notes 15 3317.pptx - ECE 3317 Prof Ji Chen Spring 2019 Notes 15 Plane Waves E x ocean z 1 Introduction to Plane Waves A plane wave is the simplest

notes 15 3317.pptx - ECE 3317 Prof Ji Chen Spring 2019...

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Prof. Ji Chen Notes 15 Plane Waves ECE 3317 1 Spring 2019 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 S E H
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The Electromagnetic Spectrum 3 0 / c f
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Source Frequency Wavelength U.S. AC Power 60 Hz 5000 km ELF Submarine Communications 500 Hz 600 km AM radio (KTRH) 740 kHz 405 m TV ch. 2 (VHF) 60 MHz 5 m FM radio (Sunny 99.1) 99.1 MHz 3 m TV ch. 8 (VHF) 180 MHz 1.7 m TV ch. 39 (UHF) 620 MHz 48 cm Cell phone (PCS) 850 MHz 35 cm Cell Phone (PCS 1900) 1.95 GHz 15 cm μ- wave oven 2.45 GHz 12 cm Police radar (X-band) 10.5 GHz 2.85 cm mm wave 100 GHz 3 mm Light 5 10 14 [Hz] 0.60 m X-ray 10 18 [Hz] 3 Å The Electromagnetic Spectrum (cont.) 4 0 / c f
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TV and Radio Spectrum 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) AM Radio: 520-1610 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 E H H J E j j      J E = 0 0 0 E H H E j j      0 0 ,
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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 0 0 0 E H E j j j         0 0 0 k   2 0 E E 0 k   0 0 E H H E j j     
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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). 2 0 E E 0 k   2 V V V      2 2 0 E E E 0 k     2 0 E E 0 k     E 0  0 0 1 1 E D 0 v   2 0 E E 0 k  
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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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