EIE 332 Electromagnetics

Transcription

1 EIE 332 Elctromagntics cturr: Dr. W.Y.Tam Room no.: DE604 Phon no.: mail: wb: Normal Offic hour: 9:00am 5:30pm (Mon-Fri) 9:00am 12:30pm (Sat) Acknowldgmnt: I would lik to thank Mr. K.Y. Tong to lt m to modify his handouts for this subjct. Intro.1

2 Assssmnt Examination (opn book) 60% Practical Mini-projct 15% Wk 9 to Wk 13 Rport, dmonstration/prsntation Tst 10% Wk 5 and Wk 10 Quis 10% Assignmnts 5% Intro.2

3 Txtbook 1. D. K. Chng, Fundamntals of Enginring Elctromagntics, Addison Wsly, D. K. Chng, Fild and Wav Elctromagntics, Addision Wsly, Intro.3

4 Why study Elctromagntics? Introduction High spd circuits - Microwav and high spd digital circuits Antnna - Wirlss communication Optical communication - ight propagation in fibrs Elctromchanical machins Elctromagntic intrfrnc and compatibility Intro.4

5 Elctromagntics startd with th xprimntal obsrvation of (i) forcs btwn lctric chargs; (ii) forcs btwn conductors carrying lctric currnts In fr spac (vacuum), Q -Q.. d F Q 2 4πε o d 2 whr ε o is prmittivity of fr spac Intro.5

6 I 1 R Forc I 2 Attraction forc btwn two paralll wirs with lngth carrying currnts I 1, I 2 in th sam dirction: µ o I1I 2 ( ) F 2 4πR whr µ o is th prmability of fr spac 2 Intro.6

7 Introduc th concpt of FIED to facilitat th manipulation of th abov forcs Elctric fild - gnratd by chargs Magntic fild - gnratd by currnts Dtrmin th forcs acting on chargs and currnts placd in lctric and magntic FIEDS Intro.7

8 Rfrnc Transmission lin (3 Wks) Chaptr 8.1 Ovrviw Chaptr 8.2 Gnralid Transmission-lin Equations Chaptr 8.3 Transmission-lin Paramtrs Chaptr 8.4 Wav Charactristics on an Infinit Transmission in Chaptr 8.5 Wav Charactristics on Finit Transmission ins Chaptr 8.6 Th Smith Chart Chaptr 8.7 Transmission-lin Impdanc Matching Intro.8

9 II. TRANSMISSION INE 2.1 Introduction Any pair of wirs and conductors carrying currnts in opposit dirctions form transmission lins. Transmission lins ar ssntial componnts in any lctrical/ communication systm. Thy includ coaxial cabls, two-wir lins, microstrip lins on printd-circuit-boards (PCB). (Not that at vry high frquncis, any conductor on a PCB must b considrd as transmission lins.) Th charactristics of transmission lins can b studid by th lctric and magntic filds propagating along th lin. But in most practical applications, it is asir to study th voltags and currnts in th lin instad. Intro.9

10 ground shild Coaxial cabl Two-wir transmission lin dilctric substrat conductor Microstrip lin Intro.10

11 Magntic fild Elctric fild Cross-sction of a coaxial cabl showing th lctric and magntic filds Intro.11

12 2.2 Rvision of Travlling Wavs Th quation rprsnts a wav travlling in th dirction with constant amplitud A, whr ω2πf, β2π/λ. Any point of constant phas P advancs towards th dirction with a phas vlocity d dt ω β Similarly a wav rprsntd by travls in th - dirction. v v A cos( ωt β) A cos( ω t β) Intro.12

13 ωt 0 P 1 sin ( β) β ωt 2 P 1 sin ( 2 β) β ωt 4 P 1 sin ( 4 β) β Intro.13

14 Oftn th amplitud of a wav varis xponntially with distanc. Th quation for such a wav is: v A α cos ( ωt β) If α is positiv, th wav amplitud is attnuatd xponntially as it travls in th v dirction. If α is ngativ, th wav amplitud incrass xponntially as it travls in th v dirction. Intro.14

15 Phasor rprsntation: Th cosin function is oftn rplacd by th complx xponntial function. A α cos ( ) j ( t ) ωt β A α ω β Th wav phasor is writtn as: A α jβ aftr dropping th trm j. ωt A α is th magnitud, and -β is th phas angl. Intro.15

16 2.3 oltag and Currnt Wavs in gnral transmission lins Equivalnt circuit of an lmnt sction (lngth ) of th transmission lin:, R ar th distributd inductanc and rsistanc (pr unit lngth) of th conductor; C,G ar th distributd capacitanc and conductanc (pr unit lngth) of th dilctric btwn th conductors. Intro.16

