HW12solution - 5a I‘d[on ECE604 Homework 12 Out Tuesday...

Info icon This preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
Image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5a I‘d-[on ECE604 Homework 12 Out: Tuesday, April 12, 2005 Due: Tuesday, April 19, 2005 Announcements: 0 Prof. Weiner will m); be available for office hours on Wednesday, April 13, 2005. 0 Final exam will be held Monday, May 2, 2005, 3:20—5:20pm, Room to be announced Problems given by number are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). ' 1. Consider the TEm,n and TMm,n modes of a metallic rectangular waveguide (m ¢ 0, 11 ¢ 0) . Show that for any specific values of m and n, the TEm,n and TMm,n modes are orthogonal (use same m,n for TB and TM mode). More precisely show that 1 ~ ~ * ~ ” * aw Mariam) flaw?) ll=° Note: It is sufficient to consider only the z—component of (P) — i.e., the component of power along the waveguide direction. 2. 10.3d _ Hint: Review section 6.4 and note equivalence between 6 and (1)8” 3. 10.4a 4. 10.40 5. In this problem we consider data transmission in two different metal waveguide structuresi (I) the parallel-plate waveguide and (11) the square waveguide. Both waveguides are taken to be air-filled and lossless. (I) 2:6;Qm I ‘ l .. . I‘- fi=30m—" p arallel metal-palates- -'$5Q‘%1?ar¢'v mctfil ‘3”de -(_‘separation.a=i6 cm) .(sliiiiie'aFSCIn) (a) We wish to transmit information using a baseband transmission scheme. Our data rate is R (units of bits/second), and each bit has duration T = UK. For each data bit equal to ‘1’, we simply transmit a constant DC voltage during time T. For each data bit equal to ‘0’, the transmitted voltage is zero during the duration T of that bit. (See sketch below, showing transmitted waveform for data ‘1011’). We desire to operate at a data rate R = 100 Mbit/sec (108 bits/sec). For very fundamental reasons, one of the two waveguide structures is expected to give much better performance. Which one, and why? (A short answer is sufficient if properly justified). (b) N ow we wish to transmit data Via binary amplitude modulation of a 6-GHz sinusoid. For each data bit equal to ‘1’ , we transmit a 6—GHz sinusoidal burst of duration T. . For each data bit equal to ‘0’, the transmitted voltage is zero during the duration T of that bit. (See sketch below, showing transmitted waveform for data ‘1011’). Again we desire to operate at a data rate R = 100 Mbit/sec. Comment again on the relative merits of the two waveguide structures for this data transmission scheme. Is there a reason to prefer one waveguide structure over the other? . 