HW4

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University of California, Los Angeles *

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101A

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Electrical Engineering

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Jan 9, 2024

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1 ECE101A Homework 4 1. Consider the plane wave given by the following phasors: ? � (? ) = ? ^ ? 0 ? −𝑗𝑗𝑘? and ? � (?) = ? ^ ? 0 ? −𝑗𝑗𝑘? . The E-field amplitude is ? 0 = 1𝑉/? and ? = 10𝑘?? . a. What direction is this wave propagating? b. If the properties of the medium are: ? = ? 0 , ? = ? 0 , and ? = 0 . Write the time averaged Poynting vector. (Give a number and make sure to get the units right). c. Write an expression for ? and ? phasors (similar to above) for a wave that is propagating in the same direction, but whose ? field direction is 90° different. d. Write an expression for ? and ? phasors (similar to above) for a wave that is propagating in z axis. And the two components of the ? field in x and y directions are with the same amplitude and 90° out of phase. What’s the polarization of ? phasor? Is it different from that of the ? phasor? 2. Consider an EM plane wave propagating in vacuum. Of the total average energy density carried by the wave, what fraction is stored in the electric field, and what fraction is stored in the magnetic field? 3. To investigate the electromagnetic coupling of cellular telephone antennas and a human head, a phantom head – a plastic container filled with a solution that approximately resembles the dielectric and conductive properties of a human head – is used for measurements. In particular, solutions are made that have the complex permittivity equal to the corresponding average head tissue parameters at two frequency bands allocated for wireless communications in North America: (i) ?′ = 44.8? 0 , ?′′ = 0.408 at ? = 850𝑀?? , and (ii) ?′ = 41.9? ?′ , ?′′ = 0.293 at ? = ?′ 1.9𝐺?? . a. Find the attenuation coefficient of a uniform plane wave propagating throughout the phantom solution and complex intrinsic impedance of the material at each of the two wireless communication frequencies. b. If the electric field intensity of the wave at its entry into the solution is 100V/m, use Poynting’s theorem in complex form to determine the time average power absorbed (lost to heat) in the first 1cm of depth into the material per 1cm 2 of cross sectional area at each frequency. 4. Find both the skin depth ( 𝛿𝛿 ? ) and the depth at which the ? field amplitude of a radio wave decreases to 1/1000 of its value at the ocean surface, at each of the frequencies 1kHz, 10kHz, 100kHz, 1MHz, and 10MHz. Take the parameters for seawater to be 0

? ? = 1 , ? ? = 80 , ? = 4𝑆/? . Explain why submarines use ULF (ultra-low- frequency) radio waves at 1kHz to communicate? 5. A 50Ω lossless transmission line is terminated in a load with impedance ? ? = 30 − 𝑗𝑗50 Ω . The wavelength is 8cm. Determine: a. The reflection coefficient at the load. b. The standing-wave ratio on the line. c. The position of the voltage maximum nearest the load. d. The position of the current maximum nearest the load. 6. A 50Ω lossless transmission line is terminated to a load ? ? = 50 + 𝑗𝑗25 Ω . Use the Smith chart to find the following: a. The reflection coefficient Γ. b. The standing-wave ratio. c. The input impedance at 0.35? from the load. d. The input admittance at 0.35? from the load. e. The shortest line length for which the input impedance is purely resistive. f. The position of the first voltage maximum from the load. 7. A 50Ω lossless line is to be matched to an antenna with ? ? = 100 + 𝑗𝑗100 Ω using a shorted stub. Use the Smith chart to determine the stub length and distance between the antenna and stub.

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