2. ELECTROMAGNETIC THEORY UNDERLYING NUMERICAL SOLUTION METHODS

Transcription

1 Chaper. LCTROMAGNTIC THORY UNDRLYING NUMRICAL SOLUTION MTHODS lecromagneic cience i eviden in modern echnology pervading everyday life. From he heavie and mo powerful moor o he malle ranior, ued in oday' microchip circui, elecromagneic cience play an imporan role in human echnological hiory. lecromagneic have heir roo in he ih cenury B.C. in ancien Greece where Thale of Mileu noed ha amber (elecron in Greek rubbed wih a dry wood produced park and araced ligh paricle. He alo noed he aracion beween naural magneic rock found in an area called Magneia. Tweny-wo cenurie laer William Gilber in a more cienific han philoophical approach invened he elecrocope for meauring elecroaic effec and he decribed he earh a a huge magne. Many cieni worked on differen properie of elecriciy and magneim in he year following. Jame C. Mawell combined all he heorie in a unified elecromagneic heory and formulaed hi well known equaion which are unaffeced by he radical change reuling from Alber inein relaiviy heory and quanum phyic in he beginning of he 0 h cenury. Mawell' equaion prediced he eience of elecromagneic wave, he naure of ligh a a form of elecromagneic wave and he propagaion peed of ligh. In he following ecion, Mawell' equaion are briefly dicued a hee are ued in he developmen of he Finie-Difference Time-Domain (FDTD mehod, which i eenially a numerical oluion of he differenial form of Mawell equaion implemened in dicree form. The decripion of he mehod of momen, which normally refer o he numerical oluion of he inegral form of Mawell equaion conclude he chaper.. Gradien, Divergence and Curl. In conra o calar, which have only magniude, vecor quaniie have boh magniude and direcion. Vecor are eenively ued in analyical elecromagneic. In engineering mahemaic i i ueful o conider he rae of change of vecor. The Page /

2 Chaper differenial form of vecor i epreed mahemaically uing gradien, divergence, and curl, which are given below in he recangular coordinae yem. The gradien of a calar quaniy f i a vecor given by gradf where f f f f ˆ yˆ ˆ (. y ˆ, yˆ and ẑ are uni vecor in he, y and direcion repecively. The divergen of a vecor A, on he oher hand, i a calar given by A div A A A y y A (. Finally, he curl of a vecor A i a vecor and i given by A curl A A ( y A A yˆ( A A ˆ( A y ˆ A y y ˆ (.3 The gradien, divergence and curl can alo be epreed in cylindrical and pherical coordinae a hown in Appendi A. yˆ y A y ˆ A. Mawell' equaion and elecromagneic heory. Mawell ha done for elecric and magneic force wha Alber inein ried and failed o do many year laer for all he force of naure. Mawell ucceeded in uing he law aribued o Gau, Amperè and Faraday in formulaing a unified elecromagneic heory. Hi heory ha mainained i validiy even afer he inroducion of he relaiviy heory in he beginning of he 0 h cenury and enabled eplanaion and predicion of many naural phenomena like he eience of elecromagneic wave in he form of radiowave and ligh. Thee equaion are ummaried in hi chaper and heir dicree form i oulined. The laer can convenienly be ued o olve elecromagneic problem numerically on digial compuer. Page /

3 Chaper.. Mawell' equaion from Gau' law for elecric field. The elecric flu deniy D and he elecric field ineniy are vecor in he ame direcion. Thi i rue for all ioropic media, i.e. media whoe properie do no change wih direcion. Gau law ae ha he elecric field flu hrough any cloed urface equal he charge encloed. Thi can be epreed a: D d where ρdυ Ddυ υ υ i inegraion over cloed urface, υ Q (.4 i inegraion over cloed volume υ, ρ i he charge deniy and Q i he charge. The divergence of he flu deniy give he free pace charge deniy a a poin. D D D y y D ρ (.5 Mawell' divergence heorem from Gau' law for elecric field ae ha he inegral of he normal componen of a vecor funcion over a cloed urface,, equal he inegral of he divergence of ha vecor hroughou he volume, υ, encloed by he urface. Conequenly preen. D ha a value whenever a charge i.. Mawell' equaion from Gau' law for magneic field. Jean Bapie Bio and Feli Savar have hown ha curren paing in a wire elemen L produce a magneic field given by (Bio-Savar law: µ IR L db 3 4 π R µ IdLinθ or db 4π r (.6 (.7 where µ i he permeabiliy of medium, and I i he curren in he conducor of lengh L. The magneic flu i given by ψ m B d (.8 and according o Gau he inegral of magneic flu deniy (B over a cloed urface i ero. Page /3