6. We are interested in the use of thin wire current sources to excite the modes of a lossless, air-filled, rectangular metal waveguide. The waveguide has dimensions a = 10 cm in the x direction and b = 5 cm in the y direction, with propagation in the z direction. All the current sources are specified to be at frequency f = 5 GHz. 0 ‘i' . a#.'10-em ---I "f0 For your convenience, the modes of a_ rectangular waveguide are written out below: 1‘an TEmn Ez = Asin kxx sin kyy Hz = BCOS'kxX COS kyy jfik .. '00 k Ex :_k2 x Acoskxxsmkyy Ex .—.J 2” y Bcoskxxsinkyy ,‘ c111,11 kcmm . ' k ‘ jwuk . Ey =— JE y ASinkxxcoskyy Ey =— k2 X BSlnkXXCOSkyy kann cm,n 'coek jBk ' 1-1;; = J 2 y Asinkxxcoskyy Hx =PiBsm kXXCOSkyy kcm,n cm,n ' ~ 'mek . ' k Hy =—J 2 X Acoskxxsmkyy H = “3 y Bcoskxxsink y k y k2 3’ cm,n cm,n 2 _ 2 2 . _ m7: . _ 1m kcm’n—kx+ky , kX—T ,ky—"6“ a) A single thin wire current source l0 is used to excite the waveguide at 5 GHz. The wire is located at y = b/2, z = 0, as shown. What waveguide mode or modes can be driven by this current source? (You must give a brief justification for your answer.) a=100m I ,0 b) Nowapair ofthin wire current sources located at x=%,z=0,and x=§f,z=0 are used to excite the waveguide at SGHz. The wires carry equal current (equal current phasors Io ) but in opposite directions, as shown. What waveguide mode or modes can be driven by this excitation scheme? (You must give a brief justification for your answer.) l E X l 3 (IOCm) 3/2 . . . . a In this questlon a set of parallel, thin w1re current sources are located at x = — and z = 0, 2 d, 2d, 3d, 4d, etc. The wires all carry equal current (equal current phasors l0 ), as shown. We wish to drive the TE10 waveguide mode to as large an amplitude as possible. Give an expression for the wire spacing (d) that meets this objective for a drive frequency of 5 GHz. (You are not required to evaluate d numerically. However, your expression must correctly indicate the dependence of d on frequency and waveguide dimensions). A simple air-filled transmission line resonantor may be constructed by shorting both ends of a transmission line, as pictured below left, for ideal (lossless) resonator. transmission line: char. impedance ZO, velocity 0 Resonator with loss Ideal (lossless) resonator The ideal resonator supports resonant modes such that m half-wavelengths fit within the In)» resonator, i.e., 7 = L, where m is a positive integer. a) What is the frequency of the mth resonance? b) Now consider a resonator with loss, which we model by replacing one of the shorted ends with a load RL, where RL << 20. Work out an approximate expression for the energy storage time and the Q of the mth resonance. You may assume that the resonant frequencies are approximately the same as for part (a). 0) Give numbers for the resonant frequencies and the Q’s for L = 3 meters, Zo = 509, and RL = 19. 1. Consider the TB,“ and TMm modes of a metallic rectangular waveguide (m ¢ 0, n 7': 0). Show that for any specific values of m and n, the TB,“ and TMm,n modes are orthogonal (use same m,n for TE and TM mode). More precisely show that ,ll <1») = I alikelmxmnr +Er’x<fiflt">*ll =0 Note: It is sufficient to consider Only the z-component of (P) - i.e., the component of power along the waveguide direction. Fnrw Job‘s 747k law/“WW" 7441“ Chet/W75 m 5 N W ’ 2’1 . ‘ "‘ k [:11 2 J15 "7 EwskaSInlc-7‘a» (v ' 6 ~ ‘ ’97. 4 ”J b.) waskxysmkyb‘ Kc t 5' kc?— ‘4 6.1246 kxk7 .; * ‘ Key 5 ’4‘ Coazkxysmz/(r/ > + wzqek \4 1 _'___._._:_7 BA; sdykxx C052k7a \LC“ m bow-aw (flaw WM WM“ CDSLC ) a..€ 5M2 )‘7qu9. 5/3“» fie kg 0,,0 [‘7 m Mm C74 MJO“ Ada/lieu?) Mr '1va fat/A flu/{- ;" f/[uum m/Ln’z‘srw/ numb/.74“? gu/w h~¥ «Ikgc’I/CC’L'MS ) CW A” {fame can m :m 1c” , 3:; M IIXKJ7 901 szX 6‘55 Z#577: «5/7 $1166 {71% v’w “LS4” WM “*‘JQM‘VW m 74w" M 4P>=o 55/40” rMmJ/‘alfi, &' a. w.“ _ wo-{J 54" ,.Q = ‘13: - VJ... But: wk- é}. (“flav = JiJV(E¢)‘1}-dv , t'f, saw Q = ”9%!vatav - w.e'_ g; {JV‘ENV ‘r °' EWP‘C" ‘5 rene, Q: Memo“: .95 Comflred NiHn ~IO‘ 4’or Qube wifiI Q23 waMs. 40'36 0 a lO-4a. Nate that wave van-shes fiwr TE»...P [E13 .104}- (0-05)} H: pao or 34’ both In ' and n=o. For TMW‘P modes : New; vgniahes H: vefiuner' m or- nae. Resonant {ire men ‘9“) . lOo (.1) ‘5, "I -wb‘ ' ’. I - 1 (:5 'm 5% 4 I 4’0." u: camba coo -.-. 3%: [m‘+n‘-+ ‘1‘ Below w: tabulate nonvcmishinfl modes, with normalitad résoncmt‘ Mama?” and (legend-0.55 . There are. '1 d'nPPer'ent. Muendea. . mm lO-4c. G= 7.7: N: use Egg» |0;4-(s-5)w3d1 M=n=P and aL=b-di. .. e ‘7 A. c. a. .‘ , U " '2- 5, 5. J. “End 4- [55‘ )dxdsd; s I a O; a. . , = '3‘ y‘fi'fi'fifi 5. Sffcmtwstmex “‘4'": sim‘———M:£)+(M‘fsh’_”:* ws‘mfi duh—”:3 )ld‘dfl‘“ But "a.1 3 2 mg)” [5% Io~4(6>] use; '60.. u ’ lb . uafi‘ For Nb, use W3 EéLH-ltl'ls 6nd 523"”“r5 ‘0 that (NAM = gunk” + 1.71445... 1- 2N4,” M1”. = géi‘ftmyuawlasaa . h = 3£Jfr{[-E&(‘¥)‘sh‘m§:9- m“l"£~l)+ [c‘a‘mgaswagsfianae = 53: 1%(%')’(%)’+ ”(+191 Likewise '-_ ' ‘ _ ' W, = %[Ea.(%t‘-)‘c%)‘+ mg»; ml... = ‘55 [$433+(“fi‘1 ; I 1.. (NI-)tM : ‘ gas—é; 9 = “1;”;22,“sp—:—w = fi%%=%gm when awe last ezxialifi".w “Om reflacins a. hus'mg , 5% .(0400. (an: 3%%) T53 Q ““9655 mu. m at wistant ‘Frequepca becausg E and H mm is constant but volume- bo-wrPace area? Mme: _Proportionul in m. .e i_ .. . J La/"JS/‘R’AAVC‘L/C‘J‘ ’I’C‘J‘WC {5.7%. 7447“,qu .... . .... .. .. .. 7’5ka '4’MM’LW’C’“L ... .. H 3... H NC M(?_ ‘ I5 ‘G‘ZFJC 7/3. )....-.. .._._. ”(9) (3 CM) "'TM“&'5""6’5TMMdoMfi¥M\5+(E"%MZ‘ M-Q h_>_____| juqu MN»? —& TMMA 5‘“ W5 ' TEHVTMH 3 5(7— “*3 A 1~°1 64”“; H ..TP. TEL “...”, .16 103‘. e M an 6H3 ‘ > )fig W/pjakg (fii’u’n 7‘6 DC , z I (70%” 1 51 IO" H; * 5‘64‘; ) . A} PAM AAA AAA... s» :4 /Mé/%«5~AAA ' AA 71mm W AA AA drum #4?“ A ~qu M414..- _W _____ _ "(A A A A AA AAA/«4A A AAAAA Am ._, A. F ‘4 2.,{J ”‘3 “5Q .’.7L.__ /'§_)1f":{ / (Sf/Ll“. % #344,... 7L kt.- ‘ 6.). /V°“ 705— Cemhé/t 4:93;)va WEYW __4 4179, J _Lu__ fit. . .4 64115141119} _9 l WCQQ Ali?— ...€f‘. _S_9_ («34444140 41:” 561,43 1455 f“ Wrtu; ‘fl/ Q 6.: *6. j‘I/f'dfio 4% >aééb; (Jaw / $9.. ié/A Mm {Sf/U»- “WC .cfyogxi/cH/M A *4 /‘*55//MM*7%¢113L0*47 . [03* 1 14440 few/“”4”“ ,. 4(‘m‘m’2sk . . . .. PM éibwfhb’evmfié £15904le /°“/€-“‘-/¢/5°,mc/IMM An ..gr/n~7‘1”"0m Ana/74 Ci/ZyLMJQJ 3(“/5 OcAALcW’vL .. V 9944‘“ E ’1'. . . % 1</..o. fim Maw/w AMA . Law/M'LM _. AAA/WIN , . - 9 FfWW/ddj‘ flWAu .fl'fi’k... .. .. fiv’fflcg héAMu/{vp I , M94014 aéew” chV/M‘ 66473 In jm«.0, 125/3 ,cu/WL .5”; ..E 7794; meeaowl/ AWCgf/W .yCUpY’v/ocfié‘a ,4/46 .0144 ._ . (/6 Y1 w fauna.) Along,“ #“57W15‘de ““4925.“ ”70% Am‘b'k; MU/fiSM—MQL dis-”145,0“ - . LUC-fia’bfiém .7’KHS7 Wm 39.44. .1. 51"“- j’é (.3 .M. 7( 43% FCfié. ’4 . W/ 141% .69: “ho/Wit?" -59.". W. S’pquw . o/wtv‘lfj 417(1“? .. . ‘ xzmx M. can: m 0%? (”www- L L A; : 2 E3 +0733 Fm TYAwm , lav-14+ vme VS Mflmzl. 3 lb "1MB “QC ‘- 6M“ + (2:07" ~Cux anew, ) , 106» b =50“: QC, " 156‘“? {ml-Hf I PMs/1030797) Mazw a} @417: w,” M°£K ‘5 cal/WI" 47 1/7?"de >4AM. N 4 W'MJMJUWA re f gxdlc J 5y Wale/w o «f y‘ré/z. H’ /\/<st:“ 235x" 0354;:ng [11% V04“- E’ Ex (3‘55) F3 My,» 01:5 gxnadw, q M‘r‘fl . o \fq‘ug (,6 Sa CD5 O‘X TS NMWAM-szo. So 0x127 TEon l/Mcbo (rt ch) CM hmawl Fu\ Waite 0-500“ 03%, 7%: AW“??- 7501 . 3W oz; goaokcosega) WWMWM. 9 WM» £§%%QK‘%>W ‘ Ea (X’é);fs M42011: Mtg mnwem So VJZ’TEQZO, T€50,TE,D 41“: wkML/H‘ % {4942. o-vJ‘a’ Ton [5 657°“ 09% . . . . a In this question a set of parallel, th1n Wire current sources are located at x = -2— and z = 0, d, 2 etc. The wires all carry equal current (equal current phasors l0 ), as shown. We wish to drive the waveguide mode to as lar e an amplitude as possible. Give g an expression for the wire spacing (d) that meets this objective for a drive frequency of T E , o 5 GHz. (You are not required to evaluate d numerically. HoWever, your expression must correctly indicate the dependence of d on frequency and waveguide dimensions). “if—Lg mafifil/‘m 14"“ WHL Jngr‘hJI' Ma #74/11 /‘J.' 7) j ‘1 .OL) L ”FM. : 2% £33 6995016130? WOW MM [035, ref/em wef, P: 23; M 7% 974% 817%,Md4 I M Ma firfimd We swat {9“ng M 4, git/m paid fia Hwe rgsmm‘m/ M9 We .‘ (4:25? is Hue WW? Wél mt~2"/E122 Wm A7 is Small WMM WM, MM 06:96? #1776, Me 57%? *6/56 pm gmée Fapfltml at Me WPWWM Mam? @pma‘mafim r9774 U; (HA ) = Ugcm WE”? A: UMP? i- W M Wm M Ugém egg :9 U = ~ (L‘zfiljt _C_£/—§? ; EC{’) LE8 :nge 2 i 57% SilfiMf/Z 79372? Z PS y/gfiw/ 45 757/29 fi‘fié & my: 72/” 05032 a 0mg 5 I/e 0/ £715 msézo/szée (15(2): aye—£1333? =Uofl ~> z=~é~ _ 3L .A flew/0“ 170‘ ‘ 0079‘) 9 W0 U5 _. '_, ‘ .» (Q ‘ WI 7%?" 6"”) We » W=A?-,§’W€%rflwerxcl~fl) : UEI‘ l 7;- X6270) [This is from WL =UE/17] ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern