1/13/016 5303 lecromagneic Analsis Using Finie Difference Time Domain Lecure #4 Review of M and Inroducion o FDTD Lecure 4These noes ma conain coprighed maerial obained under fair use rules. Disribuion of hese maerials is sricl prohibied Slide 1 Lecure Ouline Review Mawell s equaions Phsical boundar condiions The consiuive relaions Parameer relaions see Balanis Chaper 1 Inroducion o FDTD Flow of Mawell s equaions Finie difference approimaions The updae equaion The FDTD algorihm for now Lecure 4 Slide 1
1/13/016 Review of Mawell s quaions and lecromagneics Lecure 4 Slide 3 Mawell s quaions Lecure 4 4
1/13/016 Four Field Terms in Mawell s quaions J D B Vm lecric field inensi Iniial elecric push Am Magneic field inensi Iniial magneic push lecric Field Quaniies + = D Displaced Charges Magneic Field Quaniies Tiled Magneic Dipoles Lecure 4 Slide 5 Cm + = B Wb m Gauss s Law D v D D D D lecric fields diverge from posiive charges and converge on negaive charges. + If here are no charges, elecric fields mus form loops. Lecure 4 6 3
1/13/016 Gauss s Law for Magneism B 0 B B B B Magneic fields alwas form loops. Lecure 4 7 Consequence of Zero Divergence The divergence heorems force he D and B fields o be perpendicular o he propagaion direcion of a plane wave. D 0 jk r de 0 d jk d 0 k d 0 no charges k D k k Lecure 4 B 0 jk r be 0 b jk b 0 k b 0 no charges k B k k 8 4
1/13/016 Ampere s Law wih Mawell s Correcion D J a a a Circulaing magneic fields induce currens and/or ime varing elecric fields. Currens and/or ime varing elecric fields induce circulaing magneic fields. Lecure 4 9 Farada s Law of Inducion B a a a Circulaing elecric fields induce ime varing magneic fields. Time varing magneic fields induce circulaing elecric fields. Lecure 4 10 5
1/13/016 Consequence of Curl quaions The curl equaions predic elecromagneic waves!! lecric Field Magneic Field Lecure 4 11 Saring Poin for lecromagneic Analsis Divergence quaions B 0 D v Curl quaions D J B Wha produces fields Consiuive Relaions D B means convoluion means ensor ow fields inerac wih maerials Lecure 4 1 6
1/13/016 Mawell s quaions in Caresian Coordinaes (1 of 4) Vecor Terms a a a D D a D a D a a a a B B a B a B a J Ja Ja Ja Divergence quaions D 0 D D D 0 B 0 B B B 0 Lecure 4 Slide 13 Mawell s quaions in Caresian Coordinaes ( of 4) Consiuive Relaions D a a D a D a Da a D D D B B B B Lecure 4 Slide 14 7
1/13/016 Mawell s quaions in Caresian Coordinaes (3 of 4) Curl quaions B a a a Ba Ba Ba B B B a a a a a a B B B Lecure 4 Slide 15 Mawell s quaions in Caresian Coordinaes (4 of 4) Curl quaions D J a a a J a J a J a D a D a D a D D D a a a J a J a J a D J D J D J Lecure 4 Slide 16 8
1/13/016 9 Lecure 4 Slide 17 Alernaive Form of Mawell s quaions in Caresian Coordinaes (1 of ) Alernae Curl quaions a a a a a a Lecure 4 Slide 18 Alernaive Form of Mawell s quaions in Caresian Coordinaes ( of ) Alernae Curl quaions a a a a a a
1/13/016 Tensors Tensors are a generaliaion of a scaling facor where he direcion of a vecor can be alered in addiion o is magniude. Scalar Relaion V av V av Tensor Relaion a a a a V a a a V V a a a V Lecure 4 Slide 19 The Consiuive Relaions Linear, isoropic and non dispersive maerials: D Dispersive maerials: D Anisoropic maerials: D Nonlinear maerials: We will use his in our FDTD codes The poin here is ha ou can wrap all of he compleiies associaed wih modeling srange maerials ino his single equaion. The implemenaion of Mawell s equaions in FDTD will never change. This will make our code more modular and easier o modif. I ma no be as efficien as i could be hough. 1 3 3 0 e 0 e 0 e D Lecure 4 0 10
