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射頻電子 - [第二章] 傳輸線理論

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傳輸線理論

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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

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