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

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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

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The Electromagnetic Spectrum 3

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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

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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

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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

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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

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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).

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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.

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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

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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

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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:

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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

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

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

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

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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:

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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

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

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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

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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.)

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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.

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Lossy Medium (cont.) 23

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

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

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

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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.

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

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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

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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

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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

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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:

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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

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

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

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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.

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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

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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.

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