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# RF Circuit Design - [Ch1-2] Transmission Line Theory

Transmission Line Theory

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### RF Circuit Design - [Ch1-2] Transmission Line Theory

1. 1. Chapter 1-2 Transmission Line Theory Chien-Jung Li Department of Electronics Engineering National Taipei University of Technology
2. 2. Department of Electronic Engineering, NTUT Common Types of Transmission Lines Two-wire line Coaxial Microstrip 2/40
3. 3. Department of Electronic Engineering, NTUT Transmission Line Theory • At high frequencies, especially when the wavelength is not longer than the dimension of circuitry, conventional circuit theory no longer holds. • Circuitry perspective on electromagnetic waves. • Transmission-line effects:  Standing waves generated (related to position)  Load impedance changes  Departs from max power transmission • Transmission-line effect is obvious as frequency or line length increases. 3/40
4. 4. Department of Electronic Engineering, NTUT Theory Conventional Circuit Theory Microwave Engineering Optics 4/40
5. 5. Department of Electronic Engineering, NTUT Electrical Model for Transmission Line Source Source impedance Load impedance • In conventional circuit theory, you can easily find the voltage appears at load by: L s s L Z v v Z Z   Transmission line sv sZ LZ l • If the transmission line is not just an ideal interconnection? 5/40
6. 6. Department of Electronic Engineering, NTUT Distributed Circuit Model • Assume that the transmission line is uniform and the length l is divided into many identical sections Δx. • R: Ω/m, L: H/m, C: F/m, G: S/m G x L xR x C x sZ sv LZ l dx dx dx 6/40
7. 7. Department of Electronic Engineering, NTUT A Section of the Transmission Line • The voltages and currents along the trans- mission line are functions of position and time. • Input: , • Output: , • R: finite conductivity G: dielectric loss  ,v x t  ,i x t R x L x G x C x    ,v x x t    ,i x x t  ,v x t  ,i x t    ,v x x t    ,i x x t 7/40
8. 8. Department of Electronic Engineering, NTUT Transmission Line Equations (I) • Apply Kirchhoff’s voltage law (KVL) • Apply Kirchhoff’s current law (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 t  ,i x t R x L x G x C x    ,v x x t    ,i x x t 8/40
9. 9. Department of Electronic Engineering, NTUT Transmission Line Equations (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 • Rearrange Δx then we have: • Assume that Δx is very small  The partial differential equations describe the voltages and currents along the transmissions, called transmission line, or telegrapher equation. 9/40
10. 10. Department of Electronic Engineering, NTUT Lossless Transmission Line             , , , 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 i x t L x t          , ,i x t v x t C x t  ,v x t  ,i x t    ,v x x t    ,i x x tL x C x • Of particular interest in microwave electronics is the lossless transmission line, i.e., R=G=0. 10/40
11. 11. Department of Electronic Engineering, NTUT Sinusoidal Steady-state Analysis f(x) and g(x) are the real functions of position, and , describe the positional dependence of the phase.                              , cos Re Re j t x j x j t v x t f x t x f x e f x e e                              , cos Re Re j t x j x j t i x t g x t x g x e g x e e        j x I x g x e        j x V x f x e      , Re j t v x t V x e      , Re j t i x t I x e • Time-domain representation   x   x • Phasor-domain representation time-domain 11/40
12. 12. Department of Electronic Engineering, NTUT Solve for the Voltage               V x I x L j LI x x t               I x V x C j CV x x t      dV x j LI x dx      dI x j CV x dx       V x dV x x dx           2 2 2 d V x dI x j L LCV x dx dx • Laplace equation            2 2 2 2 2 2 d V x d V x LCV x V x dx dx   LC• Propagation constant • General solution      j x j x V x Ae Be ,where A, B are complex constant • Lossless transmission line equations (Think that if approaches zero?) ( is also called the “phase constant” that represents the phase change per meter for the wave traveling along the path)  12/40
