2 Electromagnetic Theory, Antennas, and Numerical Methods.

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1 lecomagneic Theo, nenna, and Numeical Mehod. The impoance of cience i eviden in eveda life. Fom he heavie and poweful moo o he malle anio, ued in oda' micochip, he elecomagneic cience pla an impoan ole in he evolvemen of human ace. The need fo uding he naue ha i oo in he ih cenu B.C. in ancien Geece. Thale of Mileu noed ha when ambe (elecon in Geek) i ubbed wih a d wood i poduce pak and ha aacion powe on ligh paicle. He alo noed he aacion beween naual magneic ock found in an aea called Magneia. Twenwo cenuie ealie William Gilbe in a moe cienific wok ha invened he elecocope fo meauing he elecoaic effec and he decibed he eah a a huge magne. Man cieni have wok on he diffeen popeie of elecici and magneim in he following ea. Jame C. Mawell ha combined all he heoie in a unied elecomagneic heo. Depie he evoluiona change wih he inoducion of he elaivi heoie b lbe inein and he quanum phic a he beginning of he h cenu, Mawell equaion poved o be unaffeced and he bae fo evoluiona pedicion like he elecomagneic wave eience, he naue of ligh a elecomagneic wave and he pedicion ligh peed. In he following ecion, he coodinae em and he baic veco heoie which of gea ignificance in elecomagneic ae inoduced. The Mawell equaion ae hen inoduced. nenna heo follow he inoducion in elecomagneic. Numeical mehod ae finall peened including he Mehod of Momen (MoM) and Finie Diffeence Time Domain (FDTD) mehod..1 Veco and Coodinae Sem. Depie he cala quani, which ha onl magniude, a veco quani ha boh magniude and diecion. In ecangula fo a veco can be analed ino hee componen. ach of he componen of he veco ma in un be epeed a he poduc of a cala magniude and a uni veco. ˆ ˆ ˆ (.1) Fom he heoem of Phagoa, he cala magniude of a veco,, i given b (.) The do poduc of wo veco i a cala B B co B co (.3) whee i he angle beween and B. Some of he do poduc popeie ae given in able.1. Pope Commen B B commuaive law B if ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 F dl F co( ) dl Table.1 Veco do poduc popeie. dl i he veco incemenal lengh angle beween F and dl

2 The co poduc of a veco, on he ohe hand, i a veco wih a magniude equal o he poduc of he veco magniude muliplied b he ine of he angle beween hem. The diecion of he new veco i pependicula o he plane conaining he wo veco. Thi i given b ˆ ˆ ˆ C B ˆ ( B B ) ˆ( B B ) ˆ( B B ) Some of he co poduc popeie ae given in able.. Pope C B in Commen magniude if C B B B (.4) ˆ ˆ ˆ, ˆ ˆ ˆ, and ˆ ˆ ˆ uni veco co poduc B C nˆbin nˆi a uni veco nomal o he plane conaining and B Table. Veco co poduc popeie. veco a a poin P can be decibed in hee coodinae em. In he ecangula coodinae em he poin P can be epeed b,, and which ae he diance fom he oigin fom -, -, and - ae epecivel. Similal, fo he pheical,, and ϕ ae ued. Table.3 hown poin anfomaion fo ecangula and pheical coodinae em and able.4 how coodinae anfomaion fomula fo veco componen. in co co in co ( ) 1 co ( π) 1 an Table.3 Poin anfomaion fo ecangula and pheical coodinae em

3 Recangula o Clindical Recangula o Speical Clindical o ecangula in co co in Spheical o ecangula in co co co in co in co in in in co Table.4 Coodinae Tanfomaion. wie ome heo fom he mahemaical handbook The gadien of a cala i a veco and i given b f f f f gadf ˆ ˆ ˆ The divegen of a veco, on he ohe hand i a cala and given b div Finall, he cul of a veco i a veco and i i given b cul ˆ ˆ ˆ ) ˆ( ) ˆ( ) ˆ( The gadien, divegence and cul fo he clindical and pheical coodinae can be found in [.1]. lecomagneic and Mawell quaion.

