Review and Walkthrough of 1D FDTD

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6//8 533 leomagnei Analsis Using Finie Diffeene Time Domain Leue #8 Review and Walhough of D FDTD Leue 8These noes ma onain opighed maeial obained unde fai use ules.

1 6//8 533 leomagnei Analsis Using Finie Diffeene Time Domain Leue #8 Review and Walhough of D FDTD Leue 8These noes ma onain opighed maeial obained unde fai use ules. Disibuion of hese maeials is sil pohibied Slide Leue #8 Ouline Fomulaion Pepae Mawell s equaions Finie diffeene appoimaions Reduion o one dimension Deivaion of Updae quaions Bells and whisles Gid esoluion, ime sep, soues, bounda ondiions, Fouie ansfoms, ansmiane and efleane Implemenaion Sequene fo Code Developmen Walhough Leue 8 Slide

2 6//8 Fomulaion of D FDTD Pepae Mawell s quaions Leue 8 Slide 3 Nomalie he Magnei Field We saisfied he divegene equaions b adoping he Yee gid sheme. We now onl have o deal wih he ul equaions. The and fields ae wo o hee odes of magniude diffeen. This will ause ounding eos in ou simulaion and i is alwas good paie o nomalie ou paamees so he ae all he same ode of magniude. ee we hoose o nomalie he magnei field. Noe: Leue 8 Slide 4

3 6//8 pand he Cul quaions Leue 8 ee we assumed linea and non dispesive maeials wih diagonal ensos. Slide 5 Fomulaion of D FDTD Finie Diffeene Appoimaions Leue 8 Slide 6 3

4 6//8 Finie Diffeene Appoimaions df f f d.5 seond ode auae fis ode deivaive This deivaive is defined o eis a he mid poin beween f and f. df d f f Leue 8 Slide 7 Finie Diffeene Appoimaion of Time Deivaives We appoimae all of he deivaives in Mawell s equaions wih finie diffeenes. The elei and magnei fields ae saggeed in ime b / so ha eve em in he finie diffeene equaions eiss a he same insan in ime. Leue 8 Slide 8 4

Impliil saisfies divegene equaions Nauall handles phsial bounda ondiions legan appoimaion of he ul equaions using finie diffeenes Consequenes Field

5 6//8 Repesening Funions on a Gid ample phsial (oninuous) D funion A gid is onsued b dividing spae ino disee ells Funion is nown onl a disee poins Repesenaion of wha is auall soed in memo Leue 8 Slide 9 Yee Cell fo D, D, and 3D Gids D Yee Gid D Yee Gids 3D Yee Gid Mode Mode Mode Mode Benefis Impliil saisfies divegene equaions Nauall handles phsial bounda ondiions legan appoimaion of he ul equaions using finie diffeenes Consequenes Field omponens ae in phsiall diffeen loaions Field omponens ma eside in diffeen maeials even if he ae in he same uni ell Field omponens will be ou of phase Leue 8 Slide 5

6 6//8 Finie Diffeene Appoimaions on a Yee Gid Finie Diffeene quaion fo Finie Diffeene quaion fo Finie Diffeene quaion fo i, j, i, j, i, j, i, j, i, j, i, j, i, j,, i, j, i, j, i j, i, j, i, j, i, j, i, j,, i j, i, j,, i, j i, j, i, j, i, j, i, j, Finie Diffeene quaion fo Finie Diffeene quaion fo Finie Diffeene quaion fo, i, j, i, j i, j, i, j,, j,, j,, j,, j, i i i i i, j, i, j, i, j,,, j, j,, j,,, i i i i j i, j, i, j, i, j, i, j, i, j, i, j, Leue 8 Slide Fomulaion of D FDTD Reduion o One Dimension Leue 8 Slide 6

6//8 Reduion o One Dimension We saw in Leue 3 ha some poblems omposed of dielei slabs an be desibed in jus one dimension. In his ase, he maeials and he fields ae unifom in wo dieions.

