Mehbub-Ur Rahman. Optimization of In-line Defect Detection by Eddy Current Techniques

Transcription

1 Mhbub-Ur Rahman Optimization of In-lin Dfct Dtction by Eddy Currnt Tchniqus

2 Mhbub-Ur Rahman Optimization of In-lin Dfct Dtction by Eddy Currnt Tchniqus kassl univrsity prss

3 This work has bn accptd by th faculty of Elctrical Enginring and Computr Scinc of th Univrsity of Kassl as a thsis for acquiring th acadmic dgr of Doktor dr Ingniurwissnschaftn (Dr.-Ing.). Suprvisor: Co-Suprvisor: Prof. Dr. rr. nat. Karl.J. Langnbrg PD Dr.-Ing. Rné Marklin Dfns day: 28 th Octobr 2010 Bibliographic information publishd by Dutsch Nationalbibliothk Th Dutsch Nationalbibliothk lists this publication in th Dutsch Nationalbibliografi; dtaild bibliographic data is availabl in th Intrnt at Zugl.: Kassl, Univ., Diss ISBN print: ISBN onlin: URN: , kassl univrsity prss GmbH, Kassl Printd by: Unidruckri, Univrsity of Kassl Printd in Grmany

4 Acknowldgmnts Th ky to th succssful compltion of any task is motivation. This thsis is th rsult of four yars of hard work. During this priod I hav bn supportd and motivatd by a numbr of popl. It would b a plasur for m to show my gratitud to all of thm. I am vry gratful to Prof. Dr. rr. nat. Karl-Jörg Langnbrg and PD Dr.-Ing. Rné Marklin for giving m th opportunity to do my Ph.D. thsis in th dpartmnt of Elctromagntic Thory, prsntly known as Computational Elctronics and Photonics. My principal advisor PD Dr.-Ing. Rné Marklin has playd a vry important rol in my thsis by guiding m during my rsarch work and by motivating m for bttr rsults. I would lik to thank him prsonally for his support and nthusiastic rol in this scintific work. I also lik to thank Dr.-Ing. Klaus Mayr, Prof. Dr. sc.tchn. Brnd Witzigmann, Mrs. Rgina Brylla and Dipl.-Ing. Yanna Mikhilina for thir coopration and hlp on concptual and softwar aspcts during this thsis. I would lik to thank my frinds Dr.-Ing. Shohrab Hossain, Nasir Uddin, M.Sc., Md. Iftkhar Alam, M.Sc. and Dominik Klaus for thir continuous motivation and to my collagus P. Kumar Chinta, M.Sc., Dipl.-Ing. Grgor Ballir and Dipl.-Ing. Ugur Akcakoca for accompanying m during my work. At th nd, I would lik to thank my parnts and sistr for thir constant support during this Ph.D. program. Kassl, Grmany July 1, 2010 Mhbub-Ur Rahman iii

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6 Contnts Acknowldgmnts Contnts iii vii Summary 1 1 Introduction to Eddy Currnt Snsors and Modling Tchniqus Statmnt of th Problm of Hot Wir Stl Inspction Numrical Modling and Simulation of th Inspction Tchniqu Modling Tools MQSFIT Opra Faraday Othr Tools Govrning Equations for Magntoquasistatic Problm Maxwll s Equations for Macroscopic Non-moving Mdium Transition and Boundary Conditions Transition Conditions Boundary Conditions Fild Equations in th Frquncy Domain Inhomognous Hlmholtz Equations Inhomognous Hlmholtz Equation for th Elctric Fild Inhomognous Hlmholtz Equation for th Magntic Fild Elctromagntic Potntials of an Elctrically Conducting Matrial Elctromagntic Potntials in Sourc-Fr Cas Elctromagntic Potntials in Cas with Elctric and Magntic Sourcs Dcoupling of Maxwll s Equations by Elctromagntic Potntials Exprssion of th Magntic Vctor Potntial in th Frquncy Domain Exprssion of th Elctric Scalar Potntial in th Frquncy Domain Exprssion of th Elctric Vctor Potntial in th Frquncy Domain Exprssion of th Magntic Scalar Potntial in th Frquncy Domain Summary of Elctromagntic Potntials in th Frquncy Domain Magntoquasistatic Fild Magntoquasistatic Fild Equations with Magntic Vctor Potntial Transition and Boundary Conditions for th Magntic Vctor Potntial Fild Equations for a Moving Conductor v

7 vi Contnts 3 Finit Intgration Tchniqu (FIT) Discrtization of th Elctromagntic Mdium in Spac Allocation of th Discrt Fild Quantitis in Cartsian Coordinats Discrt Fild Quantitis in Cartsian Coordinats in 2-D cas Drivation of th Discrt Grid Equation in Cartsian Coordinats in 2-D cas Discrt Fild Quantitis in Cylindrical Coordinats in 2-D cas Drivation of th Discrt Grid Equation in Cylindrical Coordinats in 2-D cas Boundary Conditions Formation of a Band Matrix Computation of Inducd Elctric Voltag Finit Elmnt Mthod (FEM) Formulation of th Magntoquasistatic Govrning Equation D Formulation in Cartsian Coordinats D Formulation in Cylindrical Coordinats Discrtization of th Domain in 2-D Spac Nod-Basd Elmntal Intrpolation Formulation by Galrkin s Mthod Formulation in Cartsian Coordinats Formulation in Cylindrical Coordinats Formulation of th Systm of Equations Computation of th Local Cofficints Rlating th Local Nods of th Elmnts to th Global Nods Arrangmnt of th local cofficints in global matrics Implmntation of th Boundary Conditions Boundary Elmnt Mthod (BEM) Formulation of th Magntoquasistatic Govrning Equation Formulation for th Conducting Rgion Formulation for th Non-Conducting Rgion Intgral Rprsntation of Elctrical and Magntic Fild Strngth Intgral Rprsntation for Conducting Rgion Intgral Rprsntation for Non-conducting Rgion Intgral Equation Formulation Elctric Fild Intgral (EFIE) Formulation Magntic Fild Intgral (MFIE) Formulation Boundary Elmnt Discrtization and Formulation of th Systm of Equations 73 6 Analytical Solution of an Eddy Currnt Problm Gomtry of a Typical Eddy Currnt Problm Analytical Exprssions Gauss-Lagurr Quadratur Gauss-Kronrod Quadratur Computation of th Intgrals Analytical Rsults Comparison with Numrical Rsults FIT Rsults FEM Rsults

8 Contnts vii BEM Rsults Comparison Simulation Rsults Simulation Rsults of Encircling Coil Snsors Encircling Coil Snsor without Frrit Cor to Simulat a Transvrsal Crack Encircling Coil Snsor with Frrit Cor to Simulat a Transvrsal Crack Encircling Coil Snsor without Frrit Cor to Simulat a Longitudinal Crack Simulation Rsults of Point Coil Snsors A 3-Coils Snsor in Unitary Mod A 3-Coils Snsor in Array Mod Simulation Rsults of a D-coils Snsor Simulation Rsults of a 4-Coils Snsor Simulation Rsults of a Pot Cor Snsor Slction of Eddy Currnt Snsors for th Final Prototyp Vlocity Analysis Simulation Rsults for Diffrnt Vlocitis Validation of th Simulation Rsults Conclusions 113 Bibliography 115 List of Publications 119

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10 Summary This thsis prsnts innovativ and advancd lctromagntic tchniqus for th in-lin inspction of hot wir stl. Th hot wir inspction procdur is prformd with th ddy currnt (EC) snsor tchniqu. Any typ of crack on th uppr surfac of th stl wir disturbs th ddy currnt distribution which can b dtctd by th ddy currnt snsors. Howvr, th ddy currnt distribution is only wakly influncd by cracks which ar paralll to th wir, so-calld longitudinal cracks. Th convntional ddy currnt snsors cannot dtct ths cracks proprly. To dtct such longitudinal cracks a numbr of nw EC snsors hav bn dvlopd and succssfully tstd. Th modling of th EC snsors and inspction tchniqus ar numrically prformd by stting up th govrning quations for ddy currnt problms and thn th diffrntial, as wll as th intgral quations ar solvd with a suitabl numrical mthod. A propr slction of th numrical tchniqu plays an important rol in succssful simulation of th ddy currnt inspction procdur. Diffrnt numrical mthods - th finit intgration tchniqu (FIT), th finit lmnt mthod (FEM) and th boundary lmnt mthod (BEM) ar usd to modl this NDT (non-dstructiv tsting) situation. Th prsntd work is a part of th so-calld INCOSTEEL projct which is sponsord by th Europan Commission through th Rsarch Fund for Coal and Stl (INCOSTEEL: In-lin quality control of hot wir stl - Towards innovativ contactlss solutions and data fusion). Evry numrical mthod has its own advantags and disadvantags that nd to b considrd bfor using thm to modl a crtain ddy currnt problm. FIT, FEM and BEM ar discussd laboratly in this thsis for this purpos. A FIT basd numrical tool MQSFIT is dvlopd as a part of th scintific work. Two othr simulation tools, basd on FEM and BEM, along with MQSFIT, ar usd to modl and optimiz th ddy currnt snsors dvlopd in INCOSTEEL projct. Howvr, th numrical rsults hav to b validatd against analytical and publishd rsults to prov th accuracy and rliability of th simulations. As a part of that, analytical solutions for a simpl ddy currnt problm ar discussd in th work. A comparison of all th numrical mthods to th analytical solutions is also prsntd. Th numrical modling, optimization and fdback modling hav mt th dmands of th INCOSTEEL projct and thus ld to th succssful compltion of th projct in Dcmbr Th obtaind rsults hav opnd th door for furthr futur rsarch and innovativ idas. 1

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12 Chaptr 1 Introduction to Eddy Currnt Snsors and Modling Tchniqus 1.1 Statmnt of th Problm of Hot Wir Stl Inspction Th production of long stl products lik stl wirs, rods and bars within th stl markt is subjctd to vry strict spcifications with rgard to quality and rliability as wll as in trms of incrasing productivity. Product quality along th diffrnt production procsss has a high impact on th quality of th nd componnt. Nondstructiv tsting (NDT) tchniqus ar widly usd for quality assuranc in stl manufacturing,.g., for th dtction and valuation of all potntial typs of intrnal and surfac dfcts. Th principal goal is to dvlop a snsor systm to optimiz th in-lin dtction of transvrsal and longitudinal dfcts producd in th manufacturing procss of hot wir stl. Existing NDT systms hav som limitations rgarding dtction and charactrization of svral typs of common dfcts. Th aim of this thsis is to discuss th lctromagntic modling tchniqus in ordr to analyz th ddy currnt inspction tchniqus. Hot wir stainlss stl (Fig. 1.1a) of a diamtr D = 6 42 mm is considrd, drivn by a guiding systm through th snsor coils at a tmpratur of C abov th Curi point with a rolling spd up to v c = 120 m/s dpnding on th diamtr of th wir. Typical longitudinal and transvrsal cracks in th hot wir stl ar shown in Fig. 1.1b and 1.1c, rspctivly. In a paramtr study w us cracks with th shortst lngth of l = 5 mm, th lowst width of w = 0.2 mm, and a minimum dpth of d = 0.2 mm. 1.2 Numrical Modling and Simulation of th Inspction Tchniqu In Chaptr 1 w dfin th problm of hot wir inspction and giv an outlin of th applid snsors. A brif dscription of th numrical tools to modl th inspction tchniqu numrically is also prsntd in Chaptr 1. Th mathmatical background, this mans th govrning quations of th applid snsor typs, ar givn in Chaptr 2. In Chaptr 3 w discuss th finit intgration tchniqu (FIT) to discrtiz th magntic diffusion quation in intgral form. Th discrtiztion is prformd in cartsian as wll as in cylindrical coordinat 3

13 4 Chaptr 1. Introduction to Eddy Currnt Snsors and Modling Tchniqus a) b) c) Figur 1.1: a) Hot wir stl in th production lin, b) a typical transvrsal and c) a typical longitudinal crack. systm. Th finit lmnt mthod (FEM) to solv th ddy currnt problm is discussd in Chaptr 4. Both FIT and FEM basd simulation tools ar usd to simulat two dimnsional (2-D) problms. Howvr, all th ddy currnt snsors cannot b modld in 2-D and hnc, a suitabl numrical tool is chosn for 3-D modling, which is basd on th mthod of momnts (MoM), also known as th boundary lmnt mthod (BEM). Th formulation of th so calld lctric fild intgral quation (EFIE) and magntic fild intgral quation (MFIE) ar important for discrtization in BEM and ar discussd in Chaptr 5. Th numrical rsults nd to b validatd against th analytical solutions to prov th accuracy and rliability of th modling. W discuss th analytical solutions of a typical ddy currnt problm in chaptr 6. A comparison of th numrical rsults with th analytical solutions is also prsntd. Th simulation rsults using dvlopd ddy currnt (EC) snsors, along with th prformanc analysis of th snsors ar prsntd in Chaptr 7. At th nd, w conclud th scintific work with th final outcom of th ddy currnt simulations togthr with th innovativ idas for futur rsarch. To fulfil th dmands of th INCOSTEEL projct w nd to dtct transvrsal and longitudinal cracks with high prcision, which includs propr idntification of th gomtry as wll as th location of th crack in th hot wir. Fiv typs of ddy currnt snsors ar dvlopd during this projct. A brif dscription of ths snsors is prsntd in Tabl 1.1. Th ncircling coil snsors ar modld by th finit intgration tchniqu. As th gomtry of th modld rgion contains a rotational symmtry, a 3-D ddy currnt problm is rducd to a 2-D problm in cylindrical coordinats. A FIT basd simulation tool is dvlopd to solv such a 2-D problm. Furthrmor, w also prsnt numrical modling of ncircling coil snsors with th finit lmnt mthod. Th point coil snsors, i.., 3-coils snsor, 4-coils snsor, D-coils snsor and pot cor snsor ar modld by th boundary lmnt mthod.

14 1.3. Modling Tools 5 Typ of Snsor Encircling Coil 3-coils Snsor D-coils Snsor 4-coils Snsor Pot cor snsor Charactristics Consists of four ncircling coils Good prformanc for dtcting transvrsal cracks Consists of thr coils Dsignd to dtct th longitudinal cracks mainly Dsignd to dtct th longitudinal cracks Limitd prformanc Modifid vrsion of th 3-coils snsor Dsignd for bttr dtction of th longitudinal cracks Modifid vrsion of th D-coils snsor Good prformanc in dtcting longitudinal cracks 1.3 Modling Tools Tabl 1.1: List of dvlopd ddy currnt snsors. Slctd numrical modling tools to simulat ddy currnt (EC) snsors ar: MQSFIT (Marklin, 2002) basd on Finit Intgration tchniqu (FIT) Opra-2d 10.5 (Opra, 10.5) basd on Finit Elmnt Mthod (FEM) Faraday 6.3 (Faraday, 6.1) basd on Boundary Elmnt Mthod (BEM) MQSFIT Magntoquasistatic Finit Intgration Tchniqu (MQSFIT) is a softwar tool, dvlopd for modling and charactrization of th EC snsors during th INCOSTEEL projct in MQSFIT is a frquncy domain solvr for a monochromatic imprssd currnt xcitation and is programmd in MATLAB. Th basic gomtry is mbddd in th cod, whras th gomtry dtails can b changd with a script fil asily. Th intgral form of th magntoquasistatic quation is discrtizd using th staggrd grid systm in cylindrical coordinats. Th obtaind matrics ar thn rformulatd to gnrat band matrics which ar quit spars and thrfor, ar comprssd to sav mmory and computational tim. Th rsulting linar matrix quation is thn solvd with th conjugat gradint (CG) mthod. Th simulation rsults, obtaind from MQSFIT, ar validatd against analytical as wll as masurd rsults Opra OPERA-2d is a finit lmnt program, dvlopd by Vctor Filds Softwar to analyz lctrical nginring dsigns in two dimnsions. It includs a pr and post-procssor nvironmnt and analysis moduls. Th 2-D vrsion of Opra is usd hr to simulat a rotational

15 6 Chaptr 1. Introduction to Eddy Currnt Snsors and Modling Tchniqus symmtric EC problm. Howvr, Opra-2D consists of a numbr of diffrnt moduls. Th usd moduls to solv th EC problms ar: AC Modul (stady stat) LM modul (linar motion) Th stady stat modul, along with FIT basd MQSFIT, ar usd to obsrv th ffct of cracks in magntic flux dnsity and lctric currnt dnsity distribution in th tst ara. Th linar motion modul is usful to obsrv th ffct of vlocity on ddy currnt distribution on th surfac of th stl wir Faraday 6.1 Faraday is an asy-to-us 3D ddy currnt fild solvr, dvlopd by th Intgratd Enginring Softwar. Using th boundary lmnt mthod (BEM) Faraday is suitabl for applications rquiring larg opn rgion analysis, xact modling of boundaris and problms whr daling with small skin dpths ar critical. Th paramtric solvrs of this tool allow automatically vary and xprimnt with gomtry, matrials and sourcs and thus rduc th tdious, rptitiv task of fin-tuning multipl dsign paramtrs. Th obtaind simulation rsults ar savd in txt format and a sparat MATLAB program is dvlopd to rad and plot th rsults Othr Tools Intrprtation of th simulation rsults plays an important rol to undrstand and analyz th prformanc of th snsors, as wll as th ffct of diffrnt paramtrs in dtction of cracks. Th bst way to do that is to dvlop a movi which shows th A-scan of th snsor togthr with th complt tst procdur. Th commrcial simulation tools, howvr, do not prpar such a movi. A movi making tool is dvlopd using MATLAB which shows th A-scan obtaind from th diffrntial signal on th top and th ddy currnt tst procdur along with th inducd currnt dnsity on th hot wir at th bottom. Snapshots of such movis ar includd in Chaptr 7 to prsnt th simulation rsults.

16 Chaptr 2 Govrning Equations for Magntoquasistatic Problm 2.1 Maxwll s Equations for Macroscopic Non-moving Mdium Maxwll s quations ar a st of quations that govrn all macroscopic lctromagntic phnomna. Th quations can b writtn in both diffrntial and intgral form. For gnral tim-varying filds and in cas of a non-moving mdium, Maxwll s quations in intgral form ar givn by Langnbrg (2003) and Rothwll & Cloud (2001): E(R, t) dr = t B(R, t) ds J m (R, t) ds (2.1) C= S C= S S= V S= V with th fild quantitis and th sourc quantitis H(R, t) dr = S V S t D(R, t) ds + S S J (R, t) ds (2.2) D(R, t) ds = ϱ dv (2.3) B(R, t) ds = ϱ m dv, (2.4) V E = lctric fild strngth (V/m) D = lctric flux dnsity (As/m 2 ) H = magntic fild strngth (A/m) B = magntic flux dnsity (Vs/m 2 ) J = lctric currnt dnsity (A/m 2 ) J m = magntic currnt dnsity (V/m 2 ) ϱ = lctric volum charg dnsity (As/m 3 ) ϱ m = magntic volum charg dnsity (Vs/m 3 ) 7

17 8 Chaptr 2. Govrning Equations for Magntoquasistatic Problm Maxwll s quations in diffrntial form can b drivd from Eqs. (2.1)-(2.4) by using Gauss and Stok s thorms. Lt us considr a point in spac whr all th fild quantitis and thir drivativs ar continuous. If w apply Gauss s and Stok s thorms to Eqs. (2.1)-(2.4), w obtain E(R, t) = t B(R, t) J m(r, t) (2.5) H(R, t) = t D(R, t) + J (R, t) (2.6) B(R, t) = ϱ m (R, t) (2.7) D(R, t) = ϱ (R, t). (2.8) Hr E, H, B and D stand for lctric fild strngth, magntic fild strngth, magntic fild and lctric flux dnsity rspctivly. Th continuity quation is givn by J (R, t) = t ϱ (R, t). (2.9) Hr ϱ and J stand for lctric charg dnsity and lctric currnt dnsity rspctivly. Th constitutiv quations for vacuum ar: B(R, t) = µ 0 H(R, t) (2.10) D(R, t) = ε 0 E(R, t), (2.11) whr µ 0 and ε 0 rprsnt prmability and prmittivity of fr spac. For an inhomognous, linar, isotropic and instantanously racting mdium having th rlativ prmability µ r and rlativ prmittivity ε r w rwrit Eq. (2.10)-(2.11) as B(R, t) = µ(r) H(R, t) = µ 0 µ r (R) H(R, t) (2.12) D(R, t) = ε(r) E(R, t) = ε 0 ε r (R) E(R, t). (2.13) Howvr, th lctric currnt dnsity J (R, t) on th right hand sid of Eq. (2.6) has two parts: J (R, t) = J con (R, t) + J imp (R, t). (2.14) Hr J con and J imp rprsnt th lctric currnt dnsity of th conductor and th xtrnal currnt sourc rspctivly. Using Ohm s law of fild quantitis th conducting part, i.. th lctric currnt dnsity of th conductor J con is associatd with th lctric fild strngth E(R, t) by J con (R, t) = σ (R) E(R, t), (2.15) whr σ (R) stands for lctric conductivity of an inhomognous mdium. Insrting Eq. (2.15) into Eq. (2.14) yilds J (R, t) = σ (R) E(R, t) + J imp (R, t). (2.16) With th hlp of this xprssion of lctric currnt dnsity J (R, t) w rwrit Eq. (2.6) as H(R, t) = t D(R, t) + σ (R) E(R, t) + J imp (R, t). (2.17)

18 2.2. Transition and Boundary Conditions Transition and Boundary Conditions Th lctric and magntic fild componnts satisfy th transition conditions at th intrfac btwn two mdia (s Fig. 2.1a), whras th boundary conditions nd to b fulfilld at th boundary of a prfct lctrically conducting (PEC) matrial (s Fig. 2.1b). a) Transition Condition b) Mdium (2) Mdium (1) n PEC Boundary Condition S Intrfac S Boundary Mdium PEC σ (R) n Figur 2.1: a) Transition intrfac btwn two matrials and b) boundary surfac Transition Conditions At th transition surfac, th jump at th normal componnt of th lctric flux dnsity D(R, t) is dfind by th surfac lctric charg dnsity η (R, t) (Langnbrg, 2003; EFT II, 2009). Using Eq. (2.8) w show that n ] D (2) (R, t) D (1) (R, t) = { η (R, t) with sourc 0 sourc-fr ; R S Intrfac. (2.18) Hr n rprsnts th normal vctor on th intrfac dfind in Fig. 2.1a. Th tangntial componnt of th lctric fild strngth at th intrfac follows { ] Km (R, t) with sourc n E (2) (R, t) E (1) (R, t) = (2.19) 0 sourc-fr. A jump in th tangntial componnt of th lctric fild strngth rprsnts magntic currnt dnsity K m (R, t) at th transition surfac. Again, for th magntic flux dnsity w dfin th jump in th normal componnt by magntic charg dnsity η m (R, t) at th transition surfac. { ] ηm (R, t) with sourc n B (2) (R, t) B (1) (R, t) = (2.20) 0 sourc-fr. Th tangntial componnt of th magntic fild strngth jumps at th transition surfac causs a surfac lctric currnt dnsity K (R, t) { ] K (R, t) with sourc n H (2) (R, t) H (1) (R, t) = (2.21) 0 sourc-fr. Now, using th constitutiv quations dscribd in Eq. (2.12)-(2.13) w formulat quations Eq. (2.18)-(2.21) as { ] η (R, t) with sourc n ε (2) E (2) (R, t) ε (1) E (1) (R, t) = (2.22) 0 sourc-fr

19 10 Chaptr 2. Govrning Equations for Magntoquasistatic Problm n 1 ε (2) D(2) (R, t) 1 ] { Km (R, t) with sourc ε (1) D(1) (R, t) = 0 sourc-fr (2.23) n n { ] ηm (R, t) with sourc µ (2) B (2) (R, t) µ (1) B (1) (R, t) = 0 sourc-fr 1 µ (2) B(2) (R, t) 1 ] { K (R, t) with sourc µ (1) B(1) (R, t) = 0 sourc-fr. (2.24) (2.25) Boundary Conditions For a prfctly lctrically conducting (PEC) boundary, th lctric conductivity insid th PEC matrial is xprssd by σ (EFT I, 2009). In Fig. 2.1b mdium (2) is a PEC matrial, whr th surfac S Boundary dnots th boundary. As a rsult, th tangntial componnt of th lctric fild strngth and th normal componnt of magntic flux dnsity at th boundary bcoms zro, i.., n E(R, t) = 0 (2.26) n B(R, t) = 0. (2.27) Furthrmor, th normal componnt of th lctric flux dnsity D(R, t) is xprssd by th lctric charg dnsity η (R, t) on th boundary surfac and thrfor, w obtain and using th constitutiv quation w writ n D(R, t) = η (R, t) (2.28) n E(R, t) = 1 ε η (R, t). (2.29) Th tangntial componnt of th magntic fild strngth H(R, t) is dtrmind by th surfac lctric currnt dnsity K (R, t) at th boundary surfac S Boundary, which lads to and is writtn using constitutiv quation as n H(R, t) = K (R, t) (2.30) n B(R, t) = µ K (R, t). (2.31) 2.3 Fild Equations in th Frquncy Domain Th ddy currnt problm can b solvd in th tim domain as wll as in th frquncy domain. Th tim domain tchniqu is oftn usd in cas of pulsd ddy currnt systm. In cas of xcitation puls of a singl frquncy th static solution of th ddy currnt problm can asily b obtaind in th frquncy domain. Thrfor, it is ncssary to transform th

20 2.4. Inhomognous Hlmholtz Equations 11 govrning quations of th prvious sction into frquncy domain using Fourir transformation. For a function f(t) w dfin th fourir transformd F (ω) as F (ω) = f(t) j ωt dt, (2.32) whr ω rprsnts th angular frquncy. Again, for a function F (ω) th invrs Fourir transformation is dfind by f(t) = 1 2π F (ω) j ωt dω. (2.33) According to th dfinition of th Fourir intgrals in Eq. (2.32) w obtain th following Fourir transforms: t j ω E(R, t) E(R, ω) B(R, t) B(R, ω) A m (R, t) A m (R, ω) J imp (R, t) J imp (R, ω). Using ths idntitis Maxwll s quations dscribd by Eq. (2.5)-(2.8) ar xprssd in frquncy domain by E(R, ω) = j ωb(r, ω) J m (R, ω) (2.34) H(R, ω) = j ωd(r, ω) + J (R, ω) = j ωd(r, ω) + σ (R) E(R, ω) + J imp (R, ω) (2.35) B(R, ω) = ϱ m (R, ω) (2.36) D(R, ω) = ϱ (R, ω) (2.37) and for a mdium having th rlativ prmability µ r and rlativ prmittivity ε r th constitutiv quations in frquncy domain ar formulatd as B(R, ω) = µ(r) H(R, ω) = µ 0 µ r (R) H(R, ω) (2.38) D(R, ω) = ε(r) E(R, ω) = ε 0 ε r (R) E(R, ω). (2.39) 2.4 Inhomognous Hlmholtz Equations Th inhomognous Hlmholtz quations dscrib th wav propagation in th frquncy domain for th lctric fild strngth E(R, ω) and th magntic fild strngth H(R, ω).

