EE692 Applied EM- FDTD Method Chapter 3 Introduction to Maxwell s Equations and the Yee Algorithm. ds dl ds

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1 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 1 of 38 EE692 Applied EM- FDTD Method Chapter 3 Itroductio to Mawell s Equatios ad the Yee Algorithm 3.2 Mawell s Equatios i Three Dimesios Assume o electric or magetic curret sources Farada s Law t t ds dl ds (3.1) S L S where is a equivalet magetic curret desit (V/m 2 ). Ampere s Law t t ds dl ds (3.2) S L S Gauss Law 0 ds 0 (3.3) S Gauss Law (equivalet for magetic fields) 0 ds 0 (3.4) S

2 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 2 of cotiued For simple media (i.e., homogeeous, isotropic, & liear), the followig costitutive relatios hold r r 0 0 (3.5) Defie the electric ad equivalet magetic curret desities as source source (3.6) where is the coductivit (S/m) ad is the equivalet magetic loss (/m), allows for electricall ad mageticall loss materials. With these relatios, Ampere s ad Farada s Laws become t 1 1 source (3.7) 1 1 source t (3.8) respectivel. Sice Gauss Laws are NOT trul idepedet of Farada s ad Ampere s Laws, the are ot ecessar for settig up FDTD update equatios for the electric ad magetic fields. However, the must be obeed whe settig up the FDTD grid(s).

3 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 3 of cotiued Equatios (3.7) ad (3.8) ca be split (3 equatios/each) out ito vectorcompoet scalar differetial equatios. These equatios are iterdepedet. While other orthogoal coordiate sstems ca be used, usuall we use Cartesia coordiates for the FDTD method. From Farada s Law t 1 source, 1 source, t 1 source, t From Ampere s Law (3.9) 1 t source, 1 source, t 1 t source, (3.10)

4 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 4 of Reductio to Two Dimesios Some problems do ot var with respect to oe dimesio. Therefore, the reduce to two dimesios. For eample, sa the problem is ifiite i the -directio with o variatios i material or structure cross-sectio. The, there are o spatial () chages with respect to, i.e., 0. The, From Farada s Law t 1 source, 1 source, t 1 source, t From Ampere s Law (3.11a) (3.11b) (3.11c) t 1 source, 1 source, t 1 t source, (3.12a) (3.12b) (3.12c) Eamiig these equatios, we see that the ca be grouped ito two groups of three iter-related equatios each. Each group has a uique combiatio of three (total) electric & magetic compoets.

5 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 5 of TM Mode This group icludes,, ad are trasverse to the -directio. where the magetic field compoets The applicable differetial equatios are t 1 source, 1 source, t 1 t source, (3.13a) (3.13b) (3.13c) ca ot eist ear metallic or PEC surfaces that ifiite i etet i the -directio ad still satisf the tagetial electric field boudar coditios. This implies that surface waves or creepig waves will ot be foud i this mode.

6 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 6 of TE Mode This group icludes,, ad are trasverse to the -directio. where the electric field compoets The applicable differetial equatios are t 1 source, 1 source, t 1 source, t (3.14a) (3.14b) (3.14c) I this mode, the electric field compoets ( ad ) ca eist ear metallic or PEC surfaces that are ifiite i etet i the -directio. This implies that surface waves or creepig waves are possible i this mode. 3.4 Reductio to Oe Dimesio Some problems do ot var with respect to two dimesios. Therefore, these problems reduce to oe dimesio. For eample, sa the problem is ifiite i the - & -directios with o variatios i material or structure cross-sectio. The, there are o spatial () () chages with respect to &, i.e., 0.

7 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 7 of directed, -polaried TEM Mode () Elimiate all the terms i the TM mode equatios (3.13). Also, assume o source i the -directio ad that ( t 0) 0 to get t 1 source, 0 1 source, t t 1 source, (3.15a) (3.15b)/(3.16a) (3.15c)/(3.16b) Ol have ad (hece -polariatio) field compoets ad wave propagatio i -directios directed, -polaried TEM Mode () Elimiate all the terms i the TE mode equatios (3.14). Also, assume o source i the -directio ad that ( t 0) 0 to get t 1 source, 0 1 source, t t 1 source, (3.17a) (3.17b)/(3.18a) (3.17c)/(3.18b) Ol have (hece -polariatio) ad field compoets ad wave propagatio i -directios.

