Diese Präsentation wurde erfolgreich gemeldet.
Wir verwenden Ihre LinkedIn Profilangaben und Informationen zu Ihren Aktivitäten, um Anzeigen zu personalisieren und Ihnen relevantere Inhalte anzuzeigen. Sie können Ihre Anzeigeneinstellungen jederzeit ändern.

射頻電子 - [第二章] 傳輸線理論

6.087 Aufrufe

Veröffentlicht am

https://www.facebook.com/eeRhapsody
傳輸線理論

Veröffentlicht in: Ingenieurwesen
  • Als Erste(r) kommentieren

射頻電子 - [第二章] 傳輸線理論

  1. 1. Department of Electronic Engineering National Taipei University of Technology
  2. 2. • • • • Department of Electronic Engineering, NTUT2/47
  3. 3. Two-wire line Coaxial ( ) Microstrip( ) Department of Electronic Engineering, NTUT3/47
  4. 4. • • • ( ) • Department of Electronic Engineering, NTUT4/47
  5. 5. ( ) Department of Electronic Engineering, NTUT5/47
  6. 6. • ( ) : L L s s L Z v v Z Z = + Source Source impedance Load impedance (Transmission line) sv sZ LZ l • ? ? Lv Department of Electronic Engineering, NTUT6/47
  7. 7. (Distributed Circuit Model) • l ∆x • R: /m, L: H/m, C: F/m, G: S/m G x∆ L x∆R x∆ C x∆ sZ sv LZ llll dx dx dx Department of Electronic Engineering, NTUT7/47
  8. 8. • • : • : • R: G: ( ) ( ), , ,v x t i x t ( ) ( ), , ,v x x t i x x t+ ∆ + ∆ ( ),v x x t+ ∆ ( ),i x x t+ ∆ R x∆ L x∆ G x∆ C x∆( ),v x t ( ),i x t Department of Electronic Engineering, NTUT8/47
  9. 9. (I) • (KVL): • (KCL): ( ) ( ) ( ) ( ) ( ) ( ), , , , i x t v x t v x x t R x i x t L x t ∂ − + ∆ = ∆ + ∆ ∂ ( ) ( ) ( ) ( ) ( ) ( ), , , , v x x t i x t i x x t G x v x x t C x t ∂ + ∆ − + ∆ = ∆ + ∆ + ∆ ∂ ( ),v x x t+ ∆ ( ),i x x t+ ∆ R x∆ L x∆ G x∆ C x∆( ),v x t ( ),i x t Department of Electronic Engineering, NTUT9/47
  10. 10. (II) ( ) ( ) ( ), , , v x t i x t Ri x t L x t ∂ ∂ = − − ∂ ∂ ( ) ( ) ( ), , , i x t v x t Gv x t C x t ∂ ∂ = − − ∂ ∂ ( ) ( ) ( ) ( ), , , , v x t v x x t i x t Ri x t L x t − + ∆ ∂ = + ∆ ∂ ( ) ( ) ( ) ( ), , , , i x t i x x t v x x t Gv x x t C x t − + ∆ ∂ + ∆ = + ∆ + ∆ ∂ • • ∆x (transmission-line equation) (telegrapher equation) Department of Electronic Engineering, NTUT10/47
  11. 11. (III) − • (KVL): • (KCL): ( ) ( ) ( ) ( ) ( ) ( )V x V x x R x I x j L x I xω− + ∆ = ∆ + ∆ ( ) ( ) ( ) ( ) ( ) ( )I x I x x G x V x x j C x V x xω− + ∆ = ∆ + ∆ + ∆ + ∆ ( )V x x+ ∆ ( )I x x+ ∆ R x∆ j L xω ∆ G x∆ 1 j C xω ∆( )V x ( )I x ( ) ( ), cosv x t V x tω= Department of Electronic Engineering, NTUT11/47
  12. 12. (IV) − ( ) ( ) ( ) dV x RI x j LI x dx ω= − − ( ) ( ) ( ) dI x GV x j CV x dx ω= − − ( ) ( ) ( ) ( ) V x V x x RI x j LI x x ω − + ∆ = + ∆ ( ) ( ) ( ) ( ) I x I x x GV x x j CV x x x ω − + ∆ = + ∆ + + ∆ ∆ • • ∆x (transmission-line equation) (telegrapher equation) Department of Electronic Engineering, NTUT12/47
