Name:GTID:ECE 6390: Satellite Communications and Navigation SystemsTEST 1 (Fall 2009)• Please read all instructions before continuing with the test.• This is a closed notes, closed book, closed friend, open mind test. On your desk you shouldonly have writing instruments and a calculator.• Show all work. (It helps me to give partial credit.) Work all problems in the spaces below theproblem statement. If you need more room, use the back of the page. DO NOT use or attachextra sheets of paper for work.• Work intelligently – read through the exam and do the easiest problems first. Save the hardones for last.• All necessary mathematical formulas are included either in the problem statements or the lastpage of this test.• You have 80 minutes to complete this examination. When the proctor announces a “last call”for examination papers, he will leave the room in 5 minutes. The fact that the proctor doesnot have your examination in hand will not stop him.• I will not grade your examination if you fail to 1) put your name and GTID number in theupper left-hand blanks on this page or 2) sign the blank below acknowledging the terms of thistest and the honor code policy.• Have a nice day!Pledge Signature:I acknowledge the above terms for taking this examination. I have neither given nor received unau-thorized help on this test. I have followed the Georgia Tech honor code in preparing and submittingthe test.11. Short Answer Section (10 points)(a)A Answer is the only feasible power source for long-lasting, deep space missions.(b)True or False: an antenna’s gain pattern is always less than its directivity.(c)Which of the 4 medium conditions used to derive the Helmholtz wave equation is mostclearly violated when we introduce a radiating source?(d)A Answer is an RF device that relays a radio signal without any regenerative repeating.(e)A common noble-gas propellant for ion drives is Answer .2. Polar Orbit: You are to design a polar orbit (90-degree inclination) for a satellite withapogee radius of 30,000 km occurring directly above the North pole. Answer the followingquestions based on this scenario. (30 points)(a) Given that the orbit must clear the earth’s radius + 500 km at perigee, what is theshortest possible p eriod for this orbit? (10 points)(b) Given the conditions in part (a), what is the maximum eccentricity for this orbit? (10points)(c) When the satellite is at apogee, what is the lowest possible latitude an earth station couldhave and still theoretically receive a radio transmission? (10 points)23. Satellite Conspiracy Theory: An engineer files a complaint with the FCC claiming that itmay be possible to interrogate RFID tags from an LEO satellite (500 km overhead), violatingthe privacy of anyone who happens to be wearing a Wal-Mart shirt with an inventory tag stillattached or driving their car with an automatic toll-paying tag on the windshield. There aretwo types of proposed RFID tags for inventory control, tollway payments, and other identi-fication purposes. The first type is a battery-assisted tag that uses a small, internal batteryto drive its modulating circuitry; all power backscattered by this type of tag is due to thepartial reflection of an incident wave. Highway toll tags are a common example of this typeof tag. The second type is a purely passive tag that actually powers its modulating circuitryfrom the incident radio wave. Inventory tags are a common example of purely passive RFIDtags. Both types of RFID tags operate in the 915 MHz ISM band and have antenna gains ofapproximately 0 dBi. Assume that a 915 MHz satellite interrogator can transmit and receivewith 10 dBi of antenna gain. Based on this scenario, answer the following questions. (30points)SatelliteRFID TagGtagGsatr= 500kmDownlinkUplinkBackscatterGsatGtag(a) For a passive tag, at least 20µW must be received by the tag from the satellite inter-rogator in order for the tag to power-up. How much transmit power must be sent bythe interrogating satellite to make this happen? Based on your knowledge of spacecraftpower systems, does this seem reasonable? (15 points)(b) At least -100 dBm of back-scattered power must be received by the satellite in order toread any RFID tag’s data. How much transmit power must be sent by the interrogatingsatellite to make this happen? Does this seem reasonable? Hint: In a backscatter radiolink, power is transferred from satellite to RFID tag for the first leg of the link; the re-ceived power then becomes the transmit power for the tag (assuming no internal losses)and is send back to the satellite on a second link. (15 points)34. Plane Waves: A UHF satellite is transmitting a circularly polarized wave to a boat on thesurface of a calm lake. In the immediate area around the boat, the radio wave behaves like aplane wave. The equation for its electric field is˜E(r) = 100 (ˆx + ˆz) exp(−j60[ˆx −ˆz] ·r) µV/mwhere the direction ˆz is the zenith (directly overhead) and ˆy corresponds to 0◦azimuth (duenorth). Answer the following questions accordingly. (30 points)(a) The boat on the surface of the water is measuring the angles-of-arrival for this radio wave(and, hence, the position in the sky of the satellite). What is the azimuth and elevationangles-of-arrival for this radio wave? (10 points)(b) What is the corresponding magnetic field for the incident plane wave? (10 points)(c) What is the frequency for this radio wave? (5 points)(d) Estimate the amount of power that a dish antenna with 2 square-meters of aperture couldreceive from this plane wave if pointed properly. (5 points)4Cheat Sheetλf = c c = 3 × 108m/s µo= 4π × 10−7H/m ϵo= 8.85 × 10−12F/mPR= PT+ GT+ GR− 20 log10(4πλ)− 20 log10(r) Logarithmic Form¨r = r˙θ2−GMPr2¨θ = −2 ˙r˙θrT2=4π2a3µµ = GMpG = 6.672 × 10−11Nm2/kg2ME= 5.974 × 1024kgV =√µRb = a√1 − e2perigee = (1 − e)a apogee = (1 + e)aG = ηA4πλ2AeG ≈30, 000θHPBWϕHPBW(angles in degrees)Earth radius: RE= 6380 kmλf = vpω = 2πf k =2πλPhasor Transform: A cos(2πft + ϕ) −→ A exp(jϕ)Reverse Transform: x(t) =−→ Real {˜x exp(j2πf t)}˜E(r) = E0ˆe exp(j[ϕ −kˆk ·r])˜H(r) = H0ˆh exp(j[ϕ − kˆk ·r])H0=E0ηη =√µϵvp=1√µϵˆe ׈h∗=ˆkˆh = (ˆk ׈e)∗−ˆk = cos φ cos θ ˆx + sin φ cos θ ˆy + sin θ ˆzθ = tan−1−kz√k2x+ k2yφ = tan−1kykx(add π if kx> 0)Cross Product: a ×b = detˆx ˆy