDADD DDDADADEAADAADAAEDA AD AADAADEDDAD EDDDAAADDDAED D D E ADA E ADA DA E DA EDDEDEDADAEA ED Htdl = Hftdl Hstdl Ht = K P P P DADAEDADEDAD DAEDADDDDEDAEA,D AADADAEEFDE EDEDDDADDAAADE DEDDAADADAD F DAADDDEAAAD ADAEEDD I DDDDADA s DADEAADDAAA ADAE I Is = I f I s = ( ) I f s DDDAA D

92 ADADAFAFDAF F DDDEDEE AAADADADADAAEADEDA DADAADADAAEEDA ADDEAADE ADADAEDADAAD AADEDEAADEAADAAA AAADDADAAEDAAAEDD ADDAE ADEDEDDE ADA DA E E DA D DADAAAADDAADADDD EAADADDDDEEAED DAADDDADADA DDEDDADA EDEADADAEDAED EAAFEA AEF DEDDAEDADDAAD DADEDAAEAAD DAAAEDEEADDD DDAAEDADAAA

93 ADADAFAFDAF EDA DA D ED DA D AD E EAADADDDDDEAED AA DEDADADAEDDAE EADDAEDEA DADDDADEDDDA AEDDAAAED D ADAEADEAA DEADAEDDAA DEADADAE ADEAA EAAEDEDDDAA EDADADAADDEADD AAEDDAAAEDDEAEDAAAADAE ADDDADAEADDE ADEAEDEAEDAA EDAEE ADAADAEDDEAED ADDEDADADE EE A ED

94 ADADAFAFDAF DAEEDAEAD AD DAADADADEEAE D ADADD AEEDADE D EADA DAEDEADAADDD Htx i J j yi y j = π ( x x ) ( y y ) i j i j EEADEADAADDED DD DEAED ADD Htx L j J j x j = i L j π x j ( x x ) ( y y ) i y y j i j i j dx j AD ADDDAAE EDD J j c dea Htxi = actan π D d E c dea actan D d E L c = x i x j d = y i y j j de = d = L i

95 ADADAFAFDAF AAAA AADADED AEAAAAADAEDD DAEADADD J c dea c dea j j lim actan actan.de = de d d π π D E D E J c d d DDDEDD DDEAE ADADD J Hty = i x x j i j π i j i j ( x x ) ( y y ) DADAED DDEAED AAAAED Hty L j J j x j = i L j π x j ( x x ) ( y y ) i j x x J j ( c de) d Htyi = ln 4π ( c de) d i j i j dx j ADDEEDAEAD EEADAEAE Htx L j J j y j = i L j π y j ( x x ) ( y y ) i j y y i j i j dy j J j ( d de) = ln 4π ( d de) c c Hty Lj J j y j = i Lj π y j ( x x ) ( y y ) i j x x i j i j dy j J j d dea = actan π D c E d dea actan D c E

96 ADADAFAFDAF D DADADAAEDADEA DDEDDEADAD ADAAE DADADAADEADEDA DEDEADAEDD DADDEADAD DEDDAAEDADA DADADDAEEEEAA AEEAEAAAA AEDDEDAAAD AAD D DAADADADADDDD EAED ADADDADE EAAAA D

97 ADADAFAFDAF DDEAAAEDDA A AD i d x d x j i j i j i dx d de x x d de x x J d Htx i i E A D E A D = actan actan π AAAAADDEDAD EAE E A D E A D E A D E A D E A D E A D = d s s d d q q d p p d s d d p d q d d J Htx j i actan..actan.actan.actan.ln. 8 π d de c p = d de c q = d de c = d de c s = AADADAAEADEDDE EAADADE ADAA E A D E A D = F F F F E A D E A D E A D E A D E A D E A D d de c d de c J d s s d d q q d p p d s d d p d q d d J j j d actan actan.actan.actan.actan.actan.ln. 8 lim π π DADADAD ADA A A A A DAEDDA

98 ADADAFAFDAF [ ] [ ] { } i d y d y j i j i j i dy y y de c y y de c J d Hty i i = ) ( ) ln ( ) ( ) ln ( 4 π ADDDADDEDADAED EADAEDAD E A D E A D E A D E A D E A D E A D E A D E A D = s q q q s p p p s q s p s q p d J Hty j i actan actan actan actan.ln.ln. 8 π DD de c p = de c q = d d = d d s = DD DAADAEADAADAA EEEAAE DEADADAD AD ADADEAAD E A D E A D E A D E A D E A D E A D E A D E A D = s q q q s p p p s q s p s q p d J Htx j i actan actan actan actan.ln.ln. 8 π de d p = de d q = d c = d c s = A A A A E A D E A D E A D E A D E A D E A D = c s s c c q q c p p c s c c p c q c d J Hty j i actan..actan.actan.actan.ln. 8 π d de d p = d de d q = d de d = d de d s =

99 ADADAFAFDAF A A A A D AEDAADAEDADA DAEDDDF ED DA AEADDEDDADDEA ADD D A A A A I A A f xi x f d xi x f d Hfx i = actan actan 4πd D yi y f E D yi y f E F A A A A Hfy i I f y = actan 4πd D i y f da y actan x i x f E D i y f da xi x f E F DAAEADED DDD ADEEAFDDE ADDDE

