Transmission Lines

1. Transmission Lines Dr. Sandra Cruz-Pol ECE Dept. UPRM

2. Iin + VL - + Vin - Zg Zo=30+j60 g=a +j b ZL Vg 40 m Exercise 11.3 • A 40-m long TL has Vg=15 Vrms, Zo=30+j60 W, and VL=5e-j48o Vrms. If the line is matched to the load and the generator, find: the input impedance Zin, the sending-end current Iin and Voltage Vin, the propagation constantg. • Answers:

3. Transmission Lines • TL parameters • TL Equations • Input Impedance, SWR, power • Smith Chart • Applications • Quarter-wave transformer • Slotted line • Single stub • Microstrips

4. Transmission Lines (TL) • TL have two conductors in parallel with a dielectric separating them • They transmit TEM waves inside the lines

5. Common Transmission Lines Two-wire (ribbon) Microstrip Stripline (Triplate) Coaxial

6. Other TL (higher order)[Chapter 11]

7. Fields inside the TL • V proportional to E, • I proportional to H

8. Distributed parameters The parameters that characterize the TL are given in terms of per length. • R = ohms/meter • L = Henries/ m • C = Farads/m • G = mhos/m

9. Common Transmission Lines R, L, G, and C depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EMAG techniques.

10. TL representation

11. Distributed line parameters Using KVL:

12. Distributed parameters • Taking the limit as Dz tends to 0 leads to • Similarly, applying KCL to the main node gives

13. Wave equation • Using phasors • The two expressions reduce to Wave Equation for voltage

14. TL Equations • Note that these are the wave eq. for voltage and current inside the lines. • The propagation constant is g and the wavelength and velocity are

15. z Waves moves through line • The general solution is • In time domain is • Similarly for current, I

16. Characteristic Impedance of a Line, Zo • Is the ratio of positively traveling voltage wave to current wave at any point on the line z

17. Example: • An air filled planar line with w=30cm, d=1.2cm, t=3mm, sc=7x107 S/m. • Find R, L, C, G for 500MHz • Answer d w

18. Exercise 11.1 A transmission line operating at 500MHz has Zo=80 W, a=0.04Np/m,b=1.5rad/m. Find the line parameters R,L,G, and C. • Answer: 3.2 W/m, 38.2nH/m, 0.0005 S/m, 5.97 pF/m

19. Transmission line Transmission line Transmission line Different cases of TL • Lossless • Distortionless • Lossy

20. Lossless Lines (R=0=G) Has perfect conductors and perfect dielectric medium between them. • Propagation: • Velocity: • Impedance

21. Distortionless line (R/L = G/C) Is one in which the attenuation is independent on frequency. • Propagation: • Velocity: • Impedance

22. Summary

23. Excersice 11.2 • A telephone line has R=30W/km, L=100 mH/km, G=0, and C= 20mF/km. At 1kHz, obtain: the characteristic impedance of the line, the propagation constant, the phase velocity. • Solution:

24. Define reflection coefficient at the load, GL

25. Terminated, Lossless TL Then, Similarly, The impedance anywhere along the line is given by The impedance at the load end, ZL, is given by

26. Terminated, Lossless TL Then, Conclusion: The reflection coefficient is a function of the load impedance and the characteristic impedance. Recall for the lossless case, Then

27. Example • A generator with 10Vrms and Rg=50, is connected to a 75W load thru a 0.8l, 50W-lossless line. • Find VL

28. Terminated, Lossless TL It is customary to change to a new coordinate system, z = - l, at this point. -z z = - l Rewriting the expressions for voltage and current, we have Rearranging,

29. Impedance (Lossy line) The impedanceanywhere along the line is given by The reflection coefficient can be modified as follows Then, the impedance can be written as After some algebra, an alternative expression for the impedance is given by Conclusion: The load impedance is “transformed” as we move away from the load.

30. Impedance (Lossless line) The impedance anywhere along the line is given by The reflection coefficient can be modified as follows Then, the impedance can be written as After some algebra, an alternative expression for the impedance is given by Conclusion: The load impedance is “transformed” as we move away from the load.

31. Iin + VL - + Vin - Zg Zo g=j b ZL Vg 2 cm Exercise : using formulas • A 2cm lossless TL has Vg=10 Vrms, Zg=60 W, ZL=100+j80 W and Zo=40W, l=10cm . find: the input impedance Zin, the sending-end Voltage Vin, Voltage Divider:

32. SWR or VSWR or s Whenever there is a reflected wave, a standing wave will form out of the combination of incident and reflected waves. The (Voltage) Standing Wave Ratio - SWR (or VSWR) is defined as

33. Power • The average input power at a distance l from the load is given by • which can be reduced to • The first term is the incident power and the second is the reflected power. Maximum power is delivered to load if G=0

34. Three common Cases of line-load combinations: • Shorted Line (ZL=0) • Open-circuited Line (ZL=∞) • Matched Line (ZL = Zo)

35. Voltage maxima |V(z)| -z -l/4 -l/2 -l Standing Waves -Short Shorted Line (ZL=0), we had • So substituting in V(z) *Voltage minima occurs at same place that impedance has a minimum on the line

36. Voltage minima -z -l/4 -l/2 -l Standing Waves -Open Open Line (ZL=∞) ,we had • So substituting in V(z) |V(z)|

37. -z -l/4 -l/2 -l Standing Waves -Matched Matched Line (ZL = Zo), we had • So substituting in V(z) |V(z)|

38. Java applets • http://www.amanogawa.com/transmission.html • http://physics.usask.ca/~hirose/ep225/ • http://www.educatorscorner.com/index.cgi?CONTENT_ID=2483

39. The Smith Chart

40. Smith Chart • Commonly used as graphical representation of a TL. • Used in hi-tech equipment for design and testing of microwave circuits • One turn (360o) around the SC = to l/2

41. Network Analyzer What can be seen on the screen?

42. Smith Chart • Suppose you use as coordinates the reflection coefficient real and imaginary parts. and define the normalized ZL: Gi |G| Gr

43. Now relating to z=r+jx • After some algebra, we obtain two eqs. • Similar to general equation of a circle of radius a, center at (h,k) Circles of r Circles of x

44. Examples of circles of r and x Circles of r Circles of x

45. Examples of circles of r and x Circles of r Circles of x

46. The joy of the SC • Numerically s = r on the +axis of Gr in the SC Proof:

47. Fun facts about the Smith Chart • A lossless TL is represented as a circle of constant radius, |G|, or constant s • Moving along the line from the load toward the generator, the phase decrease, therefore, in the SC equals to moves clockwisely. To generator

48. Fun facts about the Smith Chart • One turn (360o) around the SC = to l/2 because in the formula below, if you substitute length for half-wavelength, the phase changes by 2p, which is one turn. • Find the point in the SC where G=+1,-1, j, -j, 0, 0.5 • What is rand x for each case?

49. Fun facts : Admittance in the SC • The admittance, y=YL/YowhereYo=1/Zo, can be found by moving ½ turn (l/4) on the TL circle

50. Fun facts about the Smith Chart • The Gr +axis, where r > 0 corresponds to Vmax • The Gr -axis, where r < 0 corresponds to Vmin Vmin Vmax (Maximum impedance)