Faculdade de Engenharia. Transmission Lines ELECTROMAGNETIC ENGINEERING MAP TELE 2008/2009

Transcript

1 Facudad d Ennharia Transmission ins EECTROMAGNETC ENGNEERNG MAP TEE 8/9

2 Transmission ins Facudad d Ennharia transmission ins wavuids supportin TEM wavs most common typs para-pat wavuids coaxia wavuids two-wir wavuids EE 89 ins

3 Transmission ins Facudad d Ennharia nra transmission in quations today tim-harmonic soutions finit transmission ins vota, currnt and impdanc aon th in transmission ins in circuits Smith chart impdanc matchin λ/4 transformr nxt wk ractiv mnts sin-stub doub-stub transints EE 89 ins 3

4 EE 89 ins 4 Facudad d Ennharia TEM wavs in para-pat wavuids b y x W x E H y E E ˆ ˆ η r r x E H y E E ˆ ˆ η r r insid th uid:

5 ota btwn th pats Facudad d Ennharia vota P r r E d P y b x W vota btwn th pats: b ( ) E y dy be insid th uid: r E E yˆ r E H η xˆ E y E EE 89 ins 5

6 EE 89 ins 6 Facudad d Ennharia Currnt dnsity on th pats b y x W x E H y E E ˆ ˆ η r r â n currnt dnsity on th pats: uppr pat: y a n ˆ ˆ insid th uid: owr pat: y a n ˆ ˆ â n ˆ H H a J n s r r r Hr x E H ˆ η r E b y J s ˆ η r Hr x E H ˆ η r E y J s ˆ η r

7 EE 89 ins 7 Facudad d Ennharia Currnt on th pats b y x W uppr pat currnt: A ds J r r E W η x E H y E E ˆ ˆ η r r insid th uid: E b y J s ˆ η r currnt W s dx J ˆ r E b y J s ˆ η r E y J s ˆ η r currnt dnsity: owr pat currnt: E W η W s dx J ˆ r

8 ossss transmission in quations Facudad d Ennharia be E η W d d d d be E W η η ω µ µε ε x y W b d µ b ω d W d εw ω d b µ b ( H/m ) d W ω d C ε W d ( C/m) ωc b d qs. for in a ossss transmission in d ω C d d ω C d EE 89 ins 8

9 Equivant circuit of a ossss transmission in Facudad d Ennharia diffrntia nth D of a transmission in: i(,t) i(,t) v(,t) C v(,t) - - i C v (, t) i t v, t C t v (, t) i (, t) (, t) i t v C v (, t) t (, t) i (, t) EE 89 ins 9

10 Equivant circuit of a ossss transmission in Facudad d Ennharia v (, t) i (, t) (, t) i t v C v (, t) t (, t) i (, t) i(,t) v(,t) - C im v i (, t ) i(, t) d (, t) v(, t ) C t t phasor notation d d d ω ωc d ω C d d ω C d sam as bfor EE 89 ins

11 Equivant circuit of a ossy transmission in Facudad d Ennharia diffrntia nth D of a transmission in: i(,t) R i(,t) C G v(,t) v(,t) - - v R R i (, t) i(, t ) v t i G v, t i G C C v(, t ) t (, t ) v i (, t ) (, t ) i Ri(, t ) t v Gv(, t ) C t (, t ) im v (, t ) R i(, t) i, t (, t) i v t v G v, t C (, t ) (, t) t i (, t ) EE 89 ins

12 Gnra transmission in quations Facudad d Ennharia (, t ) v i (, t ) (, t ) i Ri(, t ) t v Gv(, t ) C t (, t ) v(,t) i(,t) - phasor notation d d d d ( R ω) ( G ωc) ( R ω)( G ωc ) d d d d nra soution propaation constant α attnuation constant phas constant EE 89 ins

13 Attnuation and phas constants Facudad d Ennharia ± v { } t ω, R t α { ( ω t ) α ( ω t )} R atnuation α α ( ωt ) cos( ωt ) cos phas if and ar ra EE 89 ins 3

14 ota and currnt in transmission in Facudad d Ennharia 4 constants rquird to dfin votaand currnt, ± d d d d ( ) R ω ( G ωc), R ω R ω R ω ony constants ar rquird EE 89 ins 4

15 Charactristic impdanc Facudad d Ennharia Charactristic impdanc ratio btwn vota andcurrnt for an infinit nth transmission in infinit in no rfctions, ±, R ω ( R ω)( G ωc ) R G ω ωc ( Ω) charactristic impdanc not: in nra EE 89 ins 5

16 Summary Facudad d Ennharia Propaation constant α ( R ω)( G ωc ) ( m ) Charactristic impdanc R ω G ωc ( Ω) Propaation vocity ω v ( ms ) Gnra cas frquncy dpndnt attnuation frquncy dpndnt vocity π Wavnth λ ( m) SGNA DSTORTON EE 89 ins 6