21 12 Chaptr 2. Govrning Equations for Magntoquasistatic Problm Inhomognous Hlmholtz Equation for th Elctric Fild Maxwll s first quation in diffrntial form in th frquncy domain stats Taking th curl of this quation rsults E(R, ω) = j ωb(r, ω) J m (R, ω). (2.40) E(R, ω) = j ω B(R, ω) J m (R, ω), (2.41) which is rformulatd using th constitutiv quation Eq. (2.38) as E(R, ω) = j ω µ(r) H(R, ω)] J m (R, ω). (2.42) Simplifying Eq. (2.42) for a mdium of homognous µ yilds E(R, ω) = j ωµ H(R, ω) J m (R, ω), (2.43) Using Maxwll s scond quation Eq. (2.35) w obtain E(R, ω) = j ωµ j ωd(r, ω) + σ (R) E(R, ω) + J imp (R, ω) ] J m (R, ω) = ω 2 µ D(R, ω) + j ωµσ (R) E(R, ω) + j ωµ J imp (R, ω) J m (R, ω). (2.44) Th oprator on th lft sid of Eq. (2.44) is xprssd by E(R, ω) = E(R, ω) E(R, ω). (2.45) For a mdium of homognous lctric prmittivity, ε(r) = ε and thrfor, Using th constitutiv quation Eq. (2.39) w writ E(R, ω) = 1 D(R, ω) E(R, ω), (2.46) ε which is rformulatd using Maxwll s fourth quation Eq. (2.37) as E(R, ω) = 1 ε ϱ (R, ω) E(R, ω), (2.47) Insrting Eq. (2.47) in Eq. (2.44) and rplacing D(R, ω) with εe(r, ω) rsults E(R, ω) + ω 2 µε E(R, ω) + j ωµσ (R) E(R, ω) = j ωµ J imp (R, ω) + 1 ε ϱ (R, ω) + J m (R, ω) (2.48) and for a conductiv mdium of homognous lctric conductivity, w writ σ (R) = σ and thus w obtain E(R, ω) + ω 2 µε E(R, ω) + j ωµσ E(R, ω) = j ωµ J imp (R, ω) + 1 ε ϱ (R, ω) + J m (R, ω) E(R, ω) + k 2 c E(R, ω) = j ωµ J imp (R, ω) + 1 ε ϱ (R, ω) + J m (R, ω), (2.49) whr k c is th complx wav numbr and is xprssd by k c = ω 2 µε + j ωµσ. (2.50)

22 2.4. Inhomognous Hlmholtz Equations Inhomognous Hlmholtz Equation for th Magntic Fild W rcall Maxwll s scond quation in diffrntial form in frquncy domain H(R, ω) = j ωd(r, ω) + σ (R) E(R, ω) + J imp (R, ω). (2.51) Lt us introduc a curl oprator ( ) on both sids H(R, ω) = j ωd(r, ω) + σ (R) E(R, ω) + J imp (R, ω) ]. (2.52) Using th constitutiv quation Eq. (2.39) for a mdium of homognous lctric prmittivity w obtain H(R, ω) = j ωε E(R, ω) + σ (R) E(R, ω)] + J imp (R, ω). Rarranging Eq. (2.53) rsults H(R, ω) (2.53) = j ωε + σ (R)] E(R, ω) E(R, ω) σ (R) + J imp (R, ω). (2.54) Th oprator of Eq. (2.54) is givn by H(R, ω) = H(R, ω) H(R, ω). (2.55) Using th constitutiv quation Eq. (2.38) w show that H(R, ω) = 1 B(R, ω) H(R, ω). (2.56) µ(r) For a mdium of homognous magntic prmability, it is assumd that µ(r) = µ, which lads to H(R, ω) = 1 B(R, ω) H(R, ω). (2.57) µ Using Maxwll s third quation Eq. (2.36) in Eq. (2.57) rsults Insrting Eq. (2.58) in Eq. (2.53) yilds 1 µ ϱ m(r, ω) H(R, ω) H(R, ω) = 1 µ ϱ m(r, ω) H(R, ω). (2.58) = j ωε + σ (R)] E(R, ω) E(R, ω) σ (R) + J imp (R, ω). (2.59) Rplacing E(R, ω) according to th Maxwll s first quation Eq. (2.34) w obtain 1 µ ϱ m(r, ω) H(R, ω) = j ωε + σ (R)] j ωb(r, ω) J m (R, ω)] E(R, ω) σ (R) + J imp (R, ω) (2.60)

23 14 Chaptr 2. Govrning Equations for Magntoquasistatic Problm and w rarrang by using th constitutiv quation Eq. (2.38) to obtain H(R, ω) + ω 2 εµ H(R, ω) + j ωσ (R)µ H(R, ω) E(R, ω) σ (R) = j ωεj m (R, ω) + σ (R)J m (R, ω) + 1 µ ϱ m(r, ω) J imp (R, ω). (2.61) For a mdium of homognous lctric conductivity σ w nglct th trm E(R, ω) σ (R), and hnc, Eq. (2.61) is simplifid to H(R, ω) + ω 2 µε H(R, ω) + j ωµσ H(R, ω) = j ωεj m (R, ω) + σ J m (R, ω) + 1 µ ϱ m(r, ω) J imp (R, ω), (2.62) which can b formulatd as H(R, ω) + k 2 c H(R, ω) = j ωεj m (R, ω) + σ J m (R, ω) + 1 µ ϱ m(r, ω) J imp (R, ω), (2.63) whr k c = ω 2 µε + j ωµσ. (2.64) 2.5 Elctromagntic Potntials of an Elctrically Conducting Matrial Firstly, w driv th potntial rprsntation for th lctric and magntic fild strngth using lctric and magntic scalar and vctor potntials for th sourc-fr cas. Using ths xprssions w driv th th potntial rprsntation with sourc in th scond part Elctromagntic Potntials in Sourc-Fr Cas Th Maxwll s quations in frquncy domain, dscribd by Eq. (2.34)-(2.37) ar writtn in sourc-fr cas as E(R, ω) = j ωb(r, ω) (2.65) H(R, ω) = j ωd(r, ω) + σ E(R, ω) (2.66) D(R, ω) = 0 (2.67) B(R, ω) = 0. (2.68) Drivation of th Potntial Rprsntation using Magntic Vctor Potntial and Elctric Scalar Potntial W know from th vctor algbra that th curl of divrgnc producs also zro, i., for a vctor A m (R, ω) w obtain A m (R, ω)] = 0. (2.69)

24 2.5. Elctromagntic Potntials of an Elctrically Conducting Matrial 15 From th quations Eq. (2.68) and Eq. (2.69) th magntic flux dnsity B(R, ω) is givn by B(R, ω) = A m (R, ω), (2.70) whr A m (R, ω) is calld magntic vctor potntial, bcaus it is rlatd to th magntic flux dnsity B(R, ω) through diffrntiation. Th first Maxwll s quation (Eq. (2.65)), also known as Faraday s law of induction in sourc-fr cas, is writtn using magntic vctor potntial A m (R, ω) as E(R, ω) = j ω A m (R, ω) E(R, ω) j ωa m (R, ω)] }{{} = Φ (R, ω) = 0, (2.71) which lads to th lctric scalar potntial Φ (R, ω) bcaus of th vctor idntity Rwriting Eq. (2.71) rsults Φ (R, ω)] = 0. (2.72) E(R, ω) j ωa m (R, ω) = Φ (R, ω), (2.73) whr Φ (R, ω) is calld lctric scalar potntial. Th ngativ sign on th right-hand sid is introducd bcaus of th historical convntion (Langnbrg, 2003) and it dfins th sign of th lctrical voltag in static cas. Rarranging Eq. (2.73) yilds E(R, ω) = Φ (R, ω) + j ωa m (R, ω). (2.74) Drivation of th Potntial Rprsntation using Magntic Scalar Potntial and Elctric Vctor Potntial For a vctor A (R, ω) w know from vctor idntity that A (R, ω)] = 0. (2.75) From th quations Eq. (2.68) and Eq. (2.75) th lctric flux dnsity D(R, ω) is dfind by D(R, ω) = A (R, ω), (2.76) whr A (R, ω) is calld lctric vctor potntial and it is rlatd to th lctric flux dnsity D(R, ω) through diffrntiation. Th ngativ sign on th right-hand sid is introducd according to th historical convntion. Maxwll s scond quation (Eq. (2.66)), also known as Ampèr-Maxwll circuital law of induction is rformulatd in sourc-fr cas using th constitutiv quation Eq. (2.39) as H(R, ω) = j ωd(r, ω) + σ D(R, ω). (2.77) ε Using lctric vctor potntial A (R, ω) w obtain H(R, ω) = j ω A (R, ω)] + σ ε A (R, ω)] H(R, ω) = j ω A σ ε A (R, ω) H(R, ω) j ωa (R, ω) + σ ] ε A (R, ω) = 0. (2.78)

25 16 Chaptr 2. Govrning Equations for Magntoquasistatic Problm For a scalar potntial Φ m (R, ω) Comparing Eq. (2.79) to Eq. (2.78) lads to Φ m (R, ω)] = 0. (2.79) H(R, ω) j ωa (R, ω) + σ ε A (R, ω) = Φ m (R, ω), (2.80) whr Φ m (R, ω) is calld magntic scalar potntial and th ngativ sign on th right-hand sid is introducd bcaus of th historical convntion. Rarranging Eq. (2.80) rsults H(R, ω) = Φ m (R, ω) + j ωa (R, ω) σ ε A (R, ω). (2.81) Elctromagntic Potntials in Cas with Elctric and Magntic Sourcs Th lctric and magntic filds in trms of th potntials in sourc-fr cas ar dscribd by quations (2.70), (2.74), (2.76) and (2.81). In cas of lctric and magntic sourcs for an lctrically conducting mdia using suprposition thorm w rformulat ths quations as E(R, ω) = Φ (R, ω) + j ωa m (R, ω) 1 ε A (R, ω) (2.82) H(R, ω) = Φ m (R, ω) + j ωa (R, ω) σ ε A (R, ω) + 1 µ A m(r, ω) (2.83) B(R, ω) = µ Φ m (R, ω) + j ωµa (R, ω) µσ ε A (R, ω) + A m (R, ω) (2.84) D(R, ω) = ε Φ (R, ω) + j ωεa m (R, ω) A (R, ω). (2.85) 2.6 Dcoupling of Maxwll s Equations by Elctromagntic Potntials Maxwll s quations rprsnt a coupld systm whr th lctric and magntic filds ar physically coupld through induction and magntomotiv forc. Mathmatically w obsrv th coupling of th filds through divrgnc and curl oprators. Howvr, th lctric and magntic filds can b dcoupld using magntic and lctric scalar and vctor potntials Exprssion of th Magntic Vctor Potntial in th Frquncy Domain W start with th Ampèr-Maxwll circuital law in frquncy domain and rcall Eq. (2.35) H(R, ω) = ( j ωε + σ ) E(R, ω) + J imp (R, ω). (2.86) Th magntic fild strngth H(R, ω) and lctric fild strngth E(R, ω) ar rprsntd by th potntials according to Eqs. (2.82)-(2.83) and thrfor, w rwrit Eq. (2.86) as Φ m (R, ω) + j ωa (R, ω) σ ε A (R, ω) + 1 ] µ A m(r, ω) = ( j ωε + σ ) Φ + j ωa m 1 ] ε A + J imp (R, ω). (2.87)

26 2.6. Dcoupling of Maxwll s Equations by Elctromagntic Potntials 17 Rarranging Eq. (2.87) yilds µ Φ m (R, ω) + j ωµ A }{{} (R, ω) µσ ε A (R, ω) + A m (R, ω) = 0 = j ωµε Φ (R, ω) + ω 2 µε A m (R, ω) + j ωµ A (R, ω) µσ Φ (R, ω) + j ωµσ A m (R, ω) µσ ε A (R, ω) + µ J imp (R, ω). (2.88) Cancling th common parts from both sids w obtain A m (R, ω) = j ωµε Φ (R, ω) + ω 2 µε A m (R, ω) µσ Φ (R, ω) + j ωµσ A m (R, ω) + µ J imp (R, ω). (2.89) Th oprator can b rplacd by, which lads to A m (R, ω) + ω 2 µε A m (R, ω) + j ωµσ A m (R, ω) = µj imp (R, ω) + A m (R, ω) + j ωµε Φ (R, ω) µσ Φ (R, ω) A m (R, ω) + k 2 c A m (R, ω) With th Lornz gaug w obtain from Eq. (2.90) = µj imp (R, ω) + A m (R, ω) j ωµεφ (R, ω) + µσ Φ (R, ω)]. (2.90) A m (R, ω) j ωµεφ (R, ω) + µσ Φ (R, ω) = 0 (2.91) A m (R, ω) + k 2 c A m (R, ω) = µj imp (R, ω). (2.92) Exprssion of th Elctric Scalar Potntial in th Frquncy Domain Lt us rcall Maxwll s fourth quation in frquncy domain D(R, ω) = ϱ (R, ω). (2.93) Using th xprssion of th lctric flux dnsity D(R, ω) by th potntials according to Eq. (2.85) rsults which is rarrangd as ε Φ (R, ω) + j ωεa m (R, ω) A (R, ω)] = ϱ (R, ω), (2.94) ε Φ (R, ω) + j ωε A m (R, ω) A (R, ω) = ϱ }{{} (R, ω) = 0 Φ (R, ω) = 1 ε ϱ (R, ω) + j ω A m (R, ω). (2.95)

27 18 Chaptr 2. Govrning Equations for Magntoquasistatic Problm Now w add on both sids of Eq. (2.95) th trm (ω 2 µε + j ωµσ ) Φ (R, ω) and thus w obtain th PDE for Φ (R, ω) Φ (R, ω) + ( ω 2 εµ + j ωµσ ) Φ (R, ω) = 1 ε ϱ (R, ω) + j ω A m (R, ω) + ω 2 µεφ (R, ω) + j ωµσ Φ (R, ω) Φ (R, ω) + k 2 c Φ (R, ω) = 1 ε ϱ (R, ω) + j ω A m (R, ω) j ωµεφ (R, ω) + j ωµσ Φ (R, ω)], (2.96) which is comparabl to th PDE for A m (R, ω) in Eq. (2.89). (Eq. (2.91)) w simplify Eq. (2.96) to With th Lornz gaug Φ (R, ω) + k 2 c Φ (R, ω) = 1 ε ϱ (R, ω). (2.97) Exprssion of th Elctric Vctor Potntial in th Frquncy Domain Faraday s law of induction in frquncy domain stats E(R, ω) = j ωb(r, ω) J m (R, ω). (2.98) Insrting th rprsntation of magntic fild strngth E(R, ω) and magntic flux dnsity strngth B(R, ω) by th potntials according to Eq. (2.82) and Eq. (2.84) w find Φ (R, ω) + j ωa m (R, ω) 1 ] ε A (R, ω) = j ω µ Φ m (R, ω) + j ωµa (R, ω) µσ ] ε A (R, ω) + A m (R, ω) J m (R, ω). Rarranging Eq. (2.99) rsults Φ (R, ω) + j ω A }{{} m (R, ω) 1 ε A (R, ω) = 0 = j ωµ Φ m (R, ω) ω 2 µ A (R, ω) j ω µσ ε A (R, ω) + j ω A m (R, ω) (2.99) J m (R, ω). (2.100) W cancl th common parts from both sids and rwrit Eq. (2.100) as A (R, ω) + ω 2 µε A (R, ω) + j ωµσ A (R, ω) = εj m (R, ω) + A (R, ω) j ωµε Φ m (R, ω) A (R, ω) + k 2 c A (R, ω) = εj m (R, ω) + A (R, ω) j ωµεφ m (R, ω)]. (2.101)

28 2.6. Dcoupling of Maxwll s Equations by Elctromagntic Potntials 19 With th Lornz gaug w obtain from Eq. (2.101) A (R, ω) j ωµεφ m (R, ω) = 0 (2.102) A (R, ω) + k 2 c A (R, ω) = εj m (R, ω). (2.103) Exprssion of th Magntic Scalar Potntial in th Frquncy Domain W rcall Maxwll s third quation in frquncy domain Eq. (2.36) B(R, ω) = ϱ m (R, ω). (2.104) Using th xprssion lctric flux dnsity D(R, ω) by th potntials according to Eq. (2.85) w obtain µ Φ m (R, ω) + j ωµa (R, ω) µσ ] ε A (R, ω) + A m (R, ω) = ϱ m (R, ω), which is rarrangd as (2.105) µ Φ m (R, ω) + j ωµ A (R, ω) µσ ε A (R, ω) A m (R, ω) }{{} = 0 = ϱ m (R, ω) Φ m (R, ω) = 1 µ ϱ m(r, ω) + j ω A (R, ω) σ ε A (R, ω). (2.106) Adding on both sids of Eq. (2.106) th trm (ω 2 µε + j ωµσ ) Φ m (R, ω) rsults Φ m (R, ω) + ( ω 2 εµ + j ωµσ ) Φm (R, ω) = 1 µ ϱ m(r, ω) + j ω A (R, ω) σ ε A (R, ω) + ω 2 µεφ m (R, ω) + j ωµσ Φ m (R, ω). (2.107) Following Lornz gaug Eq. (2.102) w show that Insrting Eq. (2.108) into Eq. (2.107) yilds Φ m (R, ω) + ( ω 2 εµ + j ωµσ ) Φm (R, ω) A (R, ω) j ωµεφ m (R, ω) = 0 A (R, ω) = j ωµεφ m (R, ω). (2.108) = 1 µ ϱ m(r, ω) + j ω A (R, ω) j ωµσ Φ m (R, ω) + ω 2 µεφ m (R, ω) + j ωµσ Φ m (R, ω) Φ m (R, ω) + k 2 c Φ m (R, ω) = 1 µ ϱ m(r, ω) + j ω A (R, ω) j ωµεφ m (R, ω)]. (2.109)

29 20 Chaptr 2. Govrning Equations for Magntoquasistatic Problm With th Lornz gaug Eq. (2.102) w obtain Φ m (R, ω) + k 2 c Φ m (R, ω) = 1 µ ϱ m(r, ω). (2.110) Summary of Elctromagntic Potntials in th Frquncy Domain Th final xprssions for th lctric and magntic vctor and scalar potntials ar listd from Eq. (2.92), Eq. (2.97), Eq. (2.103) and Eq. (2.110) as A m (R, ω) + k 2 c A m (R, ω) = µj imp (R, ω) (2.111) Φ (R, ω) + k 2 c Φ (R, ω) = 1 ε ϱ (R, ω) (2.112) A (R, ω) + k 2 c A (R, ω) = εj m (R, ω) (2.113) Φ m (R, ω) + k 2 c Φ m (R, ω) = 1 µ ϱ m(r, ω). (2.114) 2.7 Magntoquasistatic Fild Th magntoquasistatic (MQS) fild is dfind by using a simpl rul of thumb: Lowr th tim rat of chang (frquncy) of th driving sourc so th filds bcom static (Haus & Mlchr, 1989). If th lctric fild vanishs in this procss, th fild is MQS. As w hav discussd arlir, th complx wav numbr is givn by For a mdium of larg lctric conductivity σ w assum which rsults k c = ω 2 µε + j ωµσ. (2.115) ωµσ ω 2 µε σ ωε, (2.116) k c j ωµσ. (2.117) Howvr, th ral part of th wav numbr is originatd from th diffusion currnt of th Ampèr-Maxwll circuital law and hnc, th diffusion part on th right hand sid of Eq. (2.35) is approximatd as zro, i.. and thus w obtain from Eq. (2.35) j ωd(r, ω) 0 (2.118) H(R, ω) = σ (R) E(R, ω) + J imp (R, ω) ] 1 B(R, ω) µ(r) = σ (R) E(R, ω) + J imp (R, ω). (2.119) Using th magntoquasistatic approximation th inhomognous Hlmholtz quation for lctric fild strngth is writtn using Eq. (2.49) as E(R, ω) + j ωµσ E(R, ω) = j ωµ J imp (R, ω) + 1 ε ϱ (R, ω) + J m (R, ω) E(R, ω) + k 2 c E(R, ω) = j ωµ J imp (R, ω) + 1 ε ϱ (R, ω) + J m (R, ω) (2.120)

30 2.8. Magntoquasistatic Fild Equations with Magntic Vctor Potntial 21 and th xprssion for magntic fild strngth is obtaind from Eq. (2.63) by nglcting th trm j ωεj m (R, ω) according to Eq. (2.116) as H(R, ω) + j ωµσ H(R, ω) = σ J m (R, ω) + 1 µ ϱ m(r, ω) J imp (R, ω) whr H(R, ω) + k 2 c H(R, ω) = σ J m (R, ω) + 1 µ ϱ m(r, ω) J imp (R, ω), (2.121) k c = j ωµσ = j ωµσ. (2.122) For th squar-root of th imaginary unit j w comput j = j π 2 = ( ) j π = j π 4 It follows for th complx wav numbr ( ) 1 1 ωµσ k c = 2 + j 2 = cos π 4 + j sin π 4 = j. (2.123) 2 2 ωµσ = 2 ωµσ + j 2 = R{k c } + j I{k c }. (2.124) Th ral part R{k c } rprsnts wav propagation and th imaginary part I{k c } stands for th attnuation of th lctromagntic wavs in conductiv mdium, whras Eq. (2.124) stats that in magntoquasistatic cas th ral part and imaginary part of th complx wav numbr k c ar th sam. Again, th lctric and magntic fild strngth ar rprsntd using lctric and magntic scalar and vctor potntials ar dscribd by Eq. (2.82) and Eq. (2.83). In magntoquasistatic cas, using th approximation Eq. (2.118) w nglct th trm j ωa (R, ω) of Eq. (2.83) and thus w obtain E(R, ω) = Φ (R, ω) + j ωa m (R, ω) 1 ε A (R, ω) (2.125) H(R, ω) = Φ m (R, ω) σ ε A (R, ω) + 1 µ A m(r, ω). (2.126) 2.8 Magntoquasistatic Fild Equations with Magntic Vctor Potntial Th magntic vctor potntial A m (R, ω) can b rlatd to th lctric fild strngth E(R, ω) in frquncy domain using Eq. (2.125). Nglcting th magntic sourcs w find E(R, ω) = j ωa m (R, ω) Φ (R, ω), (2.127) which shows th rlationship btwn th lctric fild strngth E(R, ω) with th magntic vctor potntial A m (R, ω). In non-moving conductor cas th lctric scalar potntial trm of Eq. (2.127) can b approximatd as zro and hnc E(R, ω) = j ωa m (R, ω). (2.128)

31 22 Chaptr 2. Govrning Equations for Magntoquasistatic Problm Using magntic vctor potntial A m (R, ω) w rwrit Eq. (2.119) as ν(r) A m (R, ω)] j ωσ (R) A m (R, ω) = J imp (R, ω), (2.129) whr ν(r) = 1/µ(R). Eq. (2.129) rprsnts th magntic diffusion quation in frquncy domain with magntic vctor potntial for a mdium of inhomognous prmability µ(r) and lctric conductivity σ (R), whras Eq. (2.111) dscribs magntic vctor potntial for a mdium of homognous µ(r) and σ (R). Rwriting Eq. (2.129) in intgral form rsults S ν(r) A m (R, ω)] ds S j ωσ (R) A m (R, ω) ds = S J imp (R, ω) ds. (2.130) Lt us apply th Stok s Thorm on th doubl Curl trm of Eq. (2.130) and thus w obtain C= S ν(r) A m (R, ω)] dr S j ωσ (R) A m (R, ω) ds = S J imp (R, ω) ds, (2.131) which dscribs th magntoquasistatic problm in th frquncy domain with magntic vctor potntial. Eq. (2.131) is considrd as th govrning quation for th solution of th ddy currnt problm by th finit intgration tchniqu (FIT) in chaptr 3 and by th finit lmnt mthod (FEM) in chaptr 4, whr w dtrmin th unknown valus of th magntic vctor potntial A m (R, ω) to solv th magntoquasistatic problm. 2.9 Transition and Boundary Conditions for th Magntic Vctor Potntial Th transition and boundary conditions hav to b considrd whil discrtizing th govrning quation Eq. (2.131) by FIT or FEM for unknown valus of A m (R, ω). For th magntic vctor potntial th Coulomb gaug stats that A m (R, t) = 0, (2.132) which is writtn in intgral form as S= V A m (R, t) ds = 0. (2.133) W solv th intgral for an lmntary surfac S and thus w obtain ] A (2) m (R, t) A (1) m (R, t) n S = 0 ] n A (2) m (R, t) A (1) m (R, t) = 0 (2.134) and in th frquncy domain: n ] A (2) m (R, ω) A (1) m (R, ω) = 0, (2.135)

