ECE 546 Lecture -04 Transmission Lines

1. ECE 546 Lecture -04 Transmission Lines Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu

2. Maxwell’s Equations Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field Constitutive Relations

3. l l z Why Transmission Line? l Wavelength : propagation velocity l = frequency

4. In Free Space l At 10 KHz : = 30 km l At 10 GHz : = 3 cm Why Transmission Line? Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.

5. Justification for Transmission Line Let d be the largest dimension of a circuit circuit z l If d << l, a lumped model for the circuit can be used

6. Justification for Transmission Line circuit z l If d ≈ l, or d > l then use transmission line model

7. Modeling Interconnections Mid-range Frequency Low Frequency High Frequency or Short Transmission Line Lumped Reactive CKT

8. Single wire near ground

9. Single wire between grounded parallel planes ground return

10. Wires in parallel near ground for d << D, h

11. Balanced, near ground for d << D, h

12. Open 2-wire line in air

13. Parallel-plate Transmission Line

14. Types of Transmission Lines

15. Coaxial Transmission Line TEM Mode of Propagation

16. Coaxial Air Lines Infinite Conductivity Finite Conductivity

17. Coaxial Connector Standards ConnectorFrequency Range 14 mm DC - 8.5 GHz GPC-7 DC - 18 GHz Type N DC - 18 GHz 3.5 mm DC - 33 GHz 2.92 mm DC - 40 GHz 2.4 mm DC - 50 GHz 1.85 mm DC - 65 GHz 1.0 mm DC - 110 GHz

18. Microstrip

19. Microstrip Characteristic Impedance 100 90 h = 21 mils 80 h = 14 mils h = 7 mils 70 Zo (ohms) 60 50 40 30 0 1 2 3 W/h Microstrip dielectric constant : 4.3.

20. Telegraphers’ Equations L: Inductance per unit length. C: Capacitance per unit length. Assume time-harmonic dependence

21. TL Solutions

22. TL Solutions (Frequency Domain) forward wave backward wave

23. TL Solutions Propagation constant Propagation velocity Wavelength Characteristic impedance

24. Reflection Coefficient At z=0, we have V(0)=ZRI(0) But from the TL equations: Which gives where is the load reflection coefficient

25. Reflection Coefficient - If ZR = Zo, GR=0, no reflection, the line is matched - If ZR = 0, short circuit at the load, GR=-1 - If ZR inf, open circuit at the load, GR=+1 V and I can be written in terms of GR

26. Generalized Impedance Impedance transformation equation

27. Generalized Impedance - Short circuit ZR=0, line appears inductive for 0 < l < l/2 - Open circuit ZR  inf, line appears capacitive for 0 < l < l/2 - If l = l/4, the line is a quarter-wave transformer

28. Generalized Reflection Coefficient Reflection coefficient transformation equation

29. Voltage Standing Wave Ratio (VSWR) We follow the magnitude of the voltage along the TL Maximum and minimum magnitudes given by

30. Voltage Standing Wave Ratio (VSWR) Define Voltage Standing Wave Ratio as: It is a measure of the interaction between forward and backward waves

31. VSWR – Arbitrary Load Shows variation of amplitude along line

32. VSWR – For Short Circuit Load Voltage minimum is reached at load

33. VSWR – For Open Circuit Load Voltage maximum is reached at load

34. VSWR – For Open Matched Load No variation in amplitude along line

35. Application: Slotted-Line Measurement • Measure VSWR = Vmax/Vmin • Measure location of first minimum

36. Application: Slotted-Line Measurement At minimum, Therefore, So, then Since and

37. Summary of TL Equations Voltage Current Impedance Transformation  Reflection Coefficient Transformation  Reflection Coefficient – to Impedance  Impedance to Reflection Coefficient 

38. Determining V+ For lossless TL, V and I are given by reflection coefficient at the load At z = -l,

39. Determining V+ this leads to or

40. Determining V+ Divide through by with and From which

41. TL Example • A signal generator having an internal resistance Zs = 25 W and an open circuit phasor voltage Vs = 1ej0 volt is connected to a 50-W lossless transmission line as shown in the above picture. The load impedance is ZR= 75 W and the line length is l/4. • Find the magnitude and phase of the load current IR.

42. TL Example – Cont’

43. TL Example – Cont’

44. Geometric Series Expansion Since V+ can be expanded in a geometric series form

45. TL Time-Domain Solution at z=0

46. TL Time-Domain Solution At z=-l

47. TL - Time-Domain Reflectometer For TDR, ZS = Zo GS = 0, and retain only k=1

48. Wave Shifting Method Forward traveling wave at port 1 (measured at near end of line) Forward traveling wave at port 2 (measured at far end of line) Backward traveling wave at port 1 (measured at near end of line) Backward traveling wave at port 2 (measured at far end of line)

49. Wave Shifting Solution* *Schutt-Aine & Mittra, Trans. Microwave Theory Tech., pp. 529-536, vol. 36 March 1988.

50. Frequency Dependence of Lumped Circuit Models • At higher frequencies, a lumped circuit model is no longer accurate for interconnects and one must use a distributed model • Transition frequency depends on the dimensions and relative magnitude of the interconnect parameters.