17 Transmission ins spcia cass Facudad d Ennharia α ossss ins R G ( R ω)( G ωc ) R ω ω v ω C α G ωc ω C v C C Distortionss ins R G C ( R ω) C C α R ω C C v C ro or constant attnuation constant vocity constant and ra charactristic impdanc NO DSTORTON EE 89 ins 7

18 Transmission-in paramtrs Facudad d Ennharia Th bhaviour of a transmission in dpnds on th opratinfrquncyand on paramtrs R,, G and C n turn, ths paramtrs dpnd on th in omtry and on th matrias that constitut th in t σ dictric conductivity σ C conductor conductiviity ε ctric prmitivitty of th dictric µ mantic prmabiity of th dictric µ C mantic prmabiity of th conductor EE 89 ins 8

19 Transmission-in paramtrs Facudad d Ennharia b a D a a h h W coaxia two-wir conductor ovr round para pat EE 89 ins 9

20 EE 89 ins Facudad d Ennharia Finit transmission ins [ ] [ ] o o o o ± o o

21 EE 89 ins Facudad d Ennharia mpdanc aon th transmission in ± [ ] [ ] tanh tanh tanh tanh x x x x x ) tanh(

22 nput impdanc ossss transmission in Facudad d Ennharia ± ( ) ( ) ( ) tanh tanh ( ) ( ) ossss in ( x) tan( x) tanh ( ) tan tan ( ) ( ) nth in tan tan ( ) ( ) EE 89 ins

23 nput impdanc of ossss transmission ins spcia cass Facudad d Ennharia ossss transmission in of nth tan( ) in tan( ) in in cot ( ) an( ) in t aways imainary λ n in λ ( ) 4 n in EE 89 ins 3

24 Rfction cofficint at th oad Facudad d Ennharia Rfction cofficint (vota) ratio btwn rfctd and incidnt votas at th oad: rf inc ( ) o ( ) ( ) ( ) Spcia cass: no rfctions MATCHED NE EE 89 ins 4

25 Rfction cofficint at th oad Facudad d Ennharia r x ( r ) x ( r ) x Nots:. For currnt rf inc. Most oftn, is compx θ, EE 89 ins 5

26 Rfction cofficint aon th in Facudad d Ennharia at th oad: rf inc ( ) o ( ) θ aon th in: rf o inc ( ) ossss in: ( ) ( θ ) absout vau is constant EE 89 ins 6

27 ota aon th in Facudad d Ennharia ( ) ( ) cos ( x) x x ( ) cos( ) propaatin wav stationary wav EE 89 ins 7

28 Not propaatin and stationary wavs Facudad d Ennharia t A ωt ( ωt ) v(, t) R{ A } R{ A } Acos( ωt ) propaatin wav t Acos( ) v( t) R Acos( ) ωt { } Acos( ) cos( ωt), stationary wav nods ( v for vry t ) EE 89 ins 8

29 EE 89 ins 9 Facudad d Ennharia ota aon th in propaatin stationary wavs cos sin cos θ θ θ priodic trm priod/ θ

30 ota aon th in - xamp Facudad d Ennharia ( ) cos( θ ) t.8.5 m π 4 ( λ π m).6.4. MAX λ 3 min EE 89 ins 3

31 ota maxima and minima Facudad d Ennharia ( ) cos( θ ) vota maxima: cos ( θ ) ocation: / / θ M nπ M ( nπ θ) n intr vau: MAX ( ) MAX vota minima: cos( θ ) / ocation: θ ( n )π / m [( n ) π θ ] m n intr vau: ( ) min min EE 89 ins 3

32 ota aon th in - xamp Facudad d Ennharia ( ) cos( θ ) t.5 m π 4 ( λ π m) π 8 MAX MAX ( ). 5.8 M min nπ 8 5 nπ 8 / π m / π ( ) λ π 4 3 5π 8 min EE 89 ins 3

33 SWR Facudad d Ennharia SWR (ota Standin Wav Ratio) ratio btwn vota maxima and minima SWR MAX min ( ) ( ) SWR Not: SWR SWR SWR EE 89 ins 33

34 SWR particuar cass Facudad d Ennharia SWR SWR SWR Particuar cass: SWR MAX min no stationary wav no rfctions SWR matchd in SWR EE 89 ins 34

35 SWR particuar cass Facudad d Ennharia SWR SWR SWR Particuar cass: SWR ( ) ( ) MAX min MAX min SWR EE 89 ins 35

36 EE 89 ins 36 Facudad d Ennharia Currnt aon th in propaatin stationary wavs cos sin cos θ θ θ priodic trm priod/ θ

37 Currnt maxima and minima Facudad d Ennharia ( ) cos( θ ) currnt maxima: cos( θ ) ocation: θ ( n )π [( n ) π θ ] n intr vau: MAX MAX ( ) currnt minima: cos ( θ ) ocation: θ nπ ( nπ θ ) n intr vau: min ( ) min EE 89 ins 37