32 2.10. Fild Equations for a Moving Conductor 23 which mans th normal componnt of th magntic vctor potntial A m (R, t) is continuous at th transition intrfac. Again, for a sourc fr transition intrfac, rwriting Eq. (2.19) in th frquncy domain rsults n Insrting Eq. (2.128) in Eq. (2.136) yilds ] E (2) (R, ω) E (1) (R, ω) = 0. (2.136) { ]} n j ω A (2) m (R, ω) A (1) m (R, ω) = 0 ] n A (2) m (R, ω) A (1) m (R, ω) = 0, (2.137) which stats that th tangntial componnt of th magntic vctor potntial A m (R, ω) is continuous at th transition intrfac. To dfin th PEC boundary condition for th magntic vctor potntial w st A (1) m (R, ω) = 0 in Eq. (2.143) which rsults n and from Eq. (2.137) w obtain ] A (2) m (R, ω) = 0 i.., n A m (R, ω) = 0 (2.138) n A m (R, t) = 0. (2.139) W sum up from Eq. (2.138) and Eq. (2.139) that th tangntial and normal componnts of th magntic vctor potntial diminish at th PEC boundary. Th transition and boundary conditions of th magntic vctor potntial will b discussd furthr in chaptr 3 and 4 for FIT and FEM, rspctivly Fild Equations for a Moving Conductor Maxwll s quations in thir diffrntial and intgral form ar valid for any stationary mdia. Howvr, Maxwll s third and fourth quation ar not affctd by th motion of th mdium (Poljak & Brbbia, 2005). On th othr hand, th xtnsion of th law of induction, xprssd by Maxwll s first quation for a moving mdium, rquirs considrabl car. Th govrning quations for th modling of an ddy currnt snsor in a moving conductor ar Maxwll s quations for a moving mdium, whr th first on rad (Rothwll & Cloud, 2001) E(R, t) = t B(R, t) J m(r, t) + v c (R, t) B(R, t)], (2.140) which is drivd from Maxwll s quations in intgral form by using th Hlmholtz transport thorm (Rothwll & Cloud, 2001; Tai, 1997). Nglcting th magntic currnt dnsity J m (R, t) rsults E(R, t) = t B(R, t) + v c(r, t) B(R, t)]. (2.141)

33 24 Chaptr 2. Govrning Equations for Magntoquasistatic Problm Using magntic vctor potntial A m (R, t) w rwrit Eq. (2.141) as E(R, t) = t A m(r, t)] + v c (R, t) A m (R, t)] E(R, t) + A m(r, t) ] v c (R, t) A m (R, t) t }{{} = Φ (R, t) Using lctric scalar potntial Φ (R, t) w obtain E(R, t) + A m(r, t) t E(R, t) = A m(r, t) t Formulating Eq. (2.143) in frquncy domain yilds v c (R, t) A m (R, t) = Φ (R, t) = 0. (2.142) + v c (R, t) A m (R, t) Φ (R, t). (2.143) E(R, ω) = j ωa m (R, ω) + v c A m (R, ω) Φ (R, ω). (2.144) Using quation Eq. (2.131) w arriv at th magntic diffusion quation for a moving conductor ν(r) A m (R, ω)] σ (R) j ω A m (R, ω) + v c A m (R, ω) Φ (R, ω)] For a mdium of homognous µ Eq. (2.145) is simplifid to = J imp (R, ω). (2.145) A m (R, ω) + µσ (R) j ω A m (R, ω) + v c A m (R, ω) Φ (R, ω)] = µj imp (R, ω). (2.146)

34 Chaptr 3 Finit Intgration Tchniqu (FIT) Historically, th finit intgration tchniqu (FIT) was first applid to th full sts of Maxwll s quations in intgral form (Wiland, 1977). Th mthod uss all six vctor componnts of lctric fild strngth and magntic flux dnsity on a dual grid systm (Marklin, 2002). In our cas, FIT is usd for th dirct discrtisation of th govrning quation of magntoquasistatics. As dscribd ddy currnt snsors ar drivn by a monochromatic xcitation puls, it is asir to solv th magntoquasistaic problm in th frquncy domain than in th tim domain. Th govrning quations in intgral form ar thn discrtizd on a staggrd voxl grid in spac. Th allocation of th vctor componnts satisfis th transition conditions automatically, if th inhomognous matrial is discrtisd (Marklin, 1997, 2002). Th following stps should b followd to solv a magntoquasistaic problm using FIT: Discrtization of lctromagntic mdium in spac (Sc. 3.1) Allocation of th discrt fild quantitis (Sc. 3.2) Stting up th discrt fild quantitis and drivation of th grid quation in cartsian coordinats (Sc ) Drivation of th grid quation in cylindrical coordinats (Sc ) Implmntation of th boundary condition (Sc. 3.7) Formation of a band matrix (Sc. 3.8) 3.1 Discrtization of th Elctromagntic Mdium in Spac Th govrning quation of a magntoquasistatic problm in th frquncy domain can b rcalld from Chaptr 2 as ν(r) A m (R, ω)] dr j ω σ (R) A m (R, ω) ds = J imp (R, ω) ds. C= S S Bfor w start with th discrtization of th govrning quation, w hav to dfin th discrtization of th mdium in matrial clls. Th discrtization of th lctromagntic mdium M m is dfind as follows: 25 S (3.1)

35 26 Chaptr 3. Finit Intgration Tchniqu (FIT) Th matrial clls m (n) of th matrial grid M has a constant matrial filling and coincids with th grid clls g (n) of th grid G, i.. m (n) = g (n). th matrial clls m (n) is addrssd with th grid nod n in th cntr point of th grid cll g (n) of th grid G {µ(r), σ (R)}, R R 3 {µ (n), σ (n) }, n R N, whr all matrial tnsors hav th form µ = µ ii i i. 3.2 Allocation of th Discrt Fild Quantitis in Cartsian Coordinats Th govrning quation, along with th transition and boundary conditions, prscrib th allocation of th discrt lctromagntic fild componnts. Instad of driving th allocation whil discrtizing th govrning quations, w dfin th allocation of th discrt lctromagntic fild componnts as it follows. Th allocation of th discrt fild componnts has to b prformd in that way, that only continuous fild componnt ar allocatd at th boundary m (n) of nxt nairbour matrial clls m (n), in ordr to nsur th transition conditions, which rad for a sourc fr intrfac n B(R, ω) = continous (3.2) and using th Coulomb gaug w writ th transition condition for th magntic vctor potntial A m (R, ω) from Eq. (2.134) as n A m (R, ω) = continous. (3.3) For a sourc-fr intrfac th tangntial componnt of th lctric fild strngth and th normal componnt of th magntic flux dnsity must b continuous. W dfin th allocation of th discrt lctromagntic fild quantitis as follows: th thr Cartsian componnts of th magntic vctor potntial vctor A (n) i ar positiond at th cntr of th dgs of th matrial cll m (n) and ach Cartsian componnt points in positiv coordinat dirction. th thr Cartsian componnts of th magntic flux dnsity vctor B (n) i ar positiond at th cntr of th dgs of th fact with th normal vctor n = i of th matrial cll m (n) and points in positiv coordinat dirction. th thr Cartsian componnts J (n) m,i and H(n) i coincid with th B (n) i componnts. th thr Cartsian componnts J (n),i, and D(n) i componnts. componnts coincid with th A (n) i Fig. 3.1 shows th allocation of th discrt lctromagntic fild componnts assignd to th global grid nod n. This allocation nsurs that only normal componnts of th magntic flux dnsity and only tangntial componnts of th lctric fild strngth ar positiond at th intrfac btwn matrial clls with diffrnt matrial proprtis. Th transition conditions Eqs. (3.2)-(3.3) ar nsurd in a natural way.

36 3.3. Discrt Fild Quantitis in Cartsian Coordinats in 2-D cas Grid Nod (n) B 1 (n) ė B 2. (n) Ạ(n) A (n) B (n) A (n) x (n + M 3 ) x (n + M 2 ).. x Matrial Cll m (n) and Grid Cll g (n) of Grid G (n + M 1 ).... Grid Cll g (n) of Grid G Figur 3.1: Allocation of th discrt magntic fild componnts A (n) 1, A(n) 2, A(n) 3, B(n) assignd to th global grid nod (n) 1, B(n) 2, and B(n) Discrt Fild Quantitis in Cartsian Coordinats in 2-D cas W hav discussd in Chaptr 2 that th ncircling coil snsors can b simulatd using th rotational symmtry and thrfor, a 3-D magntoquasistatic problm can b rducd into a 2-D problm in cylindrical coordinats. As th first stp, th discrt fild quantitis will b discussd in cartsian coordinats in this sction. Fig. 3.1 can b rducd to Fig. 3.2 for 2-D cas, which shows th intgration cll I (n) A y. W dfin th allocation of th discrt fild quantitis as follows: Th y componnt of magntic vctor potntial A y is positiond at th cornrs of th matrial clls, as for xampl, A (n) y is positiond at th common nod of th four matrial clls m (n), m (n+mx), m (n+mz), and m (n+mx+mz) and it points th normal dirction of th xz plan. Th x and z componnts of th magntic flux dnsity vctor B x (n), B z (n), B x (n+mz), and ar positiond at th cntr of th dgs of th fact and ach componnt B (n+mx) z points th tangntial dirctions to th surfac. Th y componnt of lctric currnt dnsity J y (n) is positiond at th cntr of th matrial cll m (n). Similarly, J y (n+mx), J y (n+mz), and J y (n+mx+mz) ar positiond at th cntr of corrsponding matrial clls.

37 28 Chaptr 3. Finit Intgration Tchniqu (FIT) x z m (n) ν (n) σ (n) A (n Mz) y B (n) x ν (n+mx) σ (n+mx) m (n+mx) z A (n Mx) y B (n) z A (n) y B (n+mx) z A (n+mx) y m (n+mz) B (n+mz) x m (n+mx+mz) ν (n+mz) σ (n+mz) ν (n+mx+mz) σ (n+mx+mz) A (n+mz) y x Figur 3.2: Allocation of th discrt lctromagntic fild componnts B x, B z, and A y assignd to th global grid nod (n) in 2-D cas in cartsian coordinats. 3.4 Drivation of th Discrt Grid Equation in Cartsian Coordinats in 2-D cas Th discrt grid quation in cartsian coordinats can b computd by discrtizing th govrning quation Eq. (3.1) using th following stps: 1. Th lin intgral contains a curl oprator which can b computd in th cartsian coordinats as: Az (ω) A m (R, ω) = A ] y(ω) Ax (ω) y z x + A ] z(ω) z x y Ay (ω) + A ] x(ω) x y z. (3.4) As w ar computing in xz plan and ar considring only th y componnt of magntic vctor potntial, Th curl oprator can b rducd to A m (R, ω) = A y(ω) z x + A y(ω) x z. (3.5)

38 3.4. Drivation of th Discrt Grid Equation in Cartsian Coordinats in 2-D cas Furthrmor, th lin intgral can b dividd into four parts, namd as: Up, down, Lft, and Right intgral, which rsults ν(r) A m (R, ω)] dr = ν (u) A y(ω) z x + A ] y(ω) x z x dx C (u) C (u) ν(r) B(R, ω)] dr = ν (u) A ] y(ω) dx. (3.6) z C (u) C (u) C (d) ν(r) A m (R, ω)] dr = C (d) ν(r) B(R, ω)] dr = C (l) ν(r) A m (R, ω)] dr = C (l) ν(r) B(R, ω)] dr = C (r) ν(r) A m (R, ω)] dr = C (r) ν(r) B(R, ω)] dr = C (d) ν (d) C (d) ν (d) C (l) ν (l) C (l) ν (l) C (r) ν (r) C (r) ν (r) A y(ω) z x + A ] y(ω) x z x dx A ] y(ω) dx. (3.7) z A y(ω) z x + A ] y(ω) x z z dz + A ] y(ω) dz. (3.8) x A y(ω) z x + A ] y(ω) x z z dz + A ] y(ω) dz. (3.9) x 3. Th componnt A (n) y is positiond in th cntr of th intgration cll I (n) A y. By using th right-hand rul, th lin intgral can b shown as a sum of ths four intgrals as ν(r) A m ((R, ω)] dr C= S = ν(r) B(R, ω)] dr ν(r) B(R, ω)] dr C (d) C (u) + ν(r) B(R, ω)] dr ν(r) B(R, ω)] dr C (l) C (r) = ν (d) B (d) x (ω) dx ν (u) B (u) x (ω) dx + C (d) C (u) C (l) ν (l) B (l) z (ω) dz 4. Th clls ar considrd to b quadratic and thrfor, z = x. C (r) ν (r) B (r) z (ω) dz. 5. Th othr two intgrals of Eq. (3.1) ar surfac intgrals and can b computd dirctly. (3.10)

39 30 Chaptr 3. Finit Intgration Tchniqu (FIT) 6. Avrag valus of matrial paramtrs ν(r) and σ (R) should b takn whil computing th intgrals. Up: Fig. 3.3a shows th uppr part of th intgration cll I (n) A y. W can writ from Eq. (3.6) that whr ν (n) H x m (n+mx) as C (u) ν (u) B (u) x (ω) dx = C (u) ν(n) + ν(n+mx) 2 = ν(n) + ν (n+mx) 2 ν(n) + ν (n+mx) 2 B x (n) (ω) dx A (n M z) y A (n M z) y (ω) A (n) y z (n) (ω) A (n) y ] (ω) x (n) + O ( x) 3] (ω) ] x z = ν (n) H x A (n M z) y (ω) A (n) y (ω) ], (3.11) will b computd by avraging th magntic imprmability of th clls m (n) and ν (n) H x = ν(n) + ν (n+mx) 2. (3.12) Down: Th lowr part of th intgration cll I (n) A y (Fig. 3.3b) will b considrd hr. a) ν (n) σ (n) A (n Mz) y B (n) x ν (n+mx) σ (n+mx) z A (n) y b) A (n) y B (n+mz) x z ν (n+mz) σ (n+mz) ν (n+mx+mz) σ (n+mx+mz) A (n+mz) y x Figur 3.3: a) Uppr part and b) lowr part of th intgration cll I (n) A y.

40 3.4. Drivation of th Discrt Grid Equation in Cartsian Coordinats in 2-D cas 31 Eq. (3.7) can b r-writtn as C (d) ν (d) B (d) x (ω) dx = C (d) ν(n+mz) + ν(n+mx+mz) 2 = ν(n+mz) + ν (n+mx+mz) 2 ν(n+mz) + ν (n+mx+mz) 2 = ν (n+mz) H x A (n) y A (n) y B x (n+mz) (ω) dx (ω) A (n+mz) y z (n+mz) (ω) A (n+mz) y ] (ω) x (n+mz) + O ( x) 3] (ω) ] x z A (n) y (ω) A (n+mz) y (ω) ], (3.13) whr th magntic imprmability of th clls m (n+mz) and m (n+mx+mz) will b avragd to comput ν (n+mz) H x as ν (n+mz) H x = ν(n+mz) + ν (n+mx+mz). (3.14) 2 Lft: Lt us now considr th lft part of th intgration cll (Fig. 3.4a). W can show from Eq. (3.8) that whr ν (n) H z m (n+mz) as C (l) ν (l) B (l) z (ω) dz = C (l) ν(n) + ν(n+mz) 2 = ν(n) + ν (n+mz) 2 ν(n) + ν (n+mz) 2 = ν (n) H z A (n) y A (n) y B (n) z (ω) dz (ω) A (n Mx) y x (n) ] (ω) z (n) + O ( z) 3] (ω) A (n Mx) y (ω) ] z x A (n) y (ω) A y (n Mx) (ω) ], (3.15) will b computd by avraging th magntic imprmability of th clls m (n) and ν (n) H z = ν(n) + ν (n+mz) 2. (3.16) Right: Th right part of th lin intgral can b ralizd from Fig. 3.4b. Eq. (3.9) yilds C (r) ν (r) B (r) z (ω) dz = C (r) ν(n+mx) + ν(n+mx+mz) 2 = ν(n+mx) + ν (n+mx+mz) 2 ν(n+mx) + ν (n+mx+mz) 2 B z (n+mx) (ω) dz A (n+m x) y A (n+m x) y (ω) A (n) y x (n+mx) (ω) A (n) y Rarranging ] (ω) z (n+mx) + O ( z) 3] (ω) ] x z = ν (n+mx) H z A (n+m x) y (ω) A y (n) (ω) ], (3.17)

41 32 Chaptr 3. Finit Intgration Tchniqu (FIT) a) b) ν (n) σ (n) ν (n+mx) σ (n+mx) A (n Mx) y B (n) z A (n) y A (n) y B (n+mx) z A (n+mx) y z ν (n+mz) σ (n+mz) ν (n+mx+mz) σ (n+mx+mz) x x Figur 3.4: a) Lft part and b) right part of th intgration cll I (n) A y. whr th magntic imprmability of th clls m (n+mx) and m (n+mx+mz) will b avragd to comput ν (n+mx) H z as ν (n+mx) H z = ν(n+mx) + ν (n+mx+mz) 2. (3.18) Using Eq. (3.10) w can sum up all th parts to comput th local grid quation for th intgration cll I (n) A y as: C= S ν(r) A m (R, ω)] dr = ν (d) B (d) x C (d) = ν (n+mz) H x + ν (n) H z (ω) dx A (n) y A (n) y C (u) ν (u) B (u) x (ω) dx + (ω) A (n+mz) y (ω) A (n Mx) y (ω) ] ν (n+mx) C (l) ν (l) B (l) z (ω) dz (ω) ] ν (n) H x A (n M z) y (ω) A y (n) (ω) ] H z A (n+m x) y (ω) A (n) y (ω) ] ν (r) B (r) z C (r) (ω) dz = ν (n) H x A (n Mz) y (ω) ν (n) H z A (n Mx) y (ω) ν (n+mx) H z A y (n+mx) (ω) ν (n+mz) H x A y (n+mz) (ω) ] + ν (n) H x + ν (n+mz) H x + ν (n) H z + ν (n+mx) H z A y (n) (ω). (3.19) Now lt us comput th first surfac intgral. Th lctric conductivity at nod n will b computd by avraging th valus of surrounding clls as: σ (n) = 1 4 σ (n) + σ (n+mx) + σ (n+mz) + σ (n+mx+mz)]. (3.20)

42 3.4. Drivation of th Discrt Grid Equation in Cartsian Coordinats in 2-D cas 33 Using th avrag lctric conductivity σ (n) th first surfac intgral of Eq. (3.1) can b writtn as: S j ω σ (R) A m (R, ω) ds = j ω S σ (n) A (n) y (ω) y y ds = j ω σ (n) A (n) y (ω) x z + O ( x) 3 z + ( z) 3 x ] j ω σ (n) A (n) y (ω) ( x) 2, (3.21) by assuming x = z which rsults ds = ( x) 2. Th scond surfac intgral of Eq. (3.1) can b computd in th similar way. S J imp (R, ω) ds = S J (n) y (ω) y y ds = J (n) y (ω) x z + O ( x) 3 z + ( z) 3 x ] J (n) y (ω) ( x) 2. (3.22) Insrting Eqs. (3.19)-(3.22) in Eq. (3.1) yilds, ν (n) H x A (n Mz) y + ν (n) H z A (n Mx) y ν (n+mx) H z A (n+mx) y ν (n+mz) H x A (n+mz) y ] ν (n) H x + ν (n+mz) H x + ν (n) H z + ν (n+mx) H z A y (n) j ω σ (n) A (n) y ( x) 2 = J y (n) ( x) 2. (3.23) Rarranging Eq. (3.23) dlivrs ν (n) H x A (n Mz) y ν (n) H z A (n Mx) y + ν (n) H x + ν (n+mz) H x + ν (n) H z + ν (n+mx) H z j ω σ (n) ( x) 2] A (n) y ν (n+mx) H z A y (n+mx) ν (n+mz) H x A y (n+mz) = J (n) y ( x) 2.(3.24) Introducing th shift oprator S ±Mi and th idntity oprator I w obtain { ] ν (n) H x S Mz ν (n) H z S Mx + ν (n) H x + ν (n+mz) H x + ν (n) H z + ν (n+mx) H z j ω σ (n) ( x) 2 I } ν (n+mx) H z S Mx ν (n+mz) H x S Mz A (n) y = J (n) y ( x) 2. (3.25) Th final quation can b writtn in th form { t1 (n) S Mz + s1 (n) S Mx + d1 (n) + d2 (n)] I + s2 (n) S +Mx + t2 (n) S +Mz } x (n) = b (n),(3.26)

43 34 Chaptr 3. Finit Intgration Tchniqu (FIT) with th cofficints t1 (n) = ν (n) H x (3.27) s1 (n) = ν (n) H z (3.28) d1 (n) = ν (n) H x + ν (n+mz) H x + ν (n) H z + ν (n+mx) H z (3.29) d2 (n) = j ω σ (n) ( x) 2 (3.30) s2 (n) = ν (n+mx) H z (3.31) t2 (n) = ν (n+mz) H x (3.32) x (n) = b (n) = A (n) y (3.33) J (n) y ( x) 2. (3.34) 3.5 Discrt Fild Quantitis in Cylindrical Coordinats in 2-D cas Th discrt fild quantitis in cylindrical coordinats can b computd in th similar way as in cartsian coordinats. W hav to considr th following fw changs in this cas: Th simulation will b prformd in rz plan, instad of xz plan. Th unknown fild quantity is A ϕ. Th lft and right intgrals of Eq. (3.15) and Eq. (3.17) will contain xtra cofficints, as th curl oprator in cylindrical coordinats diffrs from that in th cartsian coordinats. Fig. 3.5 shows th intgration cll I (n) A ϕ listd blow: for 2-D cas. Th allocation of th fild quantitis is Th ϕ componnt of magntic vctor potntial A ϕ is positiond at th cornrs of th matrial clls and point th normal dirction of th rz plan. Th r and z componnts of th magntic flux dnsity vctor B r (n), B z (n), B r (n+mz), and ar positiond at th cntr of th dgs of th fact and ach componnt B (n+mr) z points th tangntial dirctions to th surfac. Th ϕ componnt of lctric currnt dnsity J ϕ (n) matrial cll m (n). is positiond at th cntr of th

44 3.6. Drivation of th Discrt Grid Equation in Cylindrical Coordinats in 2-D cas 35 z r ν (n) σ (n) A (n Mz) ϕ ν (n+mr) σ (n+mr) m (n) B (n) r m (n+mr) z A (n Mr) ϕ B (n) z A (n) ϕ B (n+mr) z A (n+mr) ϕ m (n+mz) B (n+mz) r m (n+mr+mz) ν (n+mz) σ (n+mz) ν (n+mr+mz) σ (n+mr+mz) A (n+mz) ϕ r r (n Mr) r (n 1 2 Mr) r (n) r (n+ 1 2 Mr) r (n+mr) Figur 3.5: Allocation of th discrt lctromagntic fild componnts B r, B z, and A ϕ assignd to th global grid nod (n) in 2-D cas in cylindrical coordinats. 3.6 Drivation of th Discrt Grid Equation in Cylindrical Coordinats in 2-D cas W can dtrmin th discrt grid quation in cylindrical coordinats by discrtizing th govrning quation Eq. (3.1) using th following stps: 1. Th curl oprator of th lin intgral can b computd in th cylindrical coordinats as: A m (R, ω) = 1 A z (ω) r ϕ A ] ϕ(ω) Ar (ω) z r + A ] z(ω) z r ϕ (r Aϕ (ω)) + 1 r r A r(ω) ϕ ] z. (3.35) As th rz plan is th computation plan, magntic vctor potntial has only th ϕ

45 36 Chaptr 3. Finit Intgration Tchniqu (FIT) componnt and thrfor, th curl oprator can b rducd to A m (R, ω) = A ϕ(ω) z r + 1 r (r A ϕ (ω)) r z. (3.36) 2. Th lin intgral of of Eq. (3.1) can b dividd into Up, Down, Lft, and Right intgrals, that rsults ν(r) A m (R, ω)] dr = ν (u) A ϕ(ω) z r + 1 ] (r A ϕ (ω)) r r z r dr C (u) C (u) ν(r) B(R, ω)] dr = ν (u) A ] ϕ(ω) dr. (3.37) z C (u) C (u) C (d) ν(r) A m (R, ω)] dr = C (d) ν(r) B(R, ω)] dr = C (l) ν(r) A m (R, ω)] dr = C (l) ν(r) B(R, ω)] dr = C (r) ν(r) A m (R, ω)] dr = C (r) ν(r) B(R, ω)] dr = C (d) ν (d) C (d) ν (d) C (l) ν (l) C (l) ν (l) C (r) ν (r) C (r) ν (r) A ϕ(ω) z r + 1 r ] (r A ϕ (ω)) r z r dr A ] ϕ(ω) dr. (3.38) z A ϕ(ω) z r + 1 r 1 r ] (r A ϕ (ω)) r z z dz ] (r A ϕ (ω)) dz. (3.39) r A ϕ(ω) z r + 1 r 1 r ] (r A ϕ (ω)) r z z dz ] (r A ϕ (ω)) dz. (3.40) r 3. A (n) ϕ is positiond in th cntr of th intgration cll I (n) A ϕ. Th lin intgral of Eq. (3.1) can b computd as a sum of ths four intgrals as ν(r) A m ((R, ω)] dr C= S = ν(r) B(R, ω)] dr ν(r) B(R, ω)] dr C (d) C (u) + ν(r) B(R, ω)] dr ν(r) B(R, ω)] dr C (l) C (r) = ν (d) B (d) r (ω) dr ν (u) B (u) r (ω) dr + C (d) C (u) C (l) ν (l) B (l) z (ω) dz C (r) ν (r) B (r) z (ω) dz. (3.41)