4 foce ac on wo o moe bodie ha have an elecic chage. Unlike chage aac bu like chage epel. Thi foce i given b Coulomd' law. Q1Q F k Thi elecic foce i inveel popoional o he quae of he diance beween he chage Q 1 and Q. k i he conan of popoionali and i given b 1 k 4π whee i he pemiivi o dielecic conan. The pemiivi of fee pace i ( F / ). In mo of he cae, he pemiivi of an objec i efeed m elaive o he pemiivi of he vacuum ( ). *e elaive pemiivi of dielecic, e pemiivi of dielecicin (F/m), and e pemiivi of he vacuum in (F/m). Round a chage, (Q 1 ) hee i an elecic field in which foce ac. The foce pe uni and i given b chage i defined a he elecic field ineni ( ) Q1 F ( Vm 1 ) 4π Q If moe han one chage i peen hen he oal o eulan field a a poin ai he veco um of he individual componen field a he poin. The volume chage deni a a poin i defined b ρ u Q u lim Q Q u he chage now can be epeed in em of he chage deni a Q ρ( R) du whee R i he veco o infinieimal chage volume du. The wok (o eneg) pe uni chage equied o anpo Q a diance i called he diffeence in elecic poenial V aco. The wok pe coulomb equied o bing a poiive e chage fom infini o a poin 1 i called he abolue poenial and i defined b 1 Q V ( Vol) 4π 1 The poenial diffeence a a field i given b he line inegal V b a dl b a co dl and fo a lamella o conevaive field i eo. dl The gadian of V i a veco wih a diecion along he field line. Since a poenial ie occu when moving again he elecic field he diecion of he gadian i oppoie o ha of he field. Tha he gadian of V i V V V V ˆ ˆ ˆ Thu he gadian of a cala poenial i a veco in he diecion of he maimum ae of change of V. In av elecic dipole he field line can be eplaed b upe. If each

5 upe epeen a conan amoun of elecic flu ψ, hen a an poin hee i a flu deni D. The elecic flu hough an uface equal he inegal of he flu deni ove he uface and i mahemaic hi i ψ D d he elecic flu deni i given b Q D and elecic flu can be epeed a ψ d 4π The flu deni and he elecic filed ineni ae veco wih he ame diecion. Thi i ue fo all ioopic media, i.e. media whoe popeie do no change wih diecion. Gau law ae ha he elecic field flu hough an cloed uface equal he chage encloed. Thi can be epeed a D d ρdu Ddu Q u u The divegence of he flu deni give he fee pace chage deni a a poin D D D D ρ Mawell divegence equaion fom Gau The divegence heoem ae ha he inegal of he nomal componen of a veco funcion ove a cloed uface equal he inegal of he divegence of ha veco houghou he volume u encloed b he uface. The D ha a value wheneve chage i peen. capacio i a device fo oing elecic chage, and hence, elecic eneg. The capaciance of a capacio i defined a Q C (F ) V d whee i he aea of a paallel plae capacio and d i he diance beween he wo plae. Beween he plae of a capacio hee i an elecic field ha i given b V/d. medium i momogeneou if i phical chaaceiic (ma deni, molecula ucue, ec.) do no ve wih fom poin o poin. On he ohe hand, maeial ha do no hi pope ae called inhomogeneou, nonhomogeneou, o heeogeneou. medioum i linea wih epec o an elecoaic field if he flu deni D i popoional o he elecic field ineni like fee pace. If i no, he maeial i called nonlinea. n ioopic maeial i he one whoe popeie ae independen of diecion. If he popeie of he maeial diffe wih diecion hen he maeial i call anioopic o nonioopic. If a dielecic i placed in an elecic field, alhough hee i no migaion of chage, hee i a ligh diplacemen of he negaive and poiive chage of he dielecic aom o molecule. The elecic field ineni in a dielecic canno be inceaed infiniel. I a ceain value i eceeded, paking occu and he dielecic i aid o beak down. The maimum field ineni ha a dielecic can uain wihou beakdown i called i dielecic engh. dielecic in he influence of an elecic field i polaied. Thi, depending on he dielecic, can be empoa o pemanen. The dielecic can change he elecic flu deni of a paallel plae capacio o D P