1/13/016 Anisoropic Maerials A generalied ensor for permiivi is wrien as D how much of conribues o D ij j i We see ha and D can be in differen direcions when he permiivi is anisoropic. I greal simplifies a finie difference mehod o consider onl diagonal ensors. Tha is, all of he off diagonal erms will be se o ero. 0 0 D D 0 0 D 0 0 D Special Noe: There are onl hree degrees of freedom for he ensor componens. The nine elemens canno be chosen arbiraril. I is alwas possible o choose a coordinae ssem ha makes he ensor diagonal. The off diagonal erms onl arise when he chosen coordinae ssem does no mach he crsal aes of he anisoropic maerial. The simplificaion above resrics us o onl be able o model anisoropic maerials ha align perfecl wih our,, and aes. Lecure 4 1 Simplifing Mawell s quaions 1. Assume no charges or curren sources: v 0 J 0 B 0 D D D 0 B B. Assume linear, isoropic, and non dispersive maerials: B 0 D D D 0 B B Convoluion becomes simple muliplicaion 3. Someimes he consiuive relaions are subsiued ino Mawell s equaions: 0 Noe: I is helpful o reain μ and ε and no replace hem wih refracive inde n. 0 Lecure 4 11
1/13/016 Phsical Inerpreaion of and D lecric Field A disurbance produced around charges or in he presence of a ime varing magneic field. Think of as a push Unis are vols per meer (V/m) D lecric Displacemen Field D sands for displacemen Includes, bu also accouns for displaced charges in a maerial (maerial polariaion) quivalen o flu densi Think of D as displaced charge Unis are dipole momens per uni volume (C m/m 3 ), or jus (C/m ) We can make look like an equivalen displaced charge hrough D=ε 0 V + d V d Lecure 4 3 Phsical Inerpreaion of and B Magneic Field A disurbance produced around currens or in he presence of a ime varing elecric field. Think of as a magneic push Unis are amperes per meer (A/m) B Magneic Displacemen Field Includes, bu also accouns for iled magneic dipoles in a maerial (magneiaion) quivalen o flu densi Think of B as reoriened magneic dipoles Unis are magneic dipole momens per uni volume (W m/m 3 ), or jus (W/m ) We can make look like an equivalen reoriened dipole hrough B=μ 0 C. A. Balanis, Advanced ngineering lecromagneics, (Wile, New York, 1989) Lecure 4 4 1
1/13/016 Phsical Boundar Condiions and 1 1 and Tangenial componens of and are coninuous across an inerface. 1,T,T 1,T,T Fields normal o he inerface are disconinuous across an inerface. Noe: Normal componens of D and B are coninuous across he inerface. 11,N 11,N,N,N These are more complicaed boundar condiions, phsicall and analicall. Tangenial componen of he wave vecor is coninuous across an inerface. k1,t k,t Lecure 4 5 Parameer Relaions Lecure 4 Slide 6 13
1/13/016 The Dielecric Consan, r The permiivi is a measure of how well a maerial sores elecric energ. A circulaing magneic field induces an elecric field a he cener of he circulaion in proporion o he permiivi. The dielecric consan of a maerial is is permiivi relaive o he permiivi of free space. 0 r 1 0 8.85418781710 F m 1 r r is he relaive permiivi or dielecric consan Lecure 4 Slide 7 Table of Dielecric Consans Lecure 4 Consanine A. Balanis, Advanced ngineering lecromagneics, Wile, 1989. Slide 8 14
1/13/016 The Relaive Permeabili, r The permeabili is a measure of how well a maerial sores magneic energ. A circulaing elecric field induces a magneic field a he cener of he circulaion in proporion o he permeabili. The relaive permeabili of a maerial is is permeabili relaive o he permeabili of free space. 0 r 6 0 1.5663706110 m 1 r r is he relaive permeabili Lecure 4 Slide 9 Table of Permeabiliies Lecure 4 Consanine A. Balanis, Advanced ngineering lecromagneics, Wile, 1989. Slide 30 15
1/13/016 The Refracive Inde The permiivi and permeabili appear in Mawell s equaions so he are he mos fundamenal maerial properies. owever, i is difficul o deermine phsical meaning from hem in erms of how waves propagae (i.e. speed, loss, ec.). In his case, he refracive inde is a more meaningful quani. n r r Mos maerials ehibi a negligible magneic response and he refracive inde and dielecric consan are relaed hrough n r in: one of he mos common misakes made in his course is using values of refracive inde direcl as permiivi. Lecure 4 Slide 31 Maerial Impedance The impedance of a maerial quanifies he relaion beween he elecric and magneic field of a wave ravelling hrough ha maerial. I is he mos fundamenal quani ha causes reflecions and scaering. The impedance can be wrien relaive o he free space impedance as r 0 0 0 376.73031346177 r 0 This shows ha he elecric field is around wo o hree orders of magniude larger han he magneic field. Lecure 4 Slide 3 16
