1. Chapter9 Theory and Applications of Transmission Lines

2. Application: Signal Transmission Models Electromagnetic theory General method Very complicate Lump-element model Applied to transmission line only

3. Common types of transmission lines Parallel-plate transmission line Each structure (including the twin lead) may have a dielectric between two conductors used to keep the separation between the metallic elements constant, so that the electrical properties would be constant. Two-wire (twin-lead) transmission line Coaxial transmission line

4. Microstrip line

5. Twin lead

6. Coaxial cable

7. 9-2 Parallel-Plate Transmission Line Assume it’s a plane wave propagate in the z with polarization in y direction.

8. Crossing the boundary from dielectric medium to the perfect conduction plates: 9-2 Parallel-Plate Transmission Line

10. impedance velocity of propagation along the line 9-2 Parallel-Plate Transmission Line

11. Dielectric medium between two conductors having a perimittivity and a conductivity Conductance C The two conductors have a very large but finite conductivity Resistance R

12. Example: E,H,V,I Cross-section of a coaxial cable showing the electric and magnetic fields Coaxial transmission line

14. Lumped-element model Small section of a transmission line (length = Δz) represented by an equivalent circuit 9-3 General Transmission-line equations

15. 1. Applying Kirchhoff’s voltage law at node N

16. 2. Taking the limit as Δz tends to zero, we have Time-domain form of the Transmission-line equations Solving this equation with the appropriate initial conditions and boundary condition, we can determined the voltage and current 9-3 General Transmission-line equations

17. For sinusoidal steady-state conditions, phasors can be used Similarly, 9-3 General Transmission-line equations

18. Therefore, the transmission-line equations becomes

19. 9-3 General Transmission-line equations

20. +V+(z)-+V+(z)- I + (z) z A wave is traveling in the positive z direction.

21. Three limiting cases 9-3 General Transmission-line equations

22. Three limiting cases

23. 9-3 General Transmission-line equations Three limiting cases

24. 9-3 General Transmission-line equations

25. guided wavelength  g phase velocity  v p Summary of Basic TL formulas 25

26. Where do we assign z = 0 ? The usual choice is at the load. Terminating impedance (load) Ampl. of voltage wave propagating in negative z direction at z = 0. Ampl. of voltage wave propagating in positive z direction at z = 0. 9-4 Terminated Transmission Line Note: The length l measures distance from the load: 26

27. What if we know Hence Can we use z = - l as a reference plane? Terminating impedance (load) 27 9-4 Terminated Transmission Line

28. Compare: Note: This is simply a change of reference plane, from z = 0 to z = - l. Terminating impedance (load) 28 9-4 Terminated Transmission Line

29. What is V(- l )? propagating forwards propagating backwards l  distance away from load The current at z = - l is then Terminating impedance (load) 29 9-4 Terminated Transmission Line

30. Total volt. at distance l from the load Ampl. of volt. wave prop. towards load, at the load position ( z = 0 ). Similarly, Ampl. of volt. wave prop. away from load, at the load position ( z = 0 ).  L  Load reflection coefficient  l  Reflection coefficient at z = - l 30 9-4 Terminated Transmission Line

31. Input impedance seen “looking” towards load at z = - l. 31 9-4 Terminated Transmission Line

32. At the load ( l = 0 ): Thus, Recall 32 9-4 Terminated Transmission Line Voltage reflection coefficient of the load impedance

33. Simplifying, we have Hence, we have 33 9-4 Terminated Transmission Line

34. Impedance is periodic with period g /2 Note: tan repeats when 34 9-4 Terminated Transmission Line

35. For the remainder of our transmission line discussion we will assume that the transmission line is lossless. 35 9-4 Terminated Transmission Line

36. Matched load: (Z L =Z 0 ) For any l No reflection from the load A 9-4 Terminated Transmission Line Matched Load 36

37. Short circuit load: ( Z L = 0 ) Always imaginary! Note: B S.C. can become an O.C. with a g /4 trans. line 9-4 Terminated Transmission Line Short-Circuit Load 37

38. Quarter-Wave Transformer so Hence This requires Z L to be real. ZLZL Z0Z0 Z 0T Z in 38 C 9-4 Terminated Transmission Line

39. Voltage Standing Wave Ratio 39 9-4 Terminated Transmission Line

40. For resistance-terminated lossless lines 9-4 Terminated Transmission Line

41. Line with arbitrary termination Z L can be determined by measuring S and the distance z′ m l m +z′ m =λ/2 Procedure 9-4 Terminated Transmission Line

42. 9-6 Smith Chart: Introduction Introduction A graphical tool used to solve transmission line problems. Today, a presentation medium in computer- aided design (CAD) software and measuring equipment for displaying the performance of microwave circuits.

43. 9-6 Smith-Chart rr ii r=0.5 r=1 r=2 x=0.5 x=-0.5 x=1 x=-1 x=2 x=-2 x=0 Normalized load impedance Voltage reflection coefficient r=0 P oc P sc

44. 9-6 Smith-Chart   rr ii r=0 r=0.5 r=1 r=2 x=0.5 x=-0.5 x=1 x=-1 x=2 x=-2 x=0  P M : SWR P m : 1/S For the constant r circles: 1.The centers of all the constant r circles are on the horizontal axis – real part of the reflection coefficient. 2.The radius of circles decreases when r increases. 3.All constant r circles pass through the point  r =1,  i = 0. 4.The normalized resistance r =  is at the point  r =1,  i = 0. For the constant x (partial) circles: 1.The centers of all the constant x circles are on the  r =1 line. The circles with x > 0 (inductive reactance) are above the  r axis; the circles with x < 0 (capacitive) are below the  r axis. 2. The radius of circles decreases when absolute value of x increases. 3. The normalized reactances x =  are at the point  r =1,  i = 0 The constant r circles are orthogonal to the constant x circles at every intersection. P oc P sc 1.All |  |-circles are centered at the origin, and their radii vary uniformly from 0 to 1. 2.The angle, measured from the positive real axis, of the line drawn from the origin through the point representing z L equals  . 3.The value of the r-circle passing through the intersection of the |  |-circle and the positive real axis equals the standing-wave ratio S To generator To load

45.  z L =2.6+j1.8 0.22 0.22+0.434-0.5=0.154 z L =0.69+j1.2 a   PMPM b) SWR=4 c) Z i =69+j120 d  Example 9-14

46.  z L =0.6+j3.0 0.20 z L =0.9-j0.6 a  Example 9-16 Lossy lines P sc  0.364 0.364+0.20-0.5=0.064 z L =0.64+j0.27 b)75(0.64+j0.27) =48.0+j20.3