1. Prof. David R. Jackson Dept. of ECE Fall 2015 Notes 7 ECE 6340 Intermediate EM Waves 1

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

3. TEM Transmission Line (cont.) + + + + + + + - - - - - xxx B 3 Physical Transmission Line + v(z,t) - RzRz LzLz GzGz CzCz i(z,t) + v(z+  z,t) - i(z +  z,t) z

4. TEM Transmission Line (cont.) 4 + v(z,t) - RzRz LzLz GzGz CzCz i(z,t) + v(z+  z,t) - i(z +  z,t) z

5. Hence Now let  z 0: “Telegrapher’s Equations” TEM Transmission Line (cont.) 5

6. To combine these, take the derivative of the first one with respect to z : Switch the order of the derivatives. TEM Transmission Line (cont.) 6 Substitute from the second Telegrapher’s equation.

7. The same equation also holds for i. Hence, collecting terms, we have: TEM Transmission Line (cont.) 7

8. Time-Harmonic Waves 8

9. Note that = series impedance/length = parallel admittance/length Then we can write: TEM Transmission Line (cont.) 9

10. This is one that is made by artificially cascading periodic sections of elements, which are arranged differently from what an actual (TEM) transmission line would have. We still have: Artificial Transmission Line (ATL) 10 CeCe LeLe CeCe LeLe... p

11. Let Convention: + z wave is Solution: To see what this implies, use Then Propagation Constant  is called the "propagation constant." 11

12. Write: There are two possible locations for  : Attenuation constant Phase constant Propagation constant 12 Propagation Constant (cont.)

13. We require Hence: The propagation constant  is always in the first quadrant. 13 Propagation Constant (cont.) Consider a wave going in the +z direction:

14. The principal branch of the square root function: 14 Propagation Constant (cont.) Branch cut This is the branch that MATLAB uses. We have the property that

15. To be more general, 15 Propagation Constant (cont.) (This holds for an ATL.) The principal branch ensures that Hence:

16. 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.) 16

17. Use the first Telegrapher’s Equation: so Hence Characteristic Impedance Z 0 (cont.) 17 (This form holds for an ATL.)

18. From this we have: Characteristic Impedance Z 0 (cont.) 18 (This also holds for an artificial TL.) We can also write Which sign is correct?

19. Characteristic Impedance Z 0 (cont.) 19 Hence (This also holds for an artificial TL.) For a physical transmission line we have:

20. Backward-Traveling Wave so + V - (z) - I - (z) z A general superposition of forward and backward traveling waves: Most general case: A wave is traveling in the negative z direction. 20

21. Power Flow Hence +V+(z)-+V+(z)- I + (z) Z0Z0 z A wave is traveling in the positive z direction. 21

22. Power Flow (cont.) +V (z)-+V (z)- I (z) Z0Z0 z Allow for waves in both directions: 22

23. pure imaginary Power Flow (cont.) +V (z)-+V (z)- I (z) Z0Z0 z 23

24. pure imaginary For a lossless line, Z 0 is pure real: In this case, so (orthogonality) Power Flow (cont.) +V (z)-+V (z)- I (z) Z0Z0 z 24

25. Traveling Wave Let's look at the traveling wave in the time domain. 25

26. Wavelength 26

27. Wavelength (cont.) The wave “repeats” when: Hence: 27

28. Phase Velocity Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest. z vpvp (phase velocity) 28

29. Phase Velocity (cont.) Set Hence 29

30. In general, This is dispersion, resulting in waveform distortion Phase Velocity (cont.) 30

31. In general, waveform distortion is caused by either of two things:Distortion 1)Phase velocity v p is a function of frequency (dispersion) 2)Attenuation  is a function of frequency In general, both effects arise when loss is present on a transmission line. 31

32. Lossless Case so no dispersion + no attenuation no distortion 32

33. Lossless Case (cont.) 33

34. Example Lossless coaxial cable a b z 34

35. Example (cont.) Using We have 35

36. Generalization Generalization to general lossless two-conductor transmission line with a homogeneous non-magnetic material filling: This relation for L follows from the requirement that the phase velocity be equal to the speed of light in the filling material – this is valid for any TEM mode in a lossless material, as discussed later in Notes 9. GF = geometrical factor (since and D =  E, with E independent of frequency) 36 Hence: Note: The shape of the fields are independent of frequency for a TEM mode (proof given in Notes 9).

37. Generalization (cont.) Consider next a lossy dielectric, but lossless conductors: 37 To see this, note that (principle of effective permittivity)

38. Generalization (cont.) This is proven in notes 9. 38 Also, for a lossy line we have

39. Determination of ( L, G, C ) Parameters Consider the general case of a lossy (dielectric loss only) transmission line: From the first two equations we have (multiplying and dividing the two equations): For the calculation of the lossless Z 0, we set  c =  c (ignore  c  ). We wish to calculate the parameters ( G, L, C ) in terms of the characteristic impedance of the lossless line and the complex permittivity of the filling material. 39

40. Determination of Parameters (cont.) Summary Note: Later we will see how to calculate R. (This involves the concept of the surface resistance of the conductors.) These results tells us how to calculate the ( L, G, C ) line parameters from the characteristic impedance of the lossless line and the filling material. 40

41. Distortionless Case Assume that the following condition holds: Then we have: This is the "Heaviside condition," discovered by Oliver Heaviside. 41

42. Distortionless Case (cont.) There is then attenuation but no dispersion. Note: There will be some distortion in practice, since the Heaviside condition cannot be satisfied for all frequencies: 42 (assuming a fixed loss tangent) Also, there will be distortion since  is a function of frequency.