1. Electromagnetics (ENGR 367) T-line Power, Reflection & SWR

2. T-line Theory: Something New or Not?! Power, Reflection and Standing Waves in T-lines act just like Uniform Plane Waves (UPW) in unbounded and layered media! Power, Reflection and Standing Waves in T-lines act just like Uniform Plane Waves (UPW) in unbounded and layered media! Once you understand UPWs, you can also see by analogy how waves behave in T-lines with a few simplifications! Once you understand UPWs, you can also see by analogy how waves behave in T-lines with a few simplifications!

3. Traveling Waves on T-lines Space-time phenomena may be described by phasor functions representing either Space-time phenomena may be described by phasor functions representing either –Voltage and current disturbances –Electromagnetic wave disturbances

4. T-line Traveling Waves Analagous to waves on a string or Analagous to waves on a string or sound waves in a tube since all these waves –carry real power –reflect at boundaries and discontinuities –exhibit impedance at each point in the medium Unique from waveguides since on T-lines they propagate in the (quasi-) Transverse Electromagnetic (TEM) mode: ~plane waves Unique from waveguides since on T-lines they propagate in the (quasi-) Transverse Electromagnetic (TEM) mode: ~plane waves

5. Power in T-lines via Circuit Model during time harmonic oscillation Instantaneous Power over a fixed line length z Instantaneous Power over a fixed line length z –Express the real parts of V, I in the (+) direction only –Apply Euler’s Identity –Thus

6. Power in T-lines via Circuit Model during time harmonic oscillation Time Averaged Power Time Averaged Power

7. Power Loss due to Attenuation Explicitly Explicitly In decibel (dB) units In decibel (dB) units

8. Power Loss due to Attenuation In terms of Voltage In terms of Voltage

9. Example of Calculating T-line Power Loss Exercise 1 (based on D11.2, H&B, 7/e, p. 350) Exercise 1 (based on D11.2, H&B, 7/e, p. 350) Given: two T-lines joined end-to-end by an adaptor. Line 1 is 30 m long and is rated at 0.1 dB/m, whereas line 2 is 45 m long and is rated at 0.15 dB/m. Due to a poor adaptor, the joint imparts another 3 dB loss. Find: the percentage (%) of the input power that reaches the output of this combination Solution:

10. Example of Calculating T-line Power Loss Exercise 1 (continued) Exercise 1 (continued) Solution:

11. Wave Reflection at T-line Discontinuity T-line discontinuity may consist of T-line discontinuity may consist of –an actual load termination: device with complex input impedance (e.g., antenna or display) –a junction between lines: connector and/or line mismatch Schematic model Schematic model

12. Wave Reflection at T-line Discontinuity Energized T-line with discontinuity Energized T-line with discontinuity –Incident Voltage phasor –Reflected Voltage phasor (where the time dependence e j  t has been supressed)

13. Wave Reflection at T-line Discontinuity Consider the situation at the load junction (z=0): Consider the situation at the load junction (z=0): –Voltages of opposite going waves add –Currents of opposite going waves add where the – sign arises due to neg. z-going current wave

14. Wave Reflection at T-line Discontinuity Define Voltage Reflection Coefficient (  ) Define Voltage Reflection Coefficient (  ) Solving for  in terms of impedances only Solving for  in terms of impedances only

15. Wave Transmission at T-line Discontinuity Define Voltage Transmission Coefficient (  ) Define Voltage Transmission Coefficient (  ) Solving in terms of impedances only Solving in terms of impedances only

16. Matching Condition at a T-line Junction An impedance match becomes a desired design condition for most practical T-line systems because it An impedance match becomes a desired design condition for most practical T-line systems because it –Maximizes power transferred to the load –Minimizes power reflected back to generator In terms of Z L and Z 0 In terms of Z L and Z 0

17. Power Reflected and Transmitted at a T-line Junction Ratio of Reflected to Incident Power Ratio of Reflected to Incident Power Ratio of Transmitted to Incident Power Ratio of Transmitted to Incident Power

18. Calculating Power In Case of a Line-Load Mismatch Exercise 2 (Ex. 11.5, H&B, 7/e, p. 352) Exercise 2 (Ex. 11.5, H&B, 7/e, p. 352) Given: a 50  lossless T-line terminated by a load impedance, Z L =50-j75 . Power incident from the T-line to the load is 100 mW. Find: the power dissipated by the load Solution: first calculate the reflection coefficient

19. Calculating Power In Case of a Line-Load Mismatch Exercise 2 (continued) Exercise 2 (continued) Solution: next calculate the transmitted power in terms of incident power and 

20. Calculating Power In Case of Both Line Loss and Line-Load Mismatch Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353) Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353) Given: two lossy lines joined end-to-end. Line 1 is 10 m long and has a 0.20 dB/m loss. Line 2 is 15 m long and has a 0.10 dB/m loss. At the junction of these two lines  = 0.30. Power input to line 1 is P i1 = 100 mW Find: a) the total loss of the line combination in dB. b) the power transmitted to the output of line 2. b) the power transmitted to the output of line 2.

