UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1F ELECTROMAGNETIC FIELDS FINAL EXAM 28 April 15 Examiner: Prof. Sean V. Hum Duration: 2 1 2 hours Calculator Type: 2 All non-programmable electronic calculators. Exam Type: D Closed book examination. Candidates may bring to the examination and use such aids (in the form of printed or written material) as the examiner may specify, in this case, the course textbook by David Cheng. Notes: Include units in your answers. Only answers that are fully justified will be given full credit. You may detach the last three pages which contain formula sheets and Smith Charts. If you detach the Smith Charts, be sure to write your name and ID number on the pages. NAME: STUDENT NUMBER: Problem 1 Problem 2 Problem 3 Problem 4 TOTAL /25 /25 /25 /25 /0
ECE357 Final Exam Page 1 of 12 PROBLEM #1. [25 POINTS] A load consisting of a parallel RC circuit has an impedance of Z L = 12.5 j37.5 Ω at f = GHz ( Hz). The load is to be matched to a characteristic impedance of Z 0 = Ω. a) Determine the values of components realizing the load impedance, R L and C L. Determine the load reflection coefficient Γ L using the Smith Chart and determine its magnitude and phase. [3 points] b) Match the load impedance to Z 0 by designing a shunt short-circuited single-stub matching circuit, specifying the lengths (in wavelengths) and characteristic impedances of all transmission line sections. Choose the solution with the shortest transmission line section connected to the load. Sketch the matching circuit. [7 points]
ECE357 Final Exam Page 2 of 12 c) Match the load impedance to Z 0 by designing a shunt short-circuited double-stub matching circuit, specifying the lengths (in wavelengths) and characteristic impedances of all transmission line sections. The spacing between the stubs is d = λ. Choose the solution such that the stub closest to the load has a positive susceptance. Sketch the matching circuit. [7 points] d) If the matching circuit of part (b) is to be implemented using parallel-plate waveguide operating using the TM 0 mode, specify the dimensions (in mm) of the width of the waveguide (a) and length of the waveguide sections composing the matching sections, if the height of the waveguide is b = 2 mm and the parallel plate waveguide is filled with a dielectric with ɛ r = 4. [4 points]
ECE357 Final Exam Page 3 of 12 e) If the matching circuit of part (b) is to be implemented using rectangular waveguide operating using the TE mode, specify the lengths of the transmission lines used to realize the matching circuit if air-filled WR-90 waveguide is used (a = 22.86 mm, b = 16 mm). [4 points]
ECE357 Final Exam Page 4 of 12 PROBLEM #2. [25 POINTS] A 1 GHz plane wave propagating in air is incident upon a slab of dielectric with ɛ r = 3, as shown in Figure 1. The angle of incidence is θ i = 60. Figure 1: Plane wave incident upon a dielectric slab a) Determine the angles of reflection (θ r ) and refraction (θ t ), and the corresponding vector wavenumbers k r and k t for the reflected and refracted waves, respectively. [5 points] b) Determine the reflection and transmission coefficients across the media interface, if the incident wave is perpendicularly-polarized (TE polarization). What percentage of power is transmitted into the dielectric? [5 points]
ECE357 Final Exam Page 5 of 12 c) Write complete time-domain expressions for the electric field in i) the air region, z > 0 and ii) the dielectric region, z < 0. Let the amplitude of electric field be E 0. [5 points] d) If the incident wave is now right-hand circularly-polarized (RHCP), write a phasor expression for the incident wave. [4 points]
ECE357 Final Exam Page 6 of 12 e) Determine the polarization and axial ratio (AR) of the reflected wave if the incident wave is RHCP as in part (d). [6 points]
