329sp09he3sol - ECE 329 Introduction to Electromagnetic...

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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Spring 09 University of Illinois Goddard, Peck, Waldrop, Kudeki Exam 3 Thursday, April 23, 2009 — 7:00-8:15 PM Name: W . ‘N‘K I Section: 9AM 12 Noon 1PM 2 PM | Please clearly PRINT your name in CAPITAL LETTERS and circle your section in the above boxes. This is a closed book exam and calculators are not allowed. You are allowed to bring notes on a 3x5 index card — both sides of the card may be used. Please Show all your work and make sure to include your reasoning for each answer. All answers should include units wherever appropriate. Problem 1 (25 points) Problem 2 (25 points) —l Problem 3 (25 points) J Problem 4 (25 points) TOTAL (100 points) 1. A plane wave field E = 20 cos(wt — 52L”? V/rn is propagating in free space in the +2 direction and is incident. on the z = 0 plane which happens to be the boundary of a perfect dielectric having permittivity 950 and permeability no in the region 2 > 0. Determine: a) (5 pts) The expression for the incident electric field in phasor form. g=2ac’”\%§c‘x_ / we» M M b) (10 pts) The phasor expressions for the reflected and transmitted electric fields. Hint: The reflection and transmission coefficients (F, T, respectively) are given by F = fl and 7' = 1+I‘. Be sure to define the respective ,8 in terms of the properties of the specific material being referenced. “11-3,,” F; 1*3 =17 L 71 v-I- . ‘60 ea rttfnaJ H m3; \l/w‘ ,— - b 7, EV toe. x / 6";3} c) (10 pts) The phasor expressions for the incident, reflective, and transmitted H fields in A/m units. /. v A g2 A ’6’ y 1 ‘1 ’70 Ta.“ é.a% r. A —' ‘3‘ Ht: "139%‘)(52 2. In a region of space electrostatic potential is known to be @(x,~y, z) = —3(y~ + 3)a:. a) (7 pts) What is the corresponding electrostatic field E? o 9‘)‘ W 97 2' 1 Saw “(7 *9 - 3X2? 3(D‘+3)X + 6X9. 9 WW ./ A A kw ~. 3? 11 '2 3 9.0 c/ 0/n'- 9/9") 0M -1; '0 36‘?!) 6x3 0 +3( ‘65) E z 0 ( c) (7 pts) What is the volumetric Charge density p in the region if e = 250‘? Hint: you can use the Poisson’s equation in this problem. 0 V'CEE'):/ ’9 f: Z?.C§§v1~%§1+9?a d) (7 pts) Repeat (c) if e 2 600. + ($2). Hint: you cannot use the Poisson’s equation in this problem . KS éa Ex [x1 ‘2’) \J 39"”) 0 v‘ z a LCHE’“) Ex 40””) 9159 +3 ) x a a? .2 W 6)‘ __ 1. : 6-0 [a é(>z+5>xé-x7—+ (He, x )éx] %, / . .t— W ZQA/ (5-0 V 3?. “(5.0 I l . Consider the transmission line circuit shown below: : Z? 3: 3 Zg= 500. a ‘ g ZL = 509 Tse ZO'O f(t)= 27u(t) V The source in the circuit is specified as f (t) = 27u(t) V, where u(t) is the unit—step function. The transmission has a length l = 200 In, propagation velocity of vp = g = 108 m/s, and characteristic impedance Z, = 100 Q. a) (4 pts) Find the initial amplitude of the forward voltage wave 11+ and current wave i+ on the line before the first bounce. b) (4 pts) Find the voltage reflection coefficients, FL and F3, for the load and source ends. w a “ "_( o _= . [is 501460 [‘5 z (7/ c 4 ts Sketch and label the volta e bounce dia ram for t < 6 . )(p)mw w“; s g #5 ’77—‘6' Gl‘fk d) (5 pts) Sketch the voltage at z = 100 In versus time and label the axes. V l? n.- “I t 1. 1. 4 r b 'a- K ‘) e) (3 pts) What will the steady state voltage be at z = 100 III. agl3'gv .. 9 f) pts) Remembering that the wave v+ is moving to the right (+2: direction) at speed '12,, = and that ~12“ moves to the left (—2 direction), also at speed up = write out, as a function of z and t, the equation for the total voltage = W” + v” for the waves that exist when t < 6 pas. wage): [a «(haw «é “(f*‘73'.(”pr)+Z “(e-,3; 4741,) M‘i'h Wot-3.": 4 .. -Z 7? ’umxco" W; \/ / 4. PART I: Voltage and current waveforms on lossless transmission lines are governed by telegrapher’s equations 8'1) 62' 62' (91) where [I and C are inductance and capacitance of the line per unit length, and Z = «LI/C and 0,, = denote the characteristic impedance and propagation speed of the line. a) (4 pts) If -i(z, t) = cos(wt — [32) — rect(t + A on a lossless transmission line, determine the corresponding expression for the voltage waveform 11(2, t) on the line assuming positive [7’ and w. V'C%.+) = 2-, cm(wt~(52)+ 2o Mat (-6 +gr)/ b) (3 pts) Does a current waveform 2'(z, t) = (cos(wt—fiz)—rect (t+%))2 also satisfy the telegrapher’s equations on the line? Justify your answer. No! becaure’fiu—M xw(wt‘/92)Md(£+$f) .S “cf” A fry/90b}. wMQ{ W1 W“ 61"“. MV L‘KM gopupoc,{h'm{ We, wow-£4 . c) pts) How about i(z, t) = cosg(wt — ,Bz) — rect2(t + Ya -—- ’hM'g is « Wyw’km “a W0 fro-AIM": W‘s) H'will Qua/mp, Tw- eve/HM. PART II: Two transmission lines of infinite lengths and equal propagation velocities up = c, but with different characteristic impedances are joined at z = 0 such that for z < 0, = —Z1, while for z > 0, = Z2 = 2Z1, with v— and v++ referring to voltage waveforms in z < 0 and z > 0 regions, respectively, propagating away from the junction at z = 0. Also, 1" and i++ denote currents accompanying v— and v++ waveforms. a) (10 pts) If 11+ = 30 cos(w[t — V is the voltage waveform in z < 0 region propagating toward the junction, determine "the explicit expressions for v‘ and v++ defined above. 21*?! L%l*%( ’ ~|tl1nzfl a U++=4060(°'I't"%j) V/ (1‘1: Z1—Ef: ZZ-%( a 3L .7 v": [0“.(wI-t-l b) (5 pts) Determine the explicit expressions for current waveforms z" and 73++ if Z1 = 10 Q. ' — 1 = —mz~re+s7>n/ Usajlwwsfi A/ 4'”: lw(w[e~%7) k/ ...
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329sp09he3sol - ECE 329 Introduction to Electromagnetic...

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