46 3.6. Drivation of th Discrt Grid Equation in Cylindrical Coordinats in 2-D cas Quadratic clls ar takn hr for as of discrtization, i.., z = r. 5. Th surfac intgrals of Eq. (3.1) can b computd dirctly. 6. Avrag valus of matrial paramtrs ν(r) and σ (R) should b takn whil computing th intgrals. Up: Th uppr part can b computd by rarranging Eq. (3.37) as C (u) ν (u) B (u) r (ω) dr = C (u) ν(n) + ν(n+mr) 2 = ν(n) + ν (n+mr) 2 ν(n) + ν (n+mr) 2 A ] ϕ(ω) dr z A (n M z) ϕ A (n M z) ϕ (ω) A (n) ϕ z (n) (ω) A (n) ϕ (ω) ] r z ] (ω) r (n) + O ( r) 3] = ν (n) H r A (n M z) ϕ (ω) A (n) ϕ (ω) ], (3.42) whr ν (n) H r will b computd by avraging th magntic imprmability of th clls m (n) and m (n+mr) as ν (n) H r = ν(n) + ν (n+mr) 2. (3.43) Down: Th lowr part of th intgration cll I (n) A ϕ C (d) ν (d) B (d) r (ω) dr = C (d) ν(n+mz) + ν(n+mr+mz) 2 = ν(n+mz) + ν (n+mr+mz) 2 ν(n+mz) + ν (n+mr+mz) 2 = ν (n+mz) H r A (n) ϕ A (n) ϕ can b computd using Eq. (3.38) as A ] ϕ(ω) dr z (ω) A (n+mz) ϕ z (n+mz) (ω) A (n+mz) ϕ ] (ω) r (n+mz) + O ( r) 3] (ω) ] r z A (n) ϕ (ω) A (n+mz) ϕ (ω) ], (3.44) whr th magntic imprmability of th clls m (n+mz) and m (n+mr+mz) will b avragd to comput ν (n+mz) H r as ν (n+mz) H x = ν(n+mz) + ν (n+mr+mz) 2. (3.45)

47 38 Chaptr 3. Finit Intgration Tchniqu (FIT) Lft: Lt us now rarrang Eq. (3.39) to comput th lft part as C (l) ν (l) B (l) z (ω) dz = C (l) ν(n) + ν(n+mz) 2 = ν(n) + ν (n+mz) 2 r (n 1 2 Mr) ν (n) = ν (n) r (n) H z r (n 1 2 r (n) H z r (n 1 2 A(n) Mr) ϕ A(n) Mr) ϕ 1 r r (n) A (n) ϕ ] (r A ϕ (ω)) dz r (ω) r (n Mr) A (n Mr) ϕ (ω) r(n Mr) r (n 1 2 (ω) r(n Mr) r (n 1 2 r (n) A(n Mr) Mr) ϕ A(n Mr) Mr) ϕ ] (ω) z (n) + O ( z) 3] ] z (ω) r ] (ω), (3.46) whr ν (n) H z will b computd by avraging th magntic imprmability of th clls m (n) and m (n+mz) as ν (n) H z = ν(n) + ν (n+mz) 2. (3.47) Right: Th right part of th lin intgral can b computd using Eq. (3.40) as C (r) ν (r) B (r) z (ω) dz = C (r) ν(n+mr) + ν(n+mr+mz) 2 = ν(n+mr) + ν (n+mr+mz) 2 r (n+ 1 2 Mr) r (n+mr) H z r (n+ 1 A(n+Mr) Mr) 2 r (n+mr) H z r (n+ 1 A(n+Mr) Mr) 2 ν (n+mr) = ν (n+mr) 1 r ] (r A ϕ (ω)) dz r r (n+m r) A (n+mr) ϕ ϕ (ω) r(n) ϕ (ω) r(n) (ω) r (n) A (n) ϕ r (n+mr) r (n+ 1 A(n) Mr) ϕ 2 r (n+ 1 A(n) Mr) ϕ 2 ] (ω) +O ( z) 3] ] z (ω) r ] (ω) z (n+mr), (3.48) whr th magntic imprmability of th clls m (n+mr) and m (n+mr+mz) will b avragd to comput ν (n+mr) H z as ν (n+mr) H z = ν(n+mr) + ν (n+mr+mz) 2. (3.49)

48 3.6. Drivation of th Discrt Grid Equation in Cylindrical Coordinats in 2-D cas 39 Using Eq. (3.41) w can sum up all th parts to comput th lin intgral for th intgration cll I (n) A ϕ as: C= S ν(r) A m (R, ω)] dr = ν (d) B (d) r C (d) = ν (n+mz) H r + ν (n) (ω) dr A (n) ϕ r (n) H z r (n 1 A(n) Mr) ϕ 2 r (n+mr) H z ν (n+mr) = ν (n) H r A (n Mz) ϕ ν (n+mr) r (n+m r) H z r (n ν (n) H r + ν (n+mz) H r C (u) ν (u) B (u) r (ω) dr + (ω) A ϕ (n+mz) (ω) ] ν (n) (ω) r(n Mr) r (n 1 2 r (n+ 1 A(n+Mr) Mr) 2 (ω) ν (n) A(n+Mr) Mr) ϕ C (l) ν (l) B (l) z (ω) dz H r A (n M z) ϕ (ω) A (n) ϕ (ω) ] ] (ω) A(n Mr) Mr) ϕ ϕ (ω) r(n) r (n+ 1 A(n) Mr) ϕ 2 r (n Mr) H z r (n 1 A(n Mr) Mr) ϕ 2 + ν (n) H z (ω) ν (n+mz) r (n) H r (ω) A (n+mz) ϕ r + r) (n 1 2 Mr) ν(n+m H z (ω) ] (ω) r (n) r (n+ 1 2 Mr) ν (r) B (r) z C (r) (ω) dz ] A (n) ϕ (ω). (3.50) Th first surfac intgral of Eq. (3.1) contains lctric conductivity σ, which will b computd at nod n by avraging th valus of surrounding clls as: σ (n) = 1 4 σ (n) + σ (n+mr) + σ (n+mz) + σ (n+mr+mz)]. (3.51) Using th avrag lctric conductivity σ (n) w can comput th first surfac intgral as: S j ω σ (R) A m (R, ω) ds = j ω S σ (n) A (n) ϕ (ω) ϕ ϕ ds = j ω σ (n) A (n) ϕ (ω) r z + O ( r) 3 z + ( z) 3 r ] j ω σ (n) A (n) ϕ (ω) ( r) 2, (3.52) whr th grid clls ar assumd to b quadratic, that rsults r = z and ds = ( r) 2. Th scond surfac intgral of Eq. (3.1) can b computd in th similar way. S J imp (R, ω) ds = S J (n) ϕ (ω) ϕ ϕ ds = J (n) ϕ (ω) r z + O ( r) 3 z + ( z) 3 r ] J (n) ϕ (ω) ( r) 2. (3.53)

49 40 Chaptr 3. Finit Intgration Tchniqu (FIT) Insrting Eqs. (3.50)-(3.53) in Eq. (3.1) yilds, ν (n) H r A (n Mz) ϕ ν (n) H z ν (n+mz) H r A (n+mz) ϕ + r (n Mr) r (n 1 A(n Mr) Mr) ϕ 2 ν (n) H r + ν (n+mz) H r ν (n+mr) r (n+m r) H z r (n+ 1 A(n+Mr) Mr) ϕ 2 + ν (n) H z r (n) r + r) (n 1 2 Mr) ν(n+m H z r (n) r (n+ 1 2 Mr) ] A (n) ϕ j ω σ (n) A (n) ϕ ( x) 2 = J (n) ϕ ( r) 2. (3.54) Eq. (3.54) can b rwrittn as ν (n) H r A (n Mz) ϕ + ν (n) H z ν (n) H r + ν (n+mz) H r r (n Mr) r (n 1 A(n Mr) Mr) ϕ 2 + ν (n) H z ] r (n) r + r) r (n) (n 1 2 Mr) ν(n+m H z r j ω (n+ 1 2 Mr) σ(n) ( x) 2 A (n) ϕ ν (n+mr) r (n+m r) H z r (n+ 1 A(n+Mr) Mr) ϕ 2 ν (n+mz) H r A ϕ (n+mz) = J (n) ϕ ( r) 2. (3.55) Using th shift oprator S ±Mi and th idntity oprator I w can rarrang Eqs. (3.55) as following { r (n Mr) ν (n) H r S Mz ν (n) H z r S (n 1 2 Mr) M r + ν (n) H r + ν (n+mz) H r + ν (n) H z ] r (n) r + r) r (n) (n 1 2 Mr) ν(n+m H z r j ω (n+ 1 2 Mr) σ(n) ( x) 2 I } ν (n+mr) r (n+m r) H z r S (n+ 1 2 Mr) +M r ν (n+m z) H r S +Mz A (n) ϕ Th final grid quation can b writtn in th form = J (n) ϕ ( r) 2. (3.56) { t1 (n) S Mz + s1 (n) S Mr + d1 (n) + d2 (n)] I + s2 (n) S +Mr + t2 (n) S +Mz } x (n) = b (n), (3.57) whr th cofficints can b computd as following t1 (n) = ν (n) H r (3.58) s1 (n) = ν (n) H z d1 (n) = ν (n) H r r (n Mr) r (n 1 2 Mr) (3.59) + ν (n+mz) H r + ν (n) H z r (n) r (n) r + r) (n 1 2 Mr) ν(n+m H z (3.60) r (n+ 1 2 Mr)

50 3.7. Boundary Conditions 41 d2 (n) = j ω σ (n) ( r) 2 (3.61) s2 (n) = ν (n+mr) H z r (n+m r) r (n+ 1 2 Mr) (3.62) t2 (n) = ν (n+mz) H r (3.63) x (n) = A (n) ϕ (3.64) b (n) = J (n) ϕ ( r) 2. (3.65) 3.7 Boundary Conditions Fig. 3.6 shows th simulation plan rz in cylindrical coordinats, which is discrtizd by N r N z nods. Th cofficints t1, t2, s1, s2, d1 and d2 ar computd at ach nod of th simulation plan using Eqs. (3.58)-(3.63). Bfor forming a band matrix from ths (1, 1) r A ϕ = 0 (N r, 1) z A ϕ = 0 A ϕ = 0 (1, N z ) A ϕ = 0 (N r, N z ) Figur 3.6: PEC boundary condition on a FIT cll. cofficints to solv th final grid quation, th boundary condition should b implmntd which can b prfctly lctric conducting (PEC) prfctly matchd layr (PML) Th prfctly lctric conducting (PEC) boundary condition is considrd hr. From th transition condition dscribd in Eq. (3.3) w know that n A m (R, ω) = continuous ] n A (2) m (R, ω) A (1) m (R, ω) = 0. (3.66)

51 42 Chaptr 3. Finit Intgration Tchniqu (FIT) At th matrial boundary Eq. (3.66) can b rwrittn as n A m (R, ω)] = 0. (3.67) As our simulation domain rsids in rz plan, th normal vctor n = ϕ and hnc, ϕ A m (R, ω) = 0 A ϕ (R, ω) = 0, (3.68) Th PEC boundary condition is applid to th outrmost nods (markd rd), and thrfor, th computation rgion will b rducd to (N r 1) (N z 1). Lt us now discuss th top, bottom, lft and right boundaris individually. Top Boundary: Fig. 3.7a shows th top sid of th simulation domain. As Aϕ (nr,1) = 0, a) b) A ϕ = 0 A (n Mz) ϕ A (n Mz) ϕ A (n Mr) ϕ A (n) ϕ A (n+mr) ϕ A (n Mr) ϕ A (n) ϕ A (n +M r) ϕ A ϕ = 0 A (n Mz) ϕ A (n+mz) ϕ Figur 3.7: PEC boundary condition on th a) top and b) bottom of th simulation domain. th discrt grid quation (Eq. (3.55)) can b rwrittn by stting Aϕ (n Mz) = 0 as ν (n) r (n Mr) H z r (n 1 A(n Mr) Mr) ϕ + ν (n) H r + ν (n+mz) H r + ν (n) r (n) H z 2 r + r) r (n) (n 1 2 Mr) ν(n+m H z r (n+ 1 2 Mr) j ω σ (n) ( x) 2] A (n) ϕ ν (n+mr) r (n+m r) H z r (n+ 1 A(n+Mr) Mr) ϕ 2 ν (n+mz) H r A (n+mz) ϕ = J (n) ϕ ( r) 2 (3.69) { s1 (n) S Mr + d1 (n) + d2 (n)] I + s2 (n) S +Mr + t2 (n) S +Mz } x (n) = b (n). Th cofficints can b computd using Eqs. (3.59)-(3.65) as mntiond in Sc (3.70)

52 3.7. Boundary Conditions 43 Bottom Boundary: Th bottom sid of th simulation domain is shown in Fig. 3.7b. Th boundary condition lads to Aϕ (nr,nz) = 0 and thrfor, th discrt grid quation (Eq. (3.55)) can b rwrittn by stting Aϕ (n+mz) = 0 as ν (n) H r A (n Mz) ϕ + ν (n+mr) H z ν (n) H z r (n Mr) r (n 1 A(n Mr) Mr) ϕ + 2 ] r (n) r j ω (n+ 1 2 Mr) σ(n) ( x) 2 ν (n) H r A (n) ϕ + ν (n+mz) H r ν (n+mr) + ν (n) H z r (n) r (n 1 2 Mr) r (n+m r) H z r (n+ 1 A(n+Mr) Mr) ϕ 2 = J (n) ϕ ( r) 2 (3.71) { t1 (n) S Mz + s1 (n) S Mr + d1 (n) + d2 (n)] I + s2 (n) S +Mr } x (n) = b (n). (3.72) Lft Boundary: Th ffct of PEC boundary condition at th lft sid of th simulation a) b) A ϕ = 0 A (n Mz) ϕ A (n Mz) ϕ A ϕ = 0 A (n Mr) ϕ A (n) ϕ A (n+mr) ϕ A (n Mr) ϕ A (n) ϕ A (n+mr) ϕ A (n+mz) ϕ A (n+mz) ϕ Figur 3.8: PEC boundary condition on th a) lft sid and b) right sid of th simulation domain. domain is shown in Fig. 3.8a. On th lft boundary A ϕ (1, n z ) = 0 and thrfor, th discrt grid quation (Eq. (3.55)) can b rwrittn by stting Aϕ (n Mr) = 0 as ν (n) H r A (n Mz) ϕ + ν (n) H r + ν (n+mz) H r + ν (n) H z r (n) r + r) (n 1 2 Mr) ν(n+m H z r (n) r (n+ 1 2 Mr) j ω σ (n) ( x) 2] A (n) ϕ ν (n+mr) r (n+m r) H z r (n+ 1 A(n+Mr) Mr) ϕ 2 ν (n+mz) H r A ϕ (n+mz) = J (n) ϕ ( r) 2 (3.73) { t1 (n) S Mz + d1 (n) + d2 (n)] I + s2 (n) S +Mr + t2 (n) S +Mz } x (n) = b (n). (3.74)

53 44 Chaptr 3. Finit Intgration Tchniqu (FIT) Right Boundary: Th right sid of th simulation domain is shown in Fig. 3.8b. Th boundary condition rsults Aϕ (Nr,nz) = 0 and thrfor, th discrt grid quation (Eq. (3.55)) can b rwrittn by stting Aϕ (n+mr) = 0 as ν (n) H r A (n Mz) ϕ ν (n) H z + ν (n+mr) H z r (n Mr) r (n 1 A(n Mr) Mr) ϕ + 2 ν (n) H r ] r (n) r j ω (n+ 1 2 Mr) σ(n) ( x) 2 + ν (n+mz) H r A (n) ϕ + ν (n) H z r (n) r (n 1 2 Mr) ν (n+mz) H r A ϕ (n+mz) = J (n) ϕ ( r) 2 (3.75) { t1 (n) S Mz + s1 (n) S Mr + d1 (n) + d2 (n)] I + t2 (n) S +Mz } x (n) = b (n). (3.76) 3.8 Formation of a Band Matrix A typical LU dcomposition is showd in Fig Now lt us tak th final grid quation in A] = L] U]. D]..... = L] + D] + U] L] : lowr matrix D] : diagonal matrix U] : uppr matrix Figur 3.9: 2-D cas in th rz plan: dcomposition of th matrix A] into 5 bands cylindrical coordinats. Using th shift oprator S ±Mi and th idntity oprator I, w can form a band matrix from Eq. (3.57) by taking th following stps: Th computation rgion is (N r 1) (N z 1). Th cofficints t1, t2, s1, s2, d1, d2 and b ar computd for ach nod. Th main diagonal of A] corrspond to th cofficints d1 and d2 with idntity oprator I, and thrfor, it contains N = (N r 1) (N z 1) valus. t1 and t2 corrspond to th first lowr band LM1 and th scond lowr band LM2 rspctivly. Th shift oprator S ±Mr = 1 and S ±Mz = N r. Using ths valus of th shift oprators LM1 and LM2 ar positiond as shown in Fig. 3.10, whr LM1 and LM2 hav N 1 and N N r componnts rspctivly. Similarly s1 and s2 corrspond to th first uppr band UM1 and th scond uppr band UM2 rspctivly. UM1 and UM2 contain N 1 and N N r componnts rspctivly, and ar positiond as shown in Fig

54 3.9. Computation of Inducd Elctric Voltag 45 A] =.... M r..... LM1. DṂ. UM1. UM LM2. M z LM2 : 2 nd lowr band LM1 : 1 st lowr band DM : diagonal band UM1 : 1 st uppr band UM2 : 2 nd uppr band Figur 3.10: 2-D cas in th rz plan: formation of bandmatrx W can rwrit Eq. (3.57) following ths stps as { } LM2 (n) S Mz + LM1 (n) S Mr + DM (n) I + UM1 (n) S Mr + UM2 (n) S Mz x (n) = b (n), (3.77) with th normalizd cofficints DM (n) = d1(n) + d2 (n) d1 (n) + d2 (n) = 1. (3.78) LM2 (n) = LM1 (n) = UM1 (n) = UM2 (n) = W can writ Eq. (3.77) in th gnral form t1 (n) t1 (n) + s1 (n) + s2 (n) + t2 (n) + d2 (n). (3.79) s1 (n) t1 (n) + s1 (n) + s2 (n) + t2 (n) + d2 (n). (3.80) s2 (n) t1 (n) + s1 (n) + s2 (n) + t2 (n) + d2 (n). (3.81) t2 (n) t1 (n) + s1 (n) + s2 (n) + t2 (n) + d2 (n). (3.82) x (n) = A (n) ϕ. (3.83) b (n) J ϕ (n) ( r) 2 =. (3.84) t1 (n) + s1 (n) + s2 (n) + t2 (n) + d2 (n) A] {x} = {b}, (3.85) which is a linar matrix quation and a suitabl linar matrix solvr will b usd to solv this quation by dtrmining th unknown valus of {x}. 3.9 Computation of Inducd Elctric Voltag Th lctric fild strngth E(R, ω) is thn computd from th magntic vctor potntial A m (R, ω) in th scond stp using Eq. (2.128) which can b rcalld from Chaptr 2 as E(R, ω) = j ωa m (R, ω). (3.86)

55 46 Chaptr 3. Finit Intgration Tchniqu (FIT) Th inducd lctric voltag V (ω) for ach conductor of th rciving coil is thn calculatd from th lctric fild strngth by V (ω) = E(R, ω) dr. (3.87) Th total inducd lctric voltag of th rciving coil is thn computd as V tot (ω) = N t V (ω), (3.88) whr th rciving coil consists of N t turns of coppr wir. Th modlling of th rciving coils will b discussd in dtail in Chaptr 7.

56 Chaptr 4 Finit Elmnt Mthod (FEM) Th finit lmnt mthod (FEM) is a numrical tchniqu for obtaining approximat solutions to boundary-valu problms of mathmatical physics. Th basis of th modrn finit lmnt mthod consists of Ritz variational mthod and Galrkin s mthod. Th principl of this mthod is to rplac an ntir continuous domain by a numbr of subdomains in which th unknown function is rprsntd by intrpolation functions with unknown cofficints and thus, th solution of th ntir domain is approximatd by a finit numbr of unknown cofficints. A systm of algbraic quations is thn obtaind by applying th Ritz variational or Galrkin procdur (Kost, 1994; Jin, 2002). Th systm of quations ar solvd to dtrmin th unknown cofficints. Th following stps will b followd to solv a magntoquasistatic problm using FEM: Formulation of th magntoquasistatic govrning quation (Sc. 4.1) Discrtization of th domain in spac (Sc. 4.2) Nod-basd lmntal intrpolation (Sc. 4.3) Formulation by Galrkin s mthod (Sc. 4.4) Formulation of th systm of quations (Sc. 4.5) Implmntation of th boundary conditions (Sc. 4.6) 4.1 Formulation of th Magntoquasistatic Govrning Equation As w hav discussd arlir, th dscribd ddy currnt snsors ar drivn by a monochromatic xcitation puls, it is asir to solv th magntoquasistatic problm in th frquncy domain than in th tim domain. Lt us rcall th govrning quation of th magntoquasistatic problm in th frquncy domain from Chaptr 2 as ν(r) A m (R, ω)] j ωσ (R) A m (R, ω) = J imp (R, ω). (4.1) Th dvlopd ncircling coil snsors ar simulatd using th rotational symmtry and hnc, a 3-D magntoquasistatic problm is rducd into a 2-D problm in cylindrical coordinats. As th first stp, th discrt fild quantitis will b discussd in Cartsian coordinats. 47

57 48 Chaptr 4. Finit Elmnt Mthod (FEM) D Formulation in Cartsian Coordinats Th Curl Oprator of Eq. (4.1) is computd in Cartsian coordinats as: Az (x, z, ω) A m (R, ω) = A ] y(x, z, ω) Ax (x, z, ω) y z x + A z(x, z, ω) z x Ay (x, z, ω) + A x(x, z, ω) x y ] y ] z. (4.2) Lt us considr that our simulation domain Ω lis in xz plan and thrfor, w hav to considr only th y componnt of magntic vctor potntial. As a rsult, th curl oprator is rducd to A m (R, ω) = A y(x, z, ω) z Th lft-hand sid of Eq. (4.1) can b computd as ν(r) A m (R, ω)] ] = x x + y y + z z x + A y(x, z, ω) x ν(x, z) A y(x, z, ω) z z. (4.3) x + ν(x, z) A ] y(x, z, ω) x z. (4.4) Th lctric currnt dnsity J imp (R, ω) has only th y componnt and thrfor, w hav to considr only th cofficints of y, which rsults ν(r) A m (R, ω)] = x ν(x, z) A y(x, z, ω) x ] y z ν(x, z) A ] y(x, z, ω) z y. (4.5) Insrting Eq. (4.5) in Eq. (4.1) and rplacing A m (R, ω) with A y (x, z, ω) y rsults ν(x, z) A ] y(x, z, ω) x x y ν(x, z) A ] y(x, z, ω) z z y which is rwrittn as x ν(x, z) A y(x, z, ω) x j ωσ (x, z)a y (x, z, ω) y = J y (x, z, ω) y, (4.6) ] z ν(x, z) A ] y(x, z, ω) z j ωσ (x, z)a y (x, z, ω) = J y (x, z, ω). (4.7) Th Dirichlt and Numann conditions ar common boundary conditions for th magntic vctor potntial. W writ th Dirichlt boundary condition as A y (x, z, ω) = constant. (4.8) At th intrfac btwn th two mdium having diffrnt magntic imprmability ν, th magntic vctor potntial satisfis th continuity conditions A (1) y (x, z, ω) = A (2) y (x, z, ω) (4.9) (1) A(1) y (x, z, ω) (2) A(2) y (x, z, ω) ν = ν n n. (4.10)

