329sp08he3sol - ECE 329 Introduction to Electromagnetic...

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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Spring 08 University of Illinois Goddard, Peck, Waldrop, Kudeki Exam 3 Thursday, April 17, 2008 ~— 7:00—8:15 PM . ' m I 9AM 12 Noon 1PM 2PM 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 one formula. sheet of 8.5 by 11 inch dimensions — both sides of the sheet 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) - Problem 3 (25 points) Problem 4 (25 points) TOTAL (100 points) 1. Consider a pair of infinite conducting plates separated by a distance d in to direction. The plates sustain equal surface charge densities with opposite algebraic signs. The plate at :1: = 0 is (by definition) at zero potential, i.e., (NO) 2 0, Where indicates the potential function such that the electric field E = —V<I> = —g—: it in between the plates. Finally, the permittivity between the plates varies with distance r as specified by 6c e(ar)=1+£. d a) (5 pts) If the electric field just above the bottom plate is specified as E(0+) = 2:2 V/m, determine the surface charge density p3 of the bottom plate at a? = O. 9 7; Z?O:B :2") $3220 £1 $1.1,— E J 0’ " Zn $0 X " 3m): 40 W €697 $0 I ' Dc») , as Z l '- E r " M 0‘ //// /‘/ ’ i‘ 7 ' b) (10 pts) iven that E E E in between the plates, express the field strength E for arbitrary a". between 0 and d in terms of E(O+) = 2 V/rn and — Hint: apply the differential form of Gauss’s law with p = 0 (no free volume charge between the capacitor plates) and 1? dependent E to obtain a simple condition that leads to the result. v,(gg);f a, if: Halfwj :o 9 2(x)E6<)=cm+: 20" /, Ed): 52% 2 201%) Wow 20‘) / c) (5 pts) Given the above information: will the electrostatic potential ‘1)(d) be positive or negative? Explain your answer qualitatively prior to calculating @(d) in the next part. E’ {new F0“ WE“ iv {We-r dyuluh'ccflr -' — _ fi 22:) (U @(93 :0 We Mal iG) <0 (AOL . ) r ,. fikr’kfi 250 %P Mjevj'nla QG d) (5 pts) Determine the electrostatic potential of the upper plate in terms of d by integrating the field strength from part (b) in an appropriate way. 0 A 7— A ‘ ’ 0).. gm): game = 20%.. )]0: 2(J+gl>: 2x53 2. Telegrapher’s equations 8U 6% (9i 6'0 -——8—Z—£E and —E—Cat for T,L.’s can be combined to obtain wave equations with traveling wave solutions, which superpose, for voltage 'v(z, t), as z 2 wet) = v+(t — —) + for + —), “p “p 1 \ffi—C‘ a) (8 pts) Write out v(z,t) explicitly in terms of radian frequency w and an appropriately defined wavenumber /3 if vi (t) = cos(wt) V. —~ Make sure to provide an expression for fl in terms of w and other line parameters. where vi“) are arbitrary waveforms and up E b) (12 pts) Given 0(2, t) from part (a), derive the associated 2’(z,t) by solving the telegrapher’s equations. Show all your work. We. Amie P3,; ; a [ (@34'n(0u't‘($%) «[9 gm(ot+/stfl; 7%: w wk av“ 224‘ [w (wt~{é%) -— awfw’H-‘éflj E w ’4) 42¢): Cw(w1t’@%) wean (wtfléil) 1R? a c) (5 pts) What are the locations of current nulls according to the result of part (b)? 3;") Q1417 “Mu Erica (b’tcgflf =17“ a I M(W£,m>: an(w’l;+TD\> 3:0 «'5 «Wu/u / a Nod/U «7f 52%;?“ 09 we?“ ' 3. A generator with internal resistance R9 = 609 puts out a. voltage f (t) = 606(t) down 21. TL. (see the ckt diagram on the left below) that has an unknown characteristic impedance Z, and an unknown resistive load termination BL at an unknown distance; L, from the generator. At a distance 2 = 300 m from the generator, the voltage waveform as a function of time is as plotted below on the right. V (in Volts at 2:300:71) =§ss= a if??? 1 (us) ~10/35(t-8) in: we? , - _ 3 w% M mans) UF;C-3XIO T . a) (4 pts) Determine the impedance of the transmission line, Z0. y; i9 a éo+%0 2 €27, a éo C "'9' £o=330£2 601*230 . b) (4 pts) Determine the load resistance, RL. ‘ f‘ :%L’Zv ;%r- , p; a 22560 <—24x30 :7 327,30 % c) (4 pts) Determine the length of the transmission line, L in meters. Ung L [email protected]/,L§)[?0’Ofi> ; (os’o w/. d) (6 pts) Sketch a bounce diagram using the axes shown below for the current waveform i(z,t) — not the voltage waveform as we have often done — for 0 < t < 10 ,us. Be sure to mark the numerical values for the amplitude of the current in the diagram. Curreni(A) e) (7 pts) What is the algebraic expression for the current waveform as a function of (z, t) for the ‘ domain 0 < z < L and 0 < t < 10 ns. diet) 3% vii) e; 50 +:: a) «fir at age) owW , , /*’“ ¢7%‘>’VWJ// 4. Consider a. lossiess T.L. shorted at both ends. If I = 20 In is the length of the line, up = gc = 2 x 108 m/s for the line, a) (10 pts) What are all the resonance frequencies of the line shorted at. both ends, expressed in MHZ units? Show your work and reasoning. gee-923 b) (10 pts) Sketch the shapes of current magnitude }I vs d for the line corresponding to the three lowest resonance frequencies. Label each plot clearly and explain each briefly. in ...
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This note was uploaded on 04/14/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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329sp08he3sol - ECE 329 Introduction to Electromagnetic...

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