Semi-analytical solution of an eddy current problem for a conducting medium with a surface flaw
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1 NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 Sm-lytl soluto of y ut olm fo outg mum wt suf flw V Kolsk stt mto of tut gfuto xsos s us t st to ostut sm-lytl soluto fo y ut olm otg suf flw flw s st y yll ol t u ly of two-ly outg mum xs of t yll flw os wt t xs of y ut ol yg tg ut ot soluto s sm-lytl s t two sts t soluto ou tt u t us of uml mtos: luo of omlx gvlus wtout o fomto o t loto of t gvlus soluto of systm of l lg utos sus of uml luos st Kywos Ey ut tstg g m sto of vls gvlus NOUCON NLCL solutos of y ut tstg olms fo t ss of uou outg m wll-kow t tu - s of flws outg mum sus mo om ouy vlu olms w gl ot solv y lytl mtos Numl mtos su s ft lmt mtos oft us su ss to lu t omgt fl t mum f y ut olm s solv fo uou mum t t s ssum tt t vto ottl ts to o s o of t gomtl oots ts to fty Fom ysl ot of vw t omgt fl s glgl t sufftly lg st fom t sou of ut s s us w t utos ssum tt t vto ottl s xtly o t lg l st fom t xs of yll ol Lt my olms solv usg t sm ssumto t s sow tt t omuttol o sly otoll y oosg t vlu of oos mto s usully ll t EE mto w t vto sts fo t ut Egfuto Exsos o s o t EE mto fo soluto of y ut tstg olms wt yll symmty s st fft tys of xsymmt flws tly os t tu: ft yll ol outg lf-s yll ol of ft lgt outg lf-s 6 yll luso lf-s 7 suf yll flw 8 t soluto of su xsymmt olms us t o to stmt l outvty of s ot ots wt yll symmty 9 t st t EE mto s us to solv y ut tstg olm fo t s w yll -o ol s lot ov two-ly outg mum otg suf flw t fom of yll ol oxl wt t ol olm s solv y t mto of sto of vls ot soluto s smlytl s t two sts t ou tt u t us of uml mtos: luo of omlx gvlus wtout o fomto o t loto of t gvlus soluto of systm of l lg utos Su mol us to stmt t fft of ooso mtl otgs MHEMCL FOMULON OF HE POBLEM Cos yll -o ol of lot ov outg two-ly mum s Fg u ly of t outg mum s t l outvty s st y gos ws t outvty of t low lf-s go ol s tg ut wt fuy f lft-off s t gt of t ol s u outg ly go ots flw t fom of yll ol of us gt xs of t yl os wt t xs of t ol tkss of t low ly s s wok ws tlly suot y t gt 6/ of t Ltv Coul of S V Kolsk s wt t tmt of Egg Mtmts of t g l Uvsty g LV 7 Ltv o: ; fx: ; -ml: vkolsk@ gmlom SSN: 998-7
2 Fg ol yg tg ut ov outg mum wt flw soluto of t olm sow Fg ot y suosto l f o kows t soluto of t osog olm fo sgl-tu ol s Fg Fg sgl-tu ol ov outg mum wt flw W ssum tt sgl-tu ol of us s lot t gt ov t outg mum sow Fg olm s xsymmt so tt t mu of t vto ottl s oly o o-o omot t to s t systm of yll ol oots t t O systm of utos fo t mu of t vto ottl gos s t fom s Fg : w x s t -futo f f s t st fom t xs of t ol w t omgt fl s ssum to xtly o f ouy otos w t susts oso to outv go stvly tf otos t v t fom to vto ottl s ou t fty gos : s s SOLUON FO SNGLE-UN COL o to ostut t soluto to w os two sugos of go mly } { } { solutos ot y stvly Usg t l of suosto w st t solutos to t fom NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 SSN: 998-7
