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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Fall 09 University of Illinois Franke, Kim, Oelze, Waldrop Exam 3 Thursday, November 19, 2009 — 7:00—8:15 PM Name: 50 L“ “O N Section: 8 AM 10 AM 12 PM 1 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 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. A TEM plane wave is propagating through a good conductor, with time—domain electric and magnetic
ﬁeld intensities given by: .c u)
/ ./ E(x, t) = 1006—4035 cos(47r X 10st — ﬁx + V/m Hm, t) = see4°z cos(47r x 10% — ,8zr + $2 A/m a.) (5 pts) What is the complex propagation constant '7? Xim+j<7. “540 S (19’? RV gonad WAMCJ'GV' K= 40+340 5 Llobﬁ) Wm]
b) (5 pts) What is the phase velocity 1),, of the wave? yo 5 'l‘ﬁ mg: 7
V? (f ‘70“ W l“bl “\ c) (5 pts) What is the complex impedance 7} of the material? .. .v _, '17/
m'wll’gg’ ' R 11:26} Li {9:}
441=Ig 1'74 Rvojmml csmduohw d) (5 pts) What is the phase ¢ of the electric ﬁeld? A! ' . 1T .
Earl—«MRS :— 2?— iszel
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l‘lg e
u 1r 1r 9 497?
<.o <l>— 754—1; 71(le
e) (5 pts) Discuss how you would verify that the material is a good conductor.
 2:.
93M dew wé >7 l
5 7 IO U“ Mg “50 . 2. An electromagnetic wave with a. co—sinusoidal time variation and an angular frequency of w :2 67r x 108
rad/s is propagating in the +2 direction in free space. It is known that E(z = 0,t = O) : —4:i: V/m. ‘10“? a) (4 pts) If the wave is linearily polarized, what is E at z = 0 and cut : 32:? Z
ﬂop): Ex? so a, > o 
?(z.k)= F,ws(w1”¢2 +401)? 3? 4> =0] 5, = «Ll.
4° ﬂaw/w) = '4ws(“/zro)§= o  73W»
RM _. 0% b) (4 pts) If the wave is righthanded circularly polarized, what is E at z = 0 and wt 2 5? . \Ex\= IE” 40 —E>(o/T/2w> = 4+3! V/m w!" 1“72
c) (4 pts) If the wave is right—handed circularly polarized, what is E at t = 0 and ﬂz = 77?
v) _.> \ f‘ “—
1 =— 1T ' 4
E( 4”») E(o) A») Zﬂvwv. lo armluft ’W .. " ,_ A V
’ "Ltcpsﬁrvﬂx ~ 4X /M
d) (10 pts) Write the general expression for the wave electric ﬁeld E(z, t) if the wave is righthanded circularly polarized. Verify that your expression satisﬁes the wave description given above as
well as your answers to (b) and €61.47) ; ED 50$(M’@&’+4>O/)2 + FDSCM(W't'(Ji’—hi>‘>9 7 whore Ep=~L¥ +0 SMASH E(0;°)"W ' A A A A
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E(o,“/u) = '4w3(1T/2_)'i< v 4—Sr5m("‘/z)y = "W
Ewe/03 = “Raﬁ/ﬂ? Ibisméfﬁ = +4)? \/ e) (3 pts) The timeaveraged power density transported by the linearin polarized wave described 8
in part (a) is (S) = —2 W/rnz. Does the circularly polarized wave expressed in part ((1) have @smaller, 07r70tlw same value of (S)? Brieﬂy explain your answer.
<97 = (all? = < Ell)
. "1°
'FW WW wmw ‘ <\€1> =<Ibw>z79L§=r ‘% M MSW/Ame. Yowar EJEXT—l W MW; RV W WM“,
M W a, (lawn ﬁvmVxW Cw Malay (:91. was. 3. Consider a T.L. with a characteristic impedance Z0 2 50 9, length l 2 400 In, and propagation
velocity v = 2 X 108 m/s. A voltage source f (t) with an internal resistance Rg = Z0 is connected to
the z = 0 end of the TL. and the z = I end is terminated by a load resistance RL 2 3Z0. a) (3 pts) What is the injection coefﬁcient 73? %°’\ TS ’ 2%., 7.
b) (3 pts) What is the reﬂection coefﬁcient FL at the load end of the TL? 93%o—%o ’ 
TL: ’ ail9+ &; 7'
c) (3 pts) What is the reﬂection coefﬁcient F3 at the source end of the TL?
(is = %a ’ 2:9
2;, 4" %o d) (6 pts) Construct a “bounce” diagram describing the voltage V(z,t) variation on the line for
O<z<land0<t<6psforf(t)=6(t). =0 £;ﬂ :2 S mew/Ade +0164 ‘
e) (6 pts) Write the expression for voltage V(%, t) for 0 < t < 6 ys as weighted sums of appropriately
delayed impulses 6(t). I o
\/(‘»c)=1:s6(Jch)+ f‘g £,2e
i} ZVr '55 L t’ZV‘, ’0‘. +T5FL = l. ,. L , AIL 0W W f) (4 pts) Repeat (e) for the current I(%,t) for 0 < t < 6 us. 4:.
g; 5 is . 1(a) ﬂay—280?.) + (gowns)
= 7.5805”) jojDSH'?) [k] /,, V;O @ Dow £44005  4. Consider a, lossless T.L which ‘ short cir uited at both ends. If the length of the TL. is l = 20 in,
and v : %0 m/s for the line, ' a) (13 pts) What are the resonant frequencies of the line — frequencies at which sourcefree oscil~
lations (standing waves) of voltage and current are sustained — expressed in MHz units? T4,. mm mm We wwwsmlu m wen/ks wowv: wmwg IL: nA/z. “*7 M‘ W ’1 L
V\ 5 $710,: 640 n e 5n MH? e; s i? := '25Y5'm‘13h :— 2' log
“ AM 40 b) (6 pts) Sketch the shapes of voltage magnitude as a. function of d (distance from the load
end) corresponding to the three lowest resonance frequencies. Be sure to label each plot clearly. hﬁ‘ so = F,
a) a 1
MW lv 1 PM”? [Vail ,ﬁ‘a—v [Q 0 c) (6 pts) Repeat (b) for current magnitude If as a. function of d. [EGO l ...
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 Goddart
 Polarization, Frequency, Speed of light, pts

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