Unformatted text preview: ECE 329 Homework 10 Due: Tuesday, November 10, 2009, 5 PM 1. A monochromatic plane wave with Ei = 5 sin(ωt − βy )ˆ V/m is propagating in a vacuum in the +ˆ x y direction towards the y = 0 plane, which happens to be the boundary of a perfect dielectric having permittivity = 4 0 and permeability µ0 in the region y > 0. Calculate: a) The reﬂection and transmission coeﬃcients Γ ≡ at y = 0.
η2 − η1 η2 + η1 and τ ≡ 2η2 η2 + η1 for the described interface ˜ ˜ b) The reﬂected and transmitted electric ﬁeld phasors Er and Et , respectively. ˜ c) The incident, reﬂected, and transmitted H phasor waveforms. d) The time-averaged power density (in W/m2 ) transported by the incident, reﬂected, and transmitted waves. Is your answer compatible with the conservation of energy? Explain. 2. 300 Ω twin-lead transmission lines are commonly used to connect TV sets and FM radios to their receiving antennas. For the twin-lead, the geometrical factor GF relating capacitance per unit length C and inductance per unit length L to permittivity and permeability µ, respectively, is GF = π
D cosh−1 ( 2a ) where 2a is the diameter of each wire (cylindrical conductor) of the twin lead and D is the distance between the centers of the wires. Assuming = 0 , µ = µ0 , and 2a = 1 mm, calculate D for twin-lead transmission lines having the following characteristic impedances Z0 ≡ a) Z0 = 75 Ω b) Z0 = 300 Ω c) Z0 = 450 Ω 3. Telegrapher’s equations − ∂V ∂I =L ∂z ∂t and − ∂I ∂V =C ∂z ∂t
L C: govern the voltage and current waveforms V (z, t) and I (z, t) that propagate on lossless transmission line (T.L.) systems. a) If V (z, t) = cos(ωt − βz ) is a solution to the telegrapher’s equations, determine the associated current waveform I (z, t) by diﬀerentiating and integrating one of the telegrapher’s equations. b) Determine the wavenumber β (a positive number) for the voltage and current waveforms above, in terms of constants L and C . Hint: You will need to substitute I (z, t) found in part (a) into the other telegrapher’s equation and solve for V (z, t). 4. Consider a T.L. with a characteristic impedance Z0 = 50 Ω, length l = 300 m, and propagation velocity v = 3 × 108 m/s. A voltage source f (t) with an internal resistance Rg = Z0 is connected to the z = 0 end of the T.L. and the z = l end is terminated by a load resistance RL = 2Z0 . a) Determine the injection coeﬃcient τS and the reﬂection coeﬃcients ΓL and ΓS at the load and source ends of the T.L., respectively. 1 ECE 329 Homework 11 Due: July 30, 2009, 5P 1. Consider a T.L. with characteristic imp edance Zo = 50 Ω, length l = 300 m, and propagation velo ci v = √1 C = c = 3 × 108 m/s. A voltage source f (t) with an internal resistance Rg = Zo is connected L one end of the T.L. (at z = 0) and the other end (z = l) is terminated by a load resistance RL = 2Z a) Construct diagrams diagram” the voltage V (z, ) and e, V (z , ), a td current, I on the b) Construct “bounce" a “b ounce describing to determine the tvoltagcurrent tI (z,n) variations (z , t), variatio line for 0on zhe line fo0 < < < < µsafor t (t)0=or(t)(t) = δ(t). < t < l and r 0 t z 4 l nd f > f δ f . l l l c) Write b) Write your expreVs( 2 nt)for V I 2 ,l t) ) nd weighteds sumshof d sums of apprdelayedly delayed impuls the expressions for s io , s and ( ( 2 , t a as I ( 2 , t) a weig te appropriately opriate impulses δ(t). δ (t). c) l , lo V ( lI l) t s a a function of t 0 0 < µ µs, if f 60 = ) u . ) Hint: u e th the d) Plot V ( 2Pt) tand 2 ,(t2 , a) asfunction of t for for< t< t6< s6if f (t) =(t) u(t60V(t—V. Hint: sUse e convoluti of h res result of t (b wi h 60 60) . convolution tofethe ult of parpart)(b)twithu(tu(t). 2. A with internal resistance re = 60 e Rg an 0 Ω nd an op en c voltage f (tu = 60δ (t e f (t) 5. A generatorgenerator with internal RgsistancΩ and = 6openacircuit output ircuit outp) t voltag) feeds = 60δ( fe ds ckt diagram on the left below) the has b e unknown characteristic impedance r s ic i a T.L. (seeethe a T.L. (see the ckt diagram on that left anlow) that has an unknown characteZi0 tandmp edan Zo resistive load termination R t rm unknown distance L now di generator. om he gen an unknown and an unknown resistive loadatean ination RL at an unkfromnthestance, L, frThetvoltageerator. L a atisaadistance 300 m from tfromethe agenerator (where L, <hL) oltage waa etime-dependenceon of tim waveform d t nce z = z = 300 m he g ner tor, smaller than z t e v exhibits v form as a functi is as pl below elo the right (tick which is plotted otted bon w on the right.marks are spaced by 1 µs). a) Determine the imp edance of the transmission line, Zo . b) Determine the load reﬂection co eﬃcient ΓL and the load resistance, RL . a) Determine etermine the lengtimpedanceaofsmissT.L.line, L in meters. c) D the characteristic h of the tr n the ion b) Determinektheh a b ounce diagram usingΓthe axes shownresistancerRL e current waveform I (z , t) — n d) S etc load reﬂection coeﬃcient L and the load b elow fo th c) Determine theolength of eform as we have lineen done — for 0 < t < 10 µs. Be sure to mark the numeric the v ltage wavthe transmission oft L in meters value for the a for tud voltage and nt in t e diagram d) Sketch bounces diagrams mplithe e of the currecurrenthwaveforms. for 0 < t < 10 µs. Be sure to label your axes and mark the numerical values for the amplitude in the diagrams. e) Write the algebraic expressions for the voltage and current waveforms as a function of (z, t) for the domain 0 < z < L and 0 < t < 10 µs. e) What is the algebraic expression for the current waveform as a function of (z , t) for the doma 0 < z < L and 0 < t < 10 µs. 1 2 ...
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- Spring '08
- Transmission line, Impedance matching, Characteristic impedance, load resistance RL, internal resistance Rg, twin-lead transmission lines