Electromagnetics: Plane Wave Propagation and Applications

1. EM WavesINEL 4151Ch 10 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR

2. Outline • Plane Wave depending on media • general (lossy), lossless…. • Power in a Wave • Applications and Concepts

3. 10.1 Plane waves

4. The wave equation for E • Start taking the curl of Faraday’s law • Then apply the vectorial identity • And you’re left with Cruz-Pol, Electromagnetics UPRM

5. A Wave • Let’s look at a special case for simplicity • without loosing generality: • The electric field has only an x-component • The field travels in z direction • Then we have Cruz-Pol, Electromagnetics UPRM

6. To change back to time domain • From phasor • …to time domain Cruz-Pol, Electromagnetics UPRM

7. 10.3-10.6 Plane wave Propagation in Several Media

8. Several Cases of Media • Free space • Lossless dielectric • Lossy dielectric (general) • Good Conductor Recall: Permittivity eo=8.854 x 10-12 [F/m] Permeability mo= 4px 10-7 [H/m] Cruz-Pol, Electromagnetics UPRM

9. 1. Free space There are no losses, e.g. Let’s define • The phase of the wave • The angular frequency • Phase constant • The phase velocity of the wave • The period T and wavelength l • In what direction does it moves? Cruz-Pol, Electromagnetics UPRM

10. EM wave Spectrum Cruz-Pol, Electromagnetics UPRM

11. Cruz-Pol, Electromagnetics UPRM

12. Where does it moves? Choose 3 times: Plot: +z Cruz-Pol, Electromagnetics UPRM

13. So… • Wave like this…Moves towards: Cruz-Pol, Electromagnetics UPRM

14. 3. General Case (Lossy Dielectrics) • In general, we had and • From these we obtain • So , for a known material and frequency, we can find g=a+jb Cruz-Pol, Electromagnetics UPRM

15. Intrinsic Impedance, h • If we divide E by H, we get units of ohms and the definition of the intrinsic impedance of a medium at a given frequency. *Not in-phase for a lossy medium Cruz-Pol, Electromagnetics UPRM

16. Note… • E and H areperpendicular to one another • Travel is perpendicular to the direction of propagation • The amplitude is related by the impedance • And so is the phase Cruz-Pol, Electromagnetics UPRM

17. Cruz-Pol, Electromagnetics UPRM

18. Example: Wave Propagation in Lossless materials • A wave in a nonmagnetic material is given by Find: • direction of wave propagation, • wavelength in the material • phase velocity Answer: +y, l=1.26m, up= 2x108 m/s, Cruz-Pol, Electromagnetics UPRM

19. Loss Tangent • If we divide the conduction current by the displacement current Cruz-Pol, Electromagnetics UPRM

20. Relation between tanq and ec Cruz-Pol, Electromagnetics UPRM

21. Summary Cruz-Pol, Electromagnetics UPRM

22. b can also be written as: If we substitute: Cruz-Pol, Electromagnetics UPRM

23. Applet Harmonic oscillator damping http://www.cabrillo.edu/~jmccullough/Applets/Flash/Fluids,%20Oscillations%20and%20Waves/DampedSHM.swf http://www.cabrillo.edu/~jmccullough/Applets/Applets_by_Topic/Oscillations.html Cruz-Pol, Electromagnetics UPRM

24. Review: 1. Free Space • Substituting in the general equations: Cruz-Pol, Electromagnetics UPRM

25. 2. Lossless dielectric • Substituting in the general equations: Cruz-Pol, Electromagnetics UPRM

26. 4. Good Conductors • Substituting in the general equations: Is water a good conductor??? Cruz-Pol, Electromagnetics UPRM

27. Summary Cruz-Pol, Electromagnetics UPRM

28. Example: Wave Propagation in Lossless materials • A wave in a nonmagnetic material is given by Find: • Relative permittivity of material • Electric field phasor 2.25, Cruz-Pol, Electromagnetics UPRM

29. Example • In a free space, • Find Jdand E (use phasors) Cruz-Pol, Electromagnetics UPRM

30. Example: Water Cruz-Pol, Electromagnetics UPRM

31. Permittivity of water vs. freq. http://www.random-science-tools.com/electronics/water_dielectric.htm

32. Example Water @ 60Hz, (low freq) Find a , b and h • For distilled water • For sea water • Attenuation const. • Phase constant