17 Intro.17 Rlation btwn instantanous voltag v and currnt i at any point along th lin: t i Ri v t v C Gv i For priodic signals, Fourir analysis can b applid and it is mor convnint to us phasors of voltag and currnt I. ( ) C j G I I j R ) ( ω ω

18 Dcoupling th abov quations, w gt I γ γ 2 2 I whr γ is calld th propagation constant, and is in gnral complx. γ ( R jω)( G jωc ) α jβ α is th attnuation constant, β is th phas constant. Intro.18

19 Intro.19 Th gnral solutions of th scond-ordr, linar diffrntial quation for, I ar : I I I γ γ γ γ, -, I, I - ar constants (complx phasors). Th trms containing -γ rprsnt wavs travlling in dirction; trms containing γ rprsnt wavs travlling in - dirction. j β α γ α dtrmins th attnuation along th lin, and β dtrmins th phas shift along th lin. Sinc

20 Intro.20 whr o is th charactristic impdanc of th lin, givn by C j G j R o ω ω Th currnt I can now b writtn as: o o I γ γ It can b shown that th ratio of voltag to currnt is givn by: o I o I

21 2.4 osslss transmission lins In losslss transmission lins, th distributd conductor rsistanc R and dilctric conductanc G ar both ro.in this cas th charactristic impdanc is ral and is qual to: o C Th propagation constant γ is also imaginary with: α 0 γ jβ jω C Intro.21

22 Exprssing th wavs in tim-domain, v( t, i( t, ) ) o cos cos ( ωt β ) cos ( ωt β ) ( ωt β ) cos ( ωt β ) Th vlocity with which a front of constant phas travls is calld th phas vlocity u p. In any transmission lin, In losslss transmission lin, Thrfor u p ω β o β ω ω 1 u p β C β C 2π λ Intro.22

23 In a coaxial cabl, C 2πε oε r a ln b (Chap.2) φ I µ o ln 2π a b (Chap.3) So u p ε o 1 ε r µ o ε o prmittivity of vacuum ε r rlativ prmittivity (dilctric constant) of dilctric µ o prmability of vacuum Intro.23

24 Exampl: Calculat th charactristic rsistanc Ro of a RG-58U coaxial cabl which has a innr conductor of radius a0.406 mm and a braidd outr conductor with radius b1.553 mm. Assum th dilctric is polythyln with dilctric constant of Solution: Th distributd capacitanc and inductanc of th cabl can b calculatd to b: µh/m C pf/m R o / C Ω Intro.24

25 2.5 Rflctions of tim-harmonic wavs: Considr a transmission lin of lngth l trminatd by an arbitrary impdanc : I in o _ -l 0 Intro.25

26 At th load 0, th voltag and currnt phasors can b writtn as: (0) I (0) 1 o ( ) oad impdanc (0)/I(0), so w can xprss th ratio of th backward to forward voltags as: Γ Γ is calld th load rflction cofficint if w considr as th incidnt wav and - as th rflctd wav. o o Intro.26

27 Intro.27 On important ffct of a transmission lin is to transform th load impdanc. t s find th input impdanc looking into th transmission lin of lngth l. l l l l o in in l l I l l γ γ γ γ Γ Γ ) ( ) ( ) ( ) ( Rplacing Γ in trms of o and, ) tanh( ) tanh( ) ( l l l o o o in γ γ

28 In losslss transmission lin, γ jβ giving: in ( l ) o o j j o tan( tan( βl ) βl ) Thr ar intrsting applications whn th lngth l is multipl of λ/4. Exampl: Calculat th input impdanc of a 1 m lngth of cabl that is trminatd in a load impdanc of 20Ω. Assum that th charactristic impdanc of th lin is 50Ω, its dilctric constant is 1.5 and th frquncy of opration is 50MH. Intro.28

29 Intro ) ( tan tan j j l l f u in o r o p β µ ε ε π ω β Ω ) ( j Solution:

30 2.6 Standing wav ratio In a losslss lin, th amplitud of th forward (or backward) voltag rmains constant as th wav propagats along, only with a shift in th phas angl. Th suprimposition of th forward wav and backward wav rsults in a standing wav pattrn. In a standing wav, thr ar positions at th lin whr th amplitud of th rsultant voltag has maximum and minimum. max min Intro.30

31 Th voltag standing wav ratio (SWR) is th ratio of th maximum and minimum voltag magnituds. Th distanc btwn two succssiv maximums is qual to λ/2. SWR max min 1 1 Γ Γ SWR is usful to find th maximum voltag magnitud on th lin du to rflction from th load. If inc is th incidnt voltag on th load, 2 SWR max inc SWR 1 Intro.31