100 ADADAFAFDAF ADA A( x, y) = J 4π x j de j x j de ln [( x x j ) ( y y j ) ] dx j ADA ( p de) ( p de).[ ln{ [( p de) q ]. [( p de) q ]} 4] A q p dea p dea A( x, y) = J j p.ln de q. actan actan 4π q D D q E D q EE p = x j x q = y y j ADAAEADD j p = y y q x x = AADDADDAD DAEEDADAADDDDA F E A AFE AEFFFE AEAAAEDADD DDEEDADADDADDA AA DA E DD D DA AEAA AEADEDDEDADEDED A j AEAAADDDAA EA F AAAAEDEEDDA DDDAEAAE DEAAEAAEAEA DAAADDAEDDADDAAAAED DADADEE

101 ADADAFAFDAF Atot, mu Atot, mu ADAEDDADADA DEDDAADEADEAEDAA AEADAE DEAEDAADADDED A DDEDEDAADDD DADAD DAE DDDEAADADDAEDADA DAEAADAAEEDDADDAE DADADAAEE DADDDADEDDADADADAD DADAEEDDDA D DDDEDDAAEA EDDADEAED ADDDDDAEDEA AADDADDAADAEAE DA

102 ADADAFAFDAF A A A AEDEDADADDE DDEDEADAEDEDDEAADAED DAADDEAEDDED DADDAADDA DEAEAEDDE A AED Densité de couant supeficiel tacée su le pacous D-A---D K ponctuelle K unifome K FLUXD DDA 5 5 Ecat elatif PEE unifome - FLUXD Ecat elatif % ED AA ED AF AEAADEDDAA DDADDAEDD DAADAEEDDAADAE DDEDDAEDDEAE DEAEDEDEAADA EADEDADDEEDDD EAAEADEAEDADEDD DAEEAEDAEADAEEED DDAAAEDEADEAEDDDDEAE ED

103 ADADAFAFDAF D DDADDAEADAEE DADEDEAEA DAEEAADDA AE F DD DAEDEAEADAE DAEEADADD DAEDDAEDDDDEDD AEEAADAEDD DADDAEADD DEDDAEDD DADAAEDADAA DEAAEDDADAEADDDA D.5 3 omposante x x PEE ponctuelle x PEE unifome x FLUXD. y PEE ponctuelle y PEE unifome y FLUXD omposante y 4 4. EA EA AE AE

104 ADADAFAFDAF AADEA DDDDAEEEAEADAE DADDADAAAD EEAADADEDFDAD DDDEEADDDAD E 6 Ecat elatif de ente PEE unifome et FLUXD E% 3 éléments E% 8 éléments 3 Ecat elatif de ente PEE unifome et FLUXD E% 3 éléments E% 8 éléments.5 DDA 8 DDA AE AA AE AA DA DDDADADDA EDAAEDAEE DEDDDADDDAE EAADAAAADEAADA DDAEAADA AADDAEAED EAEDAAEDDEDDAEAEEA EAEDEAEDADAAEADEDDEA EADEAEDAEEAAEADED EDAEEA

105 ADADAFAFDAF A D DADEAEDAADEADE DDEADADDEADDDDA EADEADDFEDD EDEAADAEADAE EDDADADADEAADA DADDDADDDDDAAADAD EAAA AED K PEE unifome de 3 éléments K PEE unifome de 8 éléments AED K PEE unifome de 3 éléments K PEE unifome de 8 éléments ADEDDEA ADEDDEA AEDDADDADA ADAEAADAEDD ADAEDEDEAD EAAEDADEDAAAEDAAE AA

106 ADADAFAFDAF DAAADAEDDEEAA EDAEDAADADADAA DDDF DEEEDADED DADAEEAADAAADD DF ADDAEDEDD D AAADDDEDA DAEDEDDAEEA EDDAAEDE ADAAEAEDEAAE ADDAEDDADDADADAEAED EDAEDAEAA

107 ADADAFAFDAF Atot, mu Atot, mu D DDAAAAAAEAD DAEAEAAEDADAEDEAEDDAAA EDAAADADDAAEDD AADDAAAEDDA A DDDADAEDEDEAEA ADADEAAEDAD DA D DD

108 ADADAFAFDAF DAEDEAEADAE DAEEEEADAADDDA EDDAEDDDDDAEEA DDADEDDA AD 3 omposante x x PEE unifome x FLUXD. y PEE unifome y FLUXD omposante y 5 4. EA 3 EA AE AE DEDDADDAE EADAEDEDEEAADADAD A.4 Ecat elatif de ente PEE unifome et FLUXD E% 5 éléments E% éléments DDA AE DA

109 ADADAFAFDAF DDDAAADA ADDADADDEDDEDEDAD DA AD A D EAA D EAADADDEDDAAEADAED AAEDEEDAADDEDDA AADDADAEEE AEDDA D ADAEDEDDAAAEAD EADADDADEADAEAAEDA EDEDDEDDD EADDAEAEADADDAEA DDAEDDAD ADEEEADAADDA EAADEEAADAEAEDADEAE EADADADEDDAEDDAEA DA DA AED E D E DAD DAAADDDE DEAADDDAED DAEADAADEDADADDAD DADAADAEDEAEDEDEEAADA DADAADAEADA A A A

