329fall09hw11 - ECE 329 1 A system of3 A system of two...

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Unformatted text preview: ECE 329 1. A system of3. A system of two connectedby ransmission and es drivenby a resistive age two connected transmission lines driven t a voltage source lin terminated by a volt load of 50 Ω is load infthe0 Ω isbelow.wnswitch he figuretime t = 0 and the positive voltages at shown o 5 figure sho A in t is closed at b elow. A switch is closed are measured for 5µs, giving the bounce diagram in the figure. Homework 11 Due: Tuesday, November 17, 2009, 5 PM s t measured for 5 µs giving the b ounce diagram in the figure. The imp edance and transmission time of line 1 are Z1 and T1 and t l The impedance and transmission time T = v of line one are Z1 and T1 and those of line 2 are Z2 the Using the identify shown in owfigures, deduce te s in appropriat in and T2 , respectively. figures, information the follthe ing parametherfollowing parameters e units appropriate units: a) b) c) d) e) f) g) h) i) j) a) Transmission time T1 Transmission time T1 Transmissionb) Tr2 nsmission time T2 time T a Impedance Z2 c) Imp edance Z2 Reflection coefficient Γ12 between lines 1 and 2 Impedance Zd) Reflection co efficient Γ12 1 Source resistance Rg eV Source voltage ) 0Imp edance Z1 −++ Transmitted voltage urce on sistance R f ) SoV re line 2 g Steady state voltage V1 on line 1 as t → ∞ Steady state g) Sou2 ce line l2 agt → ∞ voltage V r on vo t as e Vo b etween lines 1 and 2 h) Transmitted voltage V −++ on line 2 j) Steady state voltage V2 on line 2 as t → ∞ i) Steady state voltage V1 on line 1 as t → ∞ 1 4. Two T.L.’s with characteristic imp edances Z1 and Z2 are joined series resistance R as shown in the diagram b elow. ECE 329 Homework 12 Due: Mon, Aug 3, 20 1. Consider a lossless TL which is op en circuited at b oth ends. If l = 20 m is the length of th v = 2 c = 2 × 108 m/s for the line, 3 a) What are all the resonance frequencies of the line — frequencies at which source-free o 2. Consider(stlosslessgT.L ves) ois voltage and curboth ends.susthened onof he line — le=pressed in M a andin wa which f open circuited at rent are If tai length t the T.L. is x 20 m, 2 and v = Hint: to satisfy the “op en” b oundary conditions at b oth ends of the line, we want t 3 c m/s for the line, a) What asor to resonant afrequenciesnof d = line − frequencies lat iwhichequency is oscillations Varia ph are the vanish t d = 0 a d the l when the oscil at on fr source-free resonant. (standingewavesance from the load enareosustainede, iexpressed−zMHz units?oad lothation is take to th dist ) of voltage and current d f the lin − .e., d = in and the l Note cat the current phasor must vanish at the end points to satisfy the “open" boundary conditions. b) Sketch the shap es of current mag˜ itude |I (d)| vs d for the line corresp onding to the t n b) Sketch the shapes of current magnitude |I (d)| as a function of d (distance from the load end) resonancetorthe encielowest b el each plot clearlyBend expllabeleeach plotefly. f equthree s. La resonance frequencies. a sure to ain ach bri clearly. corresponding c) c) Rep(b)tfor voltage oltage mag˜ (tu|de |Vfunction of. d. Repeat ea (b) for v magnitude |V id) as a (d)| vs d n 2 3. Consider a transmission line segment having propagation velocity v = 3 c m/s, characteristic impedance 2. Consider a transmission line segment of propagation velo city v = 2 c = 2 × 108 m/s, cha Z0 = 50 Ω, and length l. As shown in the figure below, the segment is connected3in parallel with a mp edan e Z an 50 Ω g = le current As sh i wn i Re g˜ jωt }, where I t 2 s A is the source u ˜ 50 iΩ resistorcand o =ideal (,Zand 0) ngth l. source o(t) =n fi{Iere (a) b elow,= he 0 egment is connected with phasorΩ reωistor×a10d rad/s. eThe equivalent circuit, i(t) = Rof{I ej ωt }, where IZ= )2of 0 A is a 50 and s = π n 8 an id al current source in terms e input impedance (l ∠ current the urrent d =asor alsodshown below. 108 rad/s; figure (b) depicts an equivalent circuit in term c T.L. at ph l, is an ω = π × imp edance of the transmission line at d = l, namely Z (l) ≡ I (l) I (l/2) I (0) IR 2 0A 50 Ω V (l) I (l) . + V (l) + V (l/2) + V (0) 2 0A IR Z (l) = 50 Ω V (l) I (l) - - - d l l/2 0 (b) Equivalent circuit (a) Transmission line circuit I answering the wavelength λ (in m) a the T.L.? a)nWhat is the signalfollowing questionsonssume that the circuit ab ove is in sinuso dial steady b) a) Wan t is thetermination vellocated λ (in m) on the=transmission line? Since ha “open" signal wa is ength on the line at d 0, what is the pertinent boundary ˜ condition involving the phasor I (0) for all possible lengths l? c) b) Sitheetransmission termination”standing ted oin theeabove circuit= 0allwhat is the p ertinent Does nc an “op en line support a is lo ca wave n th line at d for , values of non-zero condit l? Explain.ion” involving the phasor I (0) for all p ossible line lengths l? ˜ d) c) Dois s the transmission line supp ort m)standing IR = 2in the Explain.circuit for all values What e the smallest non-zero value of l (in a if phasor wave 0 A? ab ove ˜ e) For ll?determined our answerwhat is phasor V (l/2)? Explain. Justify y in part (d), . ˜ f) For l determined in part (d), is I (l/2) = 0 possible? Explain. d) What is the smallest non-zero value of l (in m) if phasor IR = 2∠0 A? Explain. ˜ g) Given that I (l/2) = 0, what is the smallest possible value of l? Explain. w h) e) For l determinpartin part (dis, V (hatExplain.sor V (l/2)? Explain. For l determined in ed (g), what ) ˜ l)? is pha i) What r ltheesmallestenon-zero rvalue)ofis if (lR 2) 0? 0 p ossible? Explain. f ) Fois d termin d in pa t (d , l I I / = = Explain. g) What is the smallest non-zero alue o l characteristic 4. A quarter-wavelength long transmission linevsection fof if IR = 0? impedance Z0 = 50 Ω is terminated by an unknown impedance ZL at one end. The phasor input voltage of the quarter wave h) Given that I (l/2˜ = 0, what is the ˜malles˜ p ossible value of l? Explain ) s t section at the other end is Vin = 100 0◦ V. Let Iin and IL denote input and load current phasors, respectively, with termined in part (h), whatwayV (l)? Explain. one another and with Vin . i) For l de current directions defined in a is compatible with ˜ 3. a) Assuming wavethe gth lis ng topen missioL =in/Zse=tionwhateismLnated by an unknown imp ed A quarter- that len load o not rans (i.e., Y n l 1 e L c 0), is t r I i ? b)nIs enpossible forp˜in =oltag0◦ of the load rter-wave section at the other end is Vin = 100∠0o V o e it d. The in V t v 100 e if the quais open? Explain. u ˜in if ZL = 100 Ω? c) oWhat n).I Also the characteristic imp edance of the line is Zo = 50 Ω. n tatio is Let Iin and IL denote input and load current phasors, resp ectively, with current directions a compatible way with one another and with Vin : 2 1 ...
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