3 w ty ostts / t oots of t uto x s t Bssl s futo of t fst k of o o vto ottl s otuous t : 6 tgtg wt st to fom to osg t lmt s t sug uto w ot 7 t follows fom -7 tt 8 9 Usg 8 w ot Mulyg 9 y tgtg t sug uto wt st to fom to usg t otogoy oto t followg uto s ot Eutos wtt t fom Susttutg to w ot 6 Usg t mto of sto of vls suosto l t soluto go wtt t fom 7 o 8 w 9 x s t Bssl s futo of t so k of o o ukow gvlus solutos gos w Usg t fst ouy otos 6 7 w ot Mulyg y tgtg t ot uto wt st to fom to usg w NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 SSN: 998-7
4 ot w ~ ~ offts omut usg t followg fomuls fom : l k t t t l k 6 l k t t t l k 7 us usg 6 7 w ot ~ } { ~ lyg t sm ou to w ot 8 Elmtg fom 8 w gt 9 Usg ouy otos 8 9 w xss ostts 6 7 tms of : 6 7 Smlly lyg t ouy otos usg w lmt to ot } { o to solv t systm 9 umlly w v to tut t um of tms t ss ot wos t u summto x 9 soul l y ommtos o t so of t vlu of gv fo xml us t systm to solv s t fom } { Usg tf otos w ot t followg uto w ' ' NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 SSN:
5 NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 Euto s us to tm omlx gvlus t osog vlus mto of omutto of t gvlus s s o t two s uls 967 uss t Cuy s tom um of os s of lyt futo s los otou C s ul to t followg otou tgl ' s C 6 ot tstg su fom t toy of omlx vls s t followg f C s los uv t omlx l w os ot ss toug o of G s t to of C t s m C m ' k m w k ll os of G m k 7 w l mul o s out og to ts muy 7 Usg 7 w ostut olyoml of g k w s t sm os k s t futo Not tt ts s t olyoml s ot oxmto to t smly s t sm oots s t futo t ts stg svl ossl sttgs vll o to f f t om G s smll oug t s oly o o of s C t k t vlu of s fom 7 of t futo s C gvs us t oly o Howv f t s mo t o oot of s C t olyoml s ostut usg 7 t oots of ts olyoml lu fo xml y ms of Mtmt omm oots s t s mto fo t oots of t osog olyoml xtly t sm s t os of s C t s kow tt t oots of g o olyomls ut sstv to t vto of t offts of t olyoml us t um of os of s C soul ot too lg k soul ot x s su f k t t om G soul v to smll su-oms so tt t um of os of su-om smll t xt tl ssu s t s of t otou C wo s ss omm fo luos : ls tgls o of t s of t otou C s tt y luo of t otou tgls 6 7 t s kow tt otou tgls ovtly lu f C s l o tgl W oos tgl s t s s of C s y tgl sly suv to smll tgls s t s ly ot out t two sts t soluto oss w uml luos ssy Fst o s to solv f t omlx gvlus t osog vlus of us t vlu of sts t um of gvlus So t systm of l utos s to solv ll t ostts v to tm Nxt w omut t offts fom t s s fom 6 tt t vto ottl go s t sum of t two tms so tm o t gt- s of 6 sts t vto ottl of sgl-tu ol lot uou f s w t outg mum s st fst tm o t gt s of 6 s t u vto ottl u to t s of t outg mum H t u vto ottl go s gv y t fomul 8 V CLCULON OF HE CHNGE N MPENCE OF SNGLE-UN COL u g m of t ol sow Fg s gv y t fomul Z 9 Cluos fom wt Mtmt usg 8 9 followg vlus of t mts of t olm os: mm mm Ms/m Ms/m mm mm mm sus sow Fg fo sv vlus of fuy f fom kh to 7 kh fom to to ottom wt t st s of kh Fg g m of t sgl-tu ol fo sv fus fom kh to 7 kh 7 mm mm SSN:
