Electromagnetics (ENGR 367)

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! • 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 • Voltage and current disturbances • Electromagnetic wave disturbances

4. T-line Traveling Waves • 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

5. Power in T-lines via Circuit Model during time harmonic oscillation • 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

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

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

9. Example of Calculating T-line Power Loss • 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) Solution:

11. Wave Reflection at T-line Discontinuity • 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

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

13. Wave Reflection at T-line Discontinuity • 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 () • Solving for  in terms of impedances only

15. Wave Transmission at T-line Discontinuity • Define Voltage Transmission Coefficient () • 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 • Maximizes power transferred to the load • Minimizes power reflected back to generator • In terms of ZL and Z0

17. Power Reflected and Transmitted at a T-line Junction • Ratio of Reflected 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) Given:a 50  lossless T-line terminated by a load impedance, ZL=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) 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) 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 Pi1 = 100 mW Find: a) the total loss of the line combination in dB. 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) Solution: a) first calculate the dB loss of the joint from  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 1) Matched termination (ZL = Z0   = 0) • All waves travel from source to load • No waves reflect back to the source • 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 completely reflect • 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 3) A partially reflective termination (0<<1) • Some waves travel from source to load and back • 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! (animated partial standing wave pattern)

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

26. Terminated Lossless T-line • Where are maximum and minimum voltages located? • In terms of wavelengths () between successive • Vmax locations • Vmin locations • Vmax to Vmin locations

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

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

29. VSWR Calculationsfor a Lossless Terminated T-line • Exercise 4 Given: = 3/5 Find: VSWR = ? Solution: • 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 • 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 • Indicates how much voltage and power will be reflected (or transmitted) at the junction • May be computed from the line impedance (Z0) on the source side and the effective input impedance (ZL = Zin) on the load side

32. Conclusions • The Voltage Standing Wave Ratio (VSWR) for a terminated T-line • Indicates the degreeof 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. • Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.