1/13/016 versus f is he angular frequenc measured in radians per second. I relaes more direcl o phase and k. Think cos(). f is he ordinar frequenc measured in ccles per second. I relaes more direcl o ime. Think cos(f) and = 1/f. f Lecure 4 Slide 33 Wavelengh and Frequenc The frequenc f and free space wavelengh 0 are relaed hrough c f 0 0 Inside a maerial, he wave slows down according o he refracive inde as follows. c 0 v n c 9979458 speed of ligh in vacuum m 0 s The frequenc is he mos fundamenal parameer because i is fied. Inside a maerial, he wave slows down so he wavelengh is reduced. v f The free space wavelengh 0 can be used inerchangeabl wih frequenc f. This is mos common in opics. Lecure 4 Slide 34 17
1/13/016 Sign Convenion ow do ou define forward wave propagaion? + Sign convenion for his course Quani + Wave Soluion j k 0e j k 0e Dielecric Funcion j j Refracive Inde N n j N n j Lecure 4 Slide 35 Summar of Parameer Relaions Permiivi 0 r 1 0 8.854187817 10 F m Permeabili 0 r 6 0 1.56637061 10 m Refracive Inde Impedance n r r 0 r r 0 0 0 376.73031346177 Wave Veloci c0 v n c0 9979458 m s ac Frequenc and Wavelengh f Wave Number c f 0 0 k0 0 Lecure 4 Slide 36 18
1/13/016 Duali Beween D and B lecric Field D P ε Magneic Field B M μ Lecure 4 37 Inroducion o Finie Difference Time Domain Lecure 4 Slide 38 19
1/13/016 Flow of Mawell s quaions B A circulaing field induces a change in he B field a he cener of circulaion. B A B field induces an field in proporion o he permeabili. D A D field induces an field in proporion o he permiivi. D A circulaing field induces a change in he D field a he cener of circulaion. Noe: In reali, his all happens simulaneousl. In FDTD, i follows his flow. Lecure 4 Slide 39 Flow of Mawell s quaions Inside Linear, Isoropic and Non Dispersive Maerials In maerials ha are linear, isoropic and non dispersive we have In his case, he flow of Mawell s equaions reduces o A circulaing field induces a change in he field a he cener of circulaion in proporion o he permeabili. A circulaing field induces a change in he field a he cener of circulaion in proporion o he permiivi. Lecure 4 Slide 40 0
1/13/016 Finie Difference Approimaions df f f d 1.5 1 second order accurae firs order derivaive This derivaive is defined o eis a he mid poin beween f 1 and f. df d f f 1 Lecure 4 Slide 41 Sable Finie Difference quaions ach erm in a finie difference equaion mus eis a he same poin in ime and space. ample: f f 0 Given f 0, f, f, iss a f f iss a f 0 iss a iss a f f f 0 Your simulaion will be unsable (i.e. eplode). f() is onl known a ineger muliples of. ow do we calculae f(+/)? iss a iss a f f f f 0 Lecure 4 Slide 4 1
1/13/016 Approimaing he Time Derivaive (1 of 3) An inuiive firs guess a approimaing he ime derivaives in Mawell s equaions is: iss a iss a iss a iss a This is an unsable formulaion. Lecure 4 Slide 43 Approimaing he Time Derivaive ( of 3) We adjus he finie difference equaions so ha each erm eiss a he same poin in ime. This works, bu we will be doing more calculaions han are necessar. Is here a simpler approach? Lecure 4 Slide 44
1/13/016 Approimaing he Time Derivaive (3 of 3) We sagger and in ime so ha eiss a ineger ime seps (0,,, ) and eiss a half ime seps (/, +/, +/, ). We will handle he spaial derivaives in ne lecure in a ver similar manner. Lecure 4 Slide 45 Derivaion of he Updae quaions The updae equaions are he equaions used inside he main FDTD loop o calculae he field values a he ne ime sep. The are derived b solving our finie difference equaions for he fields a he fuure ime values. Lecure 4 Slide 46 3
1/13/016 Anaom of he FDTD Updae quaion Updae coefficien To speed up simulaion, we calculae hese before ieraing. Field a he fuure ime sep. Field a he previous ime sep. Curl of he oher field a an inermediae ime sep Lecure 4 Slide 47 The FDTD Algorihm for now Iniialie Fields o Zero 0 0 0 Done? es Finished! Loop over ime Updae from Updae from no 3 3 5 5 3 Lecure 4 Slide 48 4