EECS 430 Wireless Link Design Department of Electrical Engineering and Computer Science The University of Michigan Problem Set #1 (Electromagnetics Review) Due January 30th, 2024 at 10:30 AM Problem 1: The electric field of a uniform plane wave propagating in a dielectric (non-magnetic) medium is given by: Ẽ = 2 cos(10°t – z/V3)x +sin(10°t –z/V3)y (V/m) (a) Determine the frequency and wavelength of the wave. (b) Express the wave as a phasor. (c) What is the dielectric permittivity, and dielectric permeability of the medium? (d) Describe the polarization of the wave. (e) Find the corresponding magnetic field: A Problem 2: A vertically polarized 5MHz plane wave propagates along the horizontal plane in a direction 30° east of north. The magnetic field of the wave is sensed by a horizontal east-west array of 30cm diameter loops. The individual loops have their axes lying in the north-south direction. Determine: (a) The minimum separation of a pair of loops such that the induced EMF in them is always identical (b) The peak value of the electric field, if the RMS induced EMF in the loops is 10 micro volts. Problem 3: (a) Calculate the attenuation of plane waves of 20 KHz and 10 GHz frequency propagating in salt water, fresh water and rock. Express the results in dB/m. The material constants of the media are listed below: Medium Salt water Fresh water Rock Relative Permittivity 80 80 10 Conductivity 4.5 10-² 2x10-3 (b) What is the skin depth in rock at 20 KHz? How does it compare to that at 10 GHz? Problem 4: A circularly polarized wave E= x-jy traveling in the +z direction is incident on an antenna. (a) Is this wave right-hand or left-hand circularly polarized? Why? (b) Find the polarization loss factor if the receiving antenna is linearly polarized: (i) in the x direction: p₁ = x What can you conclude? Problem 5: (a) Derive the reflection and transmission coefficients for a perpendicularly polarized (TE) wave in medium 1 that is incident at an oblique angle onto medium 2. The first medium has material parameters E₁, μ₁, and the second medium has material parameters Ɛ2, M₂. HINT: a simple way to do this is to find the surface impedances of the incident, reflected and transmitted and then treat the problem as a transmission-line problem. wave, The surface impedance is the ratio of the tangential electric field to the tangential magnetic field. 1x1 17x2 - (ii) at 45 degrees from the x-axis: pr = + 9 Îr 2 E 'tan gential H tan ential E H tangential tangential = E¹ H² = Et -jk₁ cos x-jk₁ sin 0₁z Ele H' cose ,−jk, cose.x–jk,sing: Elei HHcose¯jk₂ cosQx− jk₂ sinė;= Eej cos ex Elejki cosex 0x1 -jk₁ cos x-jk₁ sin 0,z El H' cose, E H' cos, Ele jk₂ cos ex 0x2 = 7₁ cos 772 cos, Note: In the diagram above, we didn't write out that the incident, reflected and transmitted waves are multiplied by 2 - jk ₁ sin 0,z jk₁ sin 0,z because according to Snell's law, these three terms are the same. e 2 - jk ₂ sin 0,z (b) Is there a Brewster angle for perpendicular polarization? If so, under what conditions does it occur? Justify your answer. (c) If a right-hand circularly polarized wave hits a very calm, very deep fresh water lake at an angle of 53° from the vertical, what is polarization of the reflected wave? What percentage of the power is lost to the lake? Note: the refractive index of water at light frequencies is 1.33.

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EECS 430 Wireless Link Design Department of Electrical Engineering and Computer Science The University of Michigan Problem Set #1 (Electromagnetics Review) Due January 30th, 2024 at 10:30 AM Problem 1: The electric field of a uniform plane wave propagating in a dielectric (non-magnetic) medium is given by: Ẽ = 2 cos(1