4 Chaper B d 0 (.9 and B 0 (.0 Thi i he Mawell' equaion derived from Gau' law for magneic field...3 Mawell' equaion from Ampère' law. The magneic field H i a vecor ha ha he ame direcion a he flu deniy B and i given by B H µ (. Ampère' law ae ha he line inegral of H around a ingle cloed pah i equal o he curren encloed. H dl J d I (. To ge he curren deniy a a poin he curl equaion i ued. D curlh H J (.3 H ha a value whenever a curren i preen...4 Mawell' equaion from Faraday' law. If he flu change wih ime hen an elecromoive force (emf in induced. From Faraday' law hi i V where dψ d m dl B d (.4 ψ m i he oal flu of an open-circuied loop. In differenial form he elecric field ineniy can be wrien a B Thi i he Mawell' equaion from Faraday' law in differenial form. (.5 Table. i a ummary of Mawell' equaion in boh heir inegral and differenial form. Page /4

5 Chaper lecromagneic heory No. Inegral Differenial Decripion D ρ Ddu Q D ρ d du u u Ψ B d 0 B 0 3 V dψ d m dl B d B Mawell equaion from G Gau law ae ha he urface equal he charge en Mawell' equaion from G According o Gau he in cloed urface i ero. Mawell' equaion from F If he magneic flu change 4 D H dl d I H D Mawell equaion from Am Ampère' law ae ha curren encloed. Table. Mawell' equaion in inegral and differenial form for free pace [.].

6 Chaper.3 Wave heory. Mawell' equaion can be ued in olving elecromagneic problem. In elecromagneic wave, a changing elecric field produce a changing magneic field, which in urn generae an elecric field. lecromagneic wave hu propagae and energy i ranferred in free pace a he peed of ligh. In a ranvere elecromagneic (TM wave, he elecric and magneic field are orhogonal o each oher, and boh are orhogonal o he direcion of propagaion. In a non-conducive medium, he Mawell' curl equaion from Ampère' law i reduced o D H (.6 Uing recangular coordinae for an elecromagneic wave propagaing in he direcion, wih an elecric field vecor aligned wih he y -ai he magneic field vecor poining in he direcion may be epreed a: H y (.7 where i he permiiviy of he medium. Similarly, from Faraday' law B (.8 and which in hi cae reduce o y H µ (.9 where µ i he permeabiliy of he medium. Differeniaing.7 and.9 wih repec o ime and pace, wave equaion for he elecric and magneic field can be obained. Thee are: y µ y (.0 and H µ H (. The conan µ on he R.H.S. of.0 and.9 repreen Page /6

7 Chaper υ µ (. where υ i he velociy of propagaion of he wave in he medium. A oluion of equaion.0 and.9 i of he following form: y 0 co( ω ± β (.3 where β i a conan. Thi oluion indicae a inuoidal wave ravelling in he direcion of..4 Mawell equaion oluion. The Finie-Difference Time-Domain or FDTD i a convenien numerical mehod for olving Mawell' equaion in he ime domain. The FDTD echnique offer many advanage a a ool for elecromagneic modeling, imulaion, and analyi. Thee capabiliie according o [.] lude: Broadband repone predicion cenered abou he yem reonance. Arbirary hree-dimenion (3D model geomeie. Ineracion of M wave wih an objec of pecific conduciviy. Capabiliy for modeling maerial wih frequency-dependen M properie i.e. µ,, and σ. Differen repone ype luding near and far-field oluion. The FDTD mehod wa inroduced by Yee [.3] in 966. In hi claic paper Yee aed: "oluion o he ime-dependen Mawell' equaion in general form are unknown ecep for few pecial cae. The difficuly i mainly due o he impoiion of he boundary condiion." One of he main problem in 966 wa he limied memory available o digial compuer. Today, hiry-four year laer, advance in peronal compuer (PC have reuled in coniderably reae in compuer proceing power and memory capaciy. Mo of he imulaion preened in hi hei uing FDTD have been run on a relaively low-co peronal compuer equipped wih 56 Mbye of memory. Page /7