58 4.2. Discrtization of th Domain in 2-D Spac D Formulation in Cylindrical Coordinats W can comput th Curl Oprator of Eq. (4.1) in cylindrical coordinats as: 1 A z (r, z, ω) A m (R, ω) = A ] ϕ(r, z, ω) Ar (r, z, ω) r ϕ z r + A z(r, z, ω) z r + 1 (r Aϕ (r, z, ω)) A r(r, z, ω) r r ϕ ] ϕ ] z. (4.11) Lt us considr th rz plan as th computation plan, thrfor, th magntic vctor potntial contains only th ϕ componnt. Th curl oprator of Eq. (4.11) is rducd to A m (R, ω) = A ϕ(r, z, ω) z r + 1 r (r A ϕ (r, z, ω)) r z. (4.12) Assuming that th lctric currnt dnsity J imp (R, ω) has only th ϕ componnt, w considr only th cofficints of ϕ, which rsults ν(r) A m (R, ω)] = ν(r, z) A ϕ (r, z, ω) r r r ν(r, z) A ϕ(r, z, ω) z z ] ϕ ] ϕ. (4.13) Insrting Eq. (4.13) in Eq. (4.1) and rplacing A m (R, ω) with A ϕ (r, z, ω) ϕ rsults r ] 1 r ν(r, z) A ϕ(r, z, ω) r ϕ z ν(r, z) A ] ϕ(r, z, ω) z ϕ j ωσ (r, z)a ϕ (r, z, ω) ϕ = J ϕ (r, z, ω) ϕ, (4.14) which is rarrangd as ] 1 r r ν(r) A ϕ(r, z, ω) r z ν(r, z) A ] ϕ(r, z, ω) j ωσ (r, z)a ϕ (r, z, ω) z = J ϕ (r, z, ω). (4.15) Bfor w mploy th finit lmnt formulation for a solution of th problm, w chck th continuity conditions for ra ϕ at th intrfac of mdium 1 and 2. 1 r (1) ν (1) r (1) A (1) ϕ (r, z, ω) = r (2) A (2) ϕ (r, z, ω) (4.16) r (1) A (1) ϕ (r, z, ω) ] = 1 n r (2) ν (2) r (2) A (2) ϕ (r, z, ω) ]. (4.17) n 4.2 Discrtization of th Domain in 2-D Spac W hav to discrtiz th simulation domain bfor w start with th discrtization of th govrning quation. FEM is basd on th spatial discrtization of th solution domain, whr a givn finit lmnt is associatd to its physical dimnsions in spac (Ida, 1995). Thrfor, th first stp of a finit lmnt analysis is to divid th domain Ω in a finit

59 50 Chaptr 4. Finit Elmnt Mthod (FEM) a) 3 (x 3, z 3) b) 1 (x 1, z1). (x 2, z2) =1 =2 4 =3 = Figur 4.1: a) 2-D triangular lmnt and b) 2-D discrtization with four lmnts. numbr of lmnts, in our cas, a finit numbr of two-dimnsional lmnts. Furthrmor, th lmnts should b connctd by thir vrtics, i.., a vrtx of an lmnt can only b th vrtx of th nighboring lmnt. W hav takn linar triangular lmnts to discrtiz th domain. A typical triangular lmnt is shown in Fig. 4.1a. Th nods and lmnts can b labld with sparat sts of intgrs for idntification. Sinc ach lmnt is rlatd to thr nods, a nod is assignd a local labl in th associatd lmnt, in addition to its global labl rlativ to th ntir systm. Fig. 4.1b shows a typical FEM discrtization with four lmnts. 4.3 Nod-Basd Elmntal Intrpolation Aftr discrtizing th simulation domain w hav to approximat th unknown function A y in cas of th cartsian coordinats. For linar triangular lmnts th unknown function A y within ach lmnt can b approximatd as A y (x, z) = a + b x + c z, (4.18) whr a, b and c ar constant cofficints to b dtrmind for ach lmnt. Fig. 4.1a shows a linar triangular lmnt with thr nods, whr th nods ar numbrd countrclockwis by numrals 1, 2 and 3, with th corrsponding valus of A y dnotd by A y1, A y2 and A y3. Using Eq. (4.18) at th thr nods rsults A y1 (x, z) = a + b x 1 + c z 1 (4.19) A y2 (x, z) = a + b x 2 + c z 2 (4.20) A y3 (x, z) = a + b x 3 + c z 3, (4.21) whr x j and zj rprsnts th coordinats of th j-th nod in th -th lmnt for j = 1, 2, 3. Eqs. (4.19)-(4.21) can b rwrittn as A y1 1 x 1 z 1 a A y2 = 1 x 2 z 2 b. (4.22) 1 x 3 z 3 c A y3

60 4.4. Formulation by Galrkin s Mthod 51 Solving for th cofficints a, b und c as functions of A yj rsults a 1 x 1 z 1 b = 1 x 2 z 2 c 1 x 3 z 3 1 A y1 A y2 A y3 Insrting Eq. (4.23) in Eq. (4.18) yilds A y (x, z) = 1 x z ] 1 x 1 z 1 1 x 2 z 2 1 x 3 z 3. (4.23) 1 A y1 A y2 A y3 which can b writtn in th form (Silvstr & Frrari, 1983) A y (x, z) =, (4.24) 3 Nj (x, z) A yj, (4.25) j=1 whr N j ar th intrpolation or xpansion functions, also known as basis functions, which can b dnotd as in which N j (x, z) = 1 2 (a j + b jx + c jz), (4.26) a 1 = x 2 z 3 x 3 z 2; b 1 = z 2 z 3; c 1 = x 3 x 2 a 2 = x 3 z 1 x 1 z 3; b 2 = z 3 z 1; c 2 = x 1 x 3 (4.27) a 3 = x 1 z 2 x 2 z 1; b 3 = z 1 z 2; c 3 = x 2 x 1 and rprsnts th ara of th -th lmnt as = 1 1 x 1 z 1 1 x 2 z2 = x 3 z3 2 (b 1c 2 b 2c 1). (4.28) 4.4 Formulation by Galrkin s Mthod Galrkin s mthod blongs to th family of wightd rsidual mthods, which sks th solution by wighting th rsidual of th diffrntial quation. Whn applying Galrkin s tsting procdur, th tsting function is idntical to th xpansion function (Volakis t al., 1998) Formulation in Cartsian Coordinats Th rsidual associatd with Eq. (4.7) is computd as r = ν(x, z) A ] y(x, z, ω) x x z ν(x, z) A y(x, z, ω) z ] j ωσ (x, z)a y (x, z, ω) J y (x, z, ω). (4.29)

61 52 Chaptr 4. Finit Elmnt Mthod (FEM) For simplicity lt us rplac A y (x, z, ω) with A y and for lmnt w rarrang Eq. (4.29) as r = ν A ] y ν A ] y j ωσ A y J y (4.30) x x z z and th wightd rsidual for lmnt is formulatd as Ri = Ni r dx dz ; i = 1, 2, 3. (4.31) Ω Insrting Eq. (4.30) in Eq. (4.31) yilds Ri = Ni ( ν A ) y x x Ω which is writtn as Ri = Ω N i j ω Ω z ( ν A ) y dx dz x x Ω Ni σ A y dx dz W formulat th first intgral of Eq. (4.33) as ( Ni ν A ) y dx dz = x x Ω and w rwrit th scond intgral of Eq. (4.33) as ( Ni ν A ) y dx dz = z z Ω Ω Ω Ω ( ν A ) ] y j ωσ A y J y dx dz, (4.32) z N i ( ν A ) y dx dz z z Ω N i J y dx dz. (4.33) N i x ν A y x + x N i z ν A y z + z Using Eqs. (4.34)-(4.35) w rarrang Eq. (4.33) as Ri = N i A x ν y x + ( Ni ν A )] y dx dz x x R i = Ω N i A z ν y z + ( Ni ν A )] y dx dz z z j ω Ni σ A y dx dz Ni J y dx dz Ω Ω Ω Ω N A i x ν y x + N i A ] z ν y dx dz z ( Ni ν A )] y dx dz x ( Ni ν A )] y dx dz. z ( Ni ν A ) y + ( Ni ν A )] y dx dz x x z z j ω Ni σ A y dx dz Ω (4.34) (4.35) Ω N i J y dx dz. (4.36)

62 4.4. Formulation by Galrkin s Mthod 53 Th scond intgral of Eq. (4.36) is rwrittn as ( Ni ν A ) y + ( Ni ν A )] y dx dz x x z z whr Ω = Ω = U(x, z) + x ] V (x, z) ds z Ω U(x, z) x + V (x, z) z ] ds, (4.37) U(x, z) = N i ν A y x (4.38) V (x, z) = Ni ν A y. (4.39) z Using Gauss thorm on th right-sid of Eq. (4.37) yilds U(x, z) x + V (x, z) z ] ds = U(x, z) x + V (x, z) z ] dr, (4.40) Ω which rsults Ω C= S ( Ni ν A ) y + ( Ni ν A )] y dx dz x x z z = C= S Insrting Eq. (4.41) in Eq. (4.36) rsults Ri = Ω { Ni ν A ] y x x + Ni ν A ] } y z z dr. (4.41) C= S N A i x ν y x + N i A z ν y z Ω ] dx dz { Ni ν A ] y x x + Ni ν A ] } y z z dr j ω Ni σ A y dx dz Ω N i J y dx dz. (4.42) Th unknown function A y is writtn with basis functions using Eq. (4.25) as A y (x, z) = 3 Nj (x, z) A yj (4.43) j=1 and th first ordr drivativs of A y is xprssd as x A y (x, z) = z A y (x, z) = 3 j=1 3 j=1 N A j (x, z) yj x N A j (x, z) yj z (4.44). (4.45)

63 54 Chaptr 4. Finit Elmnt Mthod (FEM) Using Eqs. (4.43)-(4.45) w rwrit Eq. (4.42) as R i = 3 A yj j=1 Ω j ω C= S 3 A yj j=1 Ω N ν i x Nj x + N i z Ω N j z ] dx dz σ Ni Nj dx dz Ni J y dx dz { N i ν A y x ] x + Ni ν A y z ] } z dr, (4.46) which is writtn in matrix form Rarranging Eq. (4.47) rsults with th cofficints in local form Sij = S] {A} j ω T] {A} {b} + {g} = 0. (4.47) Ω C= S S] {A} j ω T] {A} = {b} {g}, (4.48) ν N i x Nj x + N i z N j z ] dx dz (4.49) σ Ni Nj dx dz (4.50) Ni J y dx dz (4.51) Tij = Ω b i = Ω gi = { Ni ν A ] y x x + Ni ν A ] } y z z dr (4.52) A = A y(x, z, ω). (4.53) In our cas, thr is no sourc at th boundary and thrfor, w st {g} = 0 and simplify Eq. (4.48) as S] {A} j ω T] {A} = {b}. (4.54) Formulation in Cylindrical Coordinats Th magntoquasistatic govrning quation in cylindrical coordinats is dscribd by Eq. (4.14). Th rsidual associatd with this quation is computd as r = r ] 1 r ν(r, z) A ϕ(r, z, ω) r z ν(r, z) A ] ϕ(r, z, ω) j ωσ (r, z)a ϕ (r, z, ω) z J ϕ (r, z, ω). (4.55)

64 4.5. Formulation of th Systm of Equations 55 If w follow th sam stps as for th cartsian coordinats, w shall obtain th matrix quation of similar form whr th cofficints ar dnotd as Sij = ν N i r r Ω C= S S] {A} j ω T] {A} = {b}, (4.56) Nj r + N i z N j z ] dr dz (4.57) σ Ni Nj dr dz (4.58) Ni J ϕ dr dz (4.59) Tij = Ω b i = Ω gi = { Ni ν A ] ϕ r r + Ni ν A ] } ϕ z z dr (4.60) A = r A ϕ(r, z, ω). (4.61) 4.5 Formulation of th Systm of Equations Lt us considr th 2-D formulation in cartsian coordinats. Th global matrics T], S] and {b} dscribd in Eq. (4.54) can b computd using th following stps: Computation of th local cofficints S ij, T ij and b i for ach lmnt Rlating th local nods of th lmnts to th global nods Arrangmnt of th local cofficints in global matrics Computation of th Local Cofficints Th local T -matrix is dscribd in Eq. (4.50) as Tij = which is computd as (s Ida, 1995, p. 316) T ] = Ω σ N i N j dx dz, (4.62) 12 σ (4.63) Furthrmor, th S-Matrix is dnotd by Eq. (4.49) Sij = Ω N ν i x Nj x + N i z N j z ] dx dz, (4.64)

65 56 Chaptr 4. Finit Elmnt Mthod (FEM) which is valuatd as (s Ida, 1995, p. 323) (b 1) 2 + (c 1) 2 b S ] = ν 1 b 2 + c 1 c 2 b 1 b 3 + c 1 c 3 b 4 2 b 1 + c 2 c 1 (b 2) 2 + (c 2) 2 b 2 b 3 + c 2 c 3. (4.65) b 3 b 1 + c 3 c 1 b 3 b 2 + c 3 c 2 (b 3) 2 + (c 3) 2 Th local b-vctor in cartsian coordinats is dscribd by b i = and is formulatd as (s Jin, 2002, p. 98) Ω N i J y dx dz (4.66) b i = 3 J y. (4.67) Rlating th Local Nods of th Elmnts to th Global Nods A typical 2-D discrtizing schm is shown in Fig. 4.2, whr th global nod numbrs ar markd in blu and th local nod numbrs in black. Each lmnt has thr nods who can b idntifid by thir local as wll as global numbrs. A rlationship btwn ths two numbrs is prsntd in Tabl 4.1. On th lft sid, th lmnts ar dscribd with global numbrs associatd to th corrsponding nods. As for xampl, an arbitrary lmnt = 7 contains thr nods with th global numbrs n(1, 7) = 5, n(2, 7) = 6 and n(3, 7) = 9. Th coordinats of th global nods ar listd on th right sid. n(1,) n(2,) n(3,) Global nod x z Tabl 4.1: Lft: 2-D global nod numbr and right: coordinats of th global nods.

66 4.5. Formulation of th Systm of Equations 57 z global nod. local nod = 14 = 16 = = 13 = 15 = = 8 = 10 = = 7 = 9 = z = 2 = 4 = = 1 = 3 = x..... x Figur 4.2: 2-D discrtization with linar triangular lmnts Arrangmnt of th local cofficints in global matrics Th local cofficints Sij, Tij and b i ar computd in cartsian coordinats using Eqs. (4.63), (4.65) and (4.67) for ach lmnt. Tij and Sij can b includd in th global matrics T] and S] by T ij = T n(i,), n(j,) (4.68) S ij = S n(i,), n(j,), (4.69) which mans, th local cofficints T ij and S ij for -th lmnt should b placd in th global T] and S] matrics, rspctivly, with th corrsponding row and column numbrs n(i, ), n(j, ). Th dimnsion of th gnratd T] and S] matrics dpnd on th numbr of global nods, and thrfor, on th numbr of lmnts. Tabl 4.1 shows an xampl with = 18 lmnts and n = 16 global nods. Th rsulting T] and S] matrics hav th dimnsions of n n =

67 58 Chaptr 4. Finit Elmnt Mthod (FEM) As a scond xampl, Eqs. (4.70)-(4.71) show th construction of T] and S] matrics rspctivly for = 6 lmnts (s Fig. 4.1b), whr th numbr of global nods is n = 8 and hnc, th siz of th matrics bcoms n n = 8 8 (Akcakoca, 2009). 6 T ] = T] = (4.70) =1 T (1) T (1) 0 0 T (1) T (1) (T (1) T (2) 11 + T (3) 11 ) T (3) 0 (T (1) T (2) (2) ) (T T (3) 13 ) T (3) (T (3) T (4) 11 + T (5) 11 ) T (5) 0 (T (3) T (4) (4) ) (T T (5) 13 ) T (5) (T (5) T (6) (5) ) 0 0 (T T (6) 13 ) T (6) 12 T (1) (T (1) T (2) (1) ) 0 0 (T T (2) 33 ) T (2) (T (2) 21 + T (3) (3) ) (T T (4) 31 ) 0 T (2) (T (2) T (3) 33 + T (4) 33 ) T (4) (T (4) 21 + T (5) (5) ) (T T (6) 31 ) 0 T (4) (T (4) T (5) 33 + T (6) 33 ) T (6) T (6) 0 0 T (6) T (6) S ] = S] = (4.71) =1 S (1) 11 S (1) 21 S (1) 31 S (1) 12 (S (1) 22 + S(2) 11 + S(3) 11 ) S(3) 12 0 S (3) S (5) S (1) 13 (S (3) 22 + S(4) 11 + S(5) 11 ) S(5) 12 (S (5) 22 + S(6) 11 ) 0 0 (S(5) 23 + S(6) 13 ) S(6) 12 (S (1) 32 + S(2) 31 ) 0 0 (S(1) 33 + S(2) 33 ) S(2) 32 0 (S (2) 21 + S(3) 31 ) (S(3) 32 + S(4) 31 ) 0 S(2) (S (4) 21 + S(5) 31 ) (S(5) 32 + S(6) 31 ) 0 S(4) S (6) (S (1) 23 + S(2) 13 ) (S(2) 12 + S(3) 13 ) (S (3) 23 + S(4) 13 ) (S(4) 12 + S(5) 13 ) (S (2) 22 + S(3) 33 + S(4) 33 ) S(4) 0 32 (S (4) 22 + S(5) 33 + S(6) 33 ) S(6) S (6) S (6) Similarly, b i can b includd in {b} vctor by b i = b n(i,), (4.72) which mans, th local cofficints b i for -th lmnt should b positiond in th global {b} vctor with th corrsponding row numbr n(i, ). As an xampl, th construction of {b}

68 4.6. Implmntation of th Boundary Conditions 59 vctor for = 6 lmnts is shown in Eq. (4.73). 6 b ] = {b} = =1 b (1) 1 b (1) 2 + b (2) 1 + b (3) 1 + b (4) 1 b (3) 2 + b (5) 1 + b (6) 1 b (5) 2 b (1) 3 + b (2) 3 b (2) 2 + b (4) 3 b (3) 3 + b (4) 2 + b (6) 3 b (5) 3 + b (6) 2. (4.73) Th boundary conditions should b applid bfor w solv th gnratd linar matrix quation Eq. (4.54). 4.6 Implmntation of th Boundary Conditions In this sction, th Dirichlt boundary conditions will b discussd. Lt us considr Fig. 4.1b and assum that th nods n = 3, n = 5 and n = 6 li on th boundary (s Fig. 4.3), whr A y is constant, i.. A yn = p n = p ; n = 3, 5, 6. (4.74) Th linar matrix quation dscribd in Eq. (4.54) is rwrittn as 1 2 =1 4 =3 =2 = A y = p 5 A y = p Figur 4.3: Dirichlt boundary condition on th simulation domain Ω. S] j ω T]{A} = {b} K]{A} = {b}, (4.75)

69 60 Chaptr 4. Finit Elmnt Mthod (FEM) whr K 11 K 12 K 13 K 14 K 15 K 16 K 21 K 22 K 23 K 24 K 25 K 26 K 31 K 32 K 33 K 34 K 35 K 36 K] = S] j ω T] = K 41 K 42 K 43 K 44 K 45 K 46 K 51 K 52 K 53 K 54 K 55 K 56 K 61 K 62 K 63 K 64 K 65 K 66 (4.76) and using this xprssion K] w rwrit Eq. (4.75) as K 11 K 12 K 13 K 14 K 15 K 16 K 21 K 22 K 23 K 24 K 25 K 26 K 31 K 32 K 33 K 34 K 35 K 36 K 41 K 42 K 43 K 44 K 45 K 46 K 51 K 52 K 53 K 54 K 55 K 56 K 61 K 62 K 63 K 64 K 65 K 66 A y1 A y2 A y3 A y4 A y5 A y6 = b 1 b 2 b 3 b 4 b 5 b 6. (4.77) To impos th conditions dscribd by Eq. (4.74), w simply st b 3 = b 5 = b 6 = p, (4.78) which rsults and K 33 = K 55 = K 66 = 1 (4.79) K 3i = K 5i = K 6i = 0 ; i = 1, 2, 3, 4, 5, 6. (4.80) Howvr, imposing boundary conditions in such a way dstroys th symmtry of th matrix K]. To rstor this proprty th K] matrix has to b modifid by stting K i3 = K i5 = K i6 = 0 ; i = 1, 2, 3, 4, 5, 6, (4.81) which causs a modification of th {b} vctor as b i b i K i3 p K i5 p K i6 p ; i = 1, 2, 4. (4.82) Using Eq. (4.78)-(4.82) th matrix K] bcoms K 11 K 12 0 K K 21 K 22 0 K K K] = K 41 K 42 0 K (4.83)

70 4.6. Implmntation of th Boundary Conditions 61 and th vctor {b} bcoms {b} = b 1 K 13 p K 15 p K 16 p b 2 K 23 p K 25 p K 26 p p b 4 K 43 p K 45 p K 46 p. (4.84) W can, howvr, dlt th third, fifth and sixth quation from th systm, which will not chang th solution. Thus, th final systm of quations will b writtn as K 11 K 12 K 14 A y1 b 1 K 13 p K 15 p K 16 p K 21 K 22 K 24 A y2 = b 2 K 23 p K 25 p K 26 p, (4.85) K 41 K 42 K 44 A y4 b 4 K 43 p K 45 p K 46 p which will b solvd by a linar matrix solvr to dtrmin th unknown valus of A y. Howvr, it is possibl to impos th Dirichlt boundary conditions following a scond approach. Lt us considr th sam boundary problm dscribd in Fig Instad of imposing th conditions (4.78)-(4.81) dirctly, w choos a vry larg numbr,.g., and thus w obtain and thrfor, p p K 33 = K 55 = K 66 = (4.86) b 3 = b 5 = b 6 = p (4.87) As a rsult, th systm of quation bcoms K 11 K 12 K 13 K 14 K 15 K 16 K 21 K 22 K 23 K 24 K 25 K 26 K 31 K K 34 K 35 K 36 K 41 K 42 K 43 K 44 K 45 K 46 K 51 K 52 K 53 K K 56 K 61 K 62 K 63 K 64 K A y1 A y2 A y3 A y4 A y5 A y6 = b 1 b 2 p b 4 p p (4.88) W obsrv from Eq. (4.88) that th symmtry of th K] matrix is rtaind and w do not rquir any additional stps. This approach is thrfor much simplr than th first approach and it nds only two oprations two impos boundary conditions: on on th diagonal lmnt and th othr on th known sourc vctor. Additionally, this tchniqu is suitabl for th spars matrics, stord in a compact form. Howvr, all th quations associatd with th boundary nods xist and hnc, a rduction of th siz of th matrics is not prmittd hr, which is th major disadvantag of this approach (Jin, 2002).

71 62 Chaptr 4. Finit Elmnt Mthod (FEM)

72 Chaptr 5 Boundary Elmnt Mthod (BEM) Th boundary lmnt mthod (BEM), also known as th mthod of momnts (MoM), approachs th ddy currnt problm by solving th undrlying boundary intgral quations (BIE), basd on quasi-variational principls, for th quivalnt sourcs, i.., for th quivalnt lctric and magntic currnt dnsity and th lctric and magntic charg dnsity. BEM is applicabl to problms for which Grn s functions can b calculatd, which usually involv filds in linar homognous mdia. This placs considrabl rstrictions on th rang and gnrality of problms to which boundary lmnts can usfully b applid (Gibson, 2007; Harrington, 1993). Th boundary intgral quation approach lads to th dvlopmnt of so-calld wk formulations, which prmit th approximat satisfaction of th diffrnt quations or conditions (Poljak & Brbbia, 2005). Th us of approximat functions for th variabls undr considration in ths wk formulations introduc rrors that ar minimizd using th wightd rsidual tchniqus, such as, Galrkin s mthod. BEM offrs a numbr of advantags ovr th othr numrical mthods, such as: Dimnsions of th problms ar ffctivly rducd by on. Th simulation domain is xtndd to infinity without th nd to discrtiz a larg portion of th body. BEM rquirs mshs, which can b asily varid without th nd to satisfy th typ of continuity rquirmnts, which is ssntial in FEM. Mor accurat wighting functions ar applid in BEM, such as, Grn s functions. Bcaus of ths advantags BEM bcoms a mor suitabl candidat than th domain typ mthods in ddy currnt analysis with high conductivity or at high frquncy. Howvr, thr ar som difficultis in analyzing ddy currnt problms with BEM (Zhang, 1997), which can b mntiond as: Th vctor natur of problms rsults in a vry larg numbr of unknowns. Loos coupling btwn lctric and magntic filds in th air at low frquncy maks it difficult to satisfy th intrfac conditions and to obtain a uniqu solution in conjunction with an appropriat choic of Gaug condition. Highly singular krnls in boundary intgral quations rquir xpnsiv numrical intgration to obtain accurat solutions. 63

73 64 Chaptr 5. Boundary Elmnt Mthod (BEM) A magntoquasistatic problm is solvd by th boundary lmnt mthod using th following stps: Formulation of th magntoquasistatic govrning quation (Sc. 5.1) Intgral quation formulation (Sc. 5.2 and 5.3) Boundary lmnt discrtization and formulation of th systm of quations (Sc. 5.4) 5.1 Formulation of th Magntoquasistatic Govrning Equation In our cas, th simulatd hot wir has a tmpratur of C, which is wll abov Curi point and thrfor, th rlativ prmability µ r can b considrd to b 1 and µ(r) = µ 0. Similarly, w can considr that th rlativ prmittivity ε = 1, which rsults ε = ε 0. As a rsult, th hot wir can b rprsntd by a linar, isotropic and homognous conducting matrial rgion V σ, which is mbddd in fr spac V 0. Th rgion V σ is boundd by th surfac S σ with th outward unit normal n σ and is charactrizd by magntic prmability µ(r) = µ 0 and lctric conductivity σ, as shown in Fig J imp V s (R, ω), J imp m (R, ω) ϱ (R, ω) ϱ m(r, ω) S c = V s R V 0 ε 0, µ 0 E in (R, ω) E sc (R, ω) Sσ = Vσ V σ ε 0, µ 0, σ 0 R n σ n Figur 5.1: Gomtry for applying Grn s functions Formulation for th Conducting Rgion Using Eq. (2.120) and Eq. (2.121) w writ th inhomognous Hlmholtz quations in magntoquasistatic cas for th conducting rgion V σ as E(R, ω) + k 2 c E(R, ω) = j ωµ 0 J imp (R, ω) + 1 ε 0 ϱ (R, ω) + J m (R, ω) (5.1) H(R, ω) + k 2 c H(R, ω) = σ J m (R, ω) + 1 µ 0 ϱ m (R, ω) J imp (R, ω), (5.2)

74 5.1. Formulation of th Magntoquasistatic Govrning Equation 65 whr k c is th complx wav numbr and is givn by k c = j ωµσ. (5.3) W xprss th lctric fild strngth E(R, ω) and magntic fild strngth H(R, ω) by th lctric and magntic scalar and vctor potntials from Eq. (2.125) and Eq. (2.126) as E(R, ω) = Φ (R, ω) + j ω A m (R, ω) 1 ε 0 A (R, ω) (5.4) H(R, ω) = Φ m (R, ω) σ ε A (R, ω) + 1 µ 0 A m (R, ω), (5.5) whr A m (R, ω), A (R, ω), Φ (R, ω) and Φ m (R, ω) rprsnt magntic vctor potntial, lctric vctor potntial, lctric scalar potntial and magntic scalar potntial rspctivly. Using Lorntz approximation w writ th following xprssions for th potntials Φ (R, ω) + k 2 cφ (R, ω) = 1 ε 0 ϱ (R, ω) (5.6) A m (R, ω) + k 2 ca m (R, ω) = µ 0 J imp (R, ω) (5.7) Φ m (R, ω) + k 2 cφ m (R, ω) = 1 µ 0 ϱ m (R, ω) (5.8) A (R, ω) + k 2 ca (R, ω) = ε 0 J m (R, ω). (5.9) Using quations Eq. (5.6)-Eq. (5.9) w nd to dfin a Grn s function which satisfis th following diffrntial quation G σ (R R, ω) + k 2 cg σ (R R, ω) = δ(r R, ω), (5.10) whr G σ (R R, ω) is th scalar Grn s function for th conducting rgion V σ dfind by G σ (R R, ω) = j kc R R 4π R R and is (5.11) and applying Grn s thorm for vctor variabls boundary intgral quations for th fild vctors ar drivd Formulation for th Non-Conducting Rgion For th non-conducting rgion V 0, σ = 0 and hnc w rwrit Eqs. (5.1)-(5.2) as E(R, ω) = j ωµ 0 J imp (R, ω) + 1 ε 0 ϱ (R, ω) + J m (R, ω) (5.12) H(R, ω) = 1 µ 0 ϱ m (R, ω) J imp (R, ω). (5.13) Th lctric fild strngth E(R, ω) and magntic fild strngth H(R, ω) ar xprssd by th lctric and magntic scalar and vctor potntials by quations Eq. (5.4) and Eq. (5.5).

