Prof. D. R. Wilton

1. ECE 3317 Prof. D. R. Wilton Notes 15 Plane Waves [Chapter 3]

2. Introduction to Plane Waves • A plane wave is the simplest solution to Maxwell’s equations for a wave that travels through free space. • The wave does not requires any conductors – it exists in free space. • A plane wave is a good model for radiation from an antenna, if we are far enough away from the antenna. E x S z H

3. The Electromagnetic Spectrum 1 fm 1 pm 1 PHz 1 THz 1 mm 1 GHz 1 MHz 1 kHz http://www.impression5.org/solarenergy/misc/emspectrum.html

4. The Electromagnetic Spectrum (cont.)

5. TV and Radio Spectrum AM Radio: 520-1610 kHz VHF TV: 55-216 MHz (channels 2-13) Band I : 55-88 MHz (channels 2-6) Band III: 175-216 MHz (channels 7-13) FM Radio: (Band II) 88-108 MHz UHF TV: 470-806 MHz (channels 14-69) Note: Digital TV broadcast takes place primarily in UHF and VHF Band III.

6. Vector Wave Equation Start with Maxwell’s equations in the phasor domain: Faraday’s law Ampere’s law Assume free space: Ohm’s law: We then have

7. Vector Wave Equation (cont.) Take the curl of the first equation and then substitute the second into the first: Define wavenumber of free space [rad/m] Hence Vector Wave Equation

8. Vector Helmholtz Equation Recall the vector Laplacian identity: Hence we have Also, from the divergence of the vector wave equation, we have: Solutions of automatically satisfy Gauss’s law

9. Vector Helmholtz Equation (cont.) Hence we have: vector Helmholtz equation Recall the property of the vector Laplacian in rectangular coordinates: Taking the x component of the vector Helmholtz equation, we have scalar Helmholtz equation

10. Plane Wave Field Assume z The electric field is polarized in the x direction, and the wave is propagating (traveling) in the z direction. y E Then x or For wave traveling in the negative z direction: Solution:

11. Plane Wave Field (cont.) Suppose we had assumed z Hence the electric field would be both polarized and vary along the z direction. Then y E x or Solution: • Plane waves are polarized transverse to the propagation direction • Longitudinally polarized plane waves are not possible.

12. Plane Wave Field (cont.) For a wave traveling in a lossless dielectric medium: or where wavenumber of dielectric medium

13. Plane Wave Field (cont.) The electric field behaves exactly as does the voltage on a transmission line. Notational change: Note: for a lossless transmission line, we have: A transmission line filled with a dielectric material has the same wavenumber as does a plane wave traveling through the same material. A wave travels with the same velocity on a transmission line as it does in space, provided the material is the same.

14. Plane Wave Field (cont.) The H field is found from: so

15. Intrinsic Impedance or Hence where Intrinsic impedance of the medium Free-space:

16. Poynting Vector The complex Poynting vector is given by: Hence we have

17. Phase Velocity From our previous discussion on phase velocity for transmission lines, we know that Hence we have so (speed of light in the dielectric material) Note: All plane waves travel at the same speed in a lossless medium, regardless of the frequency. This implies that there is no dispersion, which in turn implies that there is no signal distortion.

18. Wavelength From our previous discussion on wavelength for transmission lines, we know that Free space: Also, we can write Free space: Hence

19. Summary (Lossless Case) z S y H E x Time domain: Denote

20. Lossy Medium Return to Maxwell’s equations: E x ocean Assume Ohm’s law: z Ampere’s law We define an effective (complex) permittivity cthat accounts for conductivity:

21. Lossy Medium Maxwell’s equations then become: The form is exactly the same as we had for the lossless case, with Hence we have (complex) (complex)

22. Lossy Medium (cont.) Examine the wavenumber: Denote: Note! Note! In order to ensure decay, the wavenumber k must be in the fourth quadrant. Hence: Compare with lossy TL:

23. Lossy Medium (cont.)

24. Lossy Medium (cont.) The “depth of penetration” dpis defined.

25. Lossy Medium (cont.) The complex Poynting vector is:

26. Example Ocean water: (These values are fairly constant up through microwave frequencies.) Assume f = 2.0 GHz

27. Example (cont.) The depth of penetration into the ocean water is shown for various frequencies. fdp[m] 1 [Hz] 251.6 10 [Hz] 79.6 100 [Hz] 25.2 1 [kHz] 7.96 10 [kHz] 2.52 100 [kHz] 0.796 1 [MHz] 0.262 10 [MHz] 0.080 100 [MHz] 0.0262 1.0 [GHz] 0.013 10.0 [GHz] 0.012 100 [GHz] 0.012

28. Loss Tangent Denote: The loss tangent is defined as: We then have: The loss tangent characterizes the nature of the material: tan << 1: low-loss medium (attenuation is small over a wavelength) tan >> 1: high-loss medium (attenuation is large over a wavelength) Note:

29. Low-Loss Limit The wavenumber may be written in terms of the loss tangent as Low-loss limit: (independent of frequency) or

30. Polarization Loss The permittivity  can also be complex, due to molecular and atomic polarization loss (friction at the molecular and atomic levels) . Example: distilled water:   0 (but heats up well in a microwave oven!). or Note: In practice, it is usually difficult to distinguish how much of the loss tangent comes from conductivity and how much comes from polarization loss.

31. Polarization Loss (cont.) Regardless of where the loss comes from, (conductivity or polarization loss), we can write or where Note: If there is no polarization loss, then

32. Polarization Loss (cont.) For modeling purposes, we can lump all of the losses into an effective conductivity term: where