8 Chaper In FDTD he differenial form of Mawell equaion i ued o deermine he elecric and magneic field. For a linear ioropic maerial D B µh (.4 (.5 The oal elecric and magneic field can be epreed a he um of he iden and ered field oal iden oal iden H H H H ered ered (.6 (.7 Se elecric and magneic loe can diipae energy a hea in maerial he magneic and elecric curren, J m and J e repecively, can be wrien a J m ρ' J e σ H (.8 (.9 Here ρ ' i an equivalen magneic reiiviy in ohm per meer and σ i he elecric conduciviy in Siemen per meer ( S m. The wo curl Mawell' equaion can be hen wrien a H µ ρ' H µ (.30 H σ (.3 Rewriing he wo Mawell' curl equaion in erm of iden and ered field, we ge µ ( H H ( ρ'( H H (.3 ( ( H H σ( (.33 The free pace iden field due o he abence of any maerial are µ 0 H (.34 H 0 (.35 and he ered field in a medium i given if we ubiue he iden field above in he oal field equaion. Page /8

9 Chaper H H µ ρ' H ( µ µ 0 ρ' H (.36 H σ ( 0 σ (.37 Ouide he ered field, he oal field equaion become oal oal H µ 0 (.38a or ( H H ( µ 0 (.38b oal oal H 0 (.39a or ( H H 0 ( and he ered field in free pace can be epreed a (.39b H µ 0 (.40 H 0 (.4 In compuing he ered field, however, only equaion.36 and.37 are needed. Inide he erer µ,, ρ' and σ are ued. Ouide ρ ' σ 0, µ µ 0 and 0 are ued. Thee equaion can be rearranged o ha he ime derivaive of he field i epreed a a funcion of he remaining erm for eae in generaing he appropriae differenial equaion, namely: H ρ' ρ' ( µ µ 0 H H H ( µ µ µ µ (.4 σ σ ( 0 ( H (.43 In a finie-difference repreenaion, derivaive are replaced wih difference i.e. f f (, f (, f (, f (, lim r 0 for ime derivaive and (.44 Page /9

10 Chaper Page /0 f f f f f, (, (, (, ( lim 0 (.45 for pace derivaive. The FDTD mehod divide he modelled pace ino cell. The ie of and mu be limied in order for he yem o be able. ach cell (Yee cell mu be ufficienly mall wih dimenion fied a maller han λ/0. Thi i o enable accurae reul a he highe frequency of inere. On he oher hand, he cell ie mu be large enough o keep he compuing requiremen a manageable level. The maerial preence alo affec he cell ie. The greaer he permiiviy or conduciviy, he horer he wavelengh a a given frequency and he maller he cell ie required. The Yee cell i hown in Fig... Fig.. Yee cell ued in he FDTD mehod. The ime ep for abiliy mu aify Couran' abiliy crieria, namely: 3 ( ( ( c y c (.46 where on he RHS y. Smaller ime ep are permiible, bu do no generally reul in improvemen o compuaional accuracy. For conducive maerial, able calculaion require ime ep maller han he Couran limi. The ame i valid for non-linear maerial. From he Yee cell diagram in Fig.. i can be een ha he magneic field, in hi mehod, ha an offe of a half-cell. Then, he ered elecric field can be wrien a ( H n n i n i n n σ σ σ σ σ,, 0,,, (.47 For high conduciviy value he equaion above become:

11 Chaper (.48 In a imilar way, he correponding equaion for updaing he oher elecric and magneic field can be obained. quaion.47 can be readily olved [.]..4. Compuer reource for FDTD. Compuer reource uch a memory ie and compuaion ime are imporan facor in eablihing an FDTD model. Before he deign of a numerical model, he ie of he required memory mu be calculaed uing he following equaion. Componen Bye ID Bye Sorage N N Cell Componen Cell ID 30 (.49 Thi equaion i ued auming ha he maerial informaion i ored in -bye array, wih boh dielecric and magneic maerial conidered. N i he oal number of cell in he problem. For inance, if a numerical model coni of or,66,480 cell, 65 Mbye of RAM i required. For he head/houlder model ued in hi reearch, coniing of he ame cell dimenion a menioned above, a minimum of 96 Mbye are ued. The number of ime ep (T required in a imulaion i given approimaely by T 0 3N / 3 (.50 and he number of floaing poin operaion i approimaely given by 3 Componen Operaion Operaion 0 3N 4 / 6 0 (.5 Cell Componen.4. Far one ranformaion. A poined ou in previou ecion he main limiaion for FDTD are he available compuer reource. Thee limiaion become more evere in far-field calculaion. The far-field region relaion o a erer may be aken o ar a diance R given by [.4] D R (.5 λ Page /