75 66 Chaptr 5. Boundary Elmnt Mthod (BEM) Again, using Eqs. (5.6)-(5.9) w writ th xprssions for th potntials as Φ (R, ω) = 1 ε 0 ϱ (R, ω) (5.14) A m (R, ω) = µ 0 J imp (R, ω) (5.15) Φ m (R, ω) = 1 µ 0 ϱ m (R, ω) (5.16) A (R, ω) = ε 0 J m (R, ω). (5.17) W dfin a Grn s function G ms (R R ) for th non-conducting rgion V 0 which satisfis th quation whr G ms (R R ) = δ(r R ), (5.18) G ms (R R ) = 1 4π R R. (5.19) 5.2 Intgral Rprsntation of Elctrical and Magntic Fild Strngth Intgral Rprsntation for Conducting Rgion Th spcial solutions of th inhomognous Hlmholtz quations Eq. (5.1) and Eq. (5.2) in magntoquasistatic cas for th conducting rgion V σ ar writtn using th scalar Grn s function as E(R, ω) = 1 ϱ (R, ω) G σ (R R, ω) d 3 R ε 0 V σ + j ωµ 0 V σ J imp (R, ω) G σ (R R, ω) d 3 R V σ J m (R, ω) G σ (R R, ω) d 3 R (5.20) H(R, ω) = 1 ϱ m (R, ω) G σ (R R, ω) d 3 R µ 0 V σ σ V σ + J imp m (R, ω) G σ (R R, ω) d 3 R V σ J (R, ω) G σ (R R, ω) d 3 R. (5.21) Lt us apply Gauss s intgral thorm to rduc th volum sourcs to surfac sourcs, which rsults ϱ,m η,m (5.22) J,m K,m. (5.23)

76 5.2. Intgral Rprsntation of Elctrical and Magntic Fild Strngth 67 Using ths xprssions on th conducting surfac S σ w formulat th boundary conditions from quations Eq. (2.18)-Eq. (2.21) as η (R, ω) = ε 0 n σ E(R, ω) = ε 0 n E(R, ω) (5.24) K (R, ω) = n σ H(R, ω) = n H(R, ω) (5.25) η m (R, ω) = µ 0 n σ H(R, ω) = µ 0 n H(R, ω) (5.26) K m (R, ω) = n σ E(R, ω) = n E(R, ω), (5.27) whr n σ is th normal vctor on th conducting surfac S σ as shown in Fig. 5.1 and n = n σ. Using quations (5.22)-(5.27) w rplac th volum intgrals of Eq. (5.20) and Eq. (5.21) with th closd surfac intgrals and thus w obtain E(R, ω) = n E(R, ω) G σ (R R, ω) ds S σ j ωµ 0 n H(R, ω) G σ (R R, ω) ds S σ n E(R, ω) G σ (R R, ω) ds (5.28) S σ H(R, ω) = n H(R, ω) G σ (R R, ω) ds S σ σ n E(R, ω) G σ (R R, ω) ds S σ n H(R, ω) G σ (R R, ω) ds. (5.29) S σ Th -diffrntiation of G σ (R R, ω) can b prformd insid th intgral and hnc, th (R R ) dpndncy of G σ can b rplacd by -diffrntiation, which yilds E(R, ω) = S σ n E(R, ω) G σ (R R, ω) ds j ωµ 0 S σ n H(R, ω) G σ (R R, ω) ds S σ n E(R, ω)] G σ (R R, ω) ds (5.30)

77 68 Chaptr 5. Boundary Elmnt Mthod (BEM) H(R, ω) = S σ n H(R, ω) G σ (R R, ω) ds σ n E(R, ω) G σ (R R, ω) ds S σ n H(R, ω)] G σ (R R, ω) ds. (5.31) S σ Th quations Eq. (5.30) and Eq. (5.31) ar calld th Stratton-Chu vrsion of th intgrals. At th surfac of th conducting rgion using th boundary valu of th Hlmholtz rprsntation w rwrit Eq. (5.30) and Eq. (5.31) as 1 2 E σ (R, ω) = 1 2 H σ (R, ω) = S σ n E σ (R, ω) G σ (R R, ω) ds j ωµ 0 S σ S σ n H σ (R, ω) G σ (R R, ω) ds n E σ (R, ω) ] G σ (R R, ω) ds (5.32) S σ n H σ (R, ω) G σ (R R, ω) ds σ n E σ (R, ω) G σ (R R, ω) ds S σ n H σ (R, ω) ] G σ (R R, ω) ds, (5.33) S σ whr E σ (R, ω) and H σ (R, ω) rprsnt th lctric and magntic fild strngth in conducting rgion, rspctivly Intgral Rprsntation for Non-conducting Rgion For th non-conducting rgion w st σ = 0 in Eqs. (5.30)-(5.31) and thus w obtain th spcial solutions for th homognous Hlmholtz quations Eq. (5.12) and Eq. (5.13). W writ th lctric fild strngth and magntic fild strngth in non-conducting rgion as E 0 (R, ω) and H 0 (R, ω), rspctivly and thus w obtain E 0 (R, ω) = S 0 = V 0 n E 0 (R, ω) G ms (R R ) ds j ωµ 0 S 0 = V 0 n H 0 (R, ω) G ms (R R ) ds S 0 = V 0 n E 0 (R, ω)] G ms (R R ) ds (5.34)

78 5.3. Intgral Equation Formulation 69 H 0 (R, ω) = n H 0 (R, ω) G ms (R R ) ds S 0 = V 0 n H 0 (R, ω)] G ms (R R ) ds. (5.35) S 0 = V 0 Howvr, th xtrnal sourcs ar locatd in th non-conducting rgion (s Fig. 5.1) and hnc, w hav to add th sourc trms on th right sid of Eq. (5.34) and Eq. (5.35), which rsults E 0 (R, ω) = E imp (R, ω) H 0 (R, ω) = j ωµ 0 S 0 = V 0 n E 0 (R, ω) G ms (R R ) ds S 0 = V 0 n H 0 (R, ω) G ms (R R ) ds n E 0 (R, ω)] G ms (R R ) ds (5.36) S 0 = V 0 H imp (R, ω) S 0 = V 0 n H 0 (R, ω) G ms (R R ) ds S 0 = V 0 n H 0 (R, ω)] G ms (R R ) ds. (5.37) Considring th lctric surfac currnt dnsity K imp (R, ω) as th sourc, w writ th xprssion of incidnt lctric fild E imp (R, ω) using Grn s thorm as E imp (R, ω) = j ωµ 0 K imp (R, ω) G ms (R R ) ds S c = j ωµ 0 S c n H 0 (R, ω) G ms (R R ) ds. (5.38) Th xprssion for incidnt magntic fild H imp (R, ω) is writtn using Maxwll s first quation as j ωb imp (R, ω) = E imp (R, ω) = j ωµ 0 K imp (R, ω) G ms (R R ) ds H imp (R, ω) = 5.3 Intgral Equation Formulation S c S c K imp (R, ω) G ms (R R ) ds. (5.39) Bfor w driv th boundary intgral quations, w hav to st th transition conditions for th lctric and magntic fild. Th transition conditions for th tangntial componnts of th lctric and magntic fild strngth ar dscribd by (Zhang, 1997) n H σ (R, ω) = n H 0 (R, ω) = K (R, ω) (5.40) n E σ (R, ω) = n E 0 (R, ω) = K m (R, ω). (5.41)

79 70 Chaptr 5. Boundary Elmnt Mthod (BEM) Lt us rcall th transition condition for magntic flux dnsity at th sourc-fr transition surfac from chaptr 2: which is writtn in frquncy domain as n B σ (R, t) B 0 (R, t) ] = 0, (5.42) n B σ (R, ω) B 0 (R, ω) ] = 0 n H σ (R, ω) H 0 (R, ω) ] = 0. (5.43) Th normal componnt of th magntic flux dnsity is xprssd using Maxwll s first quation as n B σ (R, ω) = B 0 (R, ω) = 1 j ω n E 0 (R, ω)] = j ω K m(r, ω). (5.44) Th transition condition for th normal componnt of lctric fild strngth is rcalld as n E σ (R, ω) E 0 (R, ω) ] = 1 ε 0 η (R, ω). (5.45) Howvr, for th low frquncy ddy currnt problms w assum ωε 0 and hnc, th normal componnt of lctric fild strngth in conductiv mdium vanishs which rsults n E σ (R, ω) = 0 (5.46) n E 0 (R, ω) = 1 ε 0 η (R, ω). (5.47) Th intgral rprsntations of th lctric and magntic fild strngth in conductiv and non-conductiv mdium ar dscribd in quations (5.32), (5.33), (5.36) and (5.37) and nd to b solvd to find th quivalnt sourcs. In ordr to obtain accurat solutions an ovr dtrmind systm has to b solvd and th transition conditions dscribd by Eqs. (5.40)- (5.47), nd to b satisfid Elctric Fild Intgral (EFIE) Formulation Th surfac of th conductiv rgion V σ acts as a transition surfac btwn th conducting and non-conducting rgion and hnc, both Eq. (5.32) and Eq. (5.36) ar valid at this surfac EFIE for Conducting Rgion Insrting th transition conditions (5.40)-(5.47) in Eq. (5.32) yilds 1 2 E σ (R, ω) = j ωµ 0 K (R, ω) G σ (R R, ω) ds S σ S σ K m (R, ω) G σ (R R, ω) ds. (5.48) W comput n E σ (R, ω) and rarrang Eq. (5.48) by using th transition condition Eq. (5.41) as 1 2 K m(r, ω) + n K m (R, ω) G σ (R R, ω) ds S σ = j ωµ 0 n S σ K (R, ω) G σ (R R, ω) ds. (5.49)

80 5.3. Intgral Equation Formulation 71 Again, w prform th opration (n ) on both sids of Eq. (5.48) and using th transition conditions Eq. (5.46) w obtain n S σ K m (R, ω) G σ (R R, ω) ds = n j ωµ EFIE for Non-conducting Rgion S σ K (R, ω) G σ (R R, ω) ds. (5.50) W insrt th transition conditions (5.40)-(5.47) in th intgral rprsntation of th lctric fild strngth Eq. (5.36) E 0 (R, ω) = E imp (R, ω) + + j ωµ 0 S σ 1 ε 0 η (R, ω) G ms (R R ) ds S σ K (R, ω) G ms (R R ) ds S σ K m (R, ω) G ms (R R ) ds. (5.51) Prforming th opration (n ) and rarranging Eq. (5.51) yilds K m (R, ω) + n K m (R, ω) G ms (R R ) ds S σ 1 n η (R, ω) G ms (R R ) ds ε 0 S σ = j ωµ 0 n Similarly, prforming th opration (n ) rsults S σ K (R, ω) G ms (R R ) ds + n E imp (R, ω). (5.52) 1 η (R, ω) + n K ε m (R, ω) G ms (R R ) ds 0 σ 1 n η (R, ω) G ms (R R ) ds ε 0 S σ = j ωµ 0 n S σ K (R, ω) G ms (R R ) ds + n E imp (R, ω). (5.53) W solv th lctric fild intgral quations, dscribd by (5.49), (5.50) (5.52) and (5.53) to find th unknown surfac sourc quantitis. Howvr, EFIE has th disadvantag that Frdholm intgral quations of th scond kind ar usd for th lctric fild quantitis only, which lads to ill-conditiond quations systms whn th complx diffusion constant incrass at highr frquncis or highr conductivity (Ruckr t al., 1995).

81 72 Chaptr 5. Boundary Elmnt Mthod (BEM) Magntic Fild Intgral (MFIE) Formulation As w hav mntiond arlir, th surfac S σ acts as a transition surfac btwn th conducting and non-conducting rgion. Thus both Eq. (5.33) and Eq. (5.37) ar valid at this surfac MFIE for Conducting Rgion Lt us insrt th transition conditions (5.40)-(5.47) in Eq. (5.33) to obtain 1 2 H j σ (R, ω) = ωµ K m (R, ω)] G σ (R R, ω) ds S σ σ K m (R, ω) G σ (R R, ω) ds S σ + K (R, ω) G σ (R R, ω) ds. (5.54) S σ W comput thn n H σ (R, ω) and rarrang Eq. (5.54) by using th transition condition Eq. (5.40) as 1 2 K (R, ω) n S σ K (R, ω) G σ (R R, ω) ds + j ωµ n K m (R, ω)] G σ (R R, ω) ds S σ = σ n S σ K m (R, ω) G σ (R R, ω) ds. (5.55) Prforming th opration (n ) on both sids of Eq. (5.54) and using th transition conditions Eq. (5.44) yilds 1 j 2 ωµ K m(r, ω) n K (R, ω) G σ (R R, ω) ds S σ + j ωµ n K m (R, ω)] G σ (R R, ω) ds S σ = σ n MFIE for Non-conducting Rgion S σ K m (R, ω) G σ (R R, ω) ds. (5.56) Insrting th transition conditions (5.40)-(5.47) in Eq. (5.37) rsults H 0 (R, ω) = H imp j (R, ω) ωµ K m (R, ω) G ms (R R ) ds + S σ S σ K (R, ω) G ms (R R ) ds. (5.57)

82 5.4. Boundary Elmnt Discrtization and Formulation of th Systm of Equations 73 W prform th opration n on both sids of Eq. (5.57) and us th transition condition Eq. (5.40) to obtain K (R, ω) + j ωµ n K m (R, ω) G ms (R R ) ds S σ n S σ K (R, ω) G ms (R R ) ds = n H imp (R, ω). (5.58) Prforming th opration (n ) on both sids of Eq. (5.54) and using th transition conditions Eq. (5.44) yilds j ωµ K m(r, ω) + j ωµ n K m (R, ω) G ms (R R ) ds S σ n S σ K (R, ω) G ms (R R ) ds = n H imp (R, ω). (5.59) Th surfac fild quantitis ar dfind by th MFIE for th conducting and th nonconducting rgion. Th MFIE rprsnts a Frdholm intgral quation of th scond kind for th magntic fild quantitis only and thrfor, this formulation has th sam disadvantags as th EFIE formulation. 5.4 Boundary Elmnt Discrtization and Formulation of th Systm of Equations Good rsults ar obtaind by using a mixd formulation, whr th unknown surfac quantitis ar dfind by EFIE for th conducting rgion and by MFIE for th non-conducting rgion. For xampl, th so-calld KHJ-formulation (Ruckr t al., 1995) is constructd with th EFIE for th conducting rgion Eq. (5.49) for K m (R, ω), th MFIE for th conducting rgion Eq. (5.56) for H 0n (R, ω) and th MFIE for non-conducting rgion Eq. (5.58) for K (R, ω) whr H 0n (R, ω) = j ωµ K m(r, ω). (5.60) Th normal componnt of th lctric fild outsid th conductiv rgion can b calculatd with Eq. (5.53) in a scond stp. Thortically, BEM is a combination of th classical boundary intgral quation mthod and th discrtization tchniqu originatd from FEM, and thrfor, th most important stp to solv a magntoquasistatic problm by BEM is to driv th intgral quation formulation of th govrning quation that w hav discussd arlir. In ordr to apply th boundary lmnt mthod (BEM) th surfac of th conducting rgion is discrtizd into so calld boundary lmnts. Numrical solutions with high accuracy ar obtaind by using boundary lmnts of highr ordr,. g. ight-nodd quadrilatral isoparamtric lmnts using scond ordr shap functions N k (α, β) (Ruckr & Richtr, 1990). r(ζ 1, ζ 2 ) = 8 N k (ζ 1, ζ 2 ) r k, (5.61) k=1

83 74 Chaptr 5. Boundary Elmnt Mthod (BEM) whr (ζ 1, ζ 2 ) is th local coordinat systm, dfind for ach boundary lmnt and th position of th k-th nod is givn by r k. W dfin two orthogonal tangntial unit vctors u and v as n = u v (5.62) in th surfac nods for th dscription of th surfac filds. In th local coordinat systm of th considrd lmnt th tangntial vctors ar givn by u(ζ 1, ζ 2 ) = r(ζ 1, ζ 2 )/ ζ 1 r(ζ 1, ζ 2 )/ ζ 1 (5.63) v(ζ 1, ζ 2 ) = r(ζ 1, ζ 2 )/ ζ 2 r(ζ 1, ζ 2 )/ ζ 2. (5.64) Thus w obtain th orthogonal tangntial vctors in th nods of th global coordinat systm as following: u k = α k u(ζ 1, ζ 2 ) + β k u(ζ 1, ζ 2 ) (5.65) v k = γ k u(ζ 1, ζ 2 ) + δ k u(ζ 1, ζ 2 ). (5.66) As for xampl, th tangntial componnt of th lctric fild is givn by K m according to Eq. (5.41) and is computd as K m (ζ 1, ζ 2 ) = 8 N k (ζ 1, ζ 2 ) K uk u k + K vk v k ]. (5.67) k=1 Th discrtization of th boundary intgral quations lad for th KHJ formulation to th following quation systm: H uu H uv 0 j ωµg uu j ωµg uv K mu 0 H vu H vv 0 j ωµg vu j ωµg vv K mv 0 σ G nu σ G nv H nn H nu H nv H 0n = 0. (5.68) 0 0 H un H uu H uv K u Ku imp 0 0 H vn H vu H vv K v K imp v Th matrix lmnts in Eq. (5.68) ar N N sub-matrics whr N is th total numbr of nods on th closd surfac S σ. Th cofficints of th lmnt matrics H] ar listd blow h k uu = (v u k ) N k G(R, R ) ds ± 1 2 δ ij (5.69) S σ h k uv = (v v k ) h k vu = (u u k ) h k vv = (u v k ) S σ N k G(R, R ) ds (5.70) S σ N k G(R, R ) ds (5.71) S σ N k G(R, R ) ds ± 1 2 δ ij (5.72)

84 5.4. Boundary Elmnt Discrtization and Formulation of th Systm of Equations 75 h k nu = (n u k ) N k G(R, R ) ds (5.73) S σ h k nv = (n v k ) N k G(R, R ) ds (5.74) S σ h k un = v N k G(R, R ) ds (5.75) S σ h k vn = u N k G(R, R ) ds (5.76) S σ h k nn = n N k G(R, R ) ds ± 1 2 δ ij (5.77) S σ and th cofficints of th lmnt matrics G] ar givn by guu k = (v u k ) guv k = (v v k ) gvu k = (u u k ) gvv k = (u v k ) gnu k = (n u k ) gnv k = (n v k ) In gnral, Eq. (5.68) is writtn as S σ N k G(R, R ) ds ± 1 2 δ ij (5.78) S σ N k G(R, R ) ds (5.79) S σ N k G(R, R ) ds (5.80) S σ N k G(R, R ) ds ± 1 2 δ ij (5.81) S σ N k G(R, R ) ds (5.82) S σ N k G(R, R ) ds. (5.83) H] {K} = {J}, (5.84) which is a linar matrix quation. A suitabl linar matrix solvr, such as, th conjugat gradint (CG) mthod (Prss t al., 2007) or th gnralizd modifid rsidual (GMRES) mthod (Mistr, 2008) is usd to solv this quation by dtrmining th unknown valus of {K}.

85 76 Chaptr 5. Boundary Elmnt Mthod (BEM)

86 Chaptr 6 Analytical Solution of an Eddy Currnt Problm Numrical modling of ddy currnt problms plays an important rol in nondstructiv valuation. Th rsults obtaind using diffrnt numrical tchniqus should b compard to th analytical rsults to prov th accuracy and consistncy of th numrical mthods. As th first stp, a simpl gomtry is chosn for this comparison and thn th analytical solution is computd. 6.1 Gomtry of a Typical Eddy Currnt Problm Th most common gomtris of intrst involv xcitation coils nar plan or cylindrical surfacs. A typical xampl is shown in Fig. 6.1 whr a currnt coil rsids ovr a conducting half-spac. Du to th xcitation in th currnt coil, an ddy currnt will b inducd on th mtal surfac. Th ddy currnt fild is prturbd by variations in th dimnsions or proprtis (.g. prmability µ and lctric conductivity σ ) of th mtal and by th prsnc of dfcts (Howr t al., 1984). Ths prturbations, in turn, produc changs in th lctromagntic filds nar th mtal surfac which can b dtctd as changs in th impdanc of th ddy currnt xcitation (or sarch) coil. Thortical calculations typically Currnt Coil z Conducting Half-Spc r Figur 6.1: A circular coil ovr a conducting half-spac. involv a full numrical solution of th fild quations using finit intgration tchniqu (FIT), finit lmnt mthod (FEM) or mthod of momnts (MoM). Such a problm can 77

87 78 Chaptr 6. Analytical Solution of an Eddy Currnt Problm also b solvd analytically by valuating th intgral xprssions, obtaind by solving th appropriat boundary valu problm. Spcifically, w considr hr a circular coil abov and paralll to th surfac of a homognous conducting half-spac (s Fig. 6.1). Th cas of a multipl-turn coil of rctangular cross-sction has bn prsntd by Dodd & Dds (1984). Th cross-sction of such a coil is shown in Fig z z r r 1 r 2 l 2 l 1 l 1 d Figur 6.2: Cross sction of a circular coil ovr a conducting half-spac. Paramtr: r 1 =innr radius, r 2 =outr radius, l 1 =distanc btwn coil to conducting half-spac and l 2 l 1 = coil hight. 6.2 Analytical Exprssions Th xprssions of th lctric fild strngth E and magntic fild strngth H for th typical ddy currnt problm, shown in Fig. 6.2, nd to b computd in cylindrical coordinat systm. Th axis of th coil shows th z-axis and th air-mtal intrfac locatd at z = 0. As mntiond in th prvious chaptrs, w comput th magntic vctor potntial A to solv an ddy currnt problm. Th gomtry is locatd in rz-plan and thrfor, th ϕ componnt of th magntic vctor potntial A ϕ has to b computd. W can writ th xprssion of A ϕ for th conducting half-spac as whr E ϕ (r, z) = j ωa ϕ (r, z) = j ωµk A ϕ (r, z) = µk α=0 K = α=0 J 1 (αr) F a (αr 2, αr 1 ) F b (α, z) dα (6.1) J 1 (αr) F a (αr 2, αr 1 ) F b (α, z) dα, (6.2) NI (r 2 r 1 )(l 2 l 1 ) (6.3) F a (αr 2, αr 1 ) = F b (α, z) = αr 2 x=αr 1 J 1 (x) dx (6.4) ( αl 1 αl 2) α 2 + j ωµσ z α 2 (α + α 2 + j ωµσ ). (6.5)