7 6//8 Reduion o One Dimension We saw in Leue 3 ha some poblems omposed of dielei slabs an be desibed in jus one dimension. In his ase, he maeials and he fields ae unifom in wo dieions. Deivaives in hese unifom dieions will be eo. We will define he unifom dieions o be he and aes. Leue 8 Slide 3 and Deivaives ae Zeo We appoimaed he deivaives wih finie diffeenes on a Yee gid. i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, Leue 8 Slide 4 i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, 7

8 6//8 8 Summa of D FDTD Modes Slide 5 Leue 8 / Mode / Mode Leue 8 Slide 6 Fomulaion of D FDTD Deivaion of Updae quaions

9 6//8 9 Leue 8 Slide 7 / Mode: Updae quaion fo Sa wih he finie diffeene equaion whih has in he ime deivaive: Solve his fo he field a he fuue ime value. Leue 8 Slide 8 / Mode: Updae quaion fo Sa wih he finie diffeene equaion whih has in he ime deivaive: Solve his fo he field a he fuue ime value.

10 6//8 Updae quaions and Updae Coeffiiens The updae oeffiiens do no hange hei value duing he simulaion. The should be ompued onl one befoe he main FDTD loop and no a eah ieaion inside he loop. The finie diffeene equaions in ems of he updae oeffiiens ae: m m m m Leue 8 Slide 9 Fomulaion of D FDTD Bells and Whisles Leue 8 Slide

6//8 Compuing Gid Resoluion ) Resolve Wavelengh min N f n

Dimensions ) Resolve Feaues N eil d d N N d N d N N 4 d d

Genealied Couan Sabili Condiion n min Fo D Gids wih Pefel

11 6//8 Compuing Gid Resoluion ) Resolve Wavelengh min N f n ma ma min N 3) Iniial Resoluion min, d 4) Snap Gid o Ciial Dimensions ) Resolve Feaues N eil d d N N d N d N N 4 d d d d N N min d d Leue 8 Slide Compuing he Time Sep Genealied Couan Sabili Condiion n min Fo D Gids wih Pefel Absobing Bounda Condiion nb n b efaive inde a boundaies Leue 8 Slide

12 6//8 The Gaussian Soue The Gaussian soue appoimaes an impulse so ha a suue an be haaeied ove an enomous ange of fequenies in a single simulaion. g ep duaion of simulaion.5 6 f ma Leue 8 Slide 3 simaing Toal Numbe of Ieaions Toal Simulaion Time T 5 pop pop nma N ime i aes fo a wave o popagae aoss he gid one ime. Toal Numbe of Ieaions Allow fo 5 bounes. ighl esonan devies will need muh moe. Allow fo he enie pulse wihou uing i off. T STPS ound This mus be an inege quani. Leue 8 4

13 6//8 The Toal Field/Saeed Field Famewo D FDTD Gid oal field saeed field s s Poblem Poins! Leue 8 Slide 5 Animaion of TF/SF in D FDTD Leue 8 6 3

14 6//8 Coeion o Finie Diffeene quaions a he Poblem Cells ( of ) On he saeed field side of he TF/SF inefae, he finie diffeene equaion onains a em fom he oal field side. Due o he saggeed naue of he Yee gid, his onl ous in he updae equaion fo a magnei field. s s s s s m s This is a oeion em ha an be implemened afe sandad updae equaion he sandad updae equaion o inje a soue. Leue 8 Slide 7 This is an equaion in he saeed field, bu s is a oal field quani. We mus suba he soue fom o mae i loo lie a saeed field quani. s s s s s s s m s s s s m s m s s s oal field saeed field s s Coeion o Finie Diffeene quaions a he Poblem Cells ( of ) On he oal field side of he TF/SF inefae, he finie diffeene equaion onains a em fom he saeed field side. Due o he saggeed naue of he Yee gid, his onl ous in he updae equaion fo an elei field. s s This is an equaion in he s s s m saeed field, bu is a oal field quani. We mus add he soue o s o mae i loo lie a oalfield quani. s s s s s s s m s s s m s s s s s m This is a oeion em ha an be implemened afe sandad updae equaion he sandad updae equaion o inje a soue. Leue 8 Slide 8 s oal field saeed field s s 4

15 6//8 The Two Soue Tems Fom he pevious slides, we now now ha we need o alulae wo soue funions befoe he main FDTD loop. These ae: s s s s We need o mae a few obsevaions ha mus be aouned fo befoe we an alulae hese soue funions oel.. The ampliude of hese funions an be diffeen as and ae elaed hough he maeial impedane.. These funions ae a half gid ell apa and have a small ime dela beween hem 3. These funions eis a diffeen ime seps. Leue 8 Slide 9 Calulaion of he Soue Funions We alulae he elei field as s s g We alulae he magnei field as s s s ns g s Ampliude due o Mawell s equaions / Mode Dela hough one half of a gid ell We alulae he elei field as s s g We alulae he magnei field as s s s ns g s Ampliude due o Mawell s equaions / Mode Dela hough one half of a gid ell alf ime sep diffeene alf ime sep diffeene s s,, ns maeial popeies whee soue is injeed Leue 8 Slide 3 5

16 6//8 6 Leue 8 Slide 3 Diihle Bounda Condiion N N N N m N m N m m Diihle bounda ondiions assume ha all field quaniies ouside of he gid ae eo. We modif he updae equaions as follows. Leue 8 Slide 3 Pefel Absobing Bounda Condiion Condiions Waves a he boundaies ae onl avelling ouwad. Maeials a he boundaies ae linea, homogeneous, isoopi and non dispesive. Time sep is hosen so phsial waves avel ell in wo ime seps. = n/( ) Implemenaion a Low Bounda A he low bounda, we need onl modif he field updae equaion. Implemenaion a igh Bounda A he high bounda, we need onl modif he field updae equaion. 3 h h h h m 3 N N N N e e e e m

17 6//8 ffiien Fouie Tansfom ( of ) The sandad Fouie ansfom is defined as j f F f f e d If he funion f() is onl nown a disee poins, he Fouie ansfom an be appoimaed numeiall as M fm F f f m e j m M Toal numbe of ime seps m ime sep This an be wien in a slighl diffeen fom. M j f m F f e f m m ample = ps f =. G K =.978 i.8 Leue 8 Slide 33 ffiien Fouie Tansfom ( of ) The final fom on he pevious slide suggess an effiien implemenaion. The Fouie ansfom is updaed eve ieaion so b he end of he main loop: M j f m e f m F f m This mulipliaion an be done afe he main FDTD loop in a pospoessing sep. e j f This is simpl he field value of inees a he uen ime sep. This enel an be ompued pio o he main FDTD loop fo eah fequen of inees. The enels an be soed in a D aa. Leue 8 Slide 34 7

6//8 Fouie Tansfoms in FDTD The easies, bu leas memo effiien, mehod o ompue a Fouie ansfom is o pefom a simulaion and eod he desied field as a funion of ime.

18 6//8 Fouie Tansfoms in FDTD The easies, bu leas memo effiien, mehod o ompue a Fouie ansfom is o pefom a simulaion and eod he desied field as a funion of ime. Afe he simulaion is finished, hese funions an be Fouie ansfomed using an FFT. efleed field ansmied field soue Leue 8 Slide 35 Pos Poessing he Fouie Tansfoms We mus nomalie he spea o alulae ansmiane and efleane. We do his b dividing he efleion and ansmission speum b he soue speum. R f FFT FFT ef s R f f C f T T f FFT FFT n s I is ALWAYS good paie o he fo eneg onsevaion b adding he efleane and ansmiane and ensuing he sum equals % (assuming no loss o gain in ou devie). Leue 8 Slide 36 8

19 6//8 Implemenaion of D FDTD Leue 8 Slide 37 D FDTD Gid Saeed Field TF/SF Inefae Toal Field Soue Injeion Poin Noe: A eal gid would have o moe poins. Refleion Reod Poin Spae Region ma Suue Being Modeled Spae Region ma Tansmission Reod Poin PAB PAB s and s n b n ma and n min n b Leue 8 Slide 38 9

20 6//8 Iniialiing he FDTD Simulaion Iniialie Simulaion Iniialie MATLAB Define unis Define onsans Compue Time Sep nb Define Simulaion Paamees Fequen ange (f ma ) Devie paamees Gid paamees (NRS, e.) Compue Gid Resoluion Iniial esoluion min d N min min, N N Nd d Snap gid o iial dimension N eil d d N Build Devie on Gid Refe o Leue 3.,s Compue Soue.5 f 6 A ma ep n s,s Aep Iniialie Fouie Tansfoms j f i K i e Fi Compue Updae Coeffiiens m m Iniialie Fields o Zeo Iniialie Bounda Tems o Zeo h h h e e e 3 3 Leue 8 Slide 39 The Main FDTD Loop es Done? Loop ove ime no Updae (Pefel Absobing Bounda) N e N m 3 m N N N Reod a Bounda Updae (Pefel Absobing Bounda) m 3 m h h h h N andle Soue s s s m Reod a Bounda N e e e s,s f andle Soue s s m s s s Updae Fouie Tansfoms m i, j, F F K i i i Visualie Simulaion Supeimpose fields on maeials Show efleane, ansmiane and onsevaion Updae onl afe some numbe of ieaions Leue 8 Slide 4

21 6//8 Pos Poessing es Done? no Compue Response Fi,ef R fi FFT T F s i,n fi FFT s i i i C f R f T f Visualie Resuls Supeimpose fields on maeials Show efleane, ansmiane and onsevaion Show esponse on linea and db sale Finished! Leue 8 Slide 4 Sequene fo Code Developmen Leue 8 Slide 4

22 6//8 Sep Basi FDTD Algoihm Basi updae equaions Leue 8 Slide 43 Sep Add Sof Soue Basi updae equaions Add a sof soue Leue 8 Slide 44

23 6//8 Sep 3 Add Absobing Bounda Basi updae equaions Add a sof soue Add pefe bounda ondiion Leue 8 Slide 45 Sep 4 Add TF/SF Basi updae equaions Add a sof soue Add pefe bounda ondiion Inopoae TF/SF onewa soue Leue 8 Slide 46 3

6//8 Sep 5 Move Soue & Add T/R Basi updae equaions Add a sof soue Add pefe bounda ondiion Inopoae TF/SF onewa soue Move posiion of soue Calulae ansmiane and efleane Leue 8 Slide 47 Sep 6 Add

24 6//8 Sep 5 Move Soue & Add T/R Basi updae equaions Add a sof soue Add pefe bounda ondiion Inopoae TF/SF onewa soue Move posiion of soue Calulae ansmiane and efleane Leue 8 Slide 47 Sep 6 Add Devie (Complee Algoihm) Basi updae equaions Add a sof soue Add pefe bounda ondiion Inopoae TF/SF onewa soue Move posiion of soue Calulae ansmiane and efleane Add a eal devie Leue 8 Slide 48 4

25 6//8 Summa of Code Developmen Sequene Sep Implemen basi FDTD algoihm Sep Add he soue Sep 3 Add absobing bounda Sep 4 Add one wa soue Sep 5 Calulae ansmiane and efleane Sep 6 Add a devie Leue 8 Slide 49 FDTD Analsis Walhough Leue 8 Slide 5 5

