UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 2 21 March 2016, 18:00 19:00 Examiner: Prof. Sean V. Hum NAME: STUDENT NUMBER: TOTAL POINTS: 20 Include units in your answers. You can use your Cheng textbook. No other reference materials allowed. All programmable and non-programmable electronic calculators are permitted. You cannot use devices with wireless communications capabilities (e.g. smartphones) as your calculator. Only answers that are fully justified will be given full credit. Unless otherwise stated, assume that all materials are non-magnetic (µ r = 1). GOOD LUCK!
USEFUL FORMULAE Refractive index of non-magnetic dielectrics: n = ɛ r Permittivity of free space: ɛ 0 = 8.854 10 12 F/m Permeability of free space: µ 0 = 4π 10 7 H/m Speed of light in vacuum: c = 3 10 8 m/s Complex permittivity: ɛ c = ɛ jɛ = ɛ j σ ω Wavenumber: k = ω µɛ Complex propagation constant in an unbounded medium: γ = jk = α + jβ Phase constant: β = ω/v p = 2π/λ Intrinsic impedance associated with a plane wave: η = µ ɛ Plane wave in 3D: E(x, y, z) = E 0 e jˆkk r, where k = k x ˆx + k y ŷ + k z ẑ Plane wave magnetic field H = 1 η ˆk E Potential difference V AB = V A V B = A B E dl Maxwell s Equations (time-harmonic form): Integral form Point form S encl D = ρ v B ds = 0 S B = 0 C S E = jωb C encl + jω D ds S H = J + jωd Divergence operator in cylindrical coordinates: Curl operator in cylindrical coordinates: A = ˆρ A = 1 (ρa ρ ) + 1 A φ ρ ρ ρ φ + A z z ( 1 A z ρ φ A ) ( φ + z ˆφ Aρ z A ) z + ẑ 1 ( (ρaφ ) A ) ρ ρ ρ ρ φ
ECE357 Midterm Page 1 PROBLEM #1. [10 POINTS] Sea water has a dielectric constant of ɛ r = 81 and a conductivity of σ = 4 S/m. In this problem we consider the characteristics of a radio frequency (RF) plane wave propagating in sea water. a) Consider a plane wave with a frequency of 20 MHz. Determine the attenuation constant and phase constant associated with the wave. [2 points] b) If the wave travels in the +z-direction, write a time-domain expression for the electric field, E(z, t), if the wave is linearly polarized in the y-direction and the amplitude of the electric field at z = 0 is 100 0 V/m. Evaluate all known constants. [1 point] c) Determine the phase and group velocities of the wave. [2 points]
ECE357 Midterm Page 2 d) Consider a plane wave with a frequency of 20 GHz. Determine the attenuation constant and phase constant associated with the wave. [2 points] e) The wave from part (d) travels in the +z-direction, is linearly polarized in the y-direction, and has electric field at z = 0 of 100 0 V/m. Write phasor expresions for i) the conduction current density, ii) the displacement current density, and iii) the time-average Poynting vector. [3 points]
ECE357 Midterm Page 3 PROBLEM #2. [10 POINTS] An infinitely long coaxial cable is shown in Figure 1. The radius of the inner conductor is a = 1 mm and the radius of the outer conductor is b = 4.95 mm. The space between the two conductors is filled with a homogenous, lossless dielectric with a dielectric constant of 3.6. The cable is excited with a 1 GHz sinusoidal TEM wave travelling in the +z-direction. The electric field in the dielectric region is given by E(ρ, φ, z) = 25 ρ exp( j39.7z) ˆρ V/m, a < ρ < b. Figure 1: Infinitely long coaxial cable a) Write the equations describing E and H in the dielectric region a < ρ < b (the source-free region.) [1 point]
ECE357 Midterm Page 4 b) Using the results of part (a), determine a phasor expression for the magnetic field in the dielectric region a < ρ < b. Show the source-free equations are satisfied with your solution. Evaluate all known constants. [2 points] c) Write a phasor expression for the displacement current density in the dielectric region. Evaluate all known constants. What direction does the displacement current flow? [2 points]
ECE357 Midterm Page 5 d) Write a phasor expression for the current wave on the inner conductor that is associated with the magnetic field. Assume a reasonable function for the magnetic field if you were unable to complete part (a). Evaluate all known constants. [2 points] e) Write a phasor expression for the voltage wave between the inner and outer conductors that is associated with the electric field, using the outer conductor as a voltage reference. Evaluate all known constants. [2 points] turn over
ECE357 Midterm Page 6 f) Using your results derived in part (d) and (e), what is the value of the characteristic impedance associated with this coaxial cable? [1 point]