21. Calculating Power In Case of Both Line Loss and Line-Load Mismatch Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353) Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353) Solution: a) first calculate the dB loss of the joint from  then calculate the total loss of the link then calculate the total loss of the link b) now calculate the output power as

22. Voltage Standing Wave Ratio (VSWR) for Terminated T-lines The status of waves on a T-line depends on the termination: 3 possibilities exist The status of waves on a T-line depends on the termination: 3 possibilities exist 1) Matched termination (Z L = Z 0   = 0) All waves travel from source to load All waves travel from source to load No waves reflect back to the source No waves reflect back to the source No standing waves exist, only pure traveling waves No standing waves exist, only pure traveling waves 2) Perfectly reflective termination (  = 1) All waves travel from source to load and back again All waves travel from source to load and back again All waves completely reflect All waves completely reflect A pure standing wave pattern exists with fixed null and maximum voltage locations along the line A pure standing wave pattern exists with fixed null and maximum voltage locations along the line

23. Voltage Standing Wave Ratio (VSWR) for Terminated T-lines The status of waves on a T-line depends on the termination: 3 possibilities exist The status of waves on a T-line depends on the termination: 3 possibilities exist 3) A partially reflective termination (0<  <1) Some waves travel from source to load and back Some waves travel from source to load and back Some waves reflect; others pass to the load Some waves reflect; others pass to the load A partial standing wave pattern exists with fixed minimum and maximum locations along the line mixed with traveling waves! A partial standing wave pattern exists with fixed minimum and maximum locations along the line mixed with traveling waves! (animated partial standing wave pattern)

24. Terminated Lossless T-line Total voltage wave phasor (w/load @ z=0) Total voltage wave phasor (w/load @ z=0) Complete space-time voltage wave function Complete space-time voltage wave function

25. Terminated Lossless T-line After applying Euler’s Identity and taking the real part the total voltage wave function becomes After applying Euler’s Identity and taking the real part the total voltage wave function becomes

26. Terminated Lossless T-line Where are maximum and minimum voltages located? Where are maximum and minimum voltages located? In terms of wavelengths ( ) between successive In terms of wavelengths ( ) between successive –V max locations –V min locations –V max to V min locations

27. Graphical Standing Wave Patterns Voltage Standing Wave Patterns for Voltage Standing Wave Patterns for Real Reflection Coefficient Complex

28. VSWR: Terminated Lossless T-line Now define as Now define as Note special cases Note special cases –Matched termination: –Perfectly reflective termination: Range: Range: Significance: indicates the degree of standing waves vs. traveling waves present on the T-line Significance: indicates the degree of standing waves vs. traveling waves present on the T-line

29. VSWR Calculations for a Lossless Terminated T-line Exercise 4 Exercise 4 Given:  = 3/5 Find: VSWR = ? Solution: Exercise 5 Exercise 5 –Given: for a good match, we desire VSWR < 2.5 –Find: the condition on  –Solution:

30. Conclusions Traveling waves on T-lines carry power subject to the losses of attenuation over distance and any mismatch of impedance at junctions Traveling waves on T-lines carry power subject to the losses of attenuation over distance and any mismatch of impedance at junctions The power output expected from a T-line may be computed from the input power by taking into account any dB loss factors The power output expected from a T-line may be computed from the input power by taking into account any dB loss factors

31. Conclusions The reflection (or transmission) coefficient (  or  ) at any T-line discontinuity The reflection (or transmission) coefficient (  or  ) at any T-line discontinuity –Indicates how much voltage and power will be reflected (or transmitted) at the junction –May be computed from the line impedance (Z 0 ) on the source side and the effective input impedance (Z L = Z in ) on the load side

32. Conclusions The Voltage Standing Wave Ratio (VSWR) for a terminated T-line The Voltage Standing Wave Ratio (VSWR) for a terminated T-line –Indicates the degree of standing waves versus traveling waves present on the line –Serves as a figure of merit for the quality of impedance match at a junction –Represents the max. to min. voltage ratio along the line, but may be calculated directly from the reflection coefficient at a junction

33. References Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, 2006. Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, 2006. Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999. Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.