Exercises

1) A telephone line has the following distributed parameters: R=30 /km L=100 mH/km G=0 C=20 F/km At 1 kHz, find: a. its characteristic impedance b. its propagation constant c. its phase velocity 1 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 2) A 2-wire air line has the following line parameters: R=0.404 m/m L=2.0 H/k G=0 C=5.56 pF/m For operation at 5 kHz, determine: a. the attenuation constant b. the phase constant c. the phase velocity d. and the characteristic impedance 1. TRANSMISSION LINES AND RADIOFREQUENCY CIRCUITS 3) A 2 km transmission line with Z0=100and =10 rad/m is connected to a load of 50. To get a voltage on the load of VL= 7V, what is the input voltage to the line? 2 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 4) A generator with 10 Vrms and RG=50, is connected to a 75load thru a 0.850-lossless line. Find the voltage on the load. 5) For a 50lossless transmission line terminated in a load impedance of ZL=100+j50 , find the fraction of the average incident power reflected by the load. 6) A 300feedline is to be connected to a 3 m long, 150line terminated in a 150resistor. Both lines are lossless and use air as the insulating material, and the operating frequency is 50 MHz. Determine: a. the input impedance of the 3 m long line b. the voltage standing wave ratio on the feedline c. the characteristic impedance of a quarter-wave transformer to be used between the two lines in order to achieve S=1 (or VSWR=1) on the feedline. 3 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 7) Find the time-domain equation corresponding to the current in a short-circuited transmission line. The current phasor I(z) is given. Remember that in general V0+ is a complex number:   z Z V zI  cos2 0 0   8) Consider a coaxial transmission line carrying a wave of 300 MHz. The voltage and current phasors along the line are described by the equations below: Being VG= -2j/3Vpeak, ZG=75, v=75%ꞏc, and L=  , answer the following questions: a. Find the equation of the wave impedance as a function of the position on the line, that is Z(z). b. Calculate the characteristic impedance Z0of the line and the wavelength inside the line. c. Find the reflection coefficient at the input port of the line, and the impedance at the load. d. Find the power dissipated in the load. 4 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC                     zjzj zjzj ee j zI ee j zV 3 8 3 8 3 8 3 8 5 1 180 5 1 18 5   9) A quarter wave transmission line is used to match a 100resistive load to a 50 line at a given frequency. a. Calculate the characteristic impedance of the matching section. b. Calculate the VSWR on the main line with matching transformer when the frequency is increased by 20%. 5 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 10) A television transmitter operating at channel133 has an internal impedance of 75 and an available power of 150W. The transmitter is connected to an antenna whose impedance (from channel 21 to 60) can be modelled by the equivalent circuit of the figure (values in , pF, and nH). The connection between the antenna and the transmitter is done by means of a 5m length coaxial cable (75 , propagation velocity 80%) having negligible losses. Calculate: a. the wavelength and the primary line constants of the cable (loop resistance, loop inductante, insulator conductance and insulator capacitance per meter); b. the impedance at the input of the transmission line (that is the impedance seen by the transmitter); c. the net power (incoming minus reflected) that the source delivers to the line (and consequently arriving to the antenna and being radiated). NOTE 1) Channel 20 is centered at 474 MHz. Channel bandwidth is 8 MHz. 6 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC (10 Cont’d) The antenna has to be matched to the transmitter. Use the following methods2: d. lumped elements connected to the antenna (provide two different solutions using only two elements); e. a coaxial cable stub (open-circuited or short-circuited) placed somewhere in the cable connecting the transmitter to the antenna (provide two solutions). NOTE 2) In any case the values of the lumped elements and the length of the transmission line should be provided. 7 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 11) A 1 GHz source having an internal impedance of 50 is connected to a load ZL being its value plotted in the Smith Chart attached: 8 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC a. Find the normalized and denormalized value of ZL. Draw all the possible matching networks having just two or less than two lumped elements (inductors or capacitors). Plot on the Smith chart the matching procedure and calculate the values of the lumped elements. b. Match ZLusing 50transmission lines. Indicate two possible solutions: one using an opencircuited stub and the other using a shortcircuited stub. Plot on the Smith chart the matching procedure and calculate the length of the lines. 