38 ota and currnt maxima and minima ocation Facudad d Ennharia ( ) cos( θ ) ( ) cos( θ ) ( θ ) cos máximos vota d maxima tnsão AND mínimos currnt d minima corrnt / M n θ ( π ) n intr cos ( θ ) vota minima AND currnt maxima / m θ [( n ) π ] n intr EE 89 ins 38

39 Transmission ins in circuits Facudad d Ennharia ± in in o o o o in in in in ( ) ( ) in in [ ] [ ] [ ( ) ( )] EE 89 ins 39

40 EE 89 ins 4 Facudad d Ennharia Transmission ins in circuits ± in in [ ] [ ] (rfction cofficint at th sourc) [ ] vota and currnt as functions of oad: in: sourc:,,,

41 EE 89 ins 4 Facudad d Ennharia Transmission ins in circuits ± in in 3 x x x x

42 EE 89 ins 4 Facudad d Ennharia Transmission ins in circuits ± in in

43 EE 89 ins 43 Facudad d Ennharia Transmission ins in circuits ± in in

44 EE 89 ins 44 Facudad d Ennharia Powr in ossss transmission ins θ θ (ossss transmission in) { } R * P av * R av P θ θ { } R θ θ { } sin R θ constant av P incidnt rfctd

45 EE 89 ins 45 Facudad d Ennharia Powr in transmission ins nra cas θ α α θ α α { } R * P av * R av P θ α α θ α α { } sin R 4 R θ α α α av R P α α, av av R P P av in av R P P α α, R if

46 Probm Facudad d Ennharia formua EE 89 ins 46

50 oad impdanc rfction cofficint Facudad d Ennharia whr (normaid oad impdanc) R X R (ossss in) r x θ r im r x ( r ) im ( r ) im r x r r ( ) im im ( ) r im im EE 89 ins 5

51 oad impdanc rfction cofficint Facudad d Ennharia r x r r ( ) im im ( ) r im im r r im r r ( x x ) ( y y ) R im circ of radius ( r ) r cntrd at r im r ( r ) r r r th rfction cofficints of a whos ra part is r ar in this circ EE 89 ins 5

52 oad impdanc rfction cofficint Facudad d Ennharia r r im r r im Not: curv dos not dpnd on x r r r for any r im r, r, r r opn circuit r r, r r, EE 89 ins 5

53 oad impdanc rfction cofficint Facudad d Ennharia r x r r ( ) im im ( ) r im im ( ) r im x x ( x x ) ( y y ) R circ of radius x im x x cntrd at r im x th rfction cofficints of a whos imainary part is x ar hr r EE 89 ins 53

54 oad impdanc rfction cofficint Facudad d Ennharia ( ) r im x x Not: curv dos not dpnd on r symmtrica curvs for x < x im x x x infinit radius r x EE 89 ins 54

55 Smith chart Facudad d Ennharia im x constant r constant r EE 89 ins 55

56 Smith chart Facudad d Ennharia EE 89 ins 56

57 Smith chart Facudad d Ennharia from: point in chart ( intrsction of curvs corrspondin to r and x ) x and θ im θ from: r and x r r EE 89 ins 57

58 Rfction cofficint aon th in Facudad d Ennharia aon th in: rf o inc ( ) ossss in: ( ) ( θ ) constant manitud phas dcrass with im Not: ( ) toward nrator Smith chart can b usd to obtain from () r toward oad EE 89 ins 58

59 Distancs in th Smith chart Facudad d Ennharia in Smith chart th distancs ar masurd as fractions of ( θ ) ( ) whn π initia position π λ a compt turn (36º) corrsponds to a distanc / im toward nrator r toward oad EE 89 ins 59

60 nput impdanc Facudad d Ennharia. draw th point corrspondin to th normaid oad impdanc point P. draw th circ cntrd at th oriin with radius OP 3. draw th straiht in from O to P 4. draw th straiht in from O that corrsponds to a rotation of toward th nrator 5. intrsction of this in with prvious circ point P 6. obtain, whr in is rad from P in in im P P r EE 89 ins 6

61 Admittanc Facudad d Ennharia tan ( ) λ tan( π ) tan( ) 4 tan( π ) ( λ 4) y ( λ ) 4 λ 36º im λ 4 8º r. draw y. rotat 8º EE 89 ins 6

62 Maxima and minima ocation Facudad d Ennharia ( ) cos( θ ) cos θ ( θ ) cos vota maxima and currnt minima cos ( θ ) vota minima and currnt maxima ( θ ) ( ) vota maxima whr ( ) nπ vota minima whr ( ) ( n )π EE 89 ins 6

63 Maxima and minima ocation Facudad d Ennharia vota maxima whr ( ) nπ vota minima whr ( ) ( n )π im vota maxima vota minima Not:. maxima and minima whr input impdanc is ra. maxima (minima) points ar sparatd by n/ r EE 89 ins 63

64 Probm Facudad d Ennharia EE 89 ins 64