Spring 2004 Mid 2 Soln

Spring 2004 Mid 2 Soln - Notes ECE604 Midterm Exam#2 April...

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Unformatted text preview: Notes: ECE604 Midterm Exam #2 April 8, 2004 4 problems Formula sheets (3 pages) is supplied separately or attached at back. You are also allowed your own one—sided 8 1/2” X 11” formula sheet Please put your name on each page. SOIDJW‘“ Name Problem 1. 20 points A pair of circular perfectly conducting plates of radius a are separated by a distance d << 21 to form a circular capacitor. Discuss the conditions under which it is valid to write the fields within the capacitor as E- — —Y&—z sin (0t, H~ 0. 53,44} 0" choNa/ke’NH-fi I u ,\_ f is 4:44? . 314V“ ’ 9 ° TL ’ > u modecj’amwm (“SWEE— Huwm: Wm” in; we M M Le/ g WQNSJCX‘ +1.4.» across Carma G C / M ) 5001‘ ”fltflL .9 ?dw'a’l st" 74"” “M” W” 7; MHA ion/wk Name Problem 2. 20 points (2 parts) An ideal lossless transmission line is terminated by a load impedance ZL equal to half the characteristic impedance Z0 of the transmission line, i.e., ZL = 20/2. An additional impedance with value ZQ is placed across the line a distance D in front of the load impedance. Transmission line Sinusoidal signal generator Characteristic ' l4— D——>| v ' impedance Z0 - ' (a) Let D = M4, where )t is the wavelength at a certain fixed frequency (00. What value of ZQ can be used to eliminate reflections on the generator side of ZQ at frequency (00? 2 a EL 1/141» W‘— (1” 2L. k? \\ 2%0 3 %& ( Sc @717” (b) Suppose the frequency is now increased to 20%. If the distance D and additional impedance ZQ are the same as from part (a), what fraction of the incident power is now reflected back toward the generator? ‘ Mob/91)) Mme‘VL (“*M‘” 349mm Name Problem 3. 30 points (3 parts) Consider a time—harmonic (elm) uniform plane wave propagating in free-space. In phasor notation the electric field is written E=Eo(jfi+z)e—jfiy (1) where E0 is a constant and B = 0) M080 . (a) Is the wave linearly polarized, circularly polarized, or elliptically polarized? Justify your answer. (No credit without justification.) fiat/W“ Name (b) Referring to figure 4.1 below, the wave of eq. (1) is incident at oblique incidence (9 #2 0) onto a dielectric medium with 8 = 880 and u = 2M0 . transmitted Is the reflected wave linearly, circularly, or elliptically polarized? Justify your answer, but no need for detailed numerical calculation. 4% 37/0, “Ha. “TE all TM “We; (0'35“ Mrowk‘ :0 ‘Q‘ 4"“) gnaw Name (c) Referring to Fig. 4.2, the wave of eq. (1) is incident at normal incidence (6 = 0) onto a dielectric medium characterized by e = 880, u = Zuo. The boundary between the air and dielectric medium is coated with a thin wire screen which is perfectly conducting in the z—direction and perfectly insulating in the X-direction. The spacing between the individual wires in the screen is much less than the wavelength of the plane wave. dielectric IL: 2‘10 ' a: 820 wire screen incident wave (normal incidence) Fig. 4.2 Is the reflected wave linearly, circularly, or elliptically polarized? How about for the transmitted wave? ' Justify your answers, but no need for detailed numerical calculation. C‘i’M‘t) TLuX QDlGA’D‘“ 50$Aqbwb‘f‘” ’Zfit f“ 3 ma,» Name Problem 4. 30 points (3 parts) switch eluses atif i=0: ' " fl t, Z RL V T "isfiiiséibnlii‘iéfl. " ' o z=w A transmission line has characteristic impedance ZO = 509, phase velocity u = c/2, and length w = 3 meters. Initially it is charged up to a voltage V0 = 10 volts and then left open—circuited at both ends. At t = O a switch suddenly closes connecting it to a load resistor RL at one end. ‘ a) Assume that the load resistor is matched to the characteristic impedance (RL = 20). Give the current in the load as a function of time. J an ‘heammswm (his. Tiugiautmfu’ ILZO _pt,\-fiz.o. ‘ \ Q hu‘vld WIN-L— U+,r+ 2) Ah f30+\ Swi’rd/x km 395% 4°” 0' (6w 6L e A‘Kk“. ; 0+ M whit smug Gwen 4%»er J” 7‘ ' 3““ “it ’ o + li—LJ/d ~20 i) “520* ’M/Wc v(5)évo «é rgwo 2&4wa -33. : box V [3.44: \IoJ‘ V+ H. u 3 a 'I-(5'+3 . r UhCMV/fl Q \A'wh' - ‘ Hut \H to ism CMM 06 ,‘ W 4.6L locu'bw“ (3:°)’ «(5° i/ww-h QOWWLWV' “W “k 50“ to “Hume/SA: \[J'K 14“} ,3un ‘ \l (5 :D\ " NW) bu tl‘k Mung} M¥MSW¢5$IM (1b) :11 - $0 ML a)? 3:0 ' RLX kw. Vb+v*(+) ~ “VH‘” USWOCZA . may (a .20 Problem 3 Additional Worksheet “ UH Cir-27:31} Tkts f5 T~L§ WV 1AM. [AM 3L: “I[O)+>, ' ._ +Ve ’ _1W‘ >‘° ‘6’”? W ‘5 41‘ ELMO: v~,,)~u—:(f afl M 1 O 76/ H"\ DW/q 2 4'0 “Ste $2 mm, b) Show that the time-integrated current passing through the load in (a) is equal to the total charge stored on the top half of the line pn'or to the closing of the switch. .__ h 1:270 01 to” ‘ _ Lg \ \ .‘ fl ULJ 2f 2 w (A ’ Q» -—~ N 6 C2 ) ‘LxC? ) 12° é: c{ %A_.\NL23\QHW (‘5 U0 . 2w 2 V0“ 12 3.: Man Name c) Now assume that RL = 259. Sketch the voltage waveform at the midpoint of the line (2 = w/Z) as a function of time. Be sure to indicate the values of the appropriate voltage amplitudes and transition times. \> I”. f:0+) U? v°+ VrUr'5/u) -V Bomiw} owl'lm i: -IL' i R1 vo+V+ 7C. 1‘ Vb} " “V4“ €%°+rlb> -V’D 117° : ‘%Vb Vr" flu ) (: kl‘b-J . _.,.. 55%; $2 a} 41;“ [K IVS ’ 3—“ Z) W - re - an“ 4"”? DVHGL ..... L be! 19 > Va + 311-4, QML‘MM mic gray”, : {cons Problem 4 Additional Worksheet 14 ...
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