110 ADADAFAFDAF FEFEEFAE AEDADAAEDEAEDAADD DEDDADADAED EDEEEEAADAADADAAA DAAAADAEEADAEDADA EDDDDEDADEDDAE DADEDAAAE DADDDAEADA AADDAAEDEAEDDA DAAADA AEDEAEEEAADADDEAE DADAEDEAEDDAEAEAEDD EADEAEDADEEADAD 3 K( x) = x DDADADAEAA DADDDEAEDAEDEDAD DEDADDADAEEAEDDAA ADEDEAADADEAAEA DDAAEAAEEAADAAE E DDEDDDAEDEAADDAEA ADEAAD DAEAADAAEADEDE DED

111 ADADAFAFDAF Densité de couant (A/m) oodonnées des extémités des éléments (m) K éféence : K(x)= x^-/3 K PEE unifome AEDADEAADAAAEDAAE EAA A DA E D D DA ADAEAADA

112 ADADAFAFDAF Densité de couant (A/m) oodonnées des extémités des éléments (m) K éféence K PEE unifome en affinant le e élément patant de l'angle ADDDDEAADA AAADDEDADADAE DEAADA Écat elatif (%).4. Écat elatif (%) oodonnées des extémités des éléments (m) Ecat elatif ente K éf et K PEE unifome non affinement A oodonnées des extémités des éléments (m) Ecat elatif ente K éf et K PEE unifome avec affinement A D

113 ADADAFAFDAF DADAAEDAAEE EAADEAEEADAEAADADD AEADADAADAED EADE AAEADADEEEAEAA ADDEDAFADE A A DEAAAD ADAEDDEEDA K x x x x i i Lagange( x) = K L( xi ) KL ( xi ) xi xi xi xi A A EDEDA DD DDE A A A E FDD DAADEAEED A A DD K moy ( xi ) = ( K L ( xi ) K L ( xi ) ) AEDEEAADAEDDE DADEAEDEDDEDEA DAADADDEDA DD A

114 ADADAFAFDAF K L ( xi ) = K( x Ne) EEDDDEED D DE D D D EEADADDE DD DDAE A K x ) K ( x ) K ( x ) DDF L ( i = moy i L i DADDADADADADAEDD DDADADDAED DDAE DDA Densité de couant (A/m) Densité de couant (A/m) oodonnées des extémités des éléments (m) K éféence K maches d'escalie K segments de doites oodonnées des extémités des éléments (m) K éféence K maches d'escalie K segments de doites A A D D DADADDDADAEEAED DDADDA ADADA A DDA D E EA ADA D AA E DADADDADDA EEDDADADDEA EDADDAEDEAE DDADADEEADAD EAEDEDEEDADADAAE DEEDDAEAAAAEDDE DADADEDAEEA

115 ADADAFAFDAF Écat elatif (%).5.5 Écat elatif (%) sion oodonnées des extémités des éléments (m) Ecat elatif ente K éf et K segm de doites non affinement oodonnées des extémités des éléments (m) Ecat elatif ente K éf et K segm de doites avec affinement A A AEAAEDADEDDAED EAEDAADDDDEDADDDAE ADDDDEDEAEDEDD AAAAEDAEDD DAAEEAEDA EAADA DADAEDEAEDDEEAAD ADAAEAADEDDE AEADA DD DAAAE EDADDAEDDA DDDDAAEDEAA EDDAAADAADAEDEDDDA ADDDEAEEAAE DDAAEDEAEDAEDAE DEDADDDADADAAEA DDAAEEAAAEEAED DADADEDADEDEAE DDADEDAEEADDAE DDDDDEDADADD ADADAADEAEDE

116 ADADAFAFDAF EADDDEDEDA EEAD EFFEEE F EDAAEDDDDA DDDADADAEDDEED DAEDAAEAE DEDADAEAEDDDAAE EDDEDDADDAEAEDDDDDA DDEDAA DDEAEDAAADAE EEDAADAADDDA DDADADDA DADDDA EDADAEDDDA DDAAEDAEDDADDADE DDADDADADA ADDADADAEEADAEA AAAEDDDDDADAEAE DDAEADDADEAAAEAAA ADADEEAADADDE DADDAADADDEDDDADD DAADEDDADEADDAD DEDDDDDEAE

117 ADADAFAFDAF DDEDEAAEDEA DDDDAAEDDAAADDADDA DEADD EDDAADA DADAADEDAEADAAAEEADED AAEAAEADADA DEDDAAAAADDDAD AADDEDADDAEEAA DDDDDADA EAADADDEDAEDDE EDDDDAEDD DAAE EAFE AAAADEADADA AEADAEEADDAEDE EAAADEDA DAEDAAEAADDDAEDADA

118 ADADAFAFDAF EADEEEADADEA AEAEDADDE EADADDADEDADAAA ADADAEEADEDAADEDD ADEAE AADA DDADDAAEADDEAADA EDDADADAEEAAAEAEAD DADEEDA DDDEAEDAAEED AEAAADADADDDDD AAADADADDAEED EAADADDADDDAAAD AEADAAEEAAADAD AA AAEAEDDEDDEDADAD EDEDAEDEAEDDAEA EADDAEDAEADADDAED EAEDAADDDAAAED DAEDEDAAADEAED AAED DADDDAADDAEDD EDAEDDAAEDDA ADEDAEEAAAEAA ADADDDDAADEDAEADAE D

119 ADADAFAFDAF

120

121

122 ADADDAFADDAFAF FEAFE EDADEDADAEDDA D DA D ED DA EAA DADADAAAADD DAEAADDAEDDE ADAEDAADDADA EDAAAA AEDDAADEA EA E ADAD D DDA E D A EA D AAEDAAADAADEDA DADADAAEDEDAA DEAAEAEAEAAAAAED DDEADADAED AADEDAAADDEAAAAEA EDEDDAEAADAADAAA AAAADEAAAEAADDAA EDADADAAD FFEDAF AEDA ADAEAEAAEDAD EA AADAE DEEAEDAA DEDEDAA AEAEAEAEDAA ADAED D

