ece320f_2010_exam - UNIVERSITY OF TORONTO FACULTY OF...

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Unformatted text preview: UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE 320H1F FIELDS AND WAVES FINAL EXAM 20 December 2010 Examiners: Prof. George V. Eleftheriades, Prof. Sean V. Hum Duration: 2% hours Calculator Type: 1 - All programmable and non-programmable electronic calculators are permit- ted. Exam Type: C - Closed book examination. Students may bring a single aid sheet on a standard form supplied by the examiner (obtained from the Office of the Registrar). Notes: Include units in your answers. Only answers that are fully justified will be given full credit. You may detach the last two pages which contain a formula sheet and Smith Chart. If you detach the Smith Chart, be sure to write your name and ID number on the page. NAME: STUDENT NUMBER: Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 TOTAL ECE320 Final Exam Page 1 of 15 PROBLEM #1. [10 POINTS] Two sections of air-filled transmission lines are cascaded as shown in Figure 1. The lines of physical lengths given by 21 = 12 m and £2 = 15 m and characteristic impedances are given by Z01 = 50 Q and Z02 = 120 Q. The first transmission line is driven by a step function generator with a source impedance of R9 = 50 Q. The leading edge of the step is generated at time t = 0, and the amplitude is 10 V. The second transmission line is terminated in a resistance RL = 30 9. 2:0 {2:31 12:32 Figure 1: Transmission line setup a) Determine the amplitude of the initial step function launched onto the first transmission line and the time it takes the signal to travel to the input of the second transmission line. Determine the amplitude of the step function launched onto the second line. [2 points] ECE320 Final Exam Page 2 of 15 b) Plot the voltage as a function of time at a position 2 = 19.5 m, i.e., at a point halfway along the second transmission line, for 0 g t g 265 ns. Indicate all time values associated with voltage transitions and amplitudes of all signals. [4 points] ECE320 Final Exam Page 3 of 15 c) Repeat (b) if the load resistor RL = Z02. [2 points] ECE320 Final Exam Page 4 of 15 d) Plot the voltage as a function of time at a position 2 = 6 m, i.e., at a point halfway along the first transmission line, for 0 g t g 265 ns, if the load resistor RL = Z02 as in part (c). Indicate all time values associated with voltage transitions and amplitudes of all signals. [2 points] ECE320 Final Exam Page 5 of 15 PROBLEM #2. [10 POINTS] A plane wave is propagating in a linear, lossless, isotropic, homogeneous medium with e = 460, and p = ,uo. The electric field associated with this wave is given by E(:1:, y, z,t) = 7cos(47r - 1010t + 18256 — 230g — kzz):i: +3 cos(47r - 1010:: + 182:1: — 230g — 1922);) —9 cos(47r - 10101: + 182:1: — 230g — kzz)z“ [V/m]. a) What is the phase velocity in the medium? [1 mark] b) What does the value(s) of kZ need to be for this wave to satisfy the vector wave equation? [2 marks] ECE320 Final Exam Page 6 of 15 c) What is the unit vector 6N describing the direction(s) of propagation of the plane wave? [1 mark] (1) Write a real-time expression for the magnetic field H (as, y, z, t) in the medium. [2 marks] ECE320 Final Exam Page 7 of 15 e) Let the wave be propagating in the half-space defined by z < 0 and 1974 > 0. If the space in the region 2 > 0 is filled with a dielectric with e, = 