1. KAVOSHCOM RF Communication Circuits Lecture 1: Transmission Lines

2. A wave guiding structure is one that carries a signal (or power) from one point to another. There are three common types:  Transmission lines  Fiber-optic guides  Waveguides Waveguiding Structures 2

3. Transmission Line  Has two conductors running parallel  Can propagate a signal at any frequency (in theory)  Becomes lossy at high frequency  Can handle low or moderate amounts of power  Does not have signal distortion, unless there is loss  May or may not be immune to interference  Does not have E z or H z components of the fields (TEM z ) Properties Coaxial cable (coax) Twin lead (shown connected to a 4:1 impedance-transforming balun) 3

4. Transmission Line (cont.) CAT 5 cable (twisted pair) The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities. 4

5. Transmission Line (cont.) Microstrip h w rr rr w Stripline h Transmission lines commonly met on printed-circuit boards Coplanar strips h rr w w Coplanar waveguide (CPW) h rr w 5

6. Transmission Line (cont.) Transmission lines are commonly met on printed-circuit boards. A microwave integrated circuit Microstrip line 6

7. Fiber-Optic Guide Properties  Uses a dielectric rod  Can propagate a signal at any frequency (in theory)  Can be made very low loss  Has minimal signal distortion  Very immune to interference  Not suitable for high power  Has both E z and H z components of the fields 7

8. Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber 2) Multi-mode fiber Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength. Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect). 8

9. Fiber-Optic Guide (cont.) http://en.wikipedia.org/wiki/Optical_fiber Higher index core region 9

10. Waveguides  Has a single hollow metal pipe  Can propagate a signal only at high frequency:  >  c  The width must be at least one-half of a wavelength  Has signal distortion, even in the lossless case  Immune to interference  Can handle large amounts of power  Has low loss (compared with a transmission line)  Has either E z or H z component of the fields (TM z or TE z ) Properties http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 10

11.  Lumped circuits: resistors, capacitors, inductors neglect time delays (phase) account for propagation and time delays (phase change) Transmission-Line Theory  Distributed circuit elements: transmission lines We need transmission-line theory whenever the length of a line is significant compared with a wavelength. 11

12. Transmission Line 2 conductors 4 per-unit-length parameters: C = capacitance/length [ F/m ] L = inductance/length [ H/m ] R = resistance/length [  /m ] G = conductance/length [ /m or S/m ]  zz 12

13. Transmission Line (cont.) + + + + + + + - - - - - xxx B 13 RzRz LzLz GzGz CzCz z v(z+z,t)v(z+z,t) + - v(z,t)v(z,t) + - i(z,t)i(z,t) i(z+z,t)i(z+z,t)

14. Transmission Line (cont.) 14 RzRz LzLz GzGz CzCz z v(z+z,t)v(z+z,t) + - v(z,t)v(z,t) + - i(z,t)i(z,t) i(z+z,t)i(z+z,t)

15. Hence Now let  z  0: “Telegrapher’s Equations” TEM Transmission Line (cont.) 15

16. To combine these, take the derivative of the first one with respect to z : Switch the order of the derivatives. TEM Transmission Line (cont.) 16

17. The same equation also holds for i. Hence, we have: TEM Transmission Line (cont.) 17

18. TEM Transmission Line (cont.) Time-Harmonic Waves: 18

19. Note that = series impedance/length = parallel admittance/length Then we can write: TEM Transmission Line (cont.) 19

20. Let Convention: Solution: Then TEM Transmission Line (cont.)  is called the "propagation constant." 20

21. TEM Transmission Line (cont.) Forward travelling wave (a wave traveling in the positive z direction): The wave “repeats” when: Hence: 21

22. Phase Velocity Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest. vpvp ( phase velocity ) 22

23. Phase Velocity (cont.) Set Hence In expanded form: 23

24. Characteristic Impedance Z 0 so +V+(z)-+V+(z)- I + (z) z A wave is traveling in the positive z direction. ( Z 0 is a number, not a function of z.) 24

25. Use Telegrapher’s Equation: so Hence Characteristic Impedance Z 0 (cont.) 25

26. From this we have: Using We have Characteristic Impedance Z 0 (cont.) Note: The principal branch of the square root is chosen, so that Re (Z 0 ) > 0. 26

27. Note: wave in +z direction wave in -z direction General Case (Waves in Both Directions) 27

28. Backward-Traveling Wave so + V - (z) - I - (z) z A wave is traveling in the negative z direction. Note: The reference directions for voltage and current are the same as for the forward wave. 28

29. General Case A general superposition of forward and backward traveling waves: Most general case: Note: The reference directions for voltage and current are the same for forward and backward waves. 29 +V (z)-+V (z)- I (z) z

30. guided wavelength  g phase velocity  v p Summary of Basic TL formulas 30

31. Lossless Case so (indep. of freq.) (real and indep. of freq.) 31

32. Lossless Case (cont.) In the medium between the two conductors is homogeneous (uniform) and is characterized by ( ,  ), then we have that The speed of light in a dielectric medium is Hence, we have that The phase velocity does not depend on the frequency, and it is always the speed of light (in the material). (proof given later) 32

33. 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. Terminated Transmission Line Note: The length l measures distance from the load: 33

34. What if we know Terminated Transmission Line (cont.) Hence Can we use z = - l as a reference plane? Terminating impedance (load) 34

35. Terminated Transmission Line (cont.) Compare: Note: This is simply a change of reference plane, from z = 0 to z = - l. Terminating impedance (load) 35

36. What is V(- l )? propagating forwards propagating backwards Terminated Transmission Line (cont.) l  distance away from load The current at z = - l is then Terminating impedance (load) 36

37. 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 Terminated Transmission Line (cont.)  l  Reflection coefficient at z = - l 37

38. Input impedance seen “looking” towards load at z = - l. Terminated Transmission Line (cont.) 38

39. At the load ( l = 0 ): Thus, Terminated Transmission Line (cont.) Recall 39

40. Simplifying, we have Terminated Transmission Line (cont.) Hence, we have 40

41. Impedance is periodic with period g /2 Terminated Lossless Transmission Line Note: tan repeats when 41

42. For the remainder of our transmission line discussion we will assume that the transmission line is lossless. Terminated Lossless Transmission Line 42

43. Matched load: (Z L =Z 0 ) For any l No reflection from the load A Matched Load 43

44. Short circuit load: ( Z L = 0 ) Always imaginary! Note: B S.C. can become an O.C. with a g /4 trans. line Short-Circuit Load 44

45. Using Transmission Lines to Synthesize Loads A microwave filter constructed from microstrip. This is very useful is microwave engineering. 45

46. Example Find the voltage at any point on the line. 46

47. Note: At l = d : Hence Example (cont.) 47

48. Some algebra: Example (cont.) 48

49. where Example (cont.) Therefore, we have the following alternative form for the result: Hence, we have 49

50. Example (cont.) Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load). 50

51. Example (cont.) Wave-bounce method (illustrated for l = d ): 51

52. Example (cont.) Geometric series: 52

53. Example (cont.) or This agrees with the previous result (setting l = d ). Note: This is a very tedious method – not recommended. Hence 53

54. Quarter-Wave Transformer so Hence This requires Z L to be real. ZLZL Z0Z0 Z 0T Z in 54

55. Voltage Standing Wave Ratio 55