123 ADADDAFADDAFAF ADADAADDEEAEA ADDEDDDA AEADA AEAEEAAAADADDDDA AADAAEE EAAADDAEAA EAAAEDEDAAEADD E AA DA D A DA E E DDDDEDDDDDDEEA EEAAE AE AADAEAAAEDEAAD EDDAAADEEDDEA AEAAADDAAEDDDEAADE AEEEADDDDDAAAADA DEEAA EAADEAE DEADD h A, h h D l E h A l ll, h DDAA D E c h DEADEEAAEAADEAED D EADD A l c ha DDAADEDA D E E, h l lc, D EEDDDDDEEAA DAAADADDDDEDDAAED

124 ADADDAFADDAFAF AEAEADEDDDEDAAE DEAADAAADA DDDAAEDEADDEAADA DDDAA AAADEAAA EADDAADDADEEADDA AEADEAAAAEAEDE EADA EEAEED AADAAADAADDAADAD DAAAAEEAAAAAA AADAEAAEAA EDADDAEAFADDED DAEDDADADD E E A AA D AEADAAEEE ADEDDAAEEEAAA DDEDDAE ADDDAAEDAA A D DAAAEDEADAE ADEADDDAADEDEDD AD

125 ADADDAFADDAFAF ( ) ( ) = c l c h b h b l l l l l l h h h h h h SegH AD ( ) ( ) ( ) ( ) = h b h b h b h b c h h h h h h h h h h h h l l l SegV DEDDDEDE DDEFF DAAEEDAEEAE EAAAAEDAEE AAAEAAE DAAAAEDDEDAA EAADDEDD AE DADAAAADEE DAA ADAEAA DAEDDEDAAFADDEA EAADAAEAADAED EAADA D E D D A A D A DDAEADDDDADA ADADEEAADDAAED

126 ADADDAFADDAFAF DEAEDAAADD AAA EDDDDAD DEAADAAADDAAEAED DA DD DADDEAEDAAAEDD DAAEADAEEADDA DAEDDADAADEAD DDD E DA AED AA E EAAA EEA EA ADEDAEDDDAEDA DDDADDAAEADDADAEDE AEDDADAAEADAE DAADDAADDDAE EAEED DAEAEDEEAEADADEADA ADEADDADAAEAAD DADADAEDADAAAAEADA ADAEADED FFEDAF DDD D AAAAEDD AEAAEDAAAAA ADADDAAAADDA DDAA

127 ADADDAFADDAFAF EADEDEDDA ADEDDDAE AEADADEDDDEAADADD

128 ADADDAFADDAFAF AADAAADEADAE DDEDEAEADADEAA EDEAAADDAD DDDAAEAAAAAAEEAAE DDADADEDEDDEDAD EA AEDEAEDDAEDD EAAAEDDAEDDD F F DADDEDA DEAAADEAAED

129 ADADDAFADDAFAF DDADDDEDDDA EAAEEDADAAAADAED DEDADAADAADEDDAEDADA FAAEEAAEDA ADDADEDDAAAEAEEAD DDDE F DEEAADAAEAAAEA AADDEDADEAAEEDD EDAAAAAD AADADA ADDDEEAADAAD AAEAAEEDDDADDDADAE AEDAAEAAEEDDEDEDEAADA DAA AA AAEDADAADEDAE AEADADAADED DDDAADDEDEDEDEA DEDDAED EEAEDEDEDDD

130 ADADDAFADDAFAF DDD F D D F DDADDDEEAEDEDDAE DDDEDDAADADA DDDDDED ADAEDAADAEADD EAEEAADDE DDADDEDEEAAEA AADEAEDE AAEDADEDDAEAE AAEDEDDE EDAEDDE EDAAEDADEDAE AEAEAADAEDE DDADAEEDEDAADEDA

131 ADADDAFADDAFAF DAAADAEDDEDADAEDADA DDDAEDAAAEEDADD EADDDDAEAA AA AADEAEAAEED ADADEDEAADED DD DD F D F DDDADD DDAEEAEDDADEDDD EDAEDAEAA DDD EDAEDEAE DEDAEADEDDDE DDAAAAAEAAADAAAD DDAAEDADAEAAEDAADA EEDAEDEDDAAE EAEDDADDAAADAD AEEDDAAEDDAEDADDADAA

132 ADADDAFADDAFAF EDADAAE DAADDDDED DDEDDADDEDDDA EDDD DDD EDE D D D E DAA E DAD DAADDADAEDEAAA DDDDEDDD DAADAEDADAAEAADDAD ADEDAADADED DEDEAAADAEAD DDEEAADAAAADD AADDDDDDAEE EADDADAEAEDAEDAAEEEDD AAAEAEDEDDDD EDEDDDADEDDD AAADDAEDE DDAEDEDDEEA EDAEADDA EDDADDAEDAEEE

133 ADADDAFADDAFAF AA E AA E AA E D DDADAEAEDAAEAA AEDDADDAADDEAEAD DDADDEADEADDD DDADAADEAAAEAED DAEDADEDAD EAAEEDADAAEDA AEA