12 Chaper where D i he large dimenion of he erer. Boh he memory requiremen and he number of operaion are dramaically reaed if far field quaniie are direcly compued. The procedure for calculaing he far field from near field daa i raighforward for ingle frequency eciaion. The near-o-far field ranformaion can be eablihed if he comple ime-harmonic elecric and magneic curren flowing on a cloed urface urrounding he ering objec are ored during he imulaion. If hee comple field or curren are wrien o dik hen in poproceing, he far-one radiaion or he ered field can be deermined. For muliple frequencie a hybrid approach i available which ue puled eciaion for FDTD calculaion. Thi give he far field reul in he frequency domain. For each frequency of inere a running dicree Fourier ranform (DFT of ime harmonic angenial field on a cloed urface urrounding he FDTD geomery i updaed a each ime ep. If he ering or radiaing objec i urrounded by a cloed urface S ' and if nˆ i he local urface uni vecor, hen vecor ime-harmonic equivalen ered urface curren J ( ω nˆ H ( ω and M ( ω nˆ ( ω ei on he urface, where H (ω and (ω are he ered magneic and elecric field a he urface. The ime harmonic vecor-poenial N (ω and L (ω are defined a N( J ( jkr ' rˆ ω ( ω e d' (.53 ' L( ω ' M ( ω e ( jkr ' rˆ d' where k i he wave number, rˆ i he uni vecor o he far one field poin, vecor o he ource poin of inegraion, and erer. The ime harmonic far one elecric field are relaed o θ φ ( jkr ( ηnθ Lφ je λr ( jkr ( ηnφ Lθ je λr (.54 r ' i he S ' i he cloed urface urrounding he (.55 (.56 where η i he characeriic impedance of medium, and R i he diance from he origin o he far one field poin. In order o develop he correponding ime domain far one ranformaion we need o ake he invere Fourier ranformaion of he Page /

13 Chaper above. In a implified noaion, he invere Fourier ranformaion proce, he ime harmonic vecor poenial equal W ( ω jωe jωr c N( ω 4π Rc (.57 U ( ω where jωe jωr c L( ω 4π Rc (.58 π f ω k. The correponding equaion for he far one elecric field c c have no frequency dependence and can be wrien a θ ηwθ Uφ (.59 φ ηwφ Uθ (.60 We can ranform W (ω and U (ω ino ime-domain o obain he vecor poenial W ( and U ( a ( r' r ˆ R W ( J ( d' 4π Rc c c ' (.6 ( r' r ˆ R U ( M ( d' (.6 4π Rc c c ' where J nˆ H( and M ( nˆ ( are he ime-domain elecric and magneic ered curren on he cloed urface urrounding he erer..4.3 Anenna imulaion wih FDTD. FDTD ha been applied recenly o anenna radiaion and oher relaed problem uch a hielding and radar cro-ecion (RCS. The ource, in anenna imulaion, i ypically driven by a ingle applied elecric field in one FDTD cell. The advanage of he FDTD mehod over oher mehod i i abiliy o calculae far-field radiaion paern over a wide frequency band wih a ingle program eecuion. Compared o mehod of momen, FDTD require more compuaion ime and more memory. However, due o he Careian naure of FDTD pirally haped anenna, like he heli, are difficul o imulae requiring very mall cell ie. In mo cae hi i impracical. Page /3

14 Chaper The eady-ae inpu power a an anenna a each frequency i given by P in ( ω Re * [ V ( ω I ( ] ω (.63 where V ( and I ( are he Fourier ranform of he inpu poenial ( V ( and ω ω he inpu curren ( I ( repecively. The eady-ae diipaed power equal P di σ ( ω v dv (.64 where he σ i he maerial conduciviy and he gain of a lole ioropic anenna in he θ, φ direcion i given by Gain ( θ, φ F ( ω, θ, φ / η P in / 4π 0 (.65 where ( ω, θ, φ i he Fourier ranform of he ranien far one ime domain F elecric field radiaion in he θ, φ direcion, and η 0 i he characeriic impedance of free pace. The efficiency of he anenna epreing he raio of radiaed power o oal power will hen given by P fficiency in P P in di (.66 In hi udy he FDTD i eenively ued for he evaluaion of he SAR ariing in a human head iue due o he ue of mobile phone hande. For anenna operaing in free pace and made ecluively by conducive maerial he mehod of momen i much faer compared o FDTD. The mehod of momen priple are ummaried in he ne ecion..5 The Mehod of Momen. The Mehod of Momen (MoM i one of he eablihed and mo eenively ued numerical mehod for olving anenna and a wide range of elecromagneic problem [.7]. The mehod ha been iniially and uccefully implemened for olving elecromagneic field problem involving wire anenna, ering and radiaion from rucure ha could be modelled by a wire grid meh. I ha been Page /4