  13. 13. ( ) ( ), ,v x t i x t L x t ∂ ∂ = − ∂ ∂ ( ) ( ), ,i x t v x t C x t ∂ ∂ = − ∂ ∂ • R = G = 0 ( ),v x x t+ ∆ ( ),i x x t+ ∆ L x∆ C x∆( ),v x t ( ),i x t ( ) ( ) ( ), , , v x t i x t Ri x t L x t ∂ ∂ = − − ∂ ∂ ( ) ( ) ( ), , , i x t v x t Gv x t C x t ∂ ∂ = − − ∂ ∂ Department of Electronic Engineering, NTUT13/47
  14. 14. ( ) ( ) ( )( ) ( ) ( )( ) { } ( ) ( ) { }, cos Re Re j t x j x j t v x t f x t x f x e f x e e ω ϕ ϕ ω ω ϕ + = + = = ( ) ( ) ( )( ) ( ) ( )( ) { } ( ) ( ) { }, cos Re Re j t x j x j t i x t g x t x g x e g x e e ω η η ω ω η + = + = = ( ) ( ) ( )j x I x g x e η = ( ) ( ) ( )j x V x f x e ϕ = ( ) ( ){ }, Re j t v x t V x e ω = ( ) ( ){ }, Re j t i x t I x e ω = • f(x) g(x) ( )xϕ ( )xη • time-domain Department of Electronic Engineering, NTUT14/47
  15. 15. ( ) ( ) V x j LI x x ω ∂ = − ∂ ( ) ( ) I x j CV x x ω ∂ = − ∂ ( ) ( ) dV x j LI x dx ω= − ( ) ( ) dI x j CV x dx ω= − ( ) ( )V x dV x x dx ∂ = ∂ ( ) ( ) ( ) 2 2 2 d V x dI x j L LCV x dx dx ω ω= − = − • ( ) ( ) ( ) ( ) 2 2 2 2 2 2 0 d V x d V x LCV x V x dx dx ω β+ = + = LCβ ω=• • ( ) j x j x V x Ae Beβ β− = + A B (complex constant) • (Phasor) (Think that if approaches zero?)ω ( )β Phasor Department of Electronic Engineering, NTUT15/47
  16. 16. ( ) ( ) ( ) ( ) 1 1 j x j xdV x I x A j e B j e j L dx j L β β β β ω ω −  = = − + − − LCβ ω= • 0 L L L Z CLC ω ω β ω = = = j x j x A e B e L L β ββ β ω ω −   = −     ( ) j x j x V x Ae Beβ β− = + ( ) ( ) dV x j LI x dx ω= − Z0 Z0 ( ) 0 0 j x j xA B I x e e Z Z β β− = − Department of Electronic Engineering, NTUT16/47
  17. 17. − (I) • 1Z 2Z 3Z 3 1 2Z Z Z= + 1Z 2Z 2Z 1Z 2Z 1Z 3Z 1Z 4Z 5Z 4 2 3||Z Z Z= Department of Electronic Engineering, NTUT17/47
  18. 18. − (II) • 2 0 1 1 0 2 0 2 0 1 1 1ab Z Z Z Z Z Z Z Z Z Z = + = + = ++ 2 1 1 0 1 2 2 4 Z Z Z Z Z   = + +    Z0 a-b Z0 c-d Z0( ) 1Z 2Z 2Z 1Z 2Z 1Z 1Z c d a b a b c d 2Z 1Z 0Z 0Z a b Department of Electronic Engineering, NTUT18/47
  19. 19. − (III) • 2 1 1 0 1 2 2 4 Z Z Z Z Z   = + +    L 2 2 0 0 lim 4L L L L Z C C ω → = − = 2 2 0 4 L L Z C ω = − 4 LCω < 4 LCω > 0, 0 lim 4 L C LCω → → = = ∞ 1Z j Lω= 2 1 Z j Cω = L C C La b CCC 2L 2L 2L2L a b a′ Department of Electronic Engineering, NTUT19/47
  20. 20. ( ) ( ){ } ( ) ( ) { }, Re Re j x t j x tj t v x t V x e Ae Beβ ω β ωω − − + = = + ( ) ( ){ } ( ) ( ) 0 0 , Re Re j x t j x tj t A B i x t I x e e e Z Z β ω β ωω − − +  = = −    ( ) ( )cos cosA x t B x tβ ω β ω= − + + ( ) ( ) 0 0 cos cos A B x t x t Z Z β ω β ω= − − + ( ) j x j x V x Ae Beβ β− = + ( ) 0 0 j x j xA B I x e e Z Z β β− = − • Phasor Department of Electronic Engineering, NTUT20/47
  21. 21. (Wavelength) • +x x (rads) x j x e β− xβ 2 x x λ π β βλ= = = 2π λ β = ( ) { } ( )1 , Re cosj x j t v x t Ae e A t xβ ω ω β− = = − x λ= ( ) j x V x Ae β− = 0x = x λ= 0t = t T= distance time phase 0x xβ = 2x x λ β π= = For simplification, assume the wave starts from x=0 and t=0. Department of Electronic Engineering, NTUT21/47