6 Q whee P i he polaiaion which i equal o he uface chage deni ρ p of he dielecic. Fo an ioopic and linea he above equaion can be wien a D P e e The capaciance of a paallel plae capacio wih dielecic i now given b C d d The eneg oed in a capacio, i Q W CV QV C Thi eneg i oed in he elecic field beween he plae of he capacio. No maeial i needed fo eneg o be oed b a field. Thu, eneg i peen even in he vacuum. neg deni i he eneg divided b he volume ha occupie (e.g. he volume of a capacio) and i given b. 1 w 1 ( P) The econd pa in he backe i giving he eneg oed in he dielecic due o polaiaion of he molecule. The eneg oed i can be now epeed in em of eneg deni a W wdu 1 du u u lecic chage in moion coniue an elecic cuen, and an cuen cuing medium i called conduco. Cuen i defined b I v () d ρ whee v d in he dif veloci. The cuen deni i given b I J v d ρ (/m) The elaion beween he volage and he cuen ha been deemined b Geog Simon Ohm. The Ohm' law ae ha poenial diffeence o volage V beween he end of a conduco i equal o he poduc of he eiance and he cuen and in mahemaical fom i given b V IR (Vol) whee R i he eiance in Ohm. Mulipling he volage b he cuen ield he wok pe uni ime, o powe: P IV I R V /R (Wa) Reiivel i he eiance of a maeial pe uni volume and ha uni of Ohm-mee (Ωm). Thu, eiivel i indened of he dimenion of he eio. One ove he eiivel give he conducivi ( σ ) of a maeial in iemen. Simila o conducivi, conducance i equal o one ove he eiance of he eio. Uing Ohm' law i can be hown ha a a poin J σ Kichhoff' volage law ae ha he algebaic um of he elecomoive foce (emf) aound a cloed cicui equal he um of he ohmic, o IR, dop aound he cicui. V I R

7 Kichhoff' cuen law ae ha he algebaic um of he cuen a a juncion i eo. I o J d The divegence of J accoding o he cuen law of Kichhoff i ρ J Fo ead cuen, he lef pa i eo. The cuen hen i given b he ae of chage change pe ime. dq I d The moo equaion, inoduced b ndè Maie mpeè in 18, give he foce on a wie in a magneic field. F ( I B) L o d F ( I B) dl whee F i he veco indicaing he foce elemen in a conduco of L lengh, I i he cuen in he conduco and B i he flu deni. Jean Bapie Bio and Feli Sava have hown ha cuen paing in a wie elemen L poduce a magneic field given b (Bio-Sava law) µ IR L µ IdLin db o db (Tela) 3 4 π R 4π L whee µ i he pemeabili of medium ( µ inducance pe uni lengh). The d pemeabili of vacum i µ 4π nhm-1. The magneic flu of a non-unifom aea i given b ψ B d m and accoding o Gau he inegal of magneic flu deni ove a cloe uface i eo. B d and B Thi i he Mawell' equaion fom gau' law fo magneic field. Induco i a device o oe eneg in a magneic field. The inducance he aio of he oal magneic linkage ( Λ ) o he cuen I hough he induco. Table.5 give he inducance fo imple geomeie. Long olenoid Tooid µ N L co-ecional l l lengh µ N L adiu of coil R R adiu of ooid Table.5 Inducance of imple geomeie. The magneic field i a veco ha ha he ame diecion a he flu deni and i given b

8 B H µ mpèe' law ae ha he line inegal of H aound a ingle cloed pah i equal o he cuen encloed. H dl J d I mpèe' law (Mawell equaion) To ge he cuen deni a a poin he cul equaion i ued. cul H H J H ha a value wheneve a cuen i peen. If he flu change wih ime hen an emf in induced. Fom Faada' law hi i dψ m B V dl d d and in diffeenial fom B Mawell' equaion fom Faada diffeenial equaion The cuen i he ae of change of chage wih ime dq dv i C d d The cuen hough a eio i called a conducance cuen, while he cuen hough he capacio i called he diplacemen cuen. Since he cuen deni, he elecic diplacemen, and he elecic field ineni ae acuall pace veco, which all having he ame diecion in ioopic media hee elaion can be epeed a d dd J di, J d d cond σ and J oal Jcond Jdip Uing he mpèe' law, which ae ha he line inegal of H i equal o he cuen encloed, we can ge D H dl J d Mawell' equaion fom mpèe' law in inegal fom and D H J Mawell' equaion fom mpèe' law in diffeenial fom Table.6 i umma of Mawell' equaion in boh hee inegal and diffeenial fom.