134 ADADDAFADDAFAF AAE AAE

135 ADADDAFADDAFAF AAE DDDDDAEDDDDA EDEEAEDED D D D AAE AAE

136 ADADDAFADDAFAF AAE DAEDDAAEDE AEAEAEDDDE DEAEDDADDAADDEADEDD DDDDDAAEAD DEEEADAEAEDA DADAEEDAEEEEDD DDEAEE ADDEAD DEDDDDDADAEAED AEDEDAAEDEDDD DDAEDEDDED AEDAEDDAEDDE AEAEAADADAEAEDED ADEDDAD FADADEADDEDAE DAEDDAEAD DDEAAAADDDDDE DDADDEDADDADED DEDDADAAAADDEE DADAADDAAEAADADED DDAADEEEDEDAADDAAE DADDADDADEDDEAE EEDAEDADADEDDDD AADADDEEEE DADEEDDE AEAAEDDAEDDADDAEAA DDEEAAADAEA DDDDDAE EDDEADEAE EDDAAEDAEDADEDAEA DAEDEAEEADADE ADAAEDDDEDADDEEDD ADADAAEADEEDAED AAADAADDE

137 ADADDAFADDAFAF ADAAAADDEAEEAAAAD DAEADEADAEDD AADDEDDD DEEDADDADDED AEDEDEDDDDED EADEDEAAEDDA DDDE AADA EDA DDADD EDA DEDDEAAAE AEAEADEDDAAEDDA ADE DDDAE EDDAFAD DEEDDE

138 ADADDAFADDAFAF D DDADDADE EDDEDDDADEAAEA EEEAEDDAD EDEAAADDAE

139 ADADDAFADDAFAF EEEADDE EEEA A DDEDDDAEDAEDEAE AEDADADEADA AEDDDEDADDEEDDA EEDEAEDDEEAEAED DDDAEDEAEADAEAE EDDADEAAAEADDE DDDDDDDEDEAADD DDEADADAEDAAADE EDAEADADDDDDA AEDAEDDDD D DDADADAEAADAAEAD DEDADEDADDADDAA DDDEDEADDADDADAAD DAADADADDADE ADD EAAEAE ED DAADADADDADEDA DE DEDDEAEDAADE DAEEDDD D

140 ADADDAFADDAFAF DDDAE EDDEEAEAAD EDDA DEDEADAEAADE DEEDAEDDDEDA

141 ADADDAFADDAFAF AEDEADAAAEADEEAD EAADEEDDAAADDD AAAEEAEDEDAEE DADEDADAA DDDE AAD DADDAEDDADED ADEAADEADDAAD DE AEADEDAED EDEEEDEADE EAAEAEAAEDDDE ADDE EED DDEDDDADEDADAEEDD EDADEDDADEA EEEDEADDAEDDDA E DA D EADAEADEAEAEDAA DAAEDEAEDDADAEA DADEDAAEADDAD AEDDADAEDDAADDDED DAAEAADEDAADEDA DDAADDDDDAEAEAE AADADA

142 ADADDAFADDAFAF DAEADADEDAAAEAA DAEDDDEDDAEDADDDDEAA DAADAAADAEAEDAAEDD EDDDEDEAADAE DDADEEDADEADAEDDDAEDAD EAEDEAEAADADEADE D

143 ADADDAFADDAFAF EAFE EDADEAADDADA DEEDADAADA DAEAADEAAEDDDEAA DAEADADADEAADDA D DDDEDEEAAEDDE AEDEEAEDAAEAEDEDEAA ADDDAEDDDAEAA EDADAEEEDEA DADAADAAA EAAAADDEDDA AADAAEADAAADADA EDEDEDAA DEEDEAD DDAEDADDAADEDADAAEA DDAEDADAADAEEAAA DEDAEAAADADADA DAAEDAAAAEDED DDDAEEAAEAAA EADADDDEDAADEDDAEAD DAEAAADDDAD

144

145

146 FFADA AEDAEAEAADAEDDE DDDADDADEDEDDADA ADAEEAEDDDEAADAA EDEDEDAEAEAD AEDAAAEEDDAADE EADAEDDEA DADADEDDAD DEDEAEAADADAEDAD DAADDDDAAEDE DAEDADDAADA EEDDAEEDA ADADDEAAADDA EAA DDADEEDEAAADA DEADA DDEDDEAAAEDED DDDADADDEA AAEAA DAEDEEDAD DADAEDEDDAAE DAAEDDADDEDDAADDAE EDDDADDEDED EEDAEDDADEADAD AAAAADDAADAEDEAED ADAEDAEDAEEEAAEDDA AEA DADEAAEAED DAEEAADAADADADAEADED DDADAEAEDAEDEEEE AAEEDADDDE DDAEDAEDDEADD ADAAAADEDDDA EEDADDAAEADADED AA DDAEEDADEADAAED AEAEDADDAEDDEDEA DDADDDDEDADADEDADA EAADAADEDDDA

147 FFADA EEDADADADD DDADAAADADDDA DAAADADDDAEAEDEDD DADDDEDAADEDDAEADDAE EEAAADDDAD DADA DA DA D AAA D EAADA DDAD E D DDA D DA A AA DAAADAA EDADAAADDADAAADEA DADDDAADDADADDA EDEADADADADAAED DADADDDDAADEAAEDE DADEDEAAAD DDAADEEEADAE EAAADAAEDDDDDA DADEDEADEEDAAD DADDEAAEDAEADADADEDA EAADADADEADADA DADDAEAAAADD DEDEDADDAA DA DA AD E A E D DA ADADA E D DAAADDAAAAEDADAADAD DEDAAADDADD ADDDDAEDAAADED DADAADAEDA EAEDAAAADADAADADDED DEDEADADEDAEDEAED DEAADAEAAADADD DAEDDADEAADDAAD EAAAADA AADDADEEDDAE DDADDEDDEADEAA AAEADEEDEAADAD DAAEEAEDAEAADDE DEADADD DADEADDAEE AEAEDEADADA DEEDDADDA