8 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 8 of Equivalece to the Wave Equatio i Oe Dimesio Assumig the source terms i the -directed, -polaried TEM Mode equatios (3.16) are ero, ad that = * = 0. The differetial equatios reduce to t 1 ad t 1. Takig aother partial derivative with respect to time ields t t ad 1 2 t t Takig aother partial derivative with respect to ields t t ad 2 t 2 1 t 2. B cross-substitutig, we the fid the oe-dimesioal, sigle variable, scalar wave equatios ad c t 1 t c (3.19c) (3.20c) where c 1 is the speed of light i the material.

9 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 9 of The Yee Algorithm Basic Ideas Semial paper (had-out) was published i Ma of Yee s FDTD equatios assumed = * = 0 ad o. Algorithm does NOT use scalar wave equatios, but istead is based o coupled/iter-related differetial equatios (i.e., the curl equatios of Ampere s ad Farada s Laws). Therefore, we have both electric ad magetic field compoets. Havig both electric ad magetic field compoets gives tremedous fleibilit i modelig shapes (e.g., ca hadle thi wires, slots, corers where the electric field varies as 1/r 2, ) as well as chagig material properties (chages ca be made o a cell-b-cell basis). A uit cell i the grid/lattice that Yee selected is show i Figure 1. Note how the field compoets are all cetered spatiall with respect to oe aother i the -, -, ad -directios. Figure 1 also illustrates how the itegral forms of Farada s ad Ampere s Laws (omit currets ad assume simple media) dl t L S ds dl t L S ds are satisfied. For Ampere s Law, ote how is cetered o the frot face of the Figure 1(b) ad how ad circulate aroud the cotour surroudig the surface of the frot face of the cube.

10 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 10 of cotiued ( i-0.5, j+ 0.5, k+ 0.5) (a) ( ijk,, ) Figure 1 Uit cell of 3D spatial Yee grid/lattice with faces cetered o (a) magetic ad (b) electric field compoets. (b)

11 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 11 of cotiued Overall both the differetial ad itegral forms of Mawell s equatios are modeled well b the uit cells show (icludig Gauss Laws). The discretiatio of the spatial derivatives will be accomplished usig secod-order accurate cetral-differece approimatios. The Yee cells make it eas to satisf tagetial electric ad magetic boudar coditios, if the material boudaries alig with the Cartesia aes. Tremedous fleibilit i modelig differet materials (e.g.,,, ad ). These material properties ca be specified o a cell-b-cell basis which gives a staircase or step approimatio to chages i the material/structure. Note that there is o divergece for either the electric or magetic field compoets i the Yee cells. Gauss Laws are implicitl satisfied. The field quatities are iterleaved i both space ad time (see Figure 2). Note how the electric field compoets are placed at iteger multiples of the time step t while the magetic field compoets are placed halfwa i betwee. This allows for a algorithm which successivel updates the electric ad magetic field compoets i a fashio referred to as time-stepig, leapfrog, recursive, or time-marchig. times (+0.5)t, update the compoets usig the prior values & spatiall adjacet compoets from time times (+1)t, update the compoets usig the prior values & spatiall adjacet compoets from time (+0.5)t.

12 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 12 of cotiued (a) Figure 2 Uit cell of 3D spatial Yee grid/lattice at (a) times t where there are ol electric field compoets, ad (b) times (+0.5)t where there are ol magetic field compoets. (b)

13 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 13 of cotiued As part of settig up these update equatios, the discretiatio of the temporal derivatives is also accomplished usig secod-order accurate cetral-differece approimatios. Note that the Yee FDTD algorithm has o matri iversios. Ulike MoM ad fiite-elemets techiques, each field compoet is idividuall ad eplicitl calculated. The Yee FDTD algorithm is odissipative, i.e., waves do ot deca due to umerical/o-phsical reasos.

14 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 14 of Fiite Differeces ad Notatio As part of settig up these update equatios, we will adopt a shorthad otatio for epressig the spatial locatios ad time. E.g., ( i, j, k, t t) U ( i, j, k) where i, j, ad k are iteger idices associated with the -, -, ad - coordiate directios ad is a iteger ide associated with time. Here,,,, ad t are the spatial ad temporal step sies. Doig a Talor s series epasio of (,,, t ) about i gives ( i, j, k, t) U ( i 0.5, j, k) U ( i 0.5, j, k) O( ) Note the similarit to the results of Chapter 2. The ke differece is that the steps forward ad backward were /2 istead of a full. The secod-order accurate cetral-differece approimatio to the spatial derivative with respect to is the (,,, ) i j k t U ( i 0.5, j, k) U ( i 0.5, j, k) Similarl, the secod-order accurate cetral-differece approimatio to the derivative with respect to time t t is the t t (,,, ) i j k t U ( i, j, k) U ( i, j, k) 2