9 No. Inegal Diffeenial Dicipio 1. D d ρdu Ddu Q D ρ Mawell u u elecic fie Gau la hough an encloed. B d B Mawell' magneic ccoding deni ov 3 dψ m B B Mawell' V dl d jωµ H d jωµ H d The flu c 4 H dl J d ( σ jω ) d I H J ( σ jw ) Mawell e mpèe' equal o h Table.6 Mawell' equaion in inegal and diffeenial fom.

10 .3 The Finie Diffeence Time Domain Mehod of lecomagneic. The Finie Diffeence Time Domain (FDTD) i a convenien mehod fo olving he Mawell' equaion in ime domain. The FDTD echnique offe man advanage a an elecomagneic modeling, imulaion, and anali ool. Thee capabiliie include [.]: Boadband epone pedicion ceneed abou he em eonance bia hee-dimenion (3D) model geomeie Ineacion wih an objec of an conducivi fom ha of a pefec conduco, o ha of a eal meal, o ha of low o eo conducivi Fequenc-depened coniuive paamee fo modeling mo maeial Lo dielecic Magneic maeial Unconvenional maeial, including anioopic plama and magneied feie n pe of epone, including fa field deived fom nea field, uch a Scaeed field nenna paen Rada co-ecion (RCS) Suface epone Cuen, powe deni Peneaion/ineio coupling The mehod ha been inoduced b Yee [.3] in Fo linea ioopic maeial D and B µh The elecic and magneic field can be epeed a oal inciden caeed and oal inciden caeed H H H H The elecic and magneic loe ha can diipae eneg a hea in maeial. The magneic and elecic cuen hen can be wien a J m ρ' H and J e σ epecivel. Hee he ρ ' i an equivalen magneic eiivi in ohm pe mee and σ i he elecic conducivi in iemen pe mee. The wo cul Mawell' equaion can be hen wien a H 1 ρ' H and µ µ σ 1 H Rewiing he wo Mawell' cul equaion in em of inciden and caeed field, we ge inc ca inc ca µ ( H H ) inc ca ( ) ρ'( H H ) inc ca inc ca ( ) inc ca ( H H ) σ ( ) The fee pace inciden field due o abence of an maeial i

11 inc inc µ H inc inc H and he caeed field in he media i given if we ubiue he inciden field above in he oal field equaion. ca ca inc H ca H inc µ ρ' H ( µ µ ) ρ' H q.1 ca inc ca ca inc H σ ( ) σ q. Ouide he caeed field, hen, he oal field will be oal inc ca oal H inc ca ( H H ) µ o ( ) µ oal inc ca oal inc ca ( ) H o ( H H ) and he caeed field in he fee pace i calculaed o be ca ca H µ ca ca H Fo he compuaion, howeve, onl equaion 1 and ae needed. Inide he caee µ,, ρ' and σ ae ued, and ouide ρ ' σ, µ µ and. Thee equaion can be eaanged o ha he ime deivaive of he field i epeed a a funcion of he emaining em fo eae in geneaing he appopiae diffeenial equaion. ca inc H ρ' ca ρ' inc ( µ µ ) H 1 ca H H ( ) µ µ µ µ inc σ ca σ inc ( ) 1 ca ( H ) In finie diffeence, deivaive ae eplace wih diffeence f f (, ) f (, 1) f (, ) f (, 1) lim fo ime deivaive and f f (, ) f ( 1, ) f (, ) f ( 1, ) lim fo pace deivaive. The ie of and mu be limied in ode fo he em o be able. The FDTD mehod ugge ha he modeled pace mu be divided ino cell. ach cell (Yee cell) mu have i ie le han λ/1. The Yee cell i hown in fig The ime ep mu be c 3 c ( ) ( ) ( ) whee fo he igh pa.

12 The magneic field, in hi mehod, ha an offe of half cell. Fig. 1.1 Yee Cell. The caeed elecic field can be hen wien a ( ) H n n i n i n n σ σ σ σ σ 1,,, 1,, Fo a high conducive value hi equaion become inc ca. In a imila wa, he coeponding equaion fo updaing he ohe elecic and magneic field can be obained. The equaion () can be eail wien a a digial compue code.

Lecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light

Lecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light Lecue 5 Chape 3 lecomagneic Theo, Phoons, and Ligh Gauss s Gauss s Faada s Ampèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d qv A q A J ds ds In mae fields ae defined hough ineacion