6 Comuttos lso fom fo t s 7 mm mm t ot mt vlus t sm s Fg g m s sow Fg Fg g m of t sgl-tu ol sow Fg fo sv fus fom kh to 7 kh mm mm V SOLUON FO COL OF FNE MENSONS u vto ottl go fo t s of ol of ft msos sow Fg s gv y t fomul ol Not tt uml soluto of t systm fo sgl-tu ol fou y susttutg t vlus of to t gt- s of Fo ol of ft msos su o ot us s w v to tgt t g m ov t oss-sto of t ol wt st to o to fom luos fo t ol of ft s w wt t mtx fom B w t offt mtx s t lmts of t lok mts gv y mts B B w H wtt t fom Solvg t so uto w ot 6 Susttutg 6 to t fst uto gvs 7 t follows fom 6 7 tt 8 Euto soul wtt mtx fom: 9 w g C E B 6 B NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 SSN:
7 C g 6 Fomul 8 fo t u vto ottl of sgl-tu ol wtt t fom f 6 w f Susttutg 6 to usg t fomuls 6 w ot t u vto ottl u to t s of t outg mum t ut mu ts s s l y t ut sty N : ol f N 6 tgl wt st to 6 omut tms of t Bssl Stuv futos s follows H H u g m of ol of ft msos s lu y ms of t followg fomul : ol V v Z 6 Usg 6 6 w ot t u g m of t ol t fom N Z 66 V CLCULON OF HE CHNGE N MPENCE FO COL OF FNE MENSONS Cluos o usg fomul 66 fo t followg vlus of t mts of t olm sow Fg : Ms/m Ms/m mm mm mm mm 6 mm N Sv vlus of fus fom kh to 7 kh wt t st s of kh us Comuttol sus sow Fgs -6 Fg lots t g m of t ol fo t followg vlus of t mts mm mm Fg g m of t ol sow Fg fo sv fft fus fom kh to ot to 7 kh ottom ot fo t s mm mm g of t g m fo t vlus of t mts 7 mm mm s sow Fg 6 fo t sm sv fus NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 SSN:
8 NENONL OUNL OF MHEMCL MOELS N MEHOS N PPLE SCENCES Volum 8 Fg 6 g m of t ol sow Fg fo sv fft fus fom kh to ot to 7 kh ottom ot fo t s 7 mm mm V CONCLUSON N ECONS OF FUUE WOK mto of tut gfuto xsos s us t st to omut t g m of ol u to yll flw outg mum olm s solv y t mto of sto of vls ot soluto s sm-lytl s t mto of sto of vls s om t wt uml mtos o to omut omlx gvlus solv systms of l lg utos mto gl fo ot olms wt xl symmty tly sml symmt olm s solv w ol s lot ov outg lf-s wt ft vtl yll ol xs of t ol os ot o wt t xs of t ol olm s solv usg t so o vto ottl fomuo of t Mxwll s utos Su o lso gl fo t s os t st Pogs of t t WSES of o l totl ms Slo tly u Kolysk Kus O Mts Polov tmto of l outvty of mtl s usg l sl ols Pogs of t t WSES ttol of o systms oos G uly - 9 V Kolsk Kolysk Sstvty of -o l sl ol to vto of l outvty of outg sml t s l mtmts fomts Pogs of t 6 t WSES ttol of o l mtmts t Euo of o l mtmts fomts Motux Swtl m 9-9 M mowt Stgu Hook of mtmtl futos wt fomuls gs mtmtl tls Wsgto: Ntol Buu of Sts 96 V Kolsk lytl us-lytl solutos of t olms y ut tstg P tss g l Uvsty Ltv LM lvs N Lyss uml mto fo lotg t os of lyt futo Mtmts of Comutto vol N Lyss Numl lgotms s o t toy of omlx vl Po CM Sos oouls Soluto to t y-ut uto olm outg lf-s wt vtl yll ool Pogs of t oyl Soty : Mtmtl Pysl Egg Ss vol EFEENCES gooulos EE Ks Ey uts l outg m mstm: Elsv 98 M tmov Kolysk Vllout Mtmtl mols y ut tstg Motl: CM 997 P oouls EE Ks Ey ut ol olms wt o to ostutv vluto ulut: S 6 N Numl molg fo omgt o-stutv vluto Loo: Sg 99 P oouls EE Ks Ss xsos y ut ostutv vluto mols oul of mtls ossg tology vol P oouls Bowl tut go gfuto xso mto fo t soluto of ouy vlu olms y ut ostutv vluto vw of Pogss Qutttv Nostutv Evluto vol 8 7 V Kolsk Cluo of t g m of ol lot ov outg mum wt flw Pogs of t ttol of o o of otmoy ostutv tstg gg Potoo Slov Stm V Kolsk g m of ol ov wt flw Ltst ts l totl ms SSN: 998-8
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