148 FFADA DAAAEE u p v q ( ) ln u v ( ) u q p INT( u, v, p, q) := sin q π A ( q ) ( q p ) D E ln u v cos q π A atan v AA E... D D E D u E v q p sin p π A E ln( u v ) cos p π A E ( p ) ( q p ) D D atan u AA D v... D EE u p v q ln( u v ) A... ( p ) ( q ) D p q E u p q sin( q k) π v k u q k v p q q q p sin( p k) π u k v p k k p k k = k = EAAAAEAAD AAAEDDDEAEDEAAA AEDDAADEEAA EAADEDAAA DEADEEAEAD DAEDEAEDEDAAA DA DAADAAEAEAEDADDAEDD E E A AD EAA EDDDAEDAEAE DAEEAAADEEAAADA AEADEEEDADDA

149 FFADA

150

151

152 DFD F AADFDE AA AAEADEA D AAA ED AAFDDAAAAA A DAD FA DA DAAA AAAAA AEAAEDA DAAAAE DADDA A A EA FDEDEAAA DEAE AA AA A DAD AE AA E AAEA DADFAFAEA DD

153 DFD A AADDDAD DFDEDEAADEDA DEDDDA AAAAA EADADAFAFDD DDED DE A EDADADA DAADAAA DADA DAAD A A DDA DAD DA EF AAAA AEAAE AAADAAAEAA EAA AAAAAAA DEDDAEAEEF D AAA D DEDA AA

154 DFD AAAAA AAA DADAE DAA AE DDEFDAA AAAAAAE ADEAD F F DDEDAAAAA ADDA EADAAA ADAAAAA ADDADA F AAA DDDEF AAAAA DDAADDEA D A DDADA F A DDADADD F AA DDADAF F F FAAAA A A DAD AE FDAAA

155 DFD AAADAADAEAAE DDAEAE FDADAEA DEDDAAAA ADD FAAAA AAAA AE EDAEDFD D ED DAAEDAED FDD ED DDE A EAA D AA DE EDADAFDAEF DDE D D FDAA DE AADA ADEFAFE FF D

156 EE A A EFA AE F FFEAAEFEFEEFA EE A A EFA A F FFEAAEEFA EEAAEAFF EE A A EFA AE F FFEAAEFAEFAFFEFEF EE A A EFA AE F FFEAAEFEAEFAFEFEF EEE

157

158 FFFAAADDFDAFDFFDFFA AEFAEFA E F D D D E D E DAD DA D D E ED AD DADAEDDADADAAEAAEAA DADAAADAAAA DAEEDAAEAAAADEAE D A = J DAEEAADDADEAED AEAEA,φ,zDAA A A A = J φ DA D A DA E E DA EAA ADA EEDEEDADDADEDAADDD ADEDADEADAAEDAD EDDAEADA ADDDAEDD

159 FFFAAADDFDAFDFFDFFA D EDAEAADAEDEDA DEDADEDAEDD A A A = φ DDDAEDADAADAA AEAEφAEAEAAADAA A = ) e n jnφ ( d d n d d n n n e jnφ DEAEEDADAEAAAD AAEAADAADAEAAADDA A DADAD m ( ) = λ n DAEEAD ( ) m m m λ m m λ m n λ = DADADAA m = ± naaeaea EAADAAEEDADDAEE E DDAEA d d d d d da = = d d d d d = d D E ADEAEAAEEDDDDAD ka ( ) = a ln ( ) a ln ( k) = a ln D E FEADAAE DAEAEAAAAEDADAEE AADDD A EADED DDAAA EEEAADDADDADDAD AEAAAAAEDAEEDDA DDADEAAAEA

160 FFFAAADDFDAFDFFDFFA AEAAEDDADADADADE DAEDEDDDADDADEDDDDD DDAEAEAAEDDD D A AA A AA AED ED AED ED A FE D FE D D AA D AA Aext(, φ ) α ln( ) π n α π cos ( nφ ) β sin( nφ ) n n n = ( ) Aint(, φ ) n a cos ( nφ ) b π ( sin( nφ ) n n ) n = DDDADDEDAAE EADAAAAEDFA DDADDEDADDED DAADAAEDADE DD

161 FFFAAADDFDAFDFFDFFA D DAEAEAEAED A = φ A φ = EDEAEAEDDA ext(, φ ) π n = ( n ) n α sin( nφ ) β cos ( nφ ) n n ( ) α ( n ) ext(, φ ) n α cos ( nφ ) β φ π π sin( nφ ) n n n = int(, φ ) π n = n n ( ) a sin( nφ ) b cos ( nφ ) n n ( ) int(, φ ) n n a φ π cos ( nφ ) b sin( nφ ) n n n = ( ) AAEFA AE F FFE A AE FE FE EFA AEEAAADDD A, φ ) EDAEAAEAEAEDAA AA ( f f EADDAA DAEDAD DDED