88 6.3. Gauss-Lagurr Quadratur 79 Hr N is th numbr of turns on th coil, I is th lctric currnt in th coil, ω is th angular frquncy and σ is th lctric conductivity of th conducting half-spac. Furthrmor, J 1 rprsnts Bssl function of th first kind and of first ordr. At th surfac of th conducting half-spac Eq. (6.5) can b rwrittn as αl 1 αl 2 F b (α, z = 0) = α 2 (α + α 2 + j ωµσ ) = αl 1 α 2 (α + α 2 + j ωµσ ) αl2 α 2 (α + α 2 + j ωµσ ). (6.6) Th right sid of Eq. (6.2) contains intgrals which nd to b computd numrically to dtrmin th valu of A ϕ. On of th following mthods can b implmntd for this purpos: Trapzoidal rul Simpson s rul Rombrg intgration Gauss quadratur Th first thr involv qually spacd sampls of th intgrand but diffr in th mthod of approximating intrmdiat valus, which causs incorrct rsults in cas of inhrnt oscillation of th intgrand,.g., in Bssl function. Such an intgration rquirs a rlativly small stp siz and has th difficulty in dciding whn to trminat th intgration. Gauss quadratur rul can b usd to ovrcom this difficulty. Howvr, it has svral approachs, such as Gauss-Lagurr quadratur Gauss-Lagndr quadratur Chbyshv-Gauss quadratur Gauss-Kronrod quadratur Lobatto quadratur tc. Th dfinit intgral in Eq. (6.4) is solvd by Gauss-Kronrod quadratur, which attmpts to approximat th intgral of a scalar-valud function fun from a to b using high-ordr global adaptiv quadratur. Furthrmor, Gauss-Lagurr quadratur ruls ar dsignd to approximat intgrals on smi-infinit intrvals, lik 0, ), and thrfor, ar suitabl to dtrmin th intgration of Eq. (6.2). 6.3 Gauss-Lagurr Quadratur Gauss-Lagurr quadratur dals with a Gaussian quadratur with an intrval 0, ) with wighting function w(x) = x (s Abramowitz & Stgun (1972)). If thr xists an intgral of th form x f(x) dx, x=0

89 80 Chaptr 6. Analytical Solution of an Eddy Currnt Problm thn it can b computd as whr x=0 x f(x) dx = w i = 1 n w i f(x i ), (6.7) i=1 x i d dx L n(x i ) ] 2. (6.8) Hr L n (x i ) ar th roots of th Lagurr polynomial of n-th ordr. A highr valu of n should b slctd whil intgrating an oscillating function. 6.4 Gauss-Kronrod Quadratur As mntiond arlir, Gauss-Kronrod quadratur is suitabl for dfinit intgrals whr th intrvals can b typically mntiond as a, b]. If thr xists a dfinit intgral of th form b x=a f(x) dx, thn it can b approximatd by n-points Gaussian quadratur as b x=a dx n w i f(x i ), (6.9) i=1 whr w i and x i ar th wights and points at which to valuat th function f(x). If th intrval a, b] is subdividd, th Gauss valuation points of th nw subintrvals nvr coincid with th prvious valuation points (xcpt at zro for odd numbrs) and thus th intgrand must b valuatd at vry point. Gauss-Kronrod formulas ar xtnsions of Gauss quadratur formulas gnratd by adding n + 1 points to an n-point rul in such a way that th rsulting rul is of ordr 3n + 1. This allows for computing highr-ordr stimats whil rusing th function valus of a lowr-ordr stimat. 6.5 Computation of th Intgrals Insrting Eq. (6.6) into Eq. (6.2) yilds, A ϕ (r, z) = µk µk α=0 α=0 αl 1 J 1 (αr) F a (αr 2, αr 1 ) α 2 (α + α 2 + j ωµσ ) dα αl 2 J 1 (αr) F a (αr 2, αr 1 ) α 2 (α + α 2 + j ωµσ ) W hav to substitut αl 1 with α to comput th first intgral as α = α l 1 α = α l 1 dα. (6.10) dα = 1 l 1 dα. (6.11)

90 6.5. Computation of th Intgrals 81 Using this substitution th first intgral of Eq. (6.10) can b rwrittn as α=0 αl 1 J 1 (αr) F a (αr 2, αr 1 ) α 2 (α + α 2 + j ωµσ ) dα = 1 l 1 α =0 = 1 l 1 α =0 ( α J 1 α ( ( ) 2 α l 1 l 1 r ) α l 1 + ( ) α F a l 1 r 2, α l 1 r 1 ( ) 2 α l 1 + j ωµσ ) dα (6.12) α f(α ) dα, (6.13) whr f(α ) = ( α l 1 ) 2 ( α J 1 ( l 1 r ) α l 1 + ( ) α F a l 1 r 2, α l 1 r 1 ( ) 2 α l 1 + j ωµσ ) dα. (6.14) Applying Eq. (6.7) w can comput th intgral as 1 l 1 α =0 α f(α ) dα = n w i f(α i). (6.15) i=1 In ordr to comput th scond intgral of Eq. (6.10) w hav to substitut αl 2 with α as α = α l 2 α = α l 2 dα = 1 l 2 dα. (6.16) Using this substitution th scond intgral can b computd in th sam way as th first on. α=0 αl 2 J 1 (αr) F a (αr 2, αr 1 ) α 2 (α + α 2 + j ωµσ ) dα = 1 l 2 α =0 = 1 l 2 = α =0 n i=1 ( α J 1 α ( ( ) 2 α l 2 l 2 r ) α l 2 + ( ) α F a l 2 r 2, α l 2 r 1 ( ) 2 α l 2 + j ωµσ ) dα (6.17) α f(α )dα, (6.18) w i f(α i ). (6.19)

91 82 Chaptr 6. Analytical Solution of an Eddy Currnt Problm Insrting Eq. (6.15) and Eq. (6.19) into Eq. (6.10) yilds A ϕ (r, z) = µk n w i f(α i) µk i=1 n = µk w i f(α i) i=1 n i=1 n i=1 w i f(α i ) w i f(α i ) ]. (6.20) 6.6 Analytical Rsults Th coil is dscribd by numbr of turns N = 200, innr radius r 1 = 12 mm, outr radius r 2 = 16 mm, distanc from conducting half-spac l 1 = 4 mm, hight of th coil l 2 l 1 = 4 mm, lctric currnt I = 1 A and angular xcitation frquncy ω = s 1. Th conducting half spac has th thicknss d = 5 mm and th lctric conductivity of σ = A/Vm. Th whol simulation rgion is considrd to hav th prmability µ = µ 0, whr µ 0 is th prmability of fr spac. Th magntic vctor potntial A ϕ at th Aϕ in Vs/m r in m Figur 6.3: Analytic rsult: magnitud of A ϕ at z = 0. Paramtr: N = 200, r 1 = 12 mm, r 2 = 16 mm, l 1 = 4 mm, l 2 l 1 = 4 mm, d = 5 mm, I = 1 A, ω = s 1, σ = A/Vm and µ = µ 0. uppr surfac of th conducting half-spac is thn computd using Eq. (6.20). Som control of th oscillations of th intgrands can b xrtd by propr slction of th ordr n for th Gauss-Lagurr quadratur. Fig. 6.3 shows th computd A ϕ at z = 0 for n = Comparison with Numrical Rsults Th ddy currnt problm mntiond in Sc. 6.1 can also b solvd using numrical tchniqus, such as, th finit intgration tchniqu (FIT), th finit lmnt mthod (FEM) and th boundary lmnt mthod (BEM).

92 6.7. Comparison with Numrical Rsults FIT Rsults A 3-D ddy currnt problm can b rducd to a 2-D problm using rotational symmtry in cylindrical coordinats, and thrfor, it is nough to simulat th right half of Fig. 6.2, as shown in Fig Th simulation rgion is discrtizd by N r N z = clls, z r 1 r 2 l 2 l 1 z = 0 r l 1 d Figur 6.4: Gomtry for FIT and FEM using rotational symmtry. Paramtr: r 1 =innr radius, r 2 =outr radius, l 1 =distanc btwn coil to conducting half-spac and l 2 l 1 = coil hight. whr th width of ach cll = 0.25 mm. Th xcitation coil is fd with an lctric currnt dnsity of J = kA/m 2, which is quivalnt to th lctric currnt of I = 1 A in th conductors. Th prfctly lctric conducting (PEC) boundary conditions ar implmntd a) b) z in m Aϕ in Vs/m r in m r in m Figur 6.5: FIT rsult: a) A ϕ distribution in rz-plan and b) A ϕ at z = 0. Paramtr: N r = 200, N z = 200, = x = z = 0.25 mm, J = ka/m 2, ω = s 1, σ = A/Vm and µ = µ 0. on all sids of th simulation plan. Th computd magnitud of A ϕ is shown in Fig. 6.5a, whr th whil lin indicats th uppr surfac of th conducting half-spac. A ϕ at this surfac is shown in Fig. 6.5b.

93 84 Chaptr 6. Analytical Solution of an Eddy Currnt Problm FEM Rsults Th rducd 2-D problm is simulatd hr using FEM whr th rz-plan is discrtizd by N = triangular lmnts. Th xcitation coil is supplid with an lctric currnt dnsity of J = ka/m 2. Similar to FIT, th PEC boundary condition is implmntd a) b) z in m Aϕ in Vs/m r in m r in m Figur 6.6: FEM rsult: a) A ϕ distribution in rz-plan and b) A ϕ at z = 0. Paramtr: N = 18000, J = ka/m 2, ω = s 1, σ = A/Vm and µ = µ 0. hr on all sids of th simulation plan. Th computd A ϕ is shown in Fig. 6.6a, whras A ϕ at z = 0 is shown in Fig. 6.6b BEM Rsults a) b) Aϕ in Vs/m r in m Figur 6.7: BEM rsult: a) A ϕ distribution on th surfac of th conducting half-spac and b) A ϕ at z = 0. Paramtr: N = 5000, NI = 622 A-T, ω = s 1, σ = A/Vm and µ = µ 0.

94 6.7. Comparison with Numrical Rsults 85 A 3-D ddy currnt problm is solvd hr using BEM by discrtizing th surfac of th conducting half spac with N = 5000 triangular lmnts. Th xcitation of th coil is dfind by th volum currnt, i.., by assigning NI = 622 A-T. Fig. 6.7 shows th simulatd A ϕ distribution at th surfac of th conducting half-spac Comparison A dirct comparison btwn all th mthods is shown in Fig. 6.8, whr th analytical, FIT, FEM and BEM rsults ar shown in black, rd, blu and magnta, rspctivly. All th rsults agr prfctly to ach othr, which provs th accuracy and consistncy of th numrical mthods. Although w obtain accurat rsults using all th thr tchniqus, th major Aϕ in Vs/m Analytic FIT FEM BEM r in m Figur 6.8: Comparison of rsults obtaind from diffrnt tchniqus. point of considration is th computation tim. Hr w hav solvd th magntoquasistatic problm with th finit intgration tchniqu for unknowns in 2 minuts, whras a finit lmnt mthod basd softwar packag Opra (Opra, 10.5) rquirs th sam tim for unknowns and a boundary lmnt basd softwar tool Faraday (Faraday, 6.1) rquirs 120 minuts for 5000 unknowns. Du to th larg computational tim 3-D simulations will b prformd only for th modls which cannot b simulatd in 2-D, such as, for point coil snsors.

95 86 Chaptr 6. Analytical Solution of an Eddy Currnt Problm

96 Chaptr 7 Simulation Rsults In this chaptr w discuss th rsults of diffrnt ddy currnt snsors dvlopd during th INCOSTEEL projct, which is an EU rsarch projct and is prformd by th Univrsity of Kassl in collaboration with Tcnatom (Tcnatom, S.A.), TU Fribrg (TU Fribrg, FB) and Sidnor (Sidnor, S.A.). At first, a traditional ncircling coil will b discussd. Such a snsor is typically usd to dtct th transvrsal cracks. Scondly, w shall discuss about th modifid ncircling coil snsor. In th nxt part, th point cor snsors, suitabl to dtct th longitudinal cracks, will b rportd. At th nd, w shall mak a vlocity analysis to simulat th ffct of th vlocity of th hot wir on th dtction procss. 7.1 Simulation Rsults of Encircling Coil Snsors Two typs of ncircling coils hav bn analyzd in this sction. Encircling coil snsor without frrit cor Encircling coil snsor with frrit cor W simulat th following simulations with th snsors mntiond abov: Encircling coil snsor without frrit cor to simulat a transvrsal crack Encircling coil snsor with frrit cor to simulat a transvrsal crack Encircling coil snsor without frrit cor to simulat a longitudinal crack Encircling Coil Snsor without Frrit Cor to Simulat a Transvrsal Crack In this sction th modling of th dtction of a transvrsal crack with an ncircling coil snsor without frrit cor is rportd. Such a snsor is typically usd in th rolling mills to dtct th surfac dfcts. This prototyp is built according to th dsign of th ncircling coil snsor which is in opration in Sidnor (Sidnor, S.A.). Th modling of this first gnration snsor nd to b prformd to undrstand and dtrmin th ways to modify it. 87

97 88 Chaptr 7. Simulation Rsults Gomtry of an Encircling Coil Snsor without Frrit Cor Th schmatic viw of an ncircling coil snsor is shown in Fig Such a snsor consists of four coils, whr th two outr coils (rd colord in Fig. 7.1b) oprat as xcitation coils and th two innr coils (blu colord in Fig. 7.1b) oprat as rciving coils. Each of th xcitation coils has th hight of 3 mm and consists of 140 turns of coppr wir and ach of th rciving coils is constructd of 180 turns of coppr wir. a) b) Coil 0.5 mm Crack Excitation Coil Hot Wir Rcivr Coil D = 31 mm 40 mm 0.5 mm 3 mm 3 mm Figur 7.1: a) Cross sction of th ncircling coil snsor with stl wir insid and b) dtaild viw of th cross sction of th snsor coils Hot Wir Excitation Coil Zoomd Viw of Snsor Excitation Coil Rcivr Coil 0.5 mm 3 mm z in m 0.02 Crack Rcivr Coil 3 mm Zoomd Viw of Crack 0.01 l = 5 mm r in m d = 0.5 mm Figur 7.2: Schmatic viw of th gomtry of th hot wir stl with a crack at z = 20 mm mbddd in fr-spac and a sktch of an ncircling coil snsor, lctric conductivity of hot wir σ = A/Vm and magntic prmability µ = µ 0.

98 7.1. Simulation Rsults of Encircling Coil Snsors 89 Th applid voltag at th driving circuit of th xcitation coils is V = V (rms) which rsults a currnt dnsity J imp (R, ω) = A/m 2 in th xcitation coil. Th obtaind voltags in th rcivr coils ar rcordd and plottd to dtrmin th rspons of th snsor whn a crack crosss th snsor. Th siz of th crack is lngth l = 5 mm and dpth d = 0.5 mm Numrical Tchniqu to Simulat an Encircling Coil Snsor without Frrit Cor Th simulations ar prformd by finit intgration tchniqu (FIT) with th hlp of MQS- FIT (Marklin, 2002). Th gomtry shown in Fig. 7.2 lis in rz-plan and is discrtizd by = rctangular lmnts. Th rsulting linar matrix quation is itrativly by bi-congugat gradint (BCG) mthod (Shwchuk t al., 1994; Prss t al., 2007) with a rlativ tolranc of An xcitation frquncy of f = 50 khz is usd and th rsulting radial componnt of th magntic flux dnsity is displayd in Fig. 7.3, th ral part of A ϕ is displayd on th lft and th imaginary part on th right. z in m a) A b) ϕ (ral) A ϕ (imaginary) r in m r in m 0 Figur 7.3: Simulation rsult using MQSFIT: a) ral part and b) imaginary part of A ϕ distribution in hot wir. Paramtr: N r = 500, N z = 500, = x = z = 0.1 mm, f = 50 khz, σ = A/Vm and µ = µ Modling of th Rciving Coils Th cross sction of a rciving coil is shown in Fig. 7.4a. Howvr, a coil consists of coppr wirs and cannot b ralizd as a block of coppr. Fig. 7.4b shows an approximatd viw of th cross sction of th rciving coil, whr th cross sctions of th coppr wirs ar drawn as blu circls and = r = z. W comput A ϕ for ach of th nod which rsids insid th coil using MQSFIT. Th lctric fild strngth E(R, ω) is thn computd in th scond stp by E ϕ (R, ω) = j ωa ϕ (R, ω). (7.1)

99 90 Chaptr 7. Simulation Rsults a) Rciving Coil b) Approximatd Coil 2 3 mm 3 mm 2 Figur 7.4: a) cross sction of a rciving coil and b) th approximatd gomtry of th coil; = r = z. Th ϕ componnt of th lctric fild strngth E ϕ (R, ω) at th cntr of ach conductor is dtrmind by Eq. (7.1). W assum that insid ach conductor E ϕ (R, ω) rmains constant and thrfor, th inducd lctric voltag V (ω) for ach conductor of th rciving coil is calculatd by V (ω) = E(R, ω) dr = 2πR con E ϕ (R, ω), (7.2) whr R con is th radius of th considrd conductor. Th total inducd lctric voltag of th rciving coil is thn computd as whr th rciving coil consists of N t turns of coppr wir. V tot (ω) = N t V (ω), (7.3) a) b) Voltag in mv z in m Voltag Chang in mv z in m Figur 7.5: Simulation rsult using MQSFIT: a) Inducd lctric voltags of an ncircling coil snsor as a function of crack position and b) chang of inducd lctric voltags with th prsnc of a crack as a function of crack position.

100 7.1. Simulation Rsults of Encircling Coil Snsors Rsults and Prformanc Analysis of an Encircling Coil Snsor without Frrit Cor A plot of th inducd lctric voltags rcordd at th snsor coil as a function of crack position is givn in Fig. 7.5a (Rahman & Marklin, 2006). Th inducd lctric voltag at th rcivr coils without any crack is mv. Taking this voltag as an offst, th chang of th inducd voltag with th prsnc of a crack is plottd in Fig. 7.5b. If w tak a clos look of th magntic flux distribution in Fig. 7.3, w find that masurs can b takn to focus th magntic flux dnsity towards th stl surfac, which will incras th inducd lctrical voltag at th snsor coil and will nsur bttr dtction. Th outcom of th modling rsults lad to th dvlopmnt of a modifid vrsion of th ncircling coil snsor which is latr manufacturd by Msstchnik, Univrsity of Kassl (Msstchnik, KS) and is prsntd in th nxt sction Encircling Coil Snsor with Frrit Cor to Simulat a Transvrsal Crack An ncircling coil snsor with frrit cor, dvlopd by th Univrsity of Kassl, is a modifid vrsion of an ncircling coil. Th simulation of th dtction procss with th prsnc of a transvrsal crack will b discussd hr. z in m Hot Wir Crack Zoomd Viw of Snsor Frrit Cor Excitation Coil Rcivr Coil 0.5 mm 3 mm 0.8 mm 3 mm Zoomd Viw of Crack 0.01 l = 5 mm r in m d = 0.5 mm Figur 7.6: Schmatic viw of th gomtry of th hot wir stl with a crack at z = 20 mm mbddd in fr-spac and a sktch of an ncircling coil snsor with frrit cor.

101 92 Chaptr 7. Simulation Rsults Gomtry of an Encircling Coil Snsor with Frrit Cor An ncircling coil snsor with frrit cor has bn shown in Fig Th xcitations coils ar markd as rd and th rciving coils ar markd as blu. Th frrit cor of th thicknss of 0.8 mm rsids in and around th snsor coils. A so calld T38 typ frrit matrial is usd hr which posss a high magntic prmability µ r = 5000 and thrfor, it is suitabl for guiding magntic flux towards th stl wir. Th othr gomtric dtails ar th sam as th ncircling coil snsor without frrit Numrical Tchniqu to Simulat an Encircling Coil Snsor with Frrit Cor Th simulations ar prformd in cylindrical coordinats by finit lmnt mthod (FEM) with th hlp of th stady-stat AC modul of Opra-2d (Opra, 10.5). W hav usd = lmnts to discrtiz th simulation domain, which rsids in rz-plan. Th radial componnt of th magntic flux dnsity B r is displayd in Fig As th magntic flux lins ar focusd towards th stl bar with th hlp of th frrit cor, w obsrv hr highr flux concntration and hnc, highr inducd voltag than in th cas without frrit cor (s Fig. 7.8). a) Without Frrit Cor Hot Wir b) With Frrit Cor Hot Wir z in m Crack Crack r in m r in m Figur 7.7: Fild distribution of th radial componnt of th magntic flux dnsity of an ncircling coil snsor systm: a) without with frrit cor and b) with frrit cor. Paramtr: no. of triangular lmnts N = 20000, σ = A/Vm, µ = µ Rsults and Prformanc Analysis of an Encircling Coil Snsor with Frrit Cor A comparison of th inducd lctric voltags rcordd at th snsor coil of th modifid ncircling coil snsor systm with that of a traditional ncircling coil snsor systm is givn in Fig. 7.8, whr w find that th modifid ncircling coil snsor with frrit cor shows a highr snsitivity than a traditional on (Rahman & Marklin, 2009). Fig. 7.9 shows a

102 7.1. Simulation Rsults of Encircling Coil Snsors 93 a) Without Frrit Cor b) With Frrit Cor Voltag in mv z in m z in m Figur 7.8: Chang of inducd lctric voltags with th prsnc of a crack as a function of crack position obtaind from a) an ncircling coil snsor systm without frrit cor and b) with frrit cor. comparison of th synthtic A-scan to th masurd A-scan, whr th synthtic A-scan is shown on th right and th simulatd A-scan on th right. a) Masurd A-scan b) Voltag in mv Simulatd A-scan z in m z in m Figur 7.9: Eddy currnt tsting of hot wir with an ncircling coil with frrit cor: a) masurd A-scan and b) simulatd A-scan Encircling Coil Snsor without Frrit Cor to Simulat a Longitudinal Crack Th dtction of th transvrsal cracks is not nough to mt th goals of th INCOSTEEL projct. Dpnding on th production procss, th prsnc of th longitudinal cracks at th uppr surfac of hot wir is quit usual and thrfor, th ncircling coil snsors nd to b tstd whthr thy ar capabl of dtcting such cracks. In this sction w rport th modling of th dtction of a longitudinal crack with an ncircling coil snsor without frrit cor.

103 94 Chaptr 7. Simulation Rsults Gomtry of an Encircling Coil Snsor without Frrit Cor Th gomtry of th ncircling coil snsor without frrit cor is alrady discussd in Sc (s Fig. 7.2). Th two outr coils of th snsor work as xcitation coils, whil th innr two markd as R1 and R2 in Fig. 7.10, function as rcivr coils. Th dimnsion of th crack is lngth l = 20 mm, dpth d = 0.5 mm and width w = 0.5 mm. A stl wir of th lngth L = 50 mm and a diamtr D = 12 mm is tstd hr. Figur 7.10: A snapshot of th hot wir inspction with a longitudinal crack. Gomtry of Crack: l = 20 mm, d = 0.5 mm, w = 0.5 mm, Paramtr: f = 50 khz, σ = A/Vm and µ = µ Numrical Tchniqu to Simulat an Encircling Coil Snsor without Frrit Cor It is not possibl to simulat a longitudinal crack using rotational symmtry and hnc, w hav to prform a 3-D modling in this cas. Th simulations ar accomplishd by boundary lmnt mthod (BEM) with th hlp of Faraday 6.1 (Faraday, 6.1). Fig shows a snapshot of th dtction procss whr th ddy currnt distribution on th stl surfac is

104 7.1. Simulation Rsults of Encircling Coil Snsors 95 displayd in th bottom and th obtaind diffrntial signal (vr1 vr2) from th rcivr coils is plottd on th top Rsults and Prformanc Analysis of an Encircling Coil Snsor without Frrit Cor A plot of th inducd lctric voltags rcordd at th snsor coil as a function of crack position is givn in Fig Th first pak indicats th bginning of th crack, whil th scond pak indicats th nd. Such a snsor dtcts only th dgs of th cracks and dos not giv continuous rspons whil a longitudinal crack is crossing th snsor coils. Thortically, it is possibl to dtct a longitudinal crack with an ncircling coil snsor, as 3.12 vr1-vr Voltag in mv y in mm Figur 7.11: Diffrntial signal vr1 vr2 obtaind from an ncircling coil snsor without frrit cor in prsnc of a longitudinal crack. w hav sn in th simulation rsults. Practically, in a rolling mill whr mor than on cracks can occur at th sam tim, it is almost impossibl to dtct th bginning and nd of ach longitudinal cracks with an ncircling coil snsor. As a rsult, nw snsors nd to b dvlopd which will not only dtct th bginning and nd of th crack, but also produc continuous rspons whil th longitudinal cracks ar passing by Innovativ Idas to Dtct Longitudinal Cracks Th modling rsults in th prvious sction show that a nw kind of ddy currnt snsor has to b dvlopd to dtct th longitudinal cracks. As w obsrv from Fig that an ncircling coil snsor producs a ddy currnt distribution which is paralll to th snsor coils and thrfor, a longitudinal crack wkly influncs such a distribution. As a rsult, an ncircling coil snsor can only dtct th crack tips and no continuous rspons can b obtaind. To solv this problm w nd to produc such an ddy currnt distribution which is mor snsitiv to th longitudinal on, i.., a distribution which is not in th prpndicular dirction to th lngth of th longitudinal crack. A circular uniform distribution on th stl surfac is a suitabl option in this sns but to produc such a distribution w rquir point coils. Basd on this thortical background th first point coil snsor, also known as a 3-coils snsor is suggstd by Tcnatom (Tcnatom, S.A.). Howvr, th inspction tchniqu nds

105 96 Chaptr 7. Simulation Rsults to b modld bfor manufacturing such a snsor systm. Furthrmor, th bhaviour of th point coil snsor systm in array mod also nd to b studid. 7.2 Simulation Rsults of Point Coil Snsors A numbr of point coil snsors ar dvlopd in INCOSTEEL (INCOSTEEL, 2004) projct to dtct th longitudinal cracks. W shall discuss in this sction th following typs of point coil snsors: 3-coils snsor in unitary mod 3-coils snsor in array mod D-coils snsor 4-coils snsor Pot cor snsor A 3-Coils Snsor in Unitary Mod A 3-coils snsor consists of thr small lmntary coils (s Fig. 7.12a) whr on of th coils oprats as th xcitation coil and th othr two coils oprat as th rciving coils. For tst purpos this typ of snsor is first opratd in unitary mod. a) Coil b) 1 mm Frrit Cor Coil 3 mm c) Hot Wir 3.1 mm 1 mm Crack 5 mm 10 mm 5 mm 5 mm Figur 7.12: a) Photo of a 3-coils snsor, b) gomtric dtail of a singl lmnt of a 3-coils snsor and c) tst stup using a 3-coils snsor.