13. 13. Department of Electronic Engineering, NTUT Solve for the Current                      1 1 j x j xdv x I x A j e B j e j L dt j L • Solve for the current   LC • Define       0 L L L Z CLC               j x j x A e B e L L      j x j x V x Ae Be       0 0 j x j xA B I x e e Z Z      dV x j LI x dx With and ,where Z0 is the characteristics impedance of the transmission line, for the lossless condition Z0 is real. 13/40
14. 14. Department of Electronic Engineering, NTUT Time-domain Results                    , Re Re j x t j x tj t v x t V x e Ae Be                      0 0 , Re Re j x t j x tj t A B i x t I x e e e Z Z          cos cosA x t B x t           0 0 cos cos A B x t x t Z Z      j x j x V x Ae Be       0 0 j x j xA B I x e e Z Z • Time-domain results can be easily drawn from the phasor.  Voltage Wave  Current Wave 14/40
15. 15. Department of Electronic Engineering, NTUT Wave Wavelength     j x V x Ae • Consider  Take a look on the term , the phase means how many radians change for the wave to travel through the distance of x. If distance x is equal to a wavelength long, i.e., :  j x e x      2 x x     2  0x x  0t t T distance time phase  0x x     2x x            1 , Re cosj x j t v x t Ae e A t x x For simplification, assume the wave starts from x=0 and t=0. 15/40
16. 16. Department of Electronic Engineering, NTUT Wave Velocity  In the vacuum,       7 0 4 10 Wb/A-mL  0 8.85419 F/mC           8 , 0 0 1 light speed 3 10 /p vacuumv c m s        0 0 0 377 L Z C   , 0 p vaccumv f is the wavelength in vacuum                   2 1 2 2 pv f T LC • Wave velocity can be found: ( the wave goes one wavelength long in a period of T) is the intrinsic/characteristic impedance of vacuum (or free-space) 16/40
17. 17. Department of Electronic Engineering, NTUT Wave Propagation in Material  0  0r            8 0 0 1 3 10 /p r r r c v m s  Propagation in material with relative dielectric constant r (non-magnetic material)        0p r g r c v f f  Take water for an example:      8 7 , 3 10 3.32 10 / 81.5 p waterv m s    0 , 00.11 81.5 g water The propagation speed is slower than that in vacuum, and the wavelength is also shorter than that in vacuum. 17/40
18. 18. Department of Electronic Engineering, NTUT Wave at a Certain Point    2t T t    2 T      1 , cosv x t A x t  0x   1 0, cosv t A t  2 t   0 0t t A A   1 0, cosv t A t  0x x l  0x x l • Consider  At position We only pay attention to this point 18/40
19. 19. Department of Electronic Engineering, NTUT Wave at a Certain Time     2x x     2      1 , cosv x t A x t  0t   1 ,0 cosv x A x  2 x   0 0x x A A   1 ,0 cosv x A x  0x x l  0x x l • Consider  At time We now pay attention to the whole line at any time instant (here, t=0) 19/40
20. 20. Department of Electronic Engineering, NTUT Wave Propagation versus x and t  2  x A A t t T  2t T x x t 20/40
21. 21. Department of Electronic Engineering, NTUT Wave at Point x=  2  x A A t t T  2t T t   0x x l We only pay attention to this point x 21/40
22. 22. Department of Electronic Engineering, NTUT Terminated Transmission Line LZ LZ0Z 0Z  j x Ae j x Be j x Be  j x Ae  0x x l  0dd l      j x j x V x Ae Be       0 0 j x j xA B I x e e Z Z  IN d      1 1 j d j d V d A e B e      1 1 0 0 j d j dA B I d e e Z Z  1 j A Ae l  1 j B Be l where and  d xl incident wave reflected wave 22/40
23. 23. Department of Electronic Engineering, NTUT Reflection Coefficient      1 1 j d j d V d A e B e  1 j A Ae l  1 j B Be l              2 21 1 0 1 1 j d j d j d IN j d B e B d e e A e A      1 0 1 0IN B A where and • Moves from the load (at d=0) toward the source (at d=l) where is the load reflection coefficient, which is the value of at d=0: 0  IN d incident wave reflected wave 23/40