32 2.7 Smith Chart: a convnint graphical mans of dtrmining voltags along transmission lins. It is ssntially a plot of th complx rflction cofficint Γ(-l) at a point with input impdanc in (-l) looking into th nd of th transmission lin. Γ( l) in in ( l) ( l) o o t th ral and imaginary parts of Γ(-l) b Γ r, Γ i rspctivly, and b th input impdanc normalid by o. Γ in ( l) o 1 1 r jx Intro.32

33 Intro.33 Aftr som manipulations, it can b shown that: ( ) Γ Γ Γ Γ x x r r r i r i r Ths quations dfin family of circls on th ( Γ r, Γ i ) plan corrsponding to constant rsistanc r, and constant ractanc x. Th rflction cofficint at a point on th lin with normalid input impdanc rjx is thn th vctor nding at th intrsction point btwn th constant r and x circls.

34 In a losslss transmission lin, thr is no attnuation and a wav travlling along th lin will only hav a phas shift. So th rflction cofficint Γ(-l) at a point of distanc l from th load at th nd of th lin is rlatd to th load rflction cofficint Γ by: Γ ( l ) Γ j 2 β l It mans th rflction cofficint has sam magnitud but only a phas shift of 2 β l if w mov a lngth l along th lin ( Γ rotats clockwis on th Smith Chart whn moving away from th load and anti-clockwis whn moving towards th load). Intro.34

35 Im Γ(-l) Γ 2βl R constant SWR Intro.35

36 Exampl: (a) If th rflction cofficint at a location on a transmission lin of 100Ω charactristic impdanc is Γ 0.4j0.2, us Smith chart to dtrmin th input impdanc at that location. (b) A load of 50-j25 Ω is attachd to th abov lin. Us Smith chart to find th input impdanc at a distanc l 0.4λ from th load. Solution: (a) Γ 0.4j xp(j26.56 o ) Find th abov point Q in th Smith chart. It corrsponds to th intrsction of r2.0 and x1.0 circls. Intro.36

37 .Q.P1.P2 Intro.37

38 Thrfor th input impdanc in 100 (2j1.0) 200 j100 (b) Th normalid load impdanc 0.5-j0.25 is rprsntd by th point P 1. To find at a distanc l 0.4λ rotat th point P 1 to point P 2 clockwis through a wavlngth of 0.4λ. W find j0.77 Thrfor th input impdanc is 95.2-j77.0 Ω aftr multiplying by th charactristic impdanc of 100Ω. Intro.38

39 END Intro.39

40 A transmission lin always has two conductors and a dilctric btwn th two conductors. Th conductors hav a rsistanc and inductanc in sris. Th dilctric has a capacitanc and rsistanc in paralll. But all th rsistanc, inductanc and capacitanc ar distributd in natur. It mans w hav to first rprsnt a small lmntal sction of th lin by th abov quivalnt circuit, and thn assum th complt lin is rprsntd by an infinit numbr of such small lmntal sction connctd togthr. Intro.40

41 You should b abl to driv ths quations from th quivalnt circuit if you rmmbr th following formula for voltag/currnt in inductors and capacitors: In tim domain, In an inductor, In a capacitor, Using phasors, v( t) i( t) C di( t) dt dv( t) dt In an inductor, jωi In a capacitor, I jωc Intro.41

42 Th gnral xprssion for a travlling wav with a timvarying amplitud is: v A α cos( ωt β) In complx rprsntation, th cos function is rplacd j ( ωt β ) by so that v A A α j ( ωt β ) ( α jβ ) j t ω In phasors, th trm is undrstood, so A ( α jβ ) A γ jωt Intro.42

43 Intro.43 I j R ) ( ω γ γ Thrfor w can writ I as: ( ) j R I γ γ γ γ ω 1 I I γ γ On simplification, w can find I, I - and hnc th ratio of /I, - /I -.

44 t inc b th forward voltag incidnt on th load (at 0), and th load rflction cofficint Th voltag at any point on th transmission lin is: inc inc jβ inc Γ jβ ( jβ j ( β φ ) Γ ) At maximum voltag points, max β β φ 2nπ 1 Γ inc ( ) At minimum voltag points, β β φ π 2nπ min 1 inc ( Γ ) Γ Γ jφ Intro.44

45 EHF (30-300GH) Radar, radio astronomy, rmot snsing SHF (3-30GH) Radar, satllit, aircraft navigation UHF (300MH-3GH) T, radar, microwav ovn, mobil phon HF (30-300MH) T, FM, mobil radio, air traffic control HF (3-30MH) Short wav broadcasting MF (300kH-3MH) AM F (30-300kH) Wathr broadcast for air navigation F (3-30kH) Navigation and position location UF (300H-3kH) Audio signals on tlphon SF (30-300H) Ionosphric snsing, submarin communication EF (3-30H) Dtction of mtal objcts Intro.45