  22. 22. (Wave Velocity) ( )7 0 4 10 Wb/A-mL µ π − = ⋅≃ ( )12 0 8.85419 10 F/mC ε − ⋅≃ ≃ ( ) ( )8 , 0 0 1 light speed 3 10 /p vacuumv c m s µ ε = = = ⋅ ( )0 0 0 377 L Z C µ ε = = = Ω , 0 p vaccumv f λ = 2 1 2 2 pv f T LC λ ω π ω ω λ λ π β π β = = ⋅ = = = = • : ( T ) ( ) Department of Electronic Engineering, NTUT22/47
  23. 23. 0µ µ≃0,rε ε ε≃ ( ) 8 0 0 1 3 10 /p r r r c v m s µ ε ε ε ε ⋅ = = = (non-magnetic material) 0p r g r c v f f ε λ λ ε = = = εr 81.5rε = ( ) 8 7 , 3 10 3.32 10 / 81.5 p waterv m s ⋅ = ⋅≃ 0 , 00.11 81.5 g water λ λ λ= ≅ Department of Electronic Engineering, NTUT23/47
  24. 24. @1 GHz * @10 GHz Air ( ) 1 30 cm − 3 cm − Alumina( ,96%) 9.6 9.68 cm 0.48 cm 0.968 cm 0.48 mm Sapphire ( ) 9.4 9.78 cm 0.49 cm 0.978 cm 0.49 mm Glass ( ) 5 13.42 cm 0.67 cm 1.342 cm 0.67 mm Polyimide ( ) 3.2 16.77 cm 0.84 cm 1.677 cm 0.84 mm Quartz ( ) 3.8 15.39 cm 0.77 cm 1.539 cm 0.77 mm FR4 ( ) 4.5 14.14 cm 0.71 cm 1.414 cm 0.71 mm RT-duroid 5880 ( ) 2.16 20.41 cm 1.02 cm 2.041 cm 1.02 mm RT-duroid 6010 ( ) 10.2 9.39 cm 0.47 cm 0.939 cm 0.47 mm Si ( ) 11.9 8.70 cm 0.44 cm 0.870 cm 0.44 mm GaAs ( ) 12.85 8.37 cm 0.42 cm 0.837 cm 0.42 mm LTCC ( ) 7.8 10.74 cm 0.54 cm 1.074 cm 0.54 mm * : 1/20 Department of Electronic Engineering, NTUT24/47
  25. 25. 2t T tω π= = 2 T π ω = ( ) ( )1 , cosv x t A x tβ ω= − 0x = ( )1 0, cosv t A tω= π 2π tω 0 0t tω = = A A− ( )1 0, cosv t A tω= 0x = x = l 0x = x = l• We only pay attention to this point Department of Electronic Engineering, NTUT25/47
  26. 26. 2x x λ β π= = 2π λ β = ( ) ( )1 , cosv x t A x tβ ω= − 0t = ( )1 ,0 cosv x A xβ= π 2π xβ 0 0x xβ = = A A− ( )1 ,0 cosv x A xβ= 0x = x = l 0x = x = l• We now pay attention to the whole line at any time instant (here, t=0) Department of Electronic Engineering, NTUT26/47
  27. 27. ( ) 2λ λ x A A− t t T= 2t T= x x t Department of Electronic Engineering, NTUT27/47
  28. 28. x = 2λ λ x A A− t t T= 2t T= t λ 0x = x = l We only pay attention to this point x λ= Department of Electronic Engineering, NTUT28/47
  29. 29. (Terminated Transmission Line) LZ LZ 0Z 0Z j x Ae β− j x Be β j x Be β j x Ae β− 0x = x = l 0d =d = l ( ) j x j x V x Ae Beβ β− = + ( ) 0 0 j x j xA B I x e e Z Z β β− = − ( )IN dΓ ( ) 1 1 j d j d V d Ae B eβ β− = + ( ) 1 1 0 0 j d j dA B I d e e Z Z β β− = − 1 j A Ae β− = l 1 j B Be β = l incident wave reflected wave d x= −l Department of Electronic Engineering, NTUT29/47
  30. 30. (Reflected Coefficient) ( ) 1 1 j d j d V d Ae B eβ β− = + ( ) 2 21 1 0 1 1 j d j d j d IN j d B e B d e e Ae A β β β β − − − Γ = = = Γ ( ) 1 0 1 0IN B A Γ = Γ = • (at d = l ) (at d = 0) d=0 :0Γ ( )0INΓ Department of Electronic Engineering, NTUT30/47