162 FFFAAADDFDAFDFFDFFA AEDA DAAE Ai AEAADA DA A ADEEAA DDA ADDDDDEAA t DEDAAEDDDADEDAA DEDAAEDAEDADAAD DEDEAA DAAAAEAEADAEDA EDDED A(, φ ) π n = n ( a cos( nφ ) b sin( nφ ) n n )

163 FFFAAADDFDAFDFFDFFA DAAAEAEDDDAD AEDAEADAEDADEDA DEDAEDDEDADAADA DAAAAADADAEAA A Af 4 π ( ) I ln d f ( ) d cos φ φ f f ( f ) ( f ) f D A E cos ( φ φ f ) f EA Af(, φ ) π I f ln( f ) 4 π I ln f D f A E ( ) cos φ φ f f DEDA ln f D A E ( ) cos φ φ f f n = n A n cos n φ φ f f D E ( ) ADADAADA Af (, φ ) π I f ln( f ) n A n cos n ( φ φ f) D f E I π f n = ED Af (, φ ) a π f ln( f ) π n = n ( a cos ( n φ ) b sin( n φ ) fn fn ) a f I f I a f fn n ( f) n I f cos ( n φ f ) b sin n φ fn f n ( f) n ( ) ADAEDAAED

164 FFFAAADDFDAFDFFDFFA A(, φ ) π ( ( ) α ln( ) ) a ln f f π n = π n = n α cos ( nφ ) β sin( nφ ) n n ( ) n a cos ( nφ ) b sin( nφ ) fn fn ( )... D D AEADADADAAEDDADAEEA EAEAEADAEDDAAA DAA A A = Hφ = φ. (, φ ) π n = H(, φ ) φ π n n n = ( ) a sin( nφ ) b cos ( n φ ) n n n n a cos ( nφ ) b sin( nφ ) n n ( ) DAA (, φ ) π H(, φ ) φ π n = ( ) n n a sin ( nφ ) b cos ( nφ ) fn fn α n = ( n ) n ( ) n n a cos ( nφ ) b sin ( nφ ) fn fn ( ) α sin( nφ ) β cos ( nφ ) n n ( n ) n α cos ( nφ ) β sin( nφ ) n n ( ) DD ADADAAEADEAADDDED DDDEAA H( R, φ ) H( R, φ ) φ φ ( R, φ ) ( R, φ )

165 FFFAAADDFDAFDFFDFFA DDAEDEEA DAAD R n a cos ( nφ ) b sin( nφ ) n n ( ) ( ) ( ) R n a cos nφ ( ) b sin ( nφ ) R ( n ) α cos ( nφ ) β sin( nφ ) fn fn n n ( ) R n a sin ( nφ ) b cos ( n φ ) R n a sin ( nφ ) b cos ( nφ ) R ( n ) n n fn fn ( ) ( ) α sin( nφ ) β cos ( nφ ) n n DD sin( nφ ) cos ( nφ )ADA AAD AD ED R n a A D n E R n a R ( n ) α fn n R n a R n a R ( n ) α n fn n R n b A R n b R ( n ) β D n E fn n R n b R n b R ( n ) β n fn n AADAAEADDDEDDDDE AADDDADAEADD A( R, φ ) A( R, φ ) R n a cos ( nφ ) b sin( nφ ) π ( n n ) a ln π f ( f ) R n π ( a cos ( nφ ) b sin( nφ ) fn fn )... n = n = R n α cos ( nφ ) β sin( nφ ) π ( n n ) n = DAEEDAAAAEDEDA A A( R, φ ) A( R, φ ) π a f ln( f ) ADAAEDDDEDDDDEAAA AEDDEEDEDDAAED AA

166 FFFAAADDFDAFDFFDFFA A(, φ ) A(, φ ) n a cos ( nφ ) b π ( sin( nφ ) n n ) n = n a cos ( nφ ) b π ( sin( nφ ) fn fn ) n = n α π cos ( nφ ) β sin( nφ ) n n n = ( ) AA a n α n EADAE EEDAEDADEDADA DEAAD ED a n I f n ( f) n α R n n I f cos ( n φ f ) t I f n ( f) n cos ( n φ f ) n ( f) n cos ( n φ f ) I f n ( ) n cos ( n φ f ) t R f DAA b n β n I f b t sin n φ n f β n n ( f) n I f n ( ) n ( ) ( ) sin n φ f DAAD DEEDAE AD cos ( n φ ) cos ( n φf ) sin( n φ ) sin( n φf ) cos[ n ( φ φf )] AAEAEDEAA

167 FFFAAADDFDAFDFFDFFA A(, φ ) n π I t A f n cos n φ φ ( f) n = D f E A(, φ ) π I f n = n A n cos n φ φ f f D E ( ) π I f n = n A cos n φ φ n f D E ( ) DEDDAE n = n = n A n E cos n ( φ φ ) f D f n A E cos n φ φ n D ( f) ln f ( ) f ( ) f ( ) cos φ φ f ln ( ) cos φ φ ( f ) DDADAEAEDEAA A(, φ ) A(, φ ) 4π I t ln f f 4π I ln f f ( ) f ( ) f ( ) f cos ( φ φ f ) ( ) f cos ( φ φ f ) 4π I ln f cos ( φ φ f ) ( )