33. Example Water @ 10GHz, (microwaves) Find a , b and h • For distilled water • For sea water • Attenuation const. • Phase constant

34. Exercise: Lossy media propagation For each of the following determine if the material is low-loss dielectric, good conductor, etc. • Glass with mr=1, er=5 and s=10-12 S/m at 10 GHZ • Animal tissue with mr=1, er=12 and s=0.3 S/m at 100 MHZ • Wood with mr=1, er=3 and s=10-4 S/m at 1 kHZ Answer: • low-loss, a= 8.4x10-11 Np/m, b=468 r/m, l= 1.34 cm, up=1.34x108, hc=168 W • general, a=9.75, b=12, l=52 cm, up=0.5x108 m/s, hc=39.5+j31.7 W • Good conductor, a= 6.3x10-4, b= 6.3x10-4, l= 10km,up=0.1x108, hc=6.28(1+j) W Cruz-Pol, Electromagnetics UPRM

35. Today’s class • Skin depth • Short cut!  • Power Cruz-Pol, Electromagnetics UPRM

36. Skin depth, d We know that a wave attenuates in a lossy medium until it vanishes, but how deep does it go? • Is defined as the depth at which the electric amplitude is decreased to 37% Cruz-Pol, Electromagnetics UPRM

37. Microwave Oven Most food are lossy media at microwave frequencies, therefore EM power is lost in the food as heat. • Find depth of penetration for potato which at 2.45 GHz has the complex permittivity given. The power reaches the inside as soon as the oven in turned on! Cruz-Pol, Electromagnetics UPRM

38. Shortcut! • You can use Maxwell’s or use where k is the direction of propagation of the wave, i.e., the direction in which the EM wave is traveling (a unitary vector). Cruz-Pol, Electromagnetics UPRM

39. Waves - Summary • Static charges > static electric field, E • Steady current > static magnetic field, H • Static magnet> static magnetic field, H • Time-varying current > time varying E(t) & H(t) that are interdependent > electromagnetic wave • Time-varying magnet > time varying E(t) & H(t) that are interdependent > electromagnetic wave Cruz-Pol, Electromagnetics UPRM

40. EM waves don’t need a medium to propagate • Sound waves need a medium like air or water to propagate • EM waves don’t. They can travel in free space in the complete absence of matter. Look at a “wind wave”; the energy moves, the plants stay at the same place. Cruz-Pol, Electromagnetics UPRM

41. 10.7-10.8 Power and Poynting Vector

42. Power in a wave • A wave carries power and transmits it wherever it goes The power density per area carried by a wave is given by the Poynting vector. Cruz-Pol, Electromagnetics UPRM

43. Power in a wave • To get Power density in W/m2 carried by an electromagnetic wave: But we need to have a vector that shows where is the power going That’s the Poynting Vector. Cruz-Pol, Electromagnetics UPRM

44. Poynting Vector Derivation • Start with E dot Ampere’s • Apply vector identity • And end up with: Cruz-Pol, Electromagnetics UPRM

45. Rearrange Poynting Vector Derivation… • Substitute Faraday in 1rst term Cruz-Pol, Electromagnetics UPRM

46. Rate of change of stored energy in E or H Ohmic losses due to conduction current Total power across surface of volume Poynting Vector Derivation… • Taking the integral wrt volume • Applying Theorem of Divergence • Which means that the total power coming out of a volume is either due to the electric or magnetic field energy variations or is lost in ohmic losses. Cruz-Pol, Electromagnetics UPRM

47. Power: Poynting Vector • Waves carry energy and information • Poynting says that the net power flowing out of a given volume is = to the decrease in time in energy stored minus the conduction losses. Represents the instantaneous power density vector associated to the electromagnetic wave. Cruz-Pol, Electromagnetics UPRM

48. Bluetooth antenna in car Cruz-Pol, Electromagnetics UPRM

49. Cellphone power radiation Source: AnSys= Fred German Cruz-Pol, Electromagnetics UPRM

50. Time Average Power Density • The Poynting vector averaged in timeis • For the general case wave: Cruz-Pol, Electromagnetics UPRM