  31. 31. (I) ( ) ( ) ( )2 1 0 1 01j d j d j d j d V d A e e Ae eβ β β β− − = + Γ = + Γ ( ) ( ) ( )21 1 0 0 0 0 1j d j d j d j dA A I d e e e e Z Z β β β β− − = − Γ = − Γ ( ) ( ) ( ) 0 0 0 j d j d IN j d j d V d e e Z d Z I d e e β β β β − − + Γ = = − Γ ( ) 0 0 0 1 0 1 IN LZ Z Z + Γ = = − Γ 0 0 0 L L Z Z Z Z − Γ = + ( ) 1 1 j d j d V d Ae B eβ β− = + ( ) 1 1 0 0 j d j dA B I d e e Z Z β β− = − d (d = 0) 0 0Γ =0LZ Z= LZ( )V d 0d =d = l ( )IN dΓ ( )INZ d ( )I d + − 1 0 1 B A Γ = Department of Electronic Engineering, NTUT 0Z 31/47
  32. 32. (II) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 j d j d L L IN j d j d L L Z Z e Z Z e Z d Z Z Z e Z Z e β β β β − − + + − = + − − 0 0 0 0 0 0 cos sin tan cos sin tan L L L L Z d jZ d Z jZ d Z Z Z d jZ d Z jZ d β β β β β β + + = = + + 0 0 0 L L Z Z Z Z − Γ = + ( ) ( ) ( ) 0 0 0 j d j d IN j d j d V d e e Z d Z I d e e β β β β − − + Γ = = − Γ d = 0 ( )0IN LZ Z= d = l ( ) 0 0 0 tan tan L IN L Z jZ Z Z Z jZ β β + = + l l l ( ) • Department of Electronic Engineering, NTUT32/47
  33. 33. (VSWR) ( ) 2 1 01 j d V d A e β− = + Γ ( ) ( ) 0max 0min 1 1 V d VSWR V d + Γ = = − Γ ( ) ( ) ( )2 1 0 1 01j d j d j d j d V d A e e Ae eβ β β β− − = + Γ = + Γ ( voltage standing-wave ratio, VSWR): 1/4 • ( ) ( )1 0max 1V d A= + Γ ( ) ( )1 0min 1V d A= − Γ Department of Electronic Engineering, NTUT33/47
  34. 34. (Matched Line) • (Matched line): ( )IN LZ d Z= d VSWR 1 0 0Γ = 1VSWR = 0Z 0LZ Z= 0d =d = l ( ) 0INZ Z=l ( ) 0INZ d Z= 0Z ( ) 0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d β β + = + 0LZ Z= Department of Electronic Engineering, NTUT34/47
  35. 35. • d ( ) VSWR 0LZ = 0 1Γ = − 4 λ =l ( )scZ = ∞l 2 λ =l ( ) 0scZ =l 0LZ = 0d =d = l ( ) 0 tanINZ jZ β=l l ( ) 0 tanscZ d jZ dβ= 0Z ( ) 0 tanscZ d jZ dβ=( ) 0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d β β + = + 0LZ = Department of Electronic Engineering, NTUT35/47
  36. 36. 4 λ =l ( ) 0ocZ =l 2 λ =l ( )ocZ = ∞l ( ) VSWR 0 1Γ = • dLZ = ∞ ( ) 0 cotocZ d jZ dβ= −( ) 0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d β β + = + LZ → ∞ LZ = ∞ 0d =d = l ( ) 0 cotINZ jZ β= −l l ( ) 0 cotocZ d jZ dβ= − 0Z Department of Electronic Engineering, NTUT36/47
  37. 37. ¼ • ¼ : ( ) 2 0 4IN L Z Z Z λ = 0 x LZ Z Z= ⋅ 1/4 4d λ= =l LZ xZ : 1/4 75LZ = Ω 50 Ω You can simply use a transmission line with 61.2 Ohm characteristic impedance! 4λ ( )0 4 50 75 61.2IN LZ Z Zλ= ⋅ = ⋅ = Ω ( ) 0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d β β + = + LZ 0d =4d λ= 2 0 4 IN L Z Z Z λ  =    0Z Department of Electronic Engineering, NTUT37/47