106 7.2. Simulation Rsults of Point Coil Snsors Gomtry of a 3-Coils Snsor in Unitary Mod Each lmnt of a 3-coils snsor has th hight of 3 mm and th diamtr of 3.1 mm (s Fig. 7.12b). 260 turns of coppr wir is wound ovr a frrit cor to form a singl lmnt. In Fig. 7.12c th lmnts markd as R1 and R2 work as th rciving coils. Th othr lon lmnt works as th xcitation coil, which posss a currnt dnsity of J imp (R, ω) = A/m 2, whr a monochromatic frquncy of f = 74 khz is usd. Th coils rsid 0.5 mm abov th stl surfac and 5 mm apart from ach othr (s Fig. 7.12c). Figur 7.13: A snapshot of th dtction procss; top: inducd voltag in th rciving coils, bottom: ddy currnt distribution on th hot wir Numrical Tchniqu to Simulat a 3-Coils Snsor in Unitary Mod Th 3-coils snsor systm is simulatd boundary lmnt mthod (BEM) with th hlp of Faraday 6.1 (Faraday, 6.1). To discrtiz th gomtry th ddy currnt skin dpth of th hot wir stl is calculatd as 1 χ = = 1.55 mm, (7.4) πfµσ whr rlativ prmittivity of hot wir stl µ = µ 0 abov th Couri point and th lctric conductivity σ = A/Vm. Th lngth of a sid of ach triangular lmnt should

107 98 Chaptr 7. Simulation Rsults b lss than th skin dpth and thrfor, triangls having th ara of 0.21 mm 2 hav bn chosn. A total numbr of N = 6000 triangular lmnts ar usd to discrtiz th stl surfac, whras ach of th frrit cor is discrtizd using 308 triangular lmnts. Th obtaind linar matrix quation has bn solvd using gnralizd minimal rsidual (GM- RES) mthod (Mistr, 2008). Fig shows a snapshot of th dtction procss. L = 50 mm of a stl wir of th diamtr D = 31 mm has bn simulatd hr. Th siz of th crack is: lngth l = 10 mm, width w = 1 mm and dpth d = 1 mm. Th ddy currnt distribution on th stl surfac is displayd in th bottom of Fig. 7.13, whr an xcitation frquncy of f = 74 khz is usd. Th obtaind voltags in th snsor coils ar rcordd and th diffrntial signal is plottd on th top of Fig to dtrmin th rspons of th snsor whn a crack crosss th snsor. a) Masurd A-scan b) Simulatd A-scan Voltag in mv y in mm y in mm Figur 7.14: Hot wir tsting with a 3-coils snsor: a) masurd A-scan and b) simulatd A-scan Rsults and Prformanc Analysis of a 3-Coils Snsor in Unitary Mod Th synthtic A-scan, obtaind from th simulation, is validatd against th masurd A- scan in Fig Th comparison shows satisfactory rsults and provs th accuracy of th simulation (Rahman & Marklin, 2008; Rickn t al., 2008). Th modling of th inspction tchniqu shows good dtction of th longitudinal cracks with a 3-coils snsor. This lads to th manufacturing of th prototyp of this snsor and tsting in th industrial nvironmnt in TU Fribrg (TU Fribrg, FB). A 3-coils snsor shows good rsults to dtct th longitudinal cracks in laboratory as wll as industrial nvironmnt. Howvr, this snsor shows a major drawback whn any longitudinal crack rsids dirct bnath th xcitation coil. In this cas, both th xcitation coils show th sam voltags, and hnc no diffrntial signal can b obtaind. A modification nds to b prformd to ovrcom this difficulty. Furthrmor, for succssful tsting of th hot wir w hav to oprat a 3-coils snsor in array mod and hnc, mor simulations nd to b prformd to valuat th array mod of such a snsor A 3-Coils Snsor in Array Mod Th us of an array of snsors shows bttr prformanc in dtcting th cracks than a singl snsor. That s why it is important to study th prformanc of snsors whn thy work in an array.

108 7.2. Simulation Rsults of Point Coil Snsors 99 Figur 7.15: A snapshot of th dtction procss with an array of 3-coils snsor consists of 2 xcitation coils and 3 rcivr coils; top: inducd voltag in th rciving coils and bottom: ddy currnt distribution on th hot stl wir Gomtry and Numrical Tchniqu to Simulat a 3-Coils Snsor in Array Mod A snsor array consisting 2 xcitation coils and 3 rcivr coils is simulatd in Fig Th diffrntial signal obtaind from th rcivr lmnts R1 and R2 is shown on top. On th bottom th ddy currnt distribution on th hot wir is shown. Th simulatd crack has th lngth of l = 10 mm and th dpth of d = 1 mm. Th xcitation is prformd using (R, ω) = A/m 2 in th xcitation coils whr a frquncy of f = 74 khz is usd. Th shadowd bar of th A-scan indicats th inducd voltag at th particular crack position. As this typ of snsor is an dg-dtctd on, w obsrv th chang in th inducd voltag only at th bginning and nd of th crack. a currnt dnsity J imp Simulation Rsults and Prformanc Analysis of a 3-Coils Snsor in Array Mod Th voltags obtaind from th rcivr coils R1, R2 and R3 ar plottd in Fig Th calculatd diffrnc signals vr1 vr2 and vr1 vr3 ar shown in Fig Comparing

109 100 Chaptr 7. Simulation Rsults a) vr1 b) vr2 c) vr Voltag in mv y in mm y in mm y in mm Figur 7.16: Inducd voltags at a) rcivr 1 (vr1), b) rcivr 2 (vr2) and c) rcivr 3 (vr3). a) vr1 vr2 b) vr1 vr3 Voltag in mv y in mm y in mm Figur 7.17: Diffrntial signal obtaind from a 3-coils snsor in array mod: a) vr1-vr2 and b) vr1-vr3. Fig to Fig w find that th inducd voltag obtaind from a 3-coils snsor in array mod is highr than that in unitary mod, which provs, a 3-coils snsor shows bttr snsitivity in array mod than in unitary mod. Howvr, it is difficult to manufactur such arrays bcaus of th angular displacmnt btwn th rings of xcitation and rcivr coils. To ovrcom this difficulty a 4-coils snsor is dvlopd which has simplr gomtry and also bttr snsitivity Simulation Rsults of a D-coils Snsor Univrsity of Kassl has dvlopd svral dsigns of surfac prob coils throughout th projct. Th aim of this dsign was th dtction of longitudinal dfcts, although othr dfct orintations wr also dtctd Gomtry of a D-coils Snsor A D-coils snsor consists of an oval shapd xcitation coil (s Fig. 7.18a), which rsids around th rciving coils. Th rciving coils ar dvlopd insid th pot formd frrit

110 7.2. Simulation Rsults of Point Coil Snsors 101 cors (s Fig. 7.18b). Th xcitation coil is built of 150 turns of coppr wir and ach of th rciving coils is formd of 40 turns of coppr wir. Th fficincy of th prototyp has to b dtrmind through numrical modling bfor fabricating it for industrial tst. a) b) Excitation Coil Rcivr Coil Frrit Cor 2.5 mm 4 mm 10.1 mm c) 14 mm 10 mm 11.9 mm Hot Wir 5.4 mm 1 mm Crack Figur 7.18: a) Photo of a D-coils cor snsor, b) frrit cor of th rcivr coils of th snsor and c) top viw of tst stup using a d-coils snsor Numrical Tchniqu to Simulat a D-coils Snsor Fig shows a snapshot of th dtction procss using a D-coils snsor. 50 mm of a stl wir of th diamtr D = 12 mm has bn simulatd hr. Th siz of th crack is: lngth l = 10 mm, width w = 1 mm and dpth d = 1 mm. An xcitation frquncy of f = 400 khz is usd and th ddy currnt distribution on th stl surfac is displayd in th bottom of Fig Th obtaind voltags in th snsor coils ar rcordd and th diffrntial signal is plottd on th top of Fig to dtrmin th rspons of th snsor whn a crack crosss th snsor. Th simulation is prformd using th boundary lmnt mthod (BEM) with th hlp of Faraday 6.1 (Faraday, 6.1) Simulation Rsults and Prformanc Analysis of a D-coils Snsor Th inducd voltags in th rcivr coils R1 and R2 ar plottd in Fig. 7.20a and Fig. 7.20b rspctivly, whras th diffrntial signal vr1 vr2 is plottd in Fig. 7.20c. Thr ar som major cons of th D-coils snsor. As this snsor is largr than th stl wir of th diamtr of D = 12 mm, it fails to produc a uniform ddy currnt distribution on th stl surfac and thrfor, offrs poor snsitivity. Furthrmor, th rcivr coils ar orintd along th axis of th stl wir, which causs difficultis in dtrmining th position and siz of th longitudinal cracks (s Fig. 7.20c). Du to ths drawbacks w conclud from numrical modling that this prototyp is not suitabl in dtcting longitudinal cracks and thrfor, modifications nd to b prformd to ovrcoms ths problms.

111 102 Chaptr 7. Simulation Rsults Figur 7.19: Hot wir tsting using a d-coils snsor; top: inducd voltag in th rciving coils, bottom: ddy currnt distribution on th hot wir. a) vr1 b) vr2 c) vr1 vr Voltag in mv y in mm y in mm y in mm Figur 7.20: Inducd voltags at a) rcivr 1 (vr1), b) rcivr 2 (vr2) and c) diffrntial signal (vr1 vr2).

112 7.2. Simulation Rsults of Point Coil Snsors 103 On ida to modify this snsor is to rduc th siz of th pot shapd frrit cors of th rciving coils and to rdsign th xcitation coils. Th scond ida is to us sam typ of xcitation coils as th rciving coils. Ths two idas lad to th dvlopmnt of th pot coil snsor which will b discussd latr Simulation Rsults of a 4-Coils Snsor Th 4-coils snsor, a modifid vrsion of th 3-coils snsor, is dvlopd by Tcnatom in ordr to ovrcom th difficultis of a 3-coils snsor. Howvr, th initial prototyp nds to b vrifid and optimizd through numrical simulations to dvlop th final prototyp. Aftr th numrical valuation of th first prototyp w hav xcutd a paramtr study to find th optimum distanc btwn th coils. a) b) 1 mm Frrit Cor Coil 0.9 mm 2.1 mm c) Hot Wir Coil Hol 4.2 mm R2 0.5 mm Crack 4.7 mm R1 10 mm 5 mm Figur 7.21: a) Photo of a 4-coils snsor, b) gomtry of a singl lmnt of a 4-coils snsor and c) top viw of tst stup using a 4-coils snsor Gomtry of a 4-Coils Snsor A 4-coils snsor consists of four coils, whr two of thm oprat as xcitation coils and th othr two (markd as R1 and R2 in Fig. 7.21c) oprat as rciving coils. 260 turns of coppr wir is wound ovr a frrit cor of th hight of 3 mm and th width of 1mm (s Fig. 7.21b). Two rings of snsor coils ar usd hr, whr ach ring contains ight coils. Th optimum distanc btwn an xcitation and a rciving coil is 6 mm which is dtrmind by a paramtric analysis.

113 104 Chaptr 7. Simulation Rsults Numrical Tchniqu to Simulat a 4-Coils Snsor A surfac crack of th lngth l = 10 mm, width w = 0.5 mm and dpth d = 0.5 mm is dtctd using a 4-coils snsor systm in Fig Th tstd stl wir has a lngth of L = 50 mm and a diamtr D = 12 mm. Th xcitation coils ar fd with a currnt dnsity of J imp (R, ω) = A/m 2 with a frquncy of f = 200 khz. Th rsulting ddy currnt distribution on th stl surfac is displayd in th bottom of Fig Th Figur 7.22: A snapshot of th dtction procss using a 4-coils snsor; top: inducd voltag in th rciving coils, bottom: ddy currnt distribution on th hot wir. obtaind voltags in th snsor coils ar rcordd and th diffrntial signal is plottd on th top of Fig to dtrmin th rspons of th snsor. Th boundary lmnt mthod (BEM) is implmntd hr to modl this ddy currnt problm with th hlp of Faraday 6.1 (Faraday, 6.1). Inducd voltags at diffrnt rciving points hav bn shown in Fig Th rcivr 1 shows highr inducd voltag than that of rcivr 2 bcaus it rsids ovr th crack (Rahman & Marklin, 2009). A diffrntial signal vr1 vr2 is plottd on th lft sid of Fig

114 7.2. Simulation Rsults of Point Coil Snsors Simulation Rsults and Prformanc Analysis of a 4-Coils Snsor Lik its prdcssor a 4-coils snsor also shows good prformanc in dtcting longitudinal cracks. Comparing th rsults of a 3-coils snsor with that of a 4-coils snsor rvals that a 4-coils snsor shows highr snsitivity in prsnc of a crack (s Fig and Fig. 7.24). Furthrmor, unlik a 3-coils snsor this snsor systm dos not show a blind zon whn a crack rsids dirct bnath th xcitation coils. Th modling rsults show that a 4- coils snsor posss high snsitivity and bttr prformanc and hnc, is suitabl for final prototyp. This lads to th manufacturing of th final prototyp of a 4-coils snsor, which has shown vry good prformanc in laboratory tsts as wll as in th rolling mills in TU Fribrg. a) b) vr vr2 Voltag in mv y in mm y in mm Figur 7.23: Inducd voltags at a) rcivr 1 (vr1) and b) rcivr 2 (vr2). a) Masurd A-scan b) 39.2 Voltag in mv Simulatd A-scan (vr1 vr2) y in mm y in mm Figur 7.24: Hot wir tsting with a 4-coils snsor: a) masurd A-scan and b) simulatd A-scan (vr1 vr2) Effct of th Lngth of a Crack on Snsor Prformanc Th ffct of th lngth of a crack on th prformanc of a 4-coils snsor nd to b dtrmind bfor final tst in Sidnor (Sidnor, S.A.). Numrical modling is obviously th bst way

115 106 Chaptr 7. Simulation Rsults for such an valuation. In our prvious simulations, w hav usd a crack lngth of l = 10 mm. A hot wir stl with a surfac crack of th lngth l = 15 mm, width w = 0.5 mm and dpth d = 0.5 mm is inspctd using a 4-coils snsor systm in Fig Th gomtrical dtails of th snsor and th hot wir rmain unchangd. Fig shows a comparison btwn th diffrntial signals for a crack of th lngth l = 10 Figur 7.25: A snapshot of th dtction procss using a 4-coils snsor with a crack of th lngth l = 15 mm; top: inducd voltag in th rciving coils, bottom: ddy currnt distribution on th hot wir. mm and that of a crack of th lngth l = 15 mm. In cas of l = 10 mm (s Fig. 7.26a) w obsrv highr oscillation at th bginning and at th nd of th voltag pak, whras, in cas of l = 15 mm (s Fig. 7.26b) th voltag ris to th pak lvl is rlativly smooth. Furthrmor, this comparison shows that a 4-coils snsor givs continuous rspons as long as th crack is thr and thrfor, is suitabl to dtct longitudinal cracks of any lngth in hot wir stl. Bcaus of good prformanc in all th th numrical valuations, a 4-coils snsor is accptd for th final dsign and latr usd to produc a combind snsor systm.

116 7.2. Simulation Rsults of Point Coil Snsors 107 a) vr1-vr2 for l = 10 mm 39.2 b) vr1-vr2 for l = 15 mm Voltag in mv y in mm y in mm Figur 7.26: Hot wir tsting with a 4-coils snsor with a crack of th lngth a) l = 10 mm and b) l = 15 mm Simulation Rsults of a Pot Cor Snsor A pot cor snsor is a modifid vrsion of a D-coils snsor and is dvlopd by Msstchnik, Univrsity of Kassl (Msstchnik, KS). Th aim of this dsign is to apply th idas, obtaind from th numrical modling of th D-coils snsor and thus dvlop a nw snsor as an altrnativ to th 4-coils snsor for th dtction of longitudinal cracks. W hav tstd a numbr of diffrnt orintation of th coils of a pot cor snsor by numrical modling and hav found th gomtry shown in Fig. 7.27d as th bst on. a) b) c) 0.75 mm 0.25 mm 1.15 mm Coil Frrit Cor Snsor 2.5 mm d) 4 mm Hot Wir 1 mm Crack mm Figur 7.27: a) Photo of a pot cor snsor, b) pot shapd frrit cor of th snsor, c) pot cor with coil insid and d) top viw of tst stup using a pot cor snsor.

117 108 Chaptr 7. Simulation Rsults Gomtry of a Pot Cor Snsor Th formation of a pot cor snsor systm is similar to a 4-coils snsor systm. It also consists of four coils, whr two of thm oprat as xcitation coils and th othr two as rciving coils. Each coil has its own pot formd frrit cor (s Fig. 7.27b), which oprats as a magntic shild from th nighboring coils and a guid for th magntic flux towards th stl surfac. 140 turns of coppr wir is wound insid a frrit cor to form a coil, which has th hight of 1.15 mm and th outr diamtr of 2.5 mm (s Fig. 7.27c). To illustrat th siz of a pot cor, a 10 cnt coin is shown by th sid of a pot cor in Fig. 7.27a. Twlv coils ar positiond with an angular coil to coil distanc of 30 to form a ring and two such rings ar usd in this snsor systm - on for th xcitation coils and th othr on for th rcivr coils. Figur 7.28: Hot wir tsting using a pot cor snsor; top: inducd voltag in th rciving coils, bottom: ddy currnt distribution on th hot wir Numrical Tchniqu to Simulat a Pot Cor Snsor A snapshot of th dtction procss using a pot cor snsor is shown in Fig Th dtction procss is simulatd using th boundary lmnt mthod (BEM). Th siz of th

118 7.3. Slction of Eddy Currnt Snsors for th Final Prototyp 109 crack is: lngth l = 10 mm, width w = 1 mm and dpth d = 1 mm. An xcitation frquncy of f = 300 khz is usd and th ddy currnt distribution on th stl surfac is displayd in th bottom of Fig W apply a currnt dnsity J imp (R, ω) = A/m 2 in th xcitation coil and th rsulting inducd lctric voltags in th rcivr coils vr1 and vr2 ar rcordd and th diffrntial signal vr1 vr2 is plottd in Fig Simulation Rsults and Prformanc Analysis of a Pot Cor Snsor Du to th pot formd frrit cor, this snsor systm shows lss snsitivity than a 4-coils snsor systm. W conclud from Fig and Fig that a pot cor snsor shows a pak of mv at th prsnc of a crack whras, a 4-coils snsor shows a 39.2 mv pak. Howvr, th compactnss of th pot cor snsor is its major advantag ovr th othr snsor systms. As a rsult, a highr numbr of coils can b positiond around th hot wir to covr th ntir circumfrnc. In snapshot Fig th coils ar positiond at 30 angular distanc and thus, an array of 12 coils ar usd for xcitation or rcption. On th othr hand, in cas of th prvious point coil snsors, such as a 4-coils snsor or a 3-coils snsor w hav usd an angular displacmnt of 45 to position th coils, which mans, only 8 coils can b positiond in an array. Th us of highr numbr of lmnts in xcitation vr1-vr2 Voltag in mv y in mm Figur 7.29: Inducd voltags at a) rcivr 1 (vr1) and b) rcivr 2 (vr2) of a pot cor snsor. and rcption array rsults lss blind zons and nsurs bttr dtction and hnc, a pot cor snsor is also a good choic for hot wir stl inspction. According to th numrical rsults w conclud that a pot cor snsor is also a suitabl candidat for final dsign and thrfor, such a snsor is also usd in th combind EC snsor systm. 7.3 Slction of Eddy Currnt Snsors for th Final Prototyp W hav discussd diffrnt ncircling coil and point coil snsors in th prvious sctions. Th advantags and disadvantags of ach of th dvlopd snsors nd to b considrd in ordr to slct suitabl snsors for final prototyp.

119 110 Chaptr 7. Simulation Rsults Encircling coil without frrit cor: + Good prformanc for dtcting transvrsal cracks Poor dtction of longitudinal cracks Encircling coil with frrit cor: + Modifid vrsion of an ncircling coil without frrit cor + Vry Good dtction of transvrsal cracks Poor prformanc in dtcting longitudinal cracks 3-coils snsor: + Dsignd to dtct th longitudinal cracks Problms xist du to blind zons and gomtric difficulty D-coils snsor: Poor dtction of th cracks du to larg siz and orintations of th coils 4-coils snsor: + Modifid vrsion of th 3-coils snsor + Offrs th highst snsitivity + Vry good dtction of th longitudinal cracks Pot cor snsor: + Modifid vrsion of th D-coils snsor + Du to vry small siz highr numbr of lmnts can b usd in th array which nsurs lss blind zons + Good dtction of th longitudinal cracks Th analysis of th modld snsors, givn abov, shows that an ncircling coil snsor with frrit cor is suitabl for dtcting th transvrsal cracks, whras a 4-coils snsor and a pot cor snsor shows good rsults in dtcting longitudinal cracks. As a rsult, a mixd snsor systm: an ncircling coil and a 4-coils snsor or an ncircling coil and a pot cor snsor ar th suitabl options to fulfill our goal. 7.4 Vlocity Analysis Th linar vlocity of th hot wir influncs magntic flux distribution and thus causs chang in th inducd voltag in th snsor coils. Th ffct of vlocity on th prformanc of th snsor systm is discussd in this sction. As mntiond bfor, th ddy currnt problm in cas of a moving conductor is solvd by dtrmining th magntic vctor potntial A m (R, ω) using Eq. (2.146). Hr w simulat th hot wir inspction procss with th hlp of th Linar Motion (LM) modul of Opra-2d (Opra, 10.5) to dtrmin th vlocity ffct. Th LM modul is a Transint Eddy Currnt Solvr, xtndd to includ th ffcts of motion. Th simulatd modl has a rotational symmtry, whr th motion is in z- dirction. Th output tim points ar slctd first and at ths tim points, th transint valu of th inducd voltags ar rcordd. Th applid voltag at th driving circuit of th xcitation coils is V = V (rms). Th construction of th simulatd ddy currnt snsor is similar to th ncircling coil snsor, dscribd in Sc. 7.1.

120 7.4. Vlocity Analysis Simulation Rsults for Diffrnt Vlocitis Th simulatd crack has th lngth l = 5 mm and th dpth d = 0.5 mm. At tim t = 0, th crack lis btwn z = 15 mm and z = 20 mm zon of th stl wir (s Fig. 7.30a). Th stl wir movs in th positiv z-dirction at a linar vlocity v c. Fig. 7.30b shows th a) Gomtry 50 z in mm Crack Excitation Coil Rcivr Coil r in mm Voltag in mv b) Inducd Voltag vs. Crack Position z in mm 0 m/s 5 m/s 10 m/s 30 m/s 50m/s Figur 7.30: a) Gomtry of th xprimntal stup for vlocity analysis and b) inducd lctric voltags at diffrnt vlocitis: v c = 0 m/s (black), 5 m/s (rd), 10 m/s (blu), 30 m/s (magnta) and 50 m/s (cyan). comparison of th rcivd voltags at th vlocity of v c = 5, 10, 20, 30, 50 m/s of th hot wir. Th ffct of th incrasing vlocity is xplaind by rcalling Eq. (2.146) as A m (R, ω) + µσ (R) j ω A m (R, ω) + v c A m (R, ω) Φ (R, ω)] = µj imp (R, ω) (7.5) and th lctric fild strngth E(R, ω) is thn computd from th magntic vctor potntial A m (R, ω) by E(R, ω) = j ωa m (R, ω) + v c A m (R, ω) Φ (R, ω), (7.6) which is writtn in tim-domain as E(R, t) t = A m(r, t) t + v c (R, t) A m (R, t) Φ (R, t). (7.7) Th addition of th vlocity trm v c (R, t) A m (R, t) causs a ngativ ffct in th inducd voltag which is shown by Eq. (7.7) mathmatically and is also visibl in Fig. 7.30b Validation of th Simulation Rsults A validation of th vlocity analysis is important to xamin th rsults of th simulation. As no masurmnts hav bn prformd with vlocity, this validation has bn don against

121 112 Chaptr 7. Simulation Rsults a) Gomtry 100 Crack b) B r distribution E1 Excitation Coil 50 z in mm 0 Conductor 0 50 R1 Rcivr Coil r in mm r in mm Figur 7.31: a) Gomtry of th xprimntal stup for validation of th rsults and b) distribution of th radial componnt of th magntic flux B r. a) Publishd A-scan b) Simulatd A-scan Flux dnsity in Vs/m z in mm z in mm Figur 7.32: a) Publishd flux dnsity and b) simulatd flux dnsity. th rsults publishd by Yu t al. (1998). Th gomtry of th xprimntal stup is shown in Fig. 7.31a. On th lft part is th conductor which has a crack on th top. Th lngth of th crack is l = 4 mm and th dpth d = 2 mm. Th conductor is moving in th z-dirction with th vlocity of v c = 50 m/s. Thr ar two coils in this figur, mntiond as E1 and R1. Hr E1 is th xcitation coil and R1 is th rciving coil. Th distribution of th radial componnt of th magntic flux dnsity B r is shown in Fig. 7.31b. Th flux dnsity at th uppr surfac of th conductor is plottd as a function of distanc in Fig. 7.32, whr th flux dnsity from th publishd papr (Yu t al., 1998) is plottd on th lft and th simulatd rsult on th right (Ebby, 2007). Th comparison shows that th simulatd rsult is idntical to th publishd on what provs th accuracy of vlocity modling.