168 FFFAAADDDAFDFADFA F AEEAAEE AEAEDAEADA, φ ), φ ) AADAE D F D ( f f ( f f AAAEAADDED D F EAA DAAEDAADA D F DE DDED D

169 FFFAAADDDAFDFADFA DDAAAEDDD D A DAEDAEADAEDADEDA DEDAEDDEDADAA DADAAAAADADAD EDEDEDAEDDAED A(, φ ) α ln( ) π π n = π n = n a cos ( nφ ) b sin( nφ ) n n ( ) n α cos ( n φ ) β sin( n φ ) n n ( )... α I f f I α f n n ( ) n I f f cos ( n φ f ) β n n ( ) n ( ) sin n φ f DDAAEADDEDA DAEDAEDDAAEEAE A(, φ ) α ln( ) π π n = π n = n α cos( nφ ) β sin( nφ ) n n n a cos( nφ ) b sin( nφ ) n n ( ) ( )... DDAAEDDD DEE DAEDEDAADEAEDD A

170 FFFAAADDDAFDFADFA A3(, φ ) π ( ( ) α ln( ) 3 ) π n = a ln 3 f π n = n α cos ( nφ ) β sin( nφ ) 3n 3n ( ) n ( a cos ( n φ ) b sin( n φ ) 3n 3n )... a 3 I 3 f I a 3 f 3n n ( f) n I 3 f cos ( n φ f ) b sin n φ 3n f n ( f) n ( ) D D AEADADADAAEDDADAEEA EAEAEADAEDDAAA A = φ DA A Hφ =. (, φ ) π H(, φ ) φ π n = ( ) n n a sin ( nφ ) b cos ( n φ ) n n α n = ( n ) n ( ) n n a cos ( nφ ) b sin ( nφ ) n n ( ) α cos ( n φ ) β sin( n φ ) n n ( n ) n α cos ( nφ ) β sin( nφ ) n n ( ) DDA (, φ ) π H(, φ ) φ π n = ( ) n n a sin ( nφ ) b cos ( n φ ) n n α n = ( n ) n ( ) n n a cos ( nφ ) b sin ( nφ ) n n ( ) α cos ( n φ ) β sin( n φ ) n n ( n ) n ( α cos ( nφ ) β sin( nφ ) n n ) DDA 3(, φ ) π H3(, φ ) φ π n = ( ) n n a sin ( nφ ) b cos ( n φ ) 3n 3n α 3 3 n = ( n ) n ( ) n n a cos ( nφ ) b sin ( nφ ) 3n 3n ( ) α cos ( n φ ) β sin( n φ ) 3n 3n ( n ) n ( α cos ( nφ ) β sin( nφ ) 3n 3n )

171 FFFAAADDDAFDFADFA D D DDEED S R S 3 R ADADA AEADEAA DD S R H( R, φ )φ H( R, φ )φ ( R, φ ) ( R, φ ) DD S 3 R H( R, φ )φ H3( R, φ )φ ( R, φ ) 3( R, φ ) DDAEDEEA α α 3 α α 3 D α ED DD sin( nφ ) cos ( nφ )EDDADAEE DAADAAAD D A D ED ( ) n R ( ) ( n ) a R α n n ( ) n R ( ) ( n ) a R α n n ( ) n R R ( ) ( ) n ( n ) a R α R n n ( ) n ( ) ( n ) a R α n n ( ) ( n ) a R α n n ( ) n R 3 ( ) ( n ) a R α 3n 3n ( ) n R ( ) ( ) n ( n ) a R α R n n ( ) ( n ) a R α 3n 3n

172 FFFAAADDDAFDFADFA ( ) n R ( ) ( n ) b R β n n ( ) n R ( ) ( n ) b R β n n ( ) n R ( ) ( ) n ( n ) b R β R n n ( ) ( n ) b R β n n ( ) n R ( ) ( n ) b R β n n ( ) n R 3 ( ) ( n ) b R β 3n 3n ( ) n R ( ) ( ) n ( n ) b R β R n n ( ) ( n ) b R β 3n 3n AADAAEADDDEDF F DD AADADEDEDED D DD S R ( ) A( R, φ ) A R, φ DD S 3 R I f π ( ) ( ) ln R ( ) A3( R, φ ) A R, φ I 3 f π ln( f ) I R f ln R π 3 ( ) ln R D D A A EE ADAAEADDDED S S 3 AA EDDEDEEDEDAA EDAAA A(, φ ) A(, φ ) α ln A π R n π ( a cos ( nφ ) b sin( nφ ) n n ) D... E n = n α cos ( n φ ) β sin( n φ ) π ( n n ) n = α ln A π R E n a π ( cos ( nφ ) b sin( nφ ) n n ) D... n = n α cos ( nφ ) β sin( nφ ) π ( n n ) n =

173 FFFAAADDDAFDFADFA A3(, φ ) R α ln A α ln A A π R D 3 R E n a D E D π cos ( n φ ) b sin( n φ ) 3n 3n E n = n α cos ( nφ ) β sin( nφ ) π ( 3n 3n ) n = ( )... AAa n a n α n α 3n ADAE EDADADEDADEAADE AA D ED a n a n α n α 3n ( ) ( ) ( ) ( 3) A α n A 3 4 A a 3n A ( ) ( ) A ( ) ( ) ( ) A a α 3n n 3 A ( ) ( ) A ( ) ( ) ( ) ( ) α A a n 3 3n A A ( ) ( ) A ( ) ( ) ( ) ( ) ( ) ( ) ( ) a A A 3n α A 3 n A ( ) ( ) A ( ) ( ) A ( R ) n A ( R ) n DAA b n b n β n β 3n