  38. 38. • 1/2 ( ) ( )2IN LZ Zλ = 2d λ= =l ( ) 0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d β β + = + LZ 0d =2d λ= 2 IN LZ Z λ  =    0Z Department of Electronic Engineering, NTUT38/47
  39. 39. ( ) ( )1 12 sinj d j d V d A e e j A dβ β β− = − = ( ) ( ){ } 2 1, Re Re 2 sin j t j t v d t V d e A d e π ω ω β   +       = = ⋅     ( ) ( )1 14 2 sin 2 2V j A j Aλ π= = ( ) ( )12 2 sin 0V j Aλ π= = 4 d λ = 2 d λ = • Department of Electronic Engineering, NTUT39/47
  40. 40. ( ) 1, 2 sin cos 2 v d t A d t π β ω  = ⋅ +    π 2 π3 2 π2π ( ) ( ) max min V d VSWR V d = = ∞ ( )V d ( )1 max 2A V d= ( )min 0 V d=dβ d 2 λ 4 λ3 4 λλ • 2 π3 2 π π2π ( ),v d t 12A ( )min 0 V d=dβ d 2 λ 4 λ3 4 λλ 12A 3 2 t πω =5 4 t πω = 3, 4 4 t π πω = 2 t πω = 0,tω π= Department of Electronic Engineering, NTUT40/47
  41. 41. (I) 100 50sZ j= + Ω 50 50LZ j= + Ω 10 0sv ° = ∠ ( ) ( ) ( ) 50 50 10 0 3.92 11.31 (V) 50 50 100 50 L L s L s jZ V V Z Z j j + = = ∠ = ∠ + + + + • Department of Electronic Engineering, NTUT41/47
  42. 42. (II) • 50 Ohm (load reflection coefficient) VSWR 10 0sv ° = ∠ 100 50sZ j= + Ω 50 50LZ j= + Ω ( ) ( ) 0 0 0 50 50 50 0.447 63.44 50 50 50 L L jZ Z Z Z j + −− Γ = = = ∠ + + + ( ) ( ) ( ) ( ) 50 50 50tan45 8 50 100 50 50 50 50 tan45 IN j j Z j j j λ ° °  + + = = − Ω  + +  0 0 1 1 0.447 2.62 1 1 0.447 VSWR + Γ + = = = − Γ − 8λ 0 50Z = Ω l 8λ= ( )8INZ λ LV Department of Electronic Engineering, NTUT42/47
  43. 43. (II) – ( ) ( ) ( ) ( ) ( ) ( ) 8 100 50 8 10 0 5.59 26.57 8 100 50 100 50 IN s IN s Z j V V Z Z j j λ λ λ − = = ∠ = ∠ − + − + + ( ) ( )2 1 01j d j d V d Ae eβ β− = + Γ ( ) 4 2 18 5.59 26.57 1 0.447 63.44 j j V Ae e π π λ − °  = ∠ − = + ∠ ⋅    1 3.95 63.44A ° = ∠ − ( )0 3.95 63.44 1.77 5 45 (V)LV V ° ° = = ∠ − + = ∠ 100 50sZ j= + Ω 10 0sv ° = ∠ ( )8 100 50INZ jλ = − Ω Equivalent circuit Department of Electronic Engineering, NTUT A1 41 43/47
  44. 44. LZ + − sE sZ V I + − source impedance load impedance Phasor • ( ) ( ) 2 2 2 2 2 2 1 1 1 2 2 2 s s L L rms L L L s L s L s L E E R P I R I R R Z Z R R X X   = = = =  + + + +  • s LX X= −s LR R= s LZ Z∗ = • ( ) ( ) ? ( ) 2 ,max 1 8 s L s E P R = s s L E I Z Z = + Department of Electronic Engineering, NTUT44/47
  45. 45. (Power Waves) • 0 L o L o Z Z Z Z − Γ = + ( ) s sV E Z I= − LZ + − sE sZ V I + − source impedance load impedance oZ d = l LZ 0Γ 0d = ( )IN dΓ Department of Electronic Engineering, NTUT45/47
  46. 46. LZ + − sE sZ V I + − LR + − sE sR V + − sjX LjX ( ) ( ) ( ) ( ) L s s LL s L ss L s s L s L s L s L s s L R R j X XR j X X RZ R Z Z Z R R j X X R Z Z R R j X X ∗ − − + + + −− −  Γ = = = = + + + + + + + +   L s L s Z Z Z Z ∗ − Γ = + • ( )* L sZ Z= Γ = 0 • (Z0) Rs LZ + − sE sR V + − sjX L LR jX+ Department of Electronic Engineering, NTUT Z 46/47
  47. 47. • ( ) • 1 • 20λ Department of Electronic Engineering, NTUT • 47/47

×