Prof. Ji Chen Notes 15 Plane Waves ECE Spring 2014 z x E ocean

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. x z E H S 2

The Electromagnetic Spectrum 3

SourceFrequencyWavelength U.S. AC Power 60 Hz 5000 km ELF Submarine Communications500 Hz 600 km AM radio (KTRH)740 kHz405 m TV ch. 2 (VHF) 60 MHz5 m FM radio (Sunny 99.1)99.1 MHz3 m TV ch. 8 (VHF)180 MHz1.7 m TV ch. 39 (UHF)620 MHz48 cm Cell phone (PCS)850 MHz35 cm Cell Phone (PCS 1900)1.95 GHz15 cm μ-wave oven2.45 GHz12 cm Police radar (X-band)10.5 GHz2.85 cm mm wave100 GHz3 mm Light 5  [Hz]0.60  m X-ray10 18 [Hz]3 Å The Electromagnetic Spectrum (cont.) 4

TV and Radio Spectrum VHF TV: MHz (channels 2-13) Band I : MHz (channels 2-6) Band III: MHz (channels 7-13) FM Radio: (Band II) MHz UHF TV: MHz (channels 14-69) AM Radio: kHz Note: Digital TV broadcast takes place primarily in UHF and VHF Band III. 5

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

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

Vector Helmholtz Equation Recall the vector Laplacian identity: Hence we have Also, from the divergence of the vector wave equation, we have: 8 Note: There can be no charge density in the sinusoidal steady state, in free space (or actually in any homogenous region of space).

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 Vector Helmholtz Equation (cont.) Scalar Helmholtz equation 9 Reminder: This only works in rectangular coordinates.

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

where For a plane wave traveling in a lossless dielectric medium (does not have to be free space): (wavenumber of dielectric medium) Plane Wave Field (cont.) 11

The electric field behaves exactly as does the voltage on a transmission line. Notational change: Plane Wave Field (cont.) 12 All of the formulas that we had for a transmission line now hold for a transmission line, using this notation change. Example:

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. Plane Wave Field (cont.) A wave travels with the same velocity on a transmission line as it does in space, provided the material is the same. 13 For a plane wave, we have: same

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

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

Poynting Vector The complex Poynting vector is given by: Hence we have 16 (no VARS) For a wave traveling in a lossless dielectric medium:

Phase Velocity From our previous discussion on phase velocity for transmission lines, we know that Hence we have (speed of light in the dielectric material) so Notes:  All plane wave 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 distortion of the signal. 17 Time domain: Phasor domain:

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

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

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

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

Examine the wavenumber: Denote: Using the principal branch of the square root, the wavenumber k lies in the fourth quadrant. Hence: Compare with lossy TL: 22 Principal branch: Note: The complex effective permittivity lies in the fourth quadrant.

Lossy Medium (cont.) 23

Lossy Medium (cont.) The “depth of penetration” d p is defined. 24 (choose E 0 = 1 )

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

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

Example (cont.) f d p [m] 1 [Hz] [Hz] [Hz] [kHz] [kHz] [kHz] [MHz] [MHz] [MHz] [GHz] [GHz] [GHz] The depth of penetration into ocean water is shown for various frequencies. 27 Note: The relative permittivity of water starts changing at very high frequencies (above about 2 GHz), but this is ignored here.

Loss Tangent Denote: The loss tangent is defined as: We then have: 28 The loss tangent characterizes how lossy the material is.

Loss Tangent The loss tangent characterizes how fast the plane wave decays relative to a wavelength. tan  << 1: low-loss medium (attenuation is small over a wavelength) tan  >> 1: high-loss medium (attenuation is large over a wavelength) 29

Low-Loss Limit: tan  << 1 Low-loss limit: The wavenumber may be written in terms of the loss tangent as (independent of frequency) or 30

f d p [m] tan  1 [Hz]  [Hz]  [Hz]  [kHz]  [kHz]  [kHz]  [MHz] [MHz] [MHz] [GHz] [GHz] [GHz] Ocean water 31 Low-Loss Limit: tan  << 1 Low-loss region

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 it heats up well in a microwave oven!). 32 (complex permittivity) Notation:

Polarization Loss (cont.) Note: In practice, it is usually difficult to determine how much of the loss tangent comes from conductivity and how much comes from polarization loss. 33

Summary of Permittivity Formulas Note: If there is no polarization loss, then 34

Polarization Loss (cont.) 35 Complex relative permittivity for pure (distilled) water

Polarization Loss (cont.) For modeling purposes, we can lump the polarization losses into an effective conductivity term and pretend that the permittivity is real: where 36 Note: In most measurements involving heating, we cannot tell if loss is really due to conductivity or polarization loss.

Polarization Loss (cont.) Practical note: 37  For some materials, it is the conductivity that is approximately constant with frequency. Ocean water:   4 [S/m]  For other materials, it is the loss tangent that is approximately constant with frequency. Teflon: tan   0.001

Note on Notation 38 They often use only the notation . Be careful: Most books do not use the notation  c. Our notation: Usual notation:  Sometimes it means   Sometimes it means  c.