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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 nonprogrammable 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 airﬁlled 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 realtime 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 halfspace 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 airﬁlled waveguide shorter than the corre
sponding wavelength in free space. T F f) If an airﬁlled 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 parallelplate 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 parallelplate 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 IEHEEEEEEﬂIEﬂﬁln 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 nonmagnetic 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
fSBdS=0 VB=0 fCEdL=—% SBdS VxE=—%’fi fCH.dL=Imd+fS%ds VxH=J+%—Lt’ o Constitutive relations in simple media: 6E
uH
0E Hwb
nu o Timeaverage 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 Parallelplate 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” cod7’?) 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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 Fall '08
 Elefthra
 Transmission line, Impedance matching, Smith chart, Final Exam Page, ECE320 Final Exam

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