Prof. David R. Jackson Dept. of ECE

1. ECE 5317-6351 Microwave Engineering Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 1 Transmission Line Theory

2. Waveguiding Structures A waveguiding 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 Note: An alternative to waveguiding structures is wireless transmission using antennas. (antenna are discussed in ECE 5318.)

3. Transmission Line Properties • 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 Ez or Hz components of the fields (TEMz) Twin lead (shown connected to a 4:1 impedance-transforming balun) Coaxial cable (coax)

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.

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

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

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 Ez and Hz components of the fields

8. Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-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. 2) Multi-mode fiber Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

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

10. Waveguides Properties • 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 Ez or Hz component of the fields (TMzorTEz) http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)

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

12. Dz 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] 

13. + + + + + + + B - - - - - - - - - - x x x Transmission Line (cont.) RDz LDz i(z,t) i(z+z,t) + + v(z+z,t) CDz v(z,t) GDz - - z

14. Transmission Line (cont.) RDz LDz i(z,t) i(z+z,t) + + v(z+z,t) CDz v(z,t) GDz - - z

15. TEM Transmission Line (cont.) Hence Now let Dz 0: “Telegrapher’sEquations”

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

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

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

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

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

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

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

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

24. I+(z) + V+(z) - z Characteristic Impedance Z0 A wave is traveling in the positive z direction. so (Z0is a number, not a function of z.)

25. Characteristic Impedance Z0 (cont.) Use Telegrapher’s Equation: so Hence

26. Characteristic Impedance Z0 (cont.) From this we have: Using We have Note: The principal branch of the square root is chosen, so that Re (Z0) > 0.

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

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

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

30. Summary of Basic TL formulas guided wavelength  g phase velocity  vp

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

32. Lossless Case (cont.) In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that (proof given later) 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).

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

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

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

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

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

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

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

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

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

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

43. Matched Load A Matched load: (ZL=Z0) No reflection from the load For any l

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

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

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

47. Example (cont.) Note: Atl= d: Hence

48. Example (cont.) Some algebra:

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

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).