15 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 15 of Fiite-Differece Epressios for Mawell s Equatios i Three Dimesios For some ukow reaso, the tet authors have reversed the usual covetio that electric field compoets are placed at times t = t while the magetic field compoets are placed at times t = (+0.5)t. I will follow the usual covetio. Remember the relative locatios of the field compoets (both spatiall ad temporall) are importat, ot the specific idices. Assume material properties (,,, ) are time-ivariat. To begi derivig the update equatios, we ll start with the scalar differetial equatio (3.10a) for the -compoet of Ampere s Law 1 t source, Per Figure 1(b), (o the frot face) is located at the spatial locatio ( = i, = (j+0.5), = (k+0.5)) ad the time derivative is approimated about time t = (+0.5)t. Whe discretied, this equatio becomes 1 E i j k E i j k (, 0.5, 0.5) (, 0.5, 0.5) t 1 ( i, j0.5, k0.5) H ( i, j 1, k 0.5) H ( i, j, k 0.5) H ( i, j 0.5, k 1) H ( i, j 0.5, k) 0.5 Jsource, ( i, j 0.5, k 0.5) 0.5 ( i, j 0.5, k 0.5) E ( i, j 0.5, k 0.5)

16 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 16 of cotiued Now the field compoet E 0.5 ( i, j 0.5, k 0.5) is ot available i the grid. Therefore, it is approimated, usig liear iterpolatio, as E ( i, j 0.5, k 0.5) E( i, j 0.5, k 0.5) (, 0.5, 0.5) E i j k 2 Usig this approimatio ad solvig for 1 E, ields the update equatio ( i, j0.5, k0.5) t 1 2 E i j k E i j k 1 2 ( i, j0.5, k0.5) 1 ( i, j0.5, k0.5) (, 0.5, 0.5) (, 0.5, 0.5) ( i, j0.5, k0.5) t t 1 2 ( i, j0.5, k0.5) ( i, j0.5, k0.5) ( i, j0.5, k0.5) H ( i, j 1, k 0.5) H ( i, j, k 0.5) H ( i, j 0.5, k 1) H ( i, j 0.5, k) t 0.5 Jsource, ( i, j 0.5, k 0.5) (3.29a) Similarl, the update equatios for, located o the right face of Figure 1(b) at ( = (i-0.5), = (j+1), = (k+0.5)), ad, located o the top face of Figure 1(b) at ( = (i-0.5), = (j+0.5), = (k+1)), with the time derivatives approimated about time t = (+0.5)t are foud from Ampere s Law usig the scalar differetial equatios (3.10b) ad (3.10c) as

17 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 17 of cotiued ( i0.5, j1, k0.5) t 1 2 E i j k E i j k 1 2 ( i0.5, j1, k0.5) 1 ( i0.5, j1, k0.5) ( 0.5, 1, 0.5) ( 0.5, 1, 0.5) ( i0.5, j1, k0.5) t H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k) t ( i0.5, j1, k0.5) H ( i, j 1, k 0.5) H ( i 1, j 1, k 0.5) ( i0.5, j1, k0.5) t ( i0.5, j1, k0.5) Jsource, ( i 0.5, j 1, k 0.5) ad (3.29b) ( i0.5, j0.5, k1) t 1 2 E i j k E i j k 1 2 ( i0.5, j0.5, k1) 1 ( i0.5, j0.5, k1) ( 0.5, 0.5, 1) ( 0.5, 0.5, 1) ( i0.5, j0.5, k1) t H ( i, j 0.5, k 1) H ( i 1, j 0.5, k 1) t ( i0.5, j0.5, k1) H ( i 0.5, j 1, k 1) H ( i 0.5, j, k 1) ( i0.5, j0.5, k1) t 1 2 J ( i 0.5, j 0.5, k 1) 0.5 ( i0.5, j0.5, k1) source, (3.29c)