15 Chaper eended and implemened in uding radiaion and ering from conducive or loy fla plae rucure. In he MoM an elecric and magneic inegral equaion i ranformed ino a e of imulaneou linear algebraic equaion (or mari equaion, which can be direcly olved by andard numerical echnique. An ouline of he echnique i given below. From baic elecromagneic he elecric poenial V of a poin P a a diance r due o a charge Q i given by Q V 4πr (.67 From a line wih linear charge deniy ρ ( Qm he poenial a a poin i given by he inegral l ρl( V d 4π 0 r L (.68 where l i he line lengh and ρ ( i he charge per uni of lengh of line. If ρ ( i available a a funcion of, hen equaion.68 can be readily inegraed. However, if ρ ( i no known, equaion.68 repreen an inegral equaion wih he problem being o find a oluion for ρ (, ubjec o given boundary condiion. The linear charge diribuion can be approimaed by egmen wih uniform charge deniy namely Q n ρ, n,,3... N (.69 L( n n and he oal charge on he wire i now given by Q N Q n n quaion.68 can hen be wrien a (.70 N l mnqn n V Taking he cae of a rod l mn mn m (.7, m,,3,4... M (.7 4πr r mn α ( ' (.73 Page /5

16 Chaper where α i he rod radiu, i he aial diance of obervaion and ' i he aial diance of ource poin a middle of egmen n. The above equaion can be wrien in mari form a [ l ][ Q ] [ V ] mn n m (.74 Thi mari can be olved numerically on a digial compuer. The more he egmen in he model he more accurae he reul. The number of egmen ued, a in he cae of FDTD, i limied by he available compuer reource. The Numerical lecromagneic Code (NC [.8-.0] i a widely ued MoM ofware programme ha ha been compiled for many operaing yem. The NC and i predeceor AMP have been ued uccefully o model a wide range of anenna luding comple rucure, e.g. hip. Simulaion wih he MoM require ha he rucure i modelled wih a meh of elecrically mall egmen or fla plae or combinaion of he wo. Proper choice of he egmen and pache for a model i one of he mo criical ep in obaining accurae reul. The number of egmen i choen o be he minimum required for accepable accuracy, e he program running ime and memory requiremen reae rapidly a hi number reae. There are everal rule for wire-grid modeling of anenna: The lengh of he wire hould be a lea /0 of he wavelengh a he deired frequency. Segmen may no overlap e he diviion of curren beween wo overlapping egmen i undeermined. A large radiu change beween conneced egmen may decreae accuracy. A egmen i required a each poin where a nework connecion or volage ource i locaed. The number of wire joining a a ingle juncion canno eceed 30. Thi limiaion, however, i due o an array dimenion definiion in NC which can be alered by reaing hi number in he ource code When wire are parallel and very cloe ogeher, he egmen hould be aligned o avoid orrec curren perurbaion from an offe mach poin and a egmen juncion. Page /6

17 Chaper In NC daa i enered in he program a card. Card mu be idenified by heir name. For inance, GW idenifie a ring of egmen repreening a raigh wire. The ar and op coordinae, he number of egmen, and he wire properie are ome of he enrie of hi card. Card for heli rucure, pache, ranmiion line and imulaion properie (i.e frequency are alo available. The card available in NC are decribed in deail in [.0]. Depie i fa compuaional ime advanage, ome of he limiaion of he NC make i inappropriae for pracical ype of elecromagneic problem uch a reamen of non-uniform rucure. Thi i due o he reaed memory and compuaional ime requiremen ha he numerical oluion of a 3D volume inegral equaion would involved..6 Inerim concluion. In hi chaper he baic Mawell equaion have been preened. The numerical oluion of hee equaion can be ued in compuer programme o olve accuraely elecromagneic problem uch a he radiaion field from anenna. The FDTD and MoM are wo of hee elecromagneic numerical mehod. Reul from he wo mehod on mobile phone anenna will be preened and compared laer in hi repor. Some of he numerical reul, like SAR predicion, are difficul o verify by meauremen, making he numerical mehod he only oluion o he problem. In Chaper 3 an overview of publihed lieraure on boh he mehod for improving PIFA bandwidh and SAR i preened. Page /7