EE 5337 Computational Electromagnetics. Maxwell s Equations

Transcription

1 9/15/217 Instucto D. Ramond Rumpf (915) Computational lectomagnetics Lectue #2 Mawell s quations Lectue 2These notes ma contain copighted mateial obtained unde fai use ules. Distibution of these mateials is stictl pohibited Slide 1 Outline Mawell s equations Phsical Bounda conditions Paamete elations Pepaing Mawell s equations fo CM The wave equation and its solutions Scaling popeties of Mawell s equations Lectue 2 Slide 2 1

2 9/15/217 Mawell s quations Bon Died June 13,1831 dinbugh, Scotland Novembe 5, 1879 Cambidge, ngland James Clek Mawell Lectue 2 Slide 3 Sign Conventions fo Waves To descibe a wave popagating the positive diection, we have two choices:, cos, cos t A t k t A t k Most common in engineeing Most common science and phsics Both ae coect, but we must choose a convention and be consistent with it. Fo time hamonic signals, this becomes ep A jk ep A jk Negative sign convention Positive sign convention Lectue 2 Slide 4 2

3 9/15/217 Summa of Sign Conventions Lectue 2 Slide 5 Summa of Sign Conventions Lectue 2 Slide 6 3

4 9/15/217 Loent Foce Law One additional equation is needed to completel descibe classical electomagnetism... F qqvb lectic Foce Magnetic Foce Lectue 2 Slide 7 Altenate Foms of Mawell s quations Mawell s quations with Gaussian Units 1 B D4v c t 1 D B4v 4 J c t Relativistic Mawell s quations F J 1 e F J 2 fee D Mawell s quations in Moving Media 1 B D 4v Bv c t 1 D B4v 4 J Dv c t 1 Lectue 2 Slide 8 4

5 9/15/217 Time amonic Mawell s quations Time Domain B D v t D B J t Fequenc Domain (e +jk convention) Dv jb B J jd Fequenc Domain (e -jk convention) Dv jb B J jd Lectue 2 Slide 9 Gauss s Law D v D D D D lectic fields divege fom positive chages and convege on negative chages. If thee ae no chages, electic fields must fom loops. Lectue 2 Slide 1 5

6 9/15/217 Gauss s Law fo Magnetism B B B B B Magnetic fields alwas fom loops. Lectue 2 Slide 11 Consequence of Zeo Divegence The divegence theoems foce the D and B fields to be pependicula to the popagation diection of a plane wave. D jk de d jk d k d no chages k D k k B jk be b jk b k b no chages k B k k Lectue 2 Slide 12 6

7 9/15/217 Ampee s Law with Mawell s Coection D J t aˆ aˆ aˆ Ciculating magnetic fields induce cuents and/o time vaing electic fields. Cuents and/o time vaing electic D fields induce ciculating magnetic fields. J t Lectue 2 Slide 13 Faada s Law of Induction B t aˆ aˆ aˆ Ciculating electic fields induce time vaing magnetic fields. Time vaing magnetic fields induce ciculating electic fields. B t Lectue 2 Slide 14 7

8 9/15/217 Consequence of Cul quations The cul equations pedict electomagnetic waves!! lectic Field Magnetic Field Lectue 2 Slide 15 The Constitutive Relations lectic Response D lectic field intensit (V/m) Initial electic push. Induced electic field. lectic eneg in vacuum. Pemittivit (F/m) Measue of how well a mateial stoes electic eneg. lectic flu densit (C/m 2 ) Petends as if all electic eneg is displaced chage. Includes electic eneg in vacuum and matte. Magnetic Response B Magnetic field intensit (A/m) Initial magnetic push. Induced magnetic field. Magnetic eneg in vacuum. Pemeabilit (/m) Measue of how well a mateial stoes magnetic eneg. Magnetic flu densit (Wb/m 2 ) Petends as if all magnetic eneg is tilted magnetic dipoles. Includes magnetic eneg in vacuum and matte. Lectue

9 9/15/217 Mateial Classifications Linea, isotopic and non dispesive mateials: D t t Dispesive mateials: D t t t Anisotopic mateials: D t t Nonlinea mateials: We will use this almost eclusivel A ke point is that ou can wap all of the compleities associated with modeling stange mateials into this single equation. This will make ou code moe modula and easie to modif. It ma not be as efficient as it could be though e e e D t t t t Lectue 2 17 Tpes of Anisotop Isotopic D t t B t t Gotopic D t B t t t 1 j2 j2 1 3 goelectic Bi Isotopic D t t B t t Anisotopic D t B t t t Bi Anisotopic Dt t B t t aa ab ac ba bb bc ca cb cc electicall anisotopic c Lectue j2 j2 1 3 gomagnetic aa ab ac ba bb bc ca cb cc magneticall anisotopic isotopic iso iso iso uniaial o o e biaial a b iso 9