18 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 18 of cotiued I a similar fashio, the update equatios for the magetic field compoets are foud usig Farada s Law usig the scalar differetial equatios (3.11a) - (3.11c) for Figure 1(b) at ( = (i-0.5), = (j+1), = (k+1)),, located o the upper right edge of, located o the upper frot edge of Figure 1(b) at ( = i, = (j+0.5), = (k+1)), ad, located o the right frot edge of Figure 1(b) at ( = i, = (j+1), = (k+0.5)) with the time derivatives approimated about time t = t. The update equatios for the magetic field compoets are * ( i0.5, j1, k1) t 1 2 H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k 1) 0.5 ( i0.5, j1, k1) 0.5 * ( i0.5, j1, k1) t t 1 2 ( i0.5, j1, k1) * ( i0.5, j1, k1) t ( i0.5, j1, k1) 1 2 ( i0.5, j1, k1) E( i 0.5, j 1, k 1.5) E( i 0.5, j 1, k 0.5) E( i 0.5, j 1.5, k 1) E( i 0.5, j 0.5, k 1) M source, ( i 0.5, j 1, k 1) (3.30a)

19 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 19 of cotiued * ( i, j0.5, k1) t 1 2 H ( i, j 0.5, k 1) H ( i, j 0.5, k 1) 0.5 ( i, j0.5, k1) 0.5 * ( i, j0.5, k1) t t 1 2 ( i, j0.5, k1) * ( i, j0.5, k1) 1 2 ( i, j0.5, k1) source, ( i, j0.5, k1) E ( i 0.5, j 0.5, k 1) E ( i 0.5, j 0.5, k 1) E( i, j 0.5, k 1.5) E( i, j 0.5, k 0.5) t M ( i, j 0.5, k 1) (3.30b) ad * ( i, j1, k0.5) t 1 2 H ( i, j 1, k 0.5) H ( i, j 1, k 0.5) 0.5 ( i, j1, k0.5) 0.5 * ( i, j1, k0.5) t t 1 2 ( i, j1, k0.5) * ( i, j1, k0.5) 1 2 ( i, j1, k0.5) source, ( i, j1, k0.5) E ( i, j 1.5, k 0.5) E ( i, j 0.5, k 0.5) E( i 0.5, j 1, k 0.5) E( i 0.5, j 1, k 0.5) t M ( i, j 1, k 0.5) (3.30c)

20 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 20 of Space Regio with a Cotiuous Variatio of Material Properties The implemetatio of the update equatios for the electric ad magetic field compoets i a regio where the material properties chage cotiuousl or frequetl ca be doe directl as give i (3.29) ad (3.30). However, from a computatioal stadpoit, it would be grossl iefficiet to re-calculate the time-ivariat coefficiets at each time step. Istead, these coefficiets are usuall pre-processed ad stored i memor. For the electric field updates, the relevat coefficiets at a geeral locatio (i, j, k) are C C a( i, j, k ) b1( i, j, k ) ( i, j, k ) t ( i, j, k ) t ( i, j, k ) ( i, j, k ) t ( i, j, k ) t 1 2 ( i, j, k ) 1 ( i, j, k ) (3.31b) (3.31a) C b2( i, j, k ) t ( i, j, k ) t 1 2 ( i, j, k ) 2 ( i, j, k ) (3.31c) where the coefficiet locatio matches the relevat electric field compoet ad 1 & 2 deote the two possible spatial icremets (out of,, ad ). For the case of a cubic lattice where = = =, C b1 = C b2 = C b.

21 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 21 of cotiued For the magetic field updates, the relevat coefficiets at a geeral locatio (i, j, k) are D D a( i, j, k ) b1( i, j, k ) ( i, j, k ) t ( i, j, k ) t ( i, j, k ) ( i, j, k ) t ( i, j, k ) t 1 2 ( i, j, k ) 1 ( i, j, k ) (3.31b) (3.31a) D b2( i, j, k ) t ( i, j, k ) t 1 2 ( i, j, k ) 2 ( i, j, k ) (3.31c) where the coefficiet locatio matches the relevat electric field compoet ad 1 & 2 deote the two possible spatial icremets (out of,, ad ). For the case of a cubic lattice where = = =, D b1 = D b2 = D b. Usig these coefficiets, the electric field update equatios, for a cubic lattice, become E ( i, j 0.5, k 0.5) C E ( i, j 0.5, k 0.5) 1 a, E ( i, j0.5, k0.5) H ( i, j 1, k 0.5) H ( i, j, k 0.5) C b, E (, 0.5, 0.5) (, 0.5, 1) (, 0.5, ) i j k H i j k H i j k 0.5 Jsource, ( i, j 0.5, k 0.5) (3.33a)