ECE357 Final Exam Page 7 of 12 PROBLEM #3. [25 POINTS] A 3 m long coaxial transmission line (l 1 = 3 m) with Z 01 = 0 Ω is connected to a generator with R g = Ω and another coaxial transmission line, as shown in Figure 2. The second transmission line has Z 02 = Ω, is 3 m long (l 2 = 3 m), and is terminated with R L = Ω. The generator launches a 2 ns pulse onto the first transmission line at t = 0. The wave velocity on the first transmission line is u 1 = 1.5 8 m/s, while the wave velocity on the second transmission line is u 2 = 3 8 m/s. V 0 = 13.5 V. + Figure 2: Transmission line setup for transient analysis a) Determine the one-way transit times through each of the transmission lines on their own. Determine the values of the dielectric constants of the dielectrics used for both transmission lines. [4 points]
ECE357 Final Exam Page 8 of 12 b) Plot the voltage observed at the middle of the first transmission line as a function of time, v(z = 1.5 m, t) for 0 t 1 ns. [6 points] c) Determine the amplitude of the first pulse launched onto the second transmission line, sketching the equivalent circuit used to determine this. [4 points]
ECE357 Final Exam Page 9 of 12 d) Plot the voltage observed at the middle of the second transmission line as a function of time, v(z = 4.5 m, t) for 0 t 1 ns. [6 points]
ECE357 Final Exam Page of 12 e) Plane waves can also carry pulses in the same way transmission lines do (e.g. a RADAR pulse). Reflections are generated by discontinuities with planar media. Based on everything you know about plane wave behaviour, sketch an equivalent system of planar materials that produces the same transient response as the transmission line above, specifying the dielectric constants and lengths of dielectric slab(s) used. Assume the wave is first launched into an air-filled region. [5 points]
ECE357 Final Exam Page 11 of 12 PROBLEM #4. [25 POINTS] A 1 GHz plane wave propagates in a lossy medium with ɛ r = 4 and σ = S/m. The wave propagates in the +x direction and is linearly-polarized in the y-direction. The electric field has an amplitude of V/m. a) Determine the attenuation and phase constant of the medium. Write a time-domain expression for the electric field. Reminder: Ae jφ = Ae jφ/2 if A R. [5 points] b) Determine the phase velocity in the medium. How does it compare to the case if the medium was lossless? [3 points] c) Calculate the wave impedance of the medium and determine a phasor expression for the magnetic field. Describe the temporal and spatial relationship between E and H in the medium. [5 points]
ECE357 Final Exam Page 12 of 12 d) How far does the plane wave propagate in the medium before its amplitude is reduced by db? [3 points] e) Determine vector, phasor expressions for the displacement current density and conduction current density. [4 points] f) If σ ωɛ, determine an expression for the group velocity of the medium in terms of ω, σ, ɛ 0, and ɛ r. Does the medium exhibit no dispersion, normal dispersion, or anomalous dispersion? Evaluate the group velocity at f = 1 GHz if σ = S/m. [5 points]
7 3 The Complete Smith Chart Black Magic Design 0.0 0.0 9 8 9 8 170 ± 180-170 7 > WAVELENGTHS TOWARD GENERATOR > < WAVELENGTHS TOWARD LOAD < 7 0.04 6 160-160 90-90 6 0.04 85-85 0.05 5 1-1 80-80 5 0.05 140 75-75 0.06-140 0.06 4 70-70 0.07 0.07 3-130 130 0.08 2 65-65 2 0.08 1-1 60-60 1 0.09 0.09 1 1-1 INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) 4 55-55 1 9 1 0-0 - 2 8 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) 3 9 8 2 90-90 45-45 3 7 1.2 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 1.2-80 4 6 80 40-40 4 6 1.4 1.4 5 5 5 5 70-35 -70 35 1.6 1.6 6 4 4 6 1.8 1.8 60-30 -60 30 7 3 2.0 2.0 3 7 25-25 8 2-8 2 3.0 9 1 3.0 40 - -40 1 9 4.0 4.0 15 5.0 5.0-15 30-30 9-1 1 ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES 8-9 2 ANGLE OF REFLECTION COEFFICIENT IN DEGREES 2 7 3 8 4 6 3 5 5 6 4 7 SWR dbs RTN. LOSS [db] RFL. COEFF, P RFL. COEFF, E or I 040 40 30 15 5 4 3 0 1 2 3 4 5 6 7 8 9 12 14 30 0 1 1 0.0 2.5 8 2 6 1.8 5 1.6 4 0.05 RADIALLY SCALED PARAMETERS TOWARD LOAD > 1.4 1.2 1.1 1 15 7 5 3 2 0.01 1 4 < TOWARD GENERATOR 2 1 1 1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 3 1 1.5 2 3 4 5 6 15 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 0 1 9 5 0 CENTER 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ATTEN. [db] S.W. LOSS COEFF RFL. LOSS [db] S.W. PEAK (CONST. P) TRANSM. COEFF, P TRANSM. COEFF, E or I ORIGIN
7 3 The Complete Smith Chart Black Magic Design 0.0 0.0 9 8 9 8 170 ± 180-170 7 > WAVELENGTHS TOWARD GENERATOR > < WAVELENGTHS TOWARD LOAD < 7 0.04 6 160-160 90-90 6 0.04 85-85 0.05 5 1-1 80-80 5 0.05 140 75-75 0.06-140 0.06 4 70-70 0.07 0.07 3-130 130 0.08 2 65-65 2 0.08 1-1 60-60 1 0.09 0.09 1 1-1 INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) 4 55-55 1 9 1 0-0 - 2 8 RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) 3 9 8 2 90-90 45-45 3 7 1.2 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 1.2-80 4 6 80 40-40 4 6 1.4 1.4 5 5 5 5 70-35 -70 35 1.6 1.6 6 4 4 6 1.8 1.8 60-30 -60 30 7 3 2.0 2.0 3 7 25-25 8 2-8 2 3.0 9 1 3.0 40 - -40 1 9 4.0 4.0 15 5.0 5.0-15 30-30 9-1 1 ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES 8-9 2 ANGLE OF REFLECTION COEFFICIENT IN DEGREES 2 7 3 8 4 6 3 5 5 6 4 7 SWR dbs RTN. LOSS [db] RFL. COEFF, P RFL. COEFF, E or I 040 40 30 15 5 4 3 0 1 2 3 4 5 6 7 8 9 12 14 30 0 1 1 0.0 2.5 8 2 6 1.8 5 1.6 4 0.05 RADIALLY SCALED PARAMETERS TOWARD LOAD > 1.4 1.2 1.1 1 15 7 5 3 2 0.01 1 4 < TOWARD GENERATOR 2 1 1 1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 3 1 1.5 2 3 4 5 6 15 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 0 1 9 5 0 CENTER 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ATTEN. [db] S.W. LOSS COEFF RFL. LOSS [db] S.W. PEAK (CONST. P) TRANSM. COEFF, P TRANSM. COEFF, E or I ORIGIN
USEFUL FORMULAE Permittivity of free space: ɛ 0 = 8.8541878176 12 F/m Permeability of free space: µ 0 = 4π 7 H/m Speed of light in vacuum: c = 3 8 m/s Complex permittivity: ɛ c = ɛ jɛ = ɛ j σ = ω ɛ (1 j tan δ) Refractive index of non-magnetic dielectrics: n = ɛ r 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: η = µ ɛ Load reflection coefficient: Γ L = Z L Z 0 Z L +Z 0 Input reflection coefficient: Γ = Γ L e j2βl e 2αl Standing wave ratio: S = 1+ Γ 1 Γ Admittance of an open stub: Y in = jy 0 tan(βl) Impedance of a shorted stub: Z in = jz 0 tan(βl) Impedance transformation: Z in = Z 0 1+Γ(l) 1 Γ(l) Curl operator F = Constitutive relations in simple media: ( Fz y F ) ( y Fx â x + z z F ) ( z Fy â y + x x F ) x â z y D = ɛe B = µh J = σe Time-average Poynting vector: S = 1 2 Re(E H ) Group velocity v g = ω β = ( β ω ) 1
Integral form Point form D ds = Q S encl D = ρ v B ds = 0 B = 0 S E dl = d B ds E = B C dt S t H dl = I C encl + D ds H = J + D S t t Table 1: Maxwell s equations Fresnel reflection / transmission coefficients Snell s Law of refraction Γ = η 2 cos θ t η 1 cos θ i η 2 cos θ t + η 1 cos θ i 2η 2 cos θ i T = η 2 cos θ t + η 1 cos θ i Γ = η 2 cos θ i η 1 cos θ t η 2 cos θ i + η 1 cos θ t 2η 2 cos θ i T = η 2 cos θ i + η 1 cos θ t sin θt sin θ i = ɛ1 ɛ 2 = n 1 n 2 = η 2 η 1 Brewster angles sin θ B = 1 µ1 ɛ 2 /µ 2 ɛ 1 1 (µ 1 /µ 2 ; sin θ ) 2 B = 1 µ2 ɛ 1 /µ 1 ɛ 2 1 (ɛ 1 /ɛ 2 ) 2 Key parameters of parallel-plate waveguide Quantity TEM mode TM n mode TE n mode h 0 nπ/b nπ/b β k = ω µɛ k2 h 2 k2 h 2 λ c 2π/h = 2b/n 2π/h = 2b/n λ g 2π/k 2π/β 2π/β v p ω/k = 1/ µɛ ω/β ω/β E z (y, z) 0 A n sin(nπy/b)e jβz 0 H z (y, z) 0 0 B n cos(nπy/b)e jβz E x (y, z) 0 0 (jωµ/h)b n sin(nπy/b)e jβz E y (y, z) E 0 /be jβz ( jβ/h)a n cos(nπy/b)e jβz 0 H x (y, z) E 0 /(ηb)e jβz (jωɛ/h)a n cos(nπy/b)e jβz 0 H y (y, z) 0 0 (jβ/h)b n sin(nπy/b)e jβz Z Z T EM = ηb/w Z T M = βη/k Z T E = kη/β
Key parameters of rectangular waveguide Quantity TE mn mode TM mn mode h ( mπ a )2 + ( nπ b )2 ( mπ a )2 + ( nπ b )2 β k2 h 2 k2 h 2 λ c 2π/h 2π/h λ g 2π/β 2π/β v p ω/β ω/β E z (x, y, z) 0 B mn sin( mπx nπy ) sin( a b )e jβz H z (x, y, z) A mn cos( mπx nπy ) cos( a b )e jβz 0 jωµnπ E x (x, y, z) A h 2 b mn cos( mπx nπy ) sin( a b )e jβz jβmπ h 2 a mn cos( mπx nπy ) sin( a b E y (x, y, z) jωµmπ h 2 a mn sin( mπx nπy ) cos( a b )e jβz jβnπ h 2 b mn sin( mπx nπy ) cos( a b jβmπ H x (x, y, z) h 2 a mn sin( mπx nπy jωɛnπ ) cos( )e jβz a b h 2 b mn sin( mπx) a b jβnπ H y (x, y, z) h 2 b mn cos( mπx nπy ) sin( a b )e jβz jωɛmπ h 2 a mn cos( mπx nπy ) sin( a b Z Z T E = kη/β Z T M = βη/k )e jβz )e jβz )e jβz )e jβz