24. 24. Department of Electronic Engineering, NTUT Reflection Coefficient at Load                 2 1 0 1 01j d j d j d j d V d A e e A e e                 21 1 0 0 0 0 1j d j d j d j dA A I d e e e e Z Z                   0 0 0 j d j d IN j d j d V d e e Z d Z I d e e         0 0 0 1 0 1 IN LZ Z Z     0 0 0 L L Z Z Z Z      1 1 j d j d V d Ae B e      1 1 0 0 j d j dA B I d e e Z Z  Input impedance of the transmission line at any position d is defined as  Use boundary condition at load (d=0)  0 0 when  0LZ ZProperly terminated (matched line): LZ V d  0dd l  IN d  INZ d  I d   24/40
25. 25. Department of Electronic Engineering, NTUT Input Impedance of a Terminated Line                        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        0 0 0 cos sin cos sin L L Z d jZ d Z Z d jZ d      0 0 0 tan tan L L Z jZ d Z Z jZ d     0 0 0 L L Z Z Z Z                   0 0 0 j d j d IN j d j d V d e e Z d Z I d e e It gives the value of the input impedance at any position d along the transmission line At d=0   0IN LZ Z At d=l        0 0 0 tan tan L IN L Z jZ Z Z Z jZ l l l A very important property of a transmission line is the ability to change a load impedance to another value of impedance as its input. 25/40
26. 26. Department of Electronic Engineering, NTUT Voltage Standing-wave Ratio (VSWR)       2 1 01 j d V d A e      1 0max 1V d A      1 0min 1V d A           0max 0min 1 1 V d VSWR V d  The addition of the two waves traveling in opposite directions in a transmission line produces a standing-wave pattern – that is, a sinusoidal function of time whose amplitude is a function of position.                 2 1 0 1 01j d j d j d j d V d A e e Ae e  Magnitude of the voltage along the line:  The voltage standing-wave ratio (VSWR): Next we will discuss 4 important cases of the terminated line, which are matched line, short-circuited line, open-circuited line, and quarter-wave line. and 26/40
27. 27. Department of Electronic Engineering, NTUT Matched Line (Properly Terminated) • Matched line: the characteristic impedance is equal to the load impedance, i.e.,   IN LZ d Z  0LZ Z  0dd l  l  0INZ Z    0INZ d Z 0Z  There is no reflection wave, the input impedance is at any location d, VSWR has its minimum value of 1, i.e.,  0 0  1VSWRand 0Z  No matter how long the line is, there is no transmission line effects . 27/40
28. 28. Department of Electronic Engineering, NTUT Short-Circuited Line • For , it follows that , , and the input impedance at a distance d from the load, , is given by    0 tanscZ d jZ d  The amplitude of the incident and reflected waves are the same since (total reflection from the load), VSWR attains its largest value of infinity.  0LZ  0dd l  l l 0 tanINZ jZ    0 tanscZ d jZ d 0Z  0LZ   0 1  VSWR  scZ d  When Short-circuited load is transformed to open-circuited.  0 1 l   4  l  scZ  When Short-circuited load is still short- circuited seen from the input. l   2  l  0scZ 28/40
29. 29. Department of Electronic Engineering, NTUT Open-Circuited Line • For , it follows that , , and the input impedance at a distance d from the load, , is given by  The amplitude of the incident and reflected waves are the same since (total reflection from the load), VSWR attains its largest value of infinity.  LZ  0 1  VSWR  ocZ d  When Open-circuited load is transformed to short-circuited.  0 1 l   4  l  0ocZ  When Open-circuited load is still open- circuited seen from the input. l   2  l  ocZ     0 cotocZ d jZ d  LZ  0dd l  l l  0 cotINZ jZ     0 cotocZ d jZ d 0Z 29/40
30. 30. Department of Electronic Engineering, NTUT Quarter-wave Line • Quarter-wave transformer:    2 0 4IN L Z Z Z   0 4IN LZ Z Z  In order to transform a real impedance to another real impedance given by , a quarter-wave line with real characteristic impedance of value can be used. LZ  0d  4 d       2 0 4 IN L Z Z Z 0Z l    4 d LZ   4INZ  Example: , how to trans- form it to at certain frequency?  75LZ 50       0 4 50 75 61.2IN LZ Z Z You can simply use a transmission line with 61.2 Ohm characteristic impedance!  4 30/40
31. 31. Department of Electronic Engineering, NTUT Half-wave Line • Half-wave line:   2IN LZ Z  No matter what the line impedance is, when a half-wave line is used and it does not affect the impedance seen from the input. LZ  0d  2 d      2 IN LZ Z 0Z   2IN LZ Z l    2 d 31/40