10 9/15/217 All Togethe Now Divegence quations B D v Cul quations D J t B t What poduces fields Constitutive Relations Dt t t B t t t means convolution means tenso ow fields inteact with mateials Lectue 2 Slide 19 Mawell s quations in Catesian Coodinates (1 of 4) Vecto Tems ˆ ˆ ˆ a a a D D aˆ D aˆ D aˆ aˆ aˆ aˆ B B aˆ B aˆ B aˆ J Jaˆ Jaˆ Jaˆ Divegence quations D D D D B B B B Lectue 4 Slide 2 1

11 9/15/217 Mawell s quations in Catesian Coodinates (2 of 4) Constitutive Relations D aˆ ˆ a D aˆ D aˆ Daˆ aˆ D D D B B B B Lectue 4 Slide 21 Mawell s quations in Catesian Coodinates (3 of 4) Cul quations B t aˆ ˆ ˆ ˆ ˆ ˆ a a Ba Ba Ba t B B B aˆ ˆ ˆ ˆ ˆ ˆ a a a a a t t t B t B t B t Lectue 4 Slide 22 11

12 9/15/217 Mawell s quations in Catesian Coodinates (4 of 4) Cul quations D J t aˆ aˆ aˆ J aˆ J aˆ J aˆ D aˆ D aˆ D aˆ t D D D aˆ ˆ ˆ ˆ ˆ ˆ a a J a J a J a t t t D J t D J t D t J Lectue 4 Slide 23 Altenative Fom of Mawell s quations in Catesian Coodinates (1 of 2) Altenate Cul quations t aˆ ˆ ˆ ˆ a a a t t t a ˆ t t t ˆ a t t t t t t t t t t t t Lectue 4 Slide 24 12

13 9/15/ Lectue 4 Slide 25 Altenative Fom of Mawell s quations in Catesian Coodinates (2 of 2) Altenate Cul quations t ˆ ˆ ˆ ˆ ˆ ˆ a a a t t t a a a t t t t t t t t t t t t t t t Lectue 2 Slide 26 Phsical Bounda Conditions

14 9/15/217 Phsical Bounda Conditions and 1 1 and 2 2 Tangential components of and ae continuous acoss an inteface. 1,T 2,T 1,T 2,T and fields nomal to the inteface ae discontinuous acoss an inteface. Note: Nomal components of D and B ae continuous acoss the inteface. 11,N 11,N D 1,N B 1,N 22,N 22,N D 2,N B 2,N These ae moe complicated bounda conditions, phsicall and analticall. Tangential components of the wave vecto ae continuous acoss an inteface. k1,t k2,t Lectue 2 Slide 27 Paamete Relations Lectue 2 Slide 28 14

15 9/15/217 Map of Paamete Relations M f c B D P n v Lectue 2 Slide 29 The Relative Pemittivit The pemittivit is a measue of how well a mateial stoes electic eneg. A ciculating magnetic field induces an electic field at the cente of the ciculation in popotion to the pemittivit. j t The dielectic constant of a mateial is its pemittivit elative to the pemittivit of fee space F m 1 is the elative pemittivit o dielectic constant Lectue 2 Slide 3 15

16 9/15/217 The Relative Pemeabilit The pemeabilit is a measue of how well a mateial stoes magnetic eneg. A ciculating electic field induces a magnetic field at the cente of the ciculation in popotion to the pemeabilit. j t The elative pemeabilit of a mateial is its pemeabilit elative to the pemeabilit of fee space m 1 is the elative pemeabilit Lectue 2 Slide 31 Conductivit Conductivit is the measue of a mateial s abilit to suppot electic cuent. This tem is esponsible fo ohmic loss in mateials. It appeas in Ampee s Cicuit Law. J jd The cuent densit J is elated to conductivit and the electic field intensit though Ohm s Law. J Lectue 2 Slide 32 16

17 9/15/217 -j Vs. and It is edundant to have a comple dielectic constant along with a conductivit tem, although it happens. We should use eithe a comple dielectic constant o a eal dielectic constant and a conductivit. j j j j j j j j Lectue 2 Slide 33 Mateial Impedance The mateial impedance is the paamete which descibes the balance between the electic and magnetic field amplitudes. k It is calculated fom the pemeabilit and pemittivit of the mateial. Phase between and Amplitude between and fee space impedance Impedance tells us that and ae thee odes of magnitude diffeent. j Reactive component Resistive component. Lectue 2 Slide 34 17

18 9/15/217 The Comple Refactive Inde The pemittivit and pemeabilit appea in Mawell s equations so the ae the most fundamental mateial popeties. oweve, it is difficult to detemine phsical meaning fom them in tems of how waves popagate (i.e. speed, loss, etc.). In this case, the efactive inde is a moe meaningful quantit. n In the fequenc domain, the efactive inde is a comple quantit. n n j o e jk n n o is the odina efactive inde. It quantifies how quickl a wave popagates. is the etinction coefficient. It quantifies loss and how quickl a wave decas. * Note: when onl the efactive inde n is specified fo a mateial, assume = 1.. Lectue 2 Slide 35 The Comple Popagation Constant, The popagation constant is ve close to the comple efactive inde. It descibes the speed and deca of a wave. e The popagation constant has a eal and imagina pat. j e e j is the attenuation coefficient. It quantifies how quickl the amplitude of a wave decas. is the popagation constant. It quantifies how quickl a wave accumulates phase. It is elated to the comple efactive inde though jk n Lectue 2 Slide 36 18

19 9/15/217 The Absoption Coefficient, The absoption coefficient descibes how quickl the powe in a wave decas. P Pe WARNING: Notice the unfotunate euse of the smbol fo two diffeent things. This is easil confused!! The attenuation coefficient and absoption coefficient ae elated though abs 2 att The absoption coefficient and etinction coefficient ae elated though 2k abs Lectue 2 Slide 37 Loss Tangent Sometimes mateial loss is given in tems of a loss tangent. tan Recall that intepeting wave popeties (velocit and loss) is not intuitive using just the comple dielectic function. To do this, we pefeed the comple efactive inde. It tuns out that the loss tangent and the etinction coefficient ae essentiall the same. 2 abs n k n P Pe kn It is called a loss tangent because it is the angle in the comple plane fomed between the esistive component and the eactive component of the electomagnetic field. o Lectue 2 Slide 38 o 19

20 9/15/217 vesus f is the angula fequenc measued in adians pe second. It elates moe diectl to phase and k. Think cos(t). f is the odina fequenc measued in ccles pe second. It elates moe diectl to time. Think cos(2ft) and =1/f. 2 f Lectue 2 Slide 39 Wavelength and Fequenc The fequenc f and fee space wavelength ae elated though c f c speed of light in vacuum m s Inside a mateial, the wave slows down accoding to the efactive inde as follows. v c n Lectue 2 Slide 4 2

21 9/15/217 Summa of Paamete Relations Pemittivit F m Pemeabilit m Refactive Inde Impedance n Wave Velocit c v n c m s act Fequenc and Wavelength 2 f Wave Numbe c f k 2 Lectue 2 Slide 41 Table of Dielectic Constants and Loss Tangents Constantine A. Balanis, Advanced ngineeing lectomagnetics, Wile, Lectue 2 Slide 42 21

22 9/15/217 Table of Pemeabilities Constantine A. Balanis, Advanced ngineeing lectomagnetics, Wile, Lectue 2 Slide 43 Dualit Between D and B lectic Field D P ε Magnetic Field B M μ Lectue

23 9/15/217 Pepaing Mawell s quations fo CM Lectue 2 Slide 45 Simplifing Mawell s quations 1. Assume no chages o cuent souces: v, J B D t D t t t D B t B t t t 2. Tansfom Mawell s equations to fequenc domain: B jd D D jb B Convolution becomes simple multiplication Note: We have chose to poceed with the negative sign convention. 3. Substitute constitutive elations into Mawell s equations: j j Note: It is useful to etain μ and ε and not eplace them with efactive inde n. Lectue 2 Slide 46 23

24 9/15/217 Isotopic Mateials Fo anisotopic mateials, the pemittivit and pemeabilit tems ae tenso quantities. Fo isotopic mateials, the tensos educe to a single scala quantit. Mawell s equations can then be witten as j j Lectue 2 Slide 47 and dopped fom these equations because the ae constants and do not va spatiall. pand Mawell s quations Divegence quations Cul quations j j j j j j j j Lectue 2 Slide 48 24

25 9/15/217 Nomalie the Magnetic Field Standad fom of Mawell s Cul quations j j Nomalied Magnetic Field 377 n j liminates j No sign inconsistenc Just have k Note: k qualies and amplitudes Nomalied Mawell s quations k k Lectue 2 Slide 49 Stating Point fo Most CM We aive at the following set of equations that ae the same egadless of the sign convention used. k k k k k k The manne in which the magnetic field is nomalied does depend on the sign convention chosen. j negative sign convnetion j positive sign convnetion Lectue 2 Slide 5 25

26 9/15/217 The Wave quation and Its Solutions Lectue 2 Slide 51 Deivation of the Wave quation We stat with Mawell s cul equations. j quation (1) is solved fo the magnetic field. j j q. (1) q. (2) q. (3) quation (3) is substituted into q. (2). j j 1 2 k Lectue 2 Slide 52 26

27 9/15/217 Two Diffeent Wave quations We can deive a wave equation fo both and. k k It is not actuall possible to simplif these equations futhe without making an appoimation. Assuming a linea homogeneous isotopic (LI) mateial, the wave equations educe to 2 k 2 2 k 2 2 k 2 k 2 2 k 2 2 k We see that these equations will have the same solution since it is the same diffeential equation! So, we onl have to solve one of them. Lectue 2 Slide 53 Plane Wave Solution in omogeneous Media Given the wave equation in an LI mateial, 2 2 k The solution is a plane wave. jk jk ep ep Lectue 2 Slide 54 27

28 9/15/217 Amplitude Relation Given plane wave functions of the fom jk jk ep ep The amplitudes ae elated though Mawell s equations. j jk e j e k jk jk e j e jk jk k k k Lectue 2 Slide 55 IMPORTANT: Plane Waves ae of Infinite tent Man times we just daw as o sometime as with pependicula lines to epesent the wave fonts. a a + pependicula lines Unfotunatel, this suggests the wave is confined spatiall. In ealit, plane waves ae of infinite etent. Think moe this wa Lectue 2 Slide 56 28

29 9/15/217 Solving the Wave quation as a Scatteing Poblem Scatteing poblems cast the wave equation into the following mati fom. A b 1 2 k g A A souce b is needed Onl one solution eists k 1 2 b g Lectue 2 Slide 57 Solving the Wave quation as an igen Value Poblem The wave equation can also be solved as an eigen value poblem. This appoach is used when modes ae being calculated. k 1 2 No souce is needed Multiple solutions eist A B 1 A B k 2 Lectue 2 Slide 58 29

30 9/15/217 Wave quation Vs. Mawell s quations Wave quation The most genealied wave equations ae k k In LI mateials, these educe to 2 2 k 2 2 k Toda, it is ae to see the wave equations solved in this fom because it leads to spuious solutions. The fies to the spuious solutions poblem ae incopoated into Mawell s equations befoe a wave equation is deived. Mawell s quations Mawell s equations epanded into Catesian coodinates ae k k k k k k These ae often witten in mati fom as k k Tpicall, fies ae incopoated hee and then a wave equation is deived. 1 k 2 Lectue 2 Slide 59 Scaling Popeties in Mawell s quations Lectue 2 Slide 6 3

31 9/15/217 Scaling Popeties of Mawell s quations Thee is no fundamental length scale in Mawell s equations. Devices ma be scaled to opeate at diffeent fequencies just b scaling the mechanical dimensions o mateial popeties in popotion to the change in fequenc. This assumes it is phsicall possible to scale sstems in this manne. In pactice, building lage o smalle featues ma not be pactical. Futhe, the popeties of the mateials ma be diffeent at the new opeating fequenc. Lectue 2 Slide 61 Scaling Dimensions We stat with the wave equation and wite the paametes dependence on position eplicitl. 1 2 Net, we scale the dimensions b a facto a. 1 a a a a a 2 a a 1 stetch dimensions a 1 compess dimensions The scale factos multipling the opeatos ae moved to multipl the fequenc tem. 1 2 a a The effect of scaling the dimensions is just a shift in fequenc. Lectue 2 Slide 62 31

32 9/15/217 Visualiation of Sie Scaling a = 1. a =.5 f c = 1 M f c = 5 M Lectue 2 Slide 63 Scaling and We appl sepaate scaling factos to and. 1 a 2 a The scale factos ae moved to multipl the fequenc tem. 1 2 aa The effect of scaling the mateial popeties is just a shift in fequenc. Lectue 2 Slide 64 32