22 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 22 of cotiued E ( i 0.5, j 1, k 0.5) C E ( i 0.5, j 1, k 0.5) 1 a, E ( i0.5, j1, k0.5) H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k) C b, E ( 0.5, 1, 0.5) (, 1, 0.5) ( 1, 1, 0.5) i j k H i j k H i j k 0.5 Jsource, ( i 0.5, j 1, k 0.5) (3.33b) ad E ( i 0.5, j 0.5, k 1) C E ( i 0.5, j 0.5, k 1) 1 a, E ( i0.5, j0.5, k1) H ( i, j 0.5, k 1) H ( i 1, j 0.5, k 1) C b, E ( 0.5, 0.5, 1) ( 0.5, 1, 1) ( 0.5,, 1) i j k H i j k H i j k 0.5 Jsource, ( i 0.5, j 0.5, k 1) (3.33c) Similarl, usig these coefficiets, the magetic field update equatios, for a cubic lattice, become H ( i 0.5, j 1, k 1) D H ( i 0.5, j 1, k 1) a, H ( i0.5, j1, k1) E( i 0.5, j 1, k 1.5) E( i 0.5, j 1, k 0.5) D b, H ( 0.5, 1, 1) ( 0.5, 1.5, 1) ( 0.5, 0.5, 1) i j k E i j k E i j k M source, ( i 0.5, j 1, k 1) (3.34a)

23 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 23 of cotiued ad H ( i, j 0.5, k 1) D H ( i, j 0.5, k 1) a, H ( i, j0.5, k1) E( i 0.5, j 0.5, k 1) E( i 0.5, j 0.5, k 1) D b, H (, 0.5, 1) (, 0.5, 1.5) (, 0.5, 0.5) i j k E i j k E i j k M source, ( i, j 0.5, k 1) (3.34b) H ( i, j 1, k 0.5) D H ( i, j 1, k 0.5) a, H ( i, j1, k0.5) E( i, j 1.5, k 0.5) E( i, j 0.5, k 0.5) D b, H (, 1, 0.5) ( 0.5, 1, 0.5) ( 0.5, 1, 0.5) i j k E i j k E i j k M source, ( i, j 1, k 0.5) (3.34c) Usig these field update equatios, the total computer memor required is approimatel (code will have overhead) 18N = (6 compoets + 2 coefficiets/compoet 6 compoets)n where N is the total umber of Yee cells i the model.

24 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 24 of Space regio with a Fiite Number of Distict Media I free space or materials where the material properties are mostl homogeeous, the total computer memor required is reduced to approimatel 6N or 12N. Oe approach is to defie the eeded costat coefficiets before the time loop ad split the electric ad magetic field spatial update loops (i.e., loops with respect to i, j, & k) up to reflect the differet material regios. The advatage of this approach is that ou get a ver efficiet & fast code. The disadvatage is that ever problem must be custom coded. Total computer memor required is reduced to approimatel 6N Aother approach is to defie a iteger poiter arras MEDIA(i, j, k) where the stored iteger m is used to referece the appropriate coefficiets, e.g., C a (m), C b (m). D a (m), ad D b (m). The advatage of this approach is that ou get a more efficiet & faster code that is quite geeral. The disadvatage is that it is ot quite as efficiet or fast. Total computer memor required is reduced to approimatel 12N = (6 compoets + 1 coefficiet/compoet 6 compoets)n I this case, the applicable update equatios are

25 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 25 of cotiued m MEDIA ( i, j 0.5, k 0.5) E ( i, j 0.5, k 0.5) C ( m) E ( i, j 0.5, k 0.5) 1 a E H ( i, j 1, k 0.5) H ( i, j, k 0.5) C m H i j k H i j k 0.5 Jsource, ( i, j 0.5, k 0.5) b( ) (, 0.5, 1) (, 0.5, ) m MEDIA ( i 0.5, j 1, k 0.5) E ( i 0.5, j 1, k 0.5) C ( m) E ( i 0.5, j 1, k 0.5) 1 a E H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k) C m H i j k H i j k 0.5 Jsource, ( i 0.5, j 1, k 0.5) b( ) (, 1, 0.5) ( 1, 1, 0.5) m MEDIA ( i 0.5, j 0.5, k 1) E ( i 0.5, j 0.5, k 1) C ( m) E ( i 0.5, j 0.5, k 1) 1 a E H ( i, j 0.5, k 1) H ( i 1, j 0.5, k 1) C m H i j k H i j k 0.5 Jsource, ( i 0.5, j 0.5, k 1) b( ) ( 0.5, 1, 1) ( 0.5,, 1) for the electric field compoets. (3.35a) (3.35b) (3.35c)

26 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 26 of cotiued ad m MEDIA ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k 1) D ( m) H ( i 0.5, j 1, k 1) a H E( i 0.5, j 1, k 1.5) E( i 0.5, j 1, k 0.5) Db ( m) E ( i 0.5, j 1.5, k 1) E ( i 0.5, j 0.5, k 1) M source, ( i 0.5, j 1, k 1) m MEDIA ( i, j 0.5, k 1) H ( i, j 0.5, k 1) D ( m) H ( i, j 0.5, k 1) a H E( i 0.5, j 0.5, k 1) E( i 0.5, j 0.5, k 1) Db ( m) E ( i, j 0.5, k 1.5) E ( i, j 0.5, k 0.5) M source, ( i, j 0.5, k 1) m MEDIA ( i, j 1, k 0.5) H ( i, j 1, k 0.5) D ( m) H ( i, j 1, k 0.5) a H E( i, j 1.5, k 0.5) E( i, j 0.5, k 0.5) Db ( m) E ( i 0.5, j 1, k 0.5) E ( i 0.5, j 1, k 0.5) M source, ( i, j 1, k 0.5) (3.36a) (3.36b) (3.36c) for the magetic field compoets.

27 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 27 of Space Regio with Nopermeable Media I this case, the media is o-magetic (i.e., = 0 ad * = 0). The electric field compoet update equatios (3.29) are uchaged. However, the magetic field compoet update equatios (3.30) become H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k 1) E( i 0.5, j 1, k 1.5) E ( i 0.5, j 1, k 0.5) t E( i 0.5, j 1.5, k 1) E( i 0.5, j 0.5, k 1) 0 M source, ( i 0.5, j 1, k 1) H ( i, j 0.5, k 1) H ( i, j 0.5, k 1) E( i 0.5, j 0.5, k 1) E( i 0.5, j 0.5, k 1) t E( i, j 0.5, k 1.5) E( i, j 0.5, k 0.5) 0 M source, ( i, j 0.5, k 1) H ( i, j 1, k 0.5) H ( i, j 1, k 0.5) E( i, j 1.5, k 0.5) E( i, j 0.5, k 0.5) t E( i 0.5, j 1, k 0.5) E( i 0.5, j 1, k 0.5) 0 M source, ( i, j 1, k 0.5)

28 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 28 of cotiued Oe possibilit for efficietl implemetig the FDTD updates is to itroduce proportioal electric fields ad magetic equivalet currets, ˆ t E E ad 0 Mˆ t M (3.37) 0 where = = =i the FDTD update equatios, ad defiig a scaled update coefficiet ˆ t Cb ( m ) Cb ( m ) (3.38) 0 The, the updates of (3.35) ad (3.36) ca be re-writte as m MEDIA ( i, j 0.5, k 0.5) Eˆ ( i, j 0.5, k 0.5) C ( m) Eˆ ( i, j 0.5, k 0.5) 1 a E H ( i, j 1, k 0.5) H ( i, j, k 0.5) C m H i j k H i j k 0.5 Jsource, ( i, j 0.5, k 0.5) ˆ ( ) 0.5 (, 0.5, 1) 0.5 b (, 0.5, ) m MEDIA ( i 0.5, j 1, k 0.5) Eˆ ( i 0.5, j 1, k 0.5) C ( m) Eˆ ( i 0.5, j 1, k 0.5) 1 a E H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k) C m H i j k H i j k 0.5 Jsource, ( i 0.5, j 1, k 0. 5) ˆ ( ) 0.5 (, 1, 0.5) 0.5 b ( 1, 1, 0.5) (3.39a) (3.39b)

29 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 29 of cotiued m MEDIA ( i 0.5, j 0.5, k 1) Eˆ ( i 0.5, j 0.5, k 1) C ( m) Eˆ ( i 0.5, j 0.5, k 1) 1 a E H ( i, j 0.5, k 1) H ( i 1, j 0.5, k 1) C m H i j k H i j k 0.5 Jsource, ( i 0.5, j 0.5, k 1) ˆ ( ) 0.5 ( 0.5, 1, 1) 0.5 b ( 0.5,, 1) for the electric field compoets, ad (3.39c) ad H ( i 0.5, j 1, k 1) H ( i 0.5, j 1, k 1) ˆ ( 0.5, 1, 1.5) ˆ E i j k E ( i 0.5, j 1.5, k 1) ˆ ( 0.5, 1, 0.5) ˆ E i j k E( i 0.5, j 0.5, k 1) Mˆ ( i 0.5, j 1, k 1) source, H ( i, j 0.5, k 1) H ( i, j 0.5, k 1) ˆ ˆ E ( i 0.5, j 0.5, k 1) E ( i, j 0.5, k 1.5) ˆ ( 0.5, 0.5, 1) ˆ E i j k E( i, j 0.5, k 0.5) Mˆ ( i, j 0.5, k 1) source, H ( i, j 1, k 0.5) H ( i, j 1, k 0.5) ˆ ˆ E (, 1.5, 0.5) ( 0.5, 1, 0.5) i j k E i j k ˆ (, 0.5, 0.5) ˆ E i j k E( i 0.5, j 1, k 0.5) Mˆ ( i, j 1, k 0.5) source, for the magetic field compoets. (3.40a) (3.40b) (3.40c)

30 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 30 of Reductio to the Two-Dimesioal TM ad TE Modes As discussed i sectio 3.3, some problems do ot var with respect to oe dimesio. Therefore, the reduce to two dimesios, ad the si scalar compoet equatios decouple i to two groups of three equatios called TE ad TM modes. For eample, sa the problem is ifiite i the -directio with o variatios i material or structure cross-sectio. The, the si scalar compoet equatios decouple ito two groups of three differetial equatios called the TM (3.13) ad TE modes (3.14). I the followig sectios, the applicable FDTD update equatios for TM ad TE modes are give. The tet gives a versio of the update equatios for the TM ad TE modes for a fiite umber of material regios i (3.41) ad (3.42) respectivel. Two-Dimesioal TM Mode I the TM mode, the applicable field compoets are,, ad. Note that the magetic field compoets are trasverse to the -directio (i.e., the are located i the - plae). The applicable 2-D spatial grid/lattice (see Figure 3) is o a - plae. To see how this is a special case of the 3D grid/lattice, eamie the top face of the 3D Yee uit cell show i Figure 1(b). Agai, the electric field compoet (i.e., ) is placed at times t = t while the magetic field compoets (i.e., ad ) are placed at times t = (+0.5)t as show i Figure 4.

31 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 31 of cotiued j+1 j j-1 i-1 i i+1 Figure 3 2D spatial lattice for TM mode j+1 j+1 j j j-1 j-1 i-1 i i+1 i-1 i i+1 (a) (b) Figure 4 2D spatial lattice for TM mode at times (a) t (ol compoet) ad (b) (+0.5)t ( ad compoets).

32 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 32 of cotiued Discretiig (3.13) for the,, ad locatios idicated i Figure 3 leads to the TM mode update equatios * ( i0.5, j) t t ( i 0.5, j) 0.5 ( i0.5, j) H ( i 0.5, j) H ( 0.5, ) * i j * ( i0.5, j1) t ( i0.5, j) t ( i0.5, j) ( i0.5, j) E( i 0.5, j 0.5) E( i 0.5, j 0.5) M source, ( i 0.5, j) * ( i, j0.5) t t ( i, j 0.5) 0.5 ( i, j0.5) H ( i, j 0.5) H (, 0.5) * i j * ( i, j0.5) t ( i, j0.5) t ( i, j0.5) ( i, j0.5) E( i 0.5, j 0.5) E( i 0.5, j 0.5) M source, ( i0.5, j0.5) t 1 2 E i j E i j 1 2 ( i0.5, j0.5) 1 ( i0.5, j0.5) ( 0.5, 0.5) ( 0.5, 0.5) ( i0.5, j0.5) t t 1 2 ( i0.5, j0.5) ( i0.5, j0.5) ( i0.5, j0.5) ( i, j 0.5) H ( i, j 0.5) H ( i1, j0.5) H ( i 0.5, j 1) H ( i 0.5, j) t 0.5 Jsource, ( i 0.5, j 0.5)

33 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 33 of cotiued Two-Dimesioal TE Mode I the TE mode, the applicable field compoets are,, ad. Note that the electric field compoets are trasverse to the -directio (i.e., the are located i the - plae). The applicable 2D spatial grid/lattice (see Figure 5) is o a - plae. To see how this is a special case of the 3D grid/lattice, eamie the top face of the 3D Yee uit cell show i Figure 1(a). Agai, the electric field compoets (i.e., ad ) are placed at times t = t while the magetic field compoet (i.e. ) is placed at times t = (+0.5)t as show i Figure 6. j+1 j j-1 i-1 i i+1 Figure 5 2D spatial lattice for TE mode

34 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 34 of cotiued j+1 j+1 j j j-1 j-1 i-1 i i+1 i-1 i i+1 (a) (b) Figure 6 2D spatial lattice for TE mode at times (a) t ( ad compoets) ad (b) (+0.5)t (ol compoet). Discretiig (3.14) for the,, ad leads to the TE mode update equatios locatios idicated i Figure 5 ( i, j0.5) t t 1 2 t ( i, j0.5) ( i, j0.5) E ( i, j 0.5) E( i, j 0.5) ( i, j0.5) t ( i, j0.5) ( i, j0.5) ( i, j0.5) H ( i, j 1) H ( i, j) 0.5 Jsource, ( i, j 0.5)

35 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 35 of cotiued ( i0.5, j) t t 1 2 t ( i0.5, j) ( i0.5, j) E ( i 0.5, j) E( i 0.5, j) ( i0.5, j) t ( i0.5, j) ( i0.5, j) ( i0.5, j) H ( i, j) H ( i 1, j) 0.5 Jsource, ( i 0.5, j) * ( i, j) t ( i, j) 0.5 H ( i, j) H (, ) * i j ( i, j) t 1 2 ( i, j) E( i, j 0.5) E( i, j 0.5) t ( i, j) E ( i 0.5, j) E ( i 0.5, j) * ( i, j) t 1 2 ( i, j) M source, ( i, j)

36 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 36 of Iterpretatio as Farada s ad Ampere s Laws i Itegral Form Usuall, the FDTD update equatios are derived directl from the differetial or poit form of Ampere s ad Farada s Laws. While this works well i free space or homogeeous material regios, it is isufficiet to deal with model features (e.g., thi slots i metal sheets, thi resistive sheets, thi wires, ) smaller tha the spatial step sies. A method for dealig with these fie features, where the uderlig phsics are uderstood, is to use the itegral form of Ampere s ad Farada s Laws over the cotours ad surface areas of the FDTD grid/lattice. This topic is discussed i more detail i Chapter 10. To illustrate this method, we ll show how the FDTD update equatio for ca be derived b applig the itegral form of Ampere s Law to the cotour ad surface show i Figure 7 where the fields are i a free space regio with o electric curret sources ad we are at time t = (+0.5)t. H ( i-1, j+0.5, k) H ( i-0.5, j, k) E ( i-0.5, j+0.5, k) H ( i, j+0.5, k) H ( i-0.5, j+1, k) L Figure 7 FDTD update cotour ad surface

37 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 37 of cotiued I this case, the geeral itegral form of Ampere s Law (3.2) reduces to 0 ds t S L dl where the surface S is the rectagular area outlied b the cotour L (dashed lie) i Figure 7 ad ds ds aˆ dd (applied right-had rule to L). To evaluate the lie itegral o the RHS, assume that the ad values are costat over the legth of each of the four sides of S. This ields L dl H ( i 0.5, j, k) H ( i, j 0.5, k) H ( i, j 1, k) H ( i 1, j 0.5, k) To evaluate the surface itegral, assume that is costat (i.e., ca be pulled outside the itegral) over the etire surface S. This ields E ( i 0.5, j 0.5, k) d s t 0 0 t S E ( i 0.5, j 0.5, k) 0 t The, use the stadard FDTD approimatio o the time-derivative to get E ( i 0.5, j 0.5, k) ds t 0 0 t S S dd 1 E i j k E i j k 0 ( 0.5, 0.5, ) ( 0.5, 0.5, ) t

38 C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 38 of cotiued Equatig the results for the two sides of the itegral form of Ampere s Law ields 1 E ( i 0.5, j 0.5, k) E( i 0.5, j 0.5, k) 0 t H ( i 0.5, j, k) H ( i, j 0.5, k) H ( i, j 1, k) H ( i 1, j 0.5, k) We ca the solve for the update equatio 1 E i j k E i j k ( 0.5, 0.5, ) ( 0.5, 0.5, ) t H ( i, j 0.5, k) H ( i 1, j 0.5, k) 0 t H ( i, j 1, k) H ( i 0.5, j, k) 0 which is what we get from (3.29c) uder the same coditios Divergece-Free Nature This sectio shows how applig the itegral form of Gauss Law (3.3) to the si sides of a Yee uit cell (see Figure 1(b)) ad the substitutig i the appropriate update equatios for the relevat electric field compoets satisfies the coditio that the et electric flu be ero. Further work o this will be reserved for the homework.