32. 32. Department of Electronic Engineering, NTUT Voltage on the Shorted-circuited Line         1 12 sinj d j d V d A e e j A d                          2 1, Re Re 2 sin j t j t v d t V d e A d e  When      1 14 2 sin 2 2V j A j A      12 2 sin 0V j A   4 d  When   2 d • Voltage on the short-circuited line: incident wave reflected wave 32/40
33. 33. Department of Electronic Engineering, NTUT Standing-wave Pattern              1, 2 sin cos 2 v d t A d t   2 3 2 2       max min V d VSWR V d  V d  1 max 2A V d   min 0 V dd d  2  4 3 4   In order to proceed we need to know the value of the complex constant A1. For simplicity let us assume that A1 is real. Hence we obtain   2 3 2 2  ,v d t 12A   min 0 V dd d  2  4 3 4  12A   3 2 t  5 4 t    3, 4 4 t   2 t   0,t 33/40
34. 34. Department of Electronic Engineering, NTUT Example(I)   100 50sZ j   50 50LZ j   10 0sv       50 50 10 0 3.92 11.31 50 50 100 50 L L s L s jZ V V Z Z j j            Consider the circuit shown below that there is no transmission line between the source and load, find the voltage at the terminal of the load. 34/40
35. 35. Department of Electronic Engineering, NTUT Example(II)  Find the load reflection coefficient, the input impedance, and the VSWR in the transmission line shown, the length of the transmission line is and its characteristic impedance is 50 Ohm.   10 0sv   100 50sZ j   50 50LZ j l               0 0 0 50 50 50 0.447 63.44 50 50 50 L L jZ Z Z Z j                       50 50 50tan45 8 50 100 50 50 50 50 tan45 IN j j Z j j j          0 0 1 1 0.447 2.62 1 1 0.447 VSWR  8  0 50Z  8   8INZ LV 35/40
36. 36. Department of Electronic Engineering, NTUT Example(III)                           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         2 1 01j d j d V d A e e                   4 2 18 5.59 26.57 1 0.447 63.44 j j V A e e    1 3.95 63.44A           0 3.95 63.44 1.77 5 45LV V   100 50sZ j   10 0sv     8 100 50INZ j Equivalent circuit Transmission-line effect makes things different! It is not always bad, since it cab be used for matching and makes maximum power transfer possible. 36/40
37. 37. Department of Electronic Engineering, NTUT  V x  I x R x  j L x G x  j C x   V x x   I x x Lossy Transmission Line         dV x R j L I x dx         dI x G j C V x dx      2 2 2 d V x V x dx           j R j L G j C  Complex propagation constant is the attenuation constant in nepers/m and propagation constant is in rads/m   37/40
38. 38. Department of Electronic Engineering, NTUT Reflection Coefficient and Input Impedance                     , Re Re j x t j x tj t x x v x t V x e Ae e Be e                       0 0 , Re Re j x t j x tj t x xA B i x t I x e e e e e Z Z  With x=l - d      1 1 d d V d A e B e      1 1 0 0 d dA B I d e e Z Z l 1A Ae l 1B Be  The reflection coefficient of the terminated lossy line            21 0 1 d d IN d B e d e A e where          01 0 1 0 0 L IN L Z ZB A Z Z        0 0 0 tanh tanh L IN L Z Z d Z d Z Z Z d where and  The impedance of the terminated lossy line 38/40
39. 39. Department of Electronic Engineering, NTUT Summary (I)        0 0 0 tanh tanh L IN L Z Z d Z d Z Z Z d       2 0 d IN d e     0 0 0 L L Z Z Z Z      0 0 1 1 VSWR        0 0 0 tan tan L IN L Z jZ d Z d Z Z jZ d       2 0 j d IN d e   pv     2   LC• Propagation constant • Reflection coefficient at load • Reflection coefficient at any position lossless • Input impedance lossless     j where • Phase velocity • Wavelength • Voltage standing wave ratio 39/40
40. 40. Department of Electronic Engineering, NTUT Summary (II) • Transmission line affects the input impedance, and you may not get the voltage you want at the load. On the other hand, this property is useful when you need impedance matching to get maximum power transfer. • Both the wave velocity and wavelength decreases when the wave travels from vacuum into the material that has a relative dielectric constant greater than 1. • Generally speaking, when the circuit dimension is under , the transmission effects can be considered negligible.  20 40/40