Prof. D. R. Wilton

1. ECE 3317 Prof. D. R. Wilton Note 2 Transmission Lines(Time Domain)

2. Note about Notes 2 Disclaimer: Transmission lines is the subject of Chapter 6 in the book. However, the subject of wave propagation on transmission lines in the time domain is not treated very thoroughly there, appearing only in the latter half of section 6.5. Therefore, the material of this Note is roughly independent of the book. Approach: Transmission line theory can be developed starting from either circuit theory or from Maxwell’s equations directly. We’ll use the former approach because it is simpler, though it doesn’t reveal the approximations or limitations of the approach.

3. Transmission Lines a z b A transmission line is a two-conductor system that is used to transmit a signal from one point to another point. Two common examples: twin line coaxial cable A transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors.

4. Transmission Lines (cont.) Here’s what they look like in real-life. coax to twin line matching section coaxial cable twin line

5. Transmission Lines (cont.) Another common example (for printed circuit boards): w r h microstrip line

6. Transmission Lines (cont.) microstrip line

7. Transmission Lines (cont.) E - l - - + + + + + - + - - -l Some practical notes: • Coaxial cable is a perfectly shielded system (no interference). • Two-wire (twin) lines do not form a shielded system – more susceptible to noise and interference. • The coupling between two-wire lines may be reduced by using a form known as a “twisted pair.” E twin line coax

8. Transmission Lines (cont.) Load • Transmission line theory must be used instead of circuit theory for any two-conductor system if the speed-of-light travel time across the line, TL, is a significant fraction of a signal’s period T or rise time for periodic or pulse signals, respectively.

9. Transmission Lines (cont.) symbols: z 4 parameters Note: We use this schematic to represent a general transmission line, no matter what the actual shape of the conductors.

10. Transmission Lines (cont.) z Capacitance/m between the two conductors Inductance/m due to stored magnetic energy Resistance/m due to the conductors Conductance/m due to the filling material between the conductors Four fundamental parameters characterize any transmission line: These are “per unit length” parameters. 4 parameters C= capacitance/length [F/m] L= inductance/length [H/m] R= resistance/length [/m] G= conductance/length [S/m]

11. Circuit Model Dz z RDz LDz CDz GDz Circuit Model: Dz z

12. Circuit Model (cont.) z CDz CDz CDz GDz GDz GDz Dz Dz Dz Dz Circuit Model: Dz RDz RDz LDZ RDz LDZ RDz LDZ LDZ CDz GDz z

13. Coaxial Cable a z b Example: coaxial cable d = conductivity of dielectric [S/m]. m = conductivity of metal [S/m]. (skin depth of metal)

14. Coaxial Cable (cont.) E - l - - + + + + + - + - - -l Overview of derivation: capacitance per unit length

15. Coaxial Cable (cont.) y x E Js         Overview of derivation: inductance per unit length

16. Coaxial Cable (cont.) Overview of derivation: conductance per unit length RC Analogy:

17. Coaxial Cable (cont.) Relation between L and C: Speed of light in dielectric medium: This is true for ALL two-conductor transmission lines. Hence:

18. Telegrapher’s Equations RDz LDz I(z+Dz,t) I(z,t) + V(z+Dz,t) - + V(z,t) - CDz GDz z z+Dz z Apply KVL and KCL laws to a small slice of line:

19. Telegrapher’s Equations (cont.) Hence Now let Dz 0: “Telegrapher’s Equations (TE)”

20. Telegrapher’s Equations (cont.) To combine these, take the derivative of the first one with respect to z: To obtain an equation in V alone, eliminate I between eqs.: • Take the derivative of the first TE with respect to z. • Substitute in from the second TE.

21. Telegrapher’s Equations (cont.) Hence, we have: There is no exact solution to this differential equation, except for the lossless case. The same equation also holds for i.

22. Telegrapher’s Equations (cont.) Lossless case: Note: The current satisfies the same differential equation: The same equation also holds for i.

23. Solution to Telegrapher's Equations Hence we have Solution: This is called the D’Alembert solution to the Telegrapher's Equations (the solution is in the form of traveling waves). The same equation also holds for i.

24. Traveling Waves Proof of solution: General solution: It is seen that the differential equation is satisfied by the general solution.

25. Traveling Waves Example: z z0 t = t2 > t1 t = 0 t = t1 > 0 V(z,t) … … z z0 z0 + cdt1 z0 + cdt2

26. Traveling Waves Example: z z0 t = 0 t = t2 > t1 t = t1 > 0 V(z,t) … … z z0 - cdt1 z0 - cdt2 z0

27. Traveling Waves (cont.) Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually becomes broader). t = 0 V(z,t) t = t1 > 0 t = t2 > t1 z z0 z0 + cdt1 z0 + cdt2 (These effects can be studied numerically.)

28. Current (first TE) lossless Our goal is to now solve for the current on the line. Assume the following forms: The derivatives are:

29. Current (cont.) This becomes Equating terms with the same space and time variation, we have Hence we have Constants C1, C2 represent time and space-independent DC voltages or currents on the line. Assuming no initial line voltage or current we conclude C1, C2=0

30. Current (cont.) Observation about term: Define (real) characteristic impedance Z0: The units of Z0are Ohms. Then or

31. Current (cont.) General solution: OR For a forward wave, the current waveform is the same as the voltage, but reduced in amplitude by a factor of Z0. For a backward traveling wave, there is a minus sign as well. Note that without this minus sign, the ratio of voltage to current would be constant rather than varying from point-to-point and over time along the line as is generally the case!

32. Current (cont.) Picture for a forward-traveling wave: forward-traveling wave z + -

33. Current (cont.) Physical interpretation of minus sign for the backward-traveling wave: backward-traveling wave z + - The minus sign arises from the reference direction for the current.

34. Coaxial Cable a z b Example: Find the characteristic impedance of a coax.

35. Coaxial Cable (cont.) a z b (intrinsic impedance of free space)

36. Twin Line d a = radius of wires

37. Twin Line (cont.) These are the common values used for TV. 75-300 [] transformer 75 [] coax 300 [] twin line twin line coaxial cable

38. Microstrip Line w r h parallel-plate formulas:

39. Microstrip Line (cont.) t = strip thickness More accurate CAD formulas: Note: the effective relative permittivity accounts for the fact that some of the field exists outside the substrate, in the air region. The effective widthw' accounts for the strip thickness.

40. Some Comments • Transmission-line theory is valid at any frequency, and for any type of waveform (assuming an ideal transmission line). • Transmission-line theory is perfectly consistent with Maxwell's equations (although we work with voltage and current, rather than electric and magnetic fields). • Circuit theory does not view two wires as a "transmission line": it cannot predict effects such as signal propagation, distortion, etc.