6, determine the amplitude and phase of the reflected components of the electric field, Em, Eng, and Em. Assume kz = +800 m‘1 if you were unable to solve part (b). Write down a unit vector describing the direction the reflected wave propagates. [4 marks] ECE320 Final Exam Page 8 of 15 ECE320 Final Exam Page 9 of 15 PROBLEM #3. [10 POINTS] Consider an infinitely long hollow rectangular waveguide of dimensions a = 7 cm and b = 3 cm as shown in Figure 2. N4 Figure 2: Infinitely long rectangular waveguide a) Calculate the cutoff frequencies of the first four TE modes, in ascending order. [2 marks] ECE320 Final Exam Page 10 of 15 Assuming that only the TElo propagates (along the +2 direction) in the guide at f = 3 GHz and that the magnitude of the electric field at a: = a/ 2 is 300 V/m, determine the following (use phasors): b) the propagation constant fl along 2. [1 mark] c) the angle 0 (with respect to the normal) at which the wave bounces off the as = 0 and a: = a walls of the guide. [1 mark] d) the ratio between the electric field E, and the magnetic field Hm. [1.5 marks] ECE320 Final Exam Page 11 of 15 e) an expression for the Poynting vector Sz(m, [2 marks] f) the total power that travels down the guide. [2.5 marks] ECE320 Final Exam Page 12 of 15 ECE320 Final Exam Page 13 of 15 PROBLEM #4. [10 POINTS] Consider the following transmission—line circuit. All lines are air filled and of characteristic impedance equal to. 50 Q. The frequency of operation is f = 3 GHz. Note: use impedance coordinates throughout this problem. R=62.5 Ohm L=2.21nH Za Figure 3: Transmission line setup a) Place the load admittance yL on the Smith Chart. [1.5 marks] b) Place the load impedance 2L on the Smith Chart [1 mark] 0) Plot the corresponding constant VSWR circle. [1.5 marks] Now we want to match the input line to this load Z L . Using the Smith Chart, determine, e) The impedance Z, (see Figure) for the minimum possible distance d. [2 marks] f) The corresponding minimum distance d in mm. [2 marks] g) The required capacitance 02 in pF. [2 marks] ECE320 Final Exam Page 14 of 15 ECE320 Final Exam Page 15 of 15 PROBLEM #5. [10 POINTS] Please circle true or false in the following statements. [1 mark each] a) The induced emf in a closed circuit produces a magnetic flux that reinforces the existing magnetic flux. T F b) At any frequency there can be no magnetic field in a parallel plate capacitor. T F c) To generate an elliptical polarized wave propagating in the +2 direction requires that the amplitude of the z and y components of the electric field be different. T F d) The normal component of the electric field at a dielectric interface is always continuous. T F c) It is possible to produce a wavelength inside an air-filled waveguide shorter than the corre- sponding wavelength in free space. T F f) If an air-filled coaxial cable is filled with a high permittivity dielectric then both the capaci- tance and the inductance per unit length will be affected. T F g) In a parallel-plate waveguide there can be an electric field component that is perpendicular to the planes of constant phase. T F h) The phase velocity of a plane wave depends on the direction in which we measure it. T F i) In a parallel-plate waveguide filled with a linear medium, the phase constant fl is always a linear function of the angular frequency w. T F j) The reflection coefficient at an arbitrary dielectric interface can vanish under suitable polar- ization and angle of incidence conditions. T F The Complete Smith Chart Black Magic Design Q. “ .5: EH: .:::.: n IEHEEEEEEflIEflfil-n I“um ah ====== . v 6 $9.11. .3. .’ u. h. \ '. a ’ ‘ '. ..v,' . :. . -- 9’9 0’! RADIALLY SCALED PARAMETERS TOWARDLOAD—> 1 1.8 1.6 1.4 1.21.11 15 10 7 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 an . 00%? o 2 3 4 5 6 7 a 9 10 12 14 0.1 0.2 0.4 05 0.3 1 1.5 2 a 4 5 6 1015~ ' (9/ (“no 4.“ 1 0.9 0.3 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.01 1.1 1.2 1.3 1.4 15 1.6 1.71.013 2 25 3 69/ "'21 0.7 0.6 0.5 0.4 0.3 0.2 0.99 0.95 0.9 048 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ORIGIN USEFUL FORMULAE o Permittivity of free space: 60 = 8.8541878176 x 10‘12 F/m o Permeability of free space: Mo = 47r x 10~7 H/m 0 Speed of light in vacuum: c = 3 x 108 m/s 0 Complex permittivity: so = e’ — je" = e — j 5 = e’(1 — 3' tan 6) o Refractive index of non-magnetic dielectrics: n = fl 0 Wavenumber: k = wfi 0 Complex propagation constant in an unbounded medium: 7 = jk = a: + jfl a Phase constant: [3 = (.u/vp = 27r/A o Intrinsic impedance: n = \/E E 0 Load reflection coefficient: I‘L = 2—122 ZL+Zo 0 Input reflection coefficient: I‘ = I‘Le_j2fiee_2ae 0 Standing wave ratio: S = 1—3;: 0 Admittance of an open stub: Yin = on tan(,6€) o Impedance of a shorted stub: Zm = jZo tan(fl€) 1+I‘(£) o Impedance transformation: Zin = Z014“) o Maxwell’s Equations: Integral form Point form fSD'dS=Qencl V'D=pv fSB-dS=0 V-B=0 fCE-dL=—% SB-dS VxE=—%’fi fCH.dL=Imd+fS%-ds VxH=J+%—Lt’ o Constitutive relations in simple media: 6E uH 0E Hwb nu o Time-average Poynting vector: S = -;-Re(E x H *) 0 Group velocity '09 = 3—3 = (3—5:)—1 . Fresnel reflection / transmission coefficients 772 cos 0t — 771 cos 0;- .5 || 172 cos 0t + m cos 01- 2172 cos 0;- TH = —9__——'_ 172 cos t + 771 cos 01 772 cos 0,- — 771 cos 0t 11 = ———— n2 cos 0i + 771 cos 0t T _ 2772/ cos 0; 'L _ 772/ cos0t + 711/ 00801- . Snell’s Law of refraction zi—E—gf = :2 = if: = 377% o Parallel-plate waveguide Quantity TEM mode TMn mode TEn mode h 0 mr / b mr/ b fl k = am? we2 — h2 x/k2 — M AC 00 27r/h = 2b/n 27r/h = 2b/n Ag 27r/k 271' / ,6 27r/,8 up w/k= l/fl w/fl w/fl E; (y, z) 0 An sin(mry/b)e‘jfiz 0 Hz (y, z) 0 0 Bn cos(mry/b)e‘jfiZ EE( ,2) 0 0 (jean/MB" sin(n1ry/b)e‘jflz E, (y, z) —Eo/be‘jpz (—j,8/h,)An cos(n7ry/b)e‘jflz 0 Hm(y, z) Eo/(nb)e‘jpz (jams/IMAn cos(mry/b)e_mz 0 ' Hy (y, z) 0 0 (jfl/h)Bn sin('n7ry/b)e‘jfiz Z ZTE‘M = 775/111 ZTM = 577/19 ZTE = 7977/5 o Rectangular waveguide Quantity TEmn mode Tan mode h ($2 + ($2 ($)2 + ($2 fl ‘/k_2—_ hz 1062 _ hz Ac 27r/h 27r/h )‘g 277/3 ZW/fl up w/IB w/fl . Ez(a:, y, z) 0 an QM?) sin(%)e“9fiz Hz(:z;, y, z) Amn cos(m;””) cos(%u)e‘jflz 0 I Eda), y, z) Egg—“Am” cod-7’?) sin(%9)e‘jfiz' wig—2% mn cod?) sin(2%u)e"-jfiz Ey(x, y, z) fw,{‘2—T:1Amn Sin(%) cos(fl%'9)e'7fiz thT’Z'an sin(m:”) cos(%u)e‘mz Hz m,y,z LZWAWL sin “7”” cos m e’jfiz Lin"an sin m” cos m e‘jfiz _ a a b I h.b a. b _ Hy(w, y, z) hgb" Amn cos(m;”‘) sin(%3)e_wz —’%an cos(—% sin $1561!” Z ZTE = kn/fi ZTM = 1617/ k ...
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This note was uploaded on 01/29/2012 for the course ECE ECE320 taught by Professor Elefthra during the Fall '08 term at University of Toronto- Toronto.

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ece320f_2010_exam - UNIVERSITY OF TORONTO FACULTY OF...

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