26 6//8 Ouline of Seps fo FDTD Analsis Sep : Define poblem Wha devie ae ou modeling? Wha is is geome? Wha maeials is i made of? Wha do ou wan o lean abou he devie? Sep : Iniialie FDTD Compue gid esoluion Assign maeials values o poins on he gid Compue ime sep Iniialie Fouie ansfoms Sep 3: Run FDTD Sep 4: Pos poess he daa Leue 8 Slide 5 Sep : Define he Poblem Wha devie ae ou modeling? Wha is is geome? Wha maeials i is made fom? Wha do ou wan o lean? A dielei slab foo hi slab =., =6. (ouside is ai) efleane and ansmiane fom o G Ai. f., 6. Ai Leue 8 Slide 5 6

27 6//8 Sep : Compue Gid ( of ) Iniial Gid Resoluion (Wavelengh) n N ma min ma ma m s m f n. G m.437 m min N Iniial Gid Resoluion (Suue) N d 4 d 3.48 m d 7.6 m N 4 d Iniial Gid Resoluion (Oveall) min,.437 m d. f., 6. Ai Ai Leue 8 Slide 53 Sep : Compue Gid ( of ) Snap Gid o Ciial Dimension(s) The numbe of gid ells epesening he hiness of he dielei slab is d 3.48 m N 7.44 ells.437 m I is impossible o epesen he hiness of he slab eal wih his gid esoluion. To epesen he hiness of he slab eal, we ound N up o he neaes inege and hen alulae he gid esoluion based on his quani. N ound N 7 ells d 3.48 m.493 m N 7. f., 6. Ai Ai Leue 8 Slide 54 7

28 6//8 Sep : Build Devie on he Gid ( of ) Deemine Sie of Gid We need o have enough gid ells o fi he devie being modeled, some spae on eihe side of he devie ( ells fo now), and ells fo injeing he soue and eoding ansmied and efleed fields. N 7 ells 3 94 ells TRN/RF/SRC ells spae egions ells fo devie spae slab will go hee spae TF/SF Soue eod efleion eod ansmission Leue 8 Slide 55 Sep : Build Devie on he Gid ( of ) Compue Posiion of Maeials on Gid n, 3,, n n ound d n n,, Add Maeials o Gid n n,, UR(n:n) = u; R(n:n) = e; Leue 8 Slide 56 8

29 6//8 Sep : Iniialie FDTD ( of ) Compue he Time Sep m nb..493 m se s Compue Soue Paamees and 5. se fma G se Compue Numbe of Time Seps, STPS m nma N m 9 pop se s 9 8 T pop 6 Rule of humb 5 5 s s.934 se T STPS ound 495 T 5 Rule of humb Leue 8 Slide 57 pop STPS mus be an inege Time i aes a wave o popagae aoss he gid. Sep : Iniialie FDTD ( of ) Compue he Soue Funions fo / Mode s 3. A s ns.74 se. ep Aep % COMPUT GAUSSIAN SOURC FUNCTIONS = [:STPS-]*d; %ime ais del = ns*d/(*) + d/; %oal dela beween and A = - sq(es/us); %ampliude of field s = ep(-((-)/au).^); % field soue s = A*ep(-((-+del)/au).^); % field soue Iniialie he Fouie Tansfoms % INITIALIZ FOURIR TRANSFORMS NFRQ = ; FRQ = linspae(,*gigahe,nfrq); K = ep(-i**pi*d.*frq); RF = eos(,nfrq); TRN = eos(,nfrq); SRC = eos(,nfrq); Fequen, f Kenel, K. M. 5.5 M. i.3. M. i M.9994 i.336. G.999 i.45 Leue 8 Slide 58 9

6//8 Sep 3: Run FDTD (3 of 3) Leue 8 Slide

30 6//8 Sep 3: Run FDTD (3 of 3) Leue 8 Slide 59 Sep 4: Pos Poess he Daa Nomalie he Daa o he Soue Speum R T C F ef f f FFT s F n f f FFT s f R f T f % COMPUT RFLCTANC % AND TRANSMITTANC RF = abs(rf./src).^; TRN = abs(trn./src).^; CON = RF + TRN; Leue 8 Slide 6 3

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay) Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens