EECS+145+2009+Final+Exam+++Solution

EECS+145+2009+Final+Exam+++Solution - EEC8145 Fall 2009,...

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Unformatted text preview: EEC8145 Fall 2009, Chin C. Lee 10% 1. Electrical Engineering Analysis Final Examination 1:30—3:30pm, Dec 9, Wednesday Find the residue of the function f(2) = 1/[z3(z-2)] at each pole. You must present your procedures. Guessing alone will not result in any credit. 15% 2. Evaluate the contour integral of the function f(z)=(22+2)(22+4)l[(zz+1)(z+3)] forthe following 20% 20% 15% 20% .A= contours, respectively: (a). Circle lz-jl = 1, (b). Circle lzl = N2, (c). Circle Izl = 4 3. Contours C1 and C2 in the figure below are electrical conductors. 01 and Cg are electrically insulated. The space between C1 and C2 is dielectric medium. C1 is set at 0 volt and Oz is fixed at 20 volts. 3. The mapping function is known to be w = 1/2, where z = x+ jy and w = u + j v. Show that C1 is mapped to v = O and Cg is mapped to v = 2, respectively, on the w—plane. b. Derive the voltage, Lp(u,v ), between line v = 2 and'line v = O on the w-plane. c. Derive the voltage, CD(x,y), between C2 and C1 on the z—plane. d. Derive the electrical field E(x,y), which is a vector function. _ y i i 2 a. Determine the eigenvalues. _ a b. Determine the corresponding eigenvectors. \ ‘ c. Form the modal matrix M and spectral matrix A. d. Prove that M'1AM = A. You must present your procedure step by step. 5. A 6x6 matrix "A" has the characteristic equation of (ix-2)6 = 0, where A is the eigenvalue. Find the Jordan form of matrix A, respectively, for the composition of the full set of independent eigenvetrors and independent generalized eigenvretrays, given below. You must write down the value of all elements of the Jordan matrix. Do not leave any element “blank”. a. 1 eigenvectorg: and 5 generalized eigenvt'zctorg. b. 2 independent eigenvectais and 4 generalized eigenVectors. The first eigen'i/Pc'tds breeds 2 generalized eigenVQt-tsrs, and the second eigenwtms breeds 2 generalized eigenvettarg . c. 2 independent eigenvectais and 4 generalized eigenmectoifi . The first eigenwcels breeds 1 generalized eigenwctsis, and the second eigenwcrm breeds 3 generalized eigenvegm t. A = 2 0 l o l O O O i To receive any credit, you must present your procedures clearly and step by step. Just guessing the answer, even the right one, will not result in any credit. , evaluate e“. [I H H (241) (£32)(2:4J (Z‘flflz-I} M2”) fl“ 2'3 (21+: )(E 1 4') (2+3)(8~j)(¥+;) 3 C (—3 {a 3 __W 3 20 x ) 3 ‘ J 3 —2+6j -.._ 32.0 If " 1in ~ 10 v ’0 {311:1 :CemTérPc/(a (0);) w/ r; H’Serg @ 2:3 3 0p — 6711;; m3 (~33- ?33) °" ~1+63 q 3 . a _ 3 Ema—Fm} Y umk‘ubg) to o S '2‘ ' 2r? ‘9“4’3’90’ @ (0J0) W/ V: 2/3. YPSMJmQ @ 3:} ,j 3 02 3 or“ 2/1} C WEB-+33} “35—5 ‘37)” oY‘ ~ 3 Jr- _3 _. J g \Ei: 4—, (Pm'fé’fiflci/ CC) (00) w/y34 Mn «‘1' “J Lezwgzygzsvszj + 7-; 287w kyl/(U‘JV): IOV E ,x/ m (L) (b :/O wxl+\/2 g _. (A) E(&y) — ‘vv y) P/ ‘17— lo flay; 3X LX+YU (X*7) fl C i) (y2y2)-(-\/)-ZY h h — I. 3% 4y E\/' 3/ wrf ) (X1+\/2,) fix my {0(X’37i”) f 2 un—i-‘h r-.-.q~.m.m-w..._ L 2' 1 L le‘ju/Z) x D f’)‘ 0 :0 O o 3% \ :) g4"/\) {3"}‘111: O :7 A: 1/ 1, 4 l; \ .-' } 3. _ '2’ a - g G I O i :7 {2— ik- ¥ } >\I :2 / A}: }”j“‘ Af % E” e 1 0‘03 4“ g {:13 If Qfi ; 1£/\l:2 :) Q : oko+2ohvv40fi2 ~® Inc/Mr! *2 at: 9,40 W 0‘; ~-- 6) l { o D f 14x, [ Q o L 9 O \ Q «Q €23 :2 6'3 gf 8%“ H 7% {I 28 “‘29 437:8” O I i S "2-{ ~15 0 ~19 we ate/i 4:. i. 0 9632336 11, {J i] Q ... e — tat O 4 26 {t 1 O G «e «H? J! at , 1; - Q ~12 O i g 8 ' g ...
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EECS+145+2009+Final+Exam+++Solution - EEC8145 Fall 2009,...

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