ECE 222b Applied Electromagnetics Notes Set 3a

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1 C b lid lcomagnics Nos S 3a Insuco: Pof. Viali Lomakin Damn of lcical and Comu ngining Univsi of Califonia San Digo

2 Unifom Plan Wavs Consid Mawll s quaions: In a losslss mdium ε and µ a al and σ : Sinc ε ρ

3 3 Unifom Plan Wavs m M i J i ρ ε µε µ Similal In souc-f gion: m i M i J ρ ρ µε µε µε o Wav quaions Lalacian oao Poagaion consan

4 Unifom Plan Wavs 3 Scial cas: Wav quaion Two soluions: Consid Dfin ininsic imdanc: µ ε hn 4

5 Unifom Plan Wavs 4 Wav imdanc: In im domain: R qui-has lan: Z w µ ε [ ] [ ] R cos Unifom Plan Wav consan d d Phas vloci: v d d µε v µε Sd of ligh 8 µ µ ε ε v 3 m/s µ ε In vacuum 5

6 6 Unifom Plan Wavs 5 Wavlngh: λ π π π λ v v f v T wavnumb cos cos cos cos cos w w m ε µ µ ε ε cos S Consid Filds ng dnsi Pow flow dnsi

7 Unifom Plan Wavs 6 ng vloci: v ow flow dnsi ng dnsi w S w m µε Consid cos cos [ ] [ ] cos cos consan Wav ack Gou vloci: v g d d d d µε 7

8 Unifom Plan Wavs 7 Phas vloci: v d d d d v v v dv d v g v dv v d No dission: dv d v g v Nomal dission: dv d < v < g v 3 nomalous dission: dv d > v > g v 8

9 Unifom Plan Wavs 8 Sanding wav: In fqunc domain In im domain sin R[ ] sin sin. Th has is indndn of o. w ε ε cos cos wm µ ε sin sin cos cos cos v 9

10 sin sin sin sin cos cos 4 S Unifom Plan Wavs 9 Tim avag sin cos sin * S R - avag ow flow Tim S sin cos sin cos sin cos Fo a mo gnal cas: cos sin cos

11 Unifom Plan Wavs ma min an [ an ] Sanding wav aio SWR: SWR ma min Γ Γ Fo a u avling wav : SWR Fo a u sanding wav : SWR Fo a gnal wav : SWR <

12 Unifom Plan Wavs Unifom lan wav in a loss mdium: µ M i ε σ J ε σ In a souc-f gion: Z w i µ ε σ µ µ ε σ siml cas: d α d J i

13 Unifom Plan Wavs Possibl soluions: ow o find α and? α α µ ε σ α Coml quaion µε α µσ α µσ α α µσ 4α µε 4 α µεα µσ 4 3

14 4 Dsid soluion ε σ ε σ µε µε α α α 4 3 α α α α α Fou soluions: Unifom Plan Wavs 3

15 Unifom Plan Wavs 4 α µε µε σ ε σ ε µ ε σ. Fo fc conduco: σ α. Fo fc dilcic: σ α µε µ ε 5

16 Unifom Plan Wavs 5 << 3. Fo good dilcic: σ ε σ 4 σ σ σ ε ε 8 ε ε α µε σ σ µ µ ε µε ε ε 4. Fo good conduco: σ >> σ σ ε ε 3 ε ε σ σ ε 8 α µε σ µσ µσ µ ε σ 6

17 Unifom Plan Wavs 6 Skin dh: Th fild amliud ducs o 36.8% δ α α µε σ ε m. Fo fc conduco:. Fo fc dilcic: 3. Fo good conduco: 4. Fo good dilcic: δ δ δ µσ ε δ σ µ 7

18 8 Polaiaion Polaiaion: Th dicion of h lcic fild. Linal olaid in h dicion. Linal olaid in h dicion.. 3. Consid hi combinaion:

19 9 In im domain: cos cos cos cos R R R b a b a a Polaiaion. If hn b a cos a Linal olaid in h dicion: - an φ φ

20 . If hn π a b sin cos sin cos lliicall olaid Lf-hand Coun-clockwis Polaiaion 3 ciculal olaid If

21 Polaiaion 4 a π b 3. If hn sin cos lliicall olaid Righ-hand Clock wis If ciculal olaid

22 Polaiaion 5 Summa. Linal olaid:. Ciculal olaid: 3. lliicall olaid:

23 3 Obsvaion #: Polaiaion 6 n lliicall o ciculal olaid wav can b dcomosd ino wo linal olaid wavs. Obsvaion #: linal olaid wav can b dcomosd ino wo lliicall o ciculal olaid wavs. - hand igh - hand lf

24 4 Obsvaion #3: Polaiaion 7 cos S Fo a linal olaid wav: Fo a ciculal olaid wav: sin cos cos sin sin cos S Sad ow flow

25 5 Unifom Plan Wavs Wav quaion: µε Vco fom Scala fom In cangula coodinas: Z Y X T saaion of vaiabls: XYZ Z XY Y XZ X YZ Z Z Y Y X X

26 6 Saaion: Z Z Y Y X X Unifom Plan Wavs Z Z Y Y X X XYZ

27 7 Similal C α α cos ] R[ α α Unifom Plan Wavs 3 qui-amliud: consan α qui-has: consan Unifom Plan Wavs qui-has lan: consan v Phas vloci

28 8 Unifom Plan Wavs 4 d d α α α α Fo Fo Usful mahmaical fomulas: Consid a unifom lan wav: µ µ ε ε µ ε

29 9 Fo a losslss lan wavs and a alwas ndicula o. µ Sinc c b a b c a c b a Unifom Plan Wavs 5 µ iad fom a k µε µ Sinc k k Dission laion

30 Unifom Plan Wavs 6 Summa: 3

31 3 Unifom Plan Wavs 7 Fo a mdium wih a ngaiv miivi and ngaiv mabili Doubl ngaiv: Sinc c b a b c a c b a Lf-handd maial LM: ackwad oagaion

32 3 Plan Wavs Gnad b Cun Sh ssum a im-hamonic cun: m m h s h J J J J δ : Fo : Fo ε µ µ ε < > Find filds gnad b his cun sh.

33 33 Fom has maching : Fom bounda condiion : h h J J n h h s µ ε Plan Wavs Gnad b Cun Sh h h ε µ J J N find h lcic fild.

34 Plan Wavs Gnad b Cun Sh 3 Fom ε ε ε h h h ε h ssum ha µ µ ε ε. Fom bounda condiion : n h h 34

35 35 J h h Plan Wavs Gnad b Cun Sh 4 h h J h J h Thfo J J h Scial cas:

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