12) Draw the schematic that generates the shifts on the Smith chart attached. For each element of the cascade, you should indicate the kind of element (resistor, inductors, capacitor, transmission line) and its value (resistance, capacitance, inductance, transmission line length and characteristic impedance in either case), and if they are shunt or in-series. The reference impedance is 50, the operating frequency is 300 MHz, and the propagation velocity c. The network does not include lossy elements. 9 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 16 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 21) Given the cascade of networks below (the loaded circulator is a two port device): a. Which are the ideal scattering matrix of each component at f0. b. Find the scattering parameters of the whole cascade (if required use the conversion table from Pozar’s book). Z0 G 17 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 22) A 10GHz wave with a power density of -97dBW/m2reaches a parabolic reflector with 40dB gain. Which is the maximum power transferred to the receiver? a) -101.5 dBW b) -100 dBW c) -98.5 dBW d) -97 dBW 23) Consider a wireless link between an ATC and an aircraft. If the distance between the tower and the aircraft is doubled, the received power (e.g. in the aircraft) is expected to be reduced by a factor of: a) 2 dB b) 3 dBW c) 4 dBW d) 6 dB 24) A parabolic reflector operating at a frequency fhas a radiation efficiency of erad. Its -3 dB-beamwidths in two orthogonal planes are Δθand Δ  . Consequently, its gain G must be close to: a) b) c) d)            4 log 10 rad e           4 log 10            4 log 20 rad e                 4 log 20 2 c f erad 2. ANTENNAS 18 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 25) Calculate the polarization loss factor of a linearly polarized antenna when receiving a linearly polarized wave, being the respective polarization vectors given by the following equations: a) 0b) c) d) yxayax ia ˆˆ ˆˆ     2 4a  2 2 2 1 2 a a   2 2 2 1 4 a a  26) The gain of an antenna is 23 dB and its input impedance is Z=100+j100 . If the reference impedance is 50 , which is the realizable gain of the antenna? a) 19 dB b) 21 dB c) 23 dB d) 25 dB 27) A parabolic reflector has a -3 dB beamwidth of, respectively, 2º in the H plane and 4º in the E plane. Its directivity is around: a) 15 dBi b) 37 dBi c) 43 dBi d) 86 dBi 19 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 28) Specify the range (r) dependency of the antenna gain. Consider its definition: a) Gdoes not depend on rb) Gincreases with r c) Gincreases with r2d) Gdecreases with r2 rad rad P U eG 4   20 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 29) An aircraft flights at constant altitude (h) approaching to an earth station. The aircraft transmits signals with constant power PTX using an (lets assume) omnidirectional antenna with gain GTX=GTX,0. On earth, a receiver at a slant range Rfrom the airplane, uses an antenna having a vertical pattern GRX=GRX,maxꞏf(  ). Find the angular dependence of the antenna receiver in order to have a constant received power on earth PRX. The graph below could help to understand the problem. NOTE: Assume neither antenna mismatch nor polarization losses. SOL: f(  )=K/sin2(  ) being K a real constant. 21 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 30)A half-wavelength dipole antenna has been characterized in terms of its input impedance versus frequency. Given its simulated performance in the graphs below answer the following questions: a. Find the frequency at wich the antenna is resonant and the frequency at wich the antenna is matched. b. Which are the input impedance and the return losses of the antenna at 4.8GHz? c. Assuming that the radiation efficiency of the antenna is 95%, which is the antenna efficiency at 4.8GHz. d. Make an estimate about the length of the dipole. 22 Aeronautical Communications C. Collado, J.M. González-Arbesú EETAC-UPC 31)Given the following array current distributions (assuming isotropic basic elements): Distribution 1: Distribution 2: Distribution 3: Distribution 4: a. Which is the one expected to have maximum directivity? Justify your response. b. Which is the one expected to have maximum radiated field instensity in the maximum direction? Justify your response.  00000 ,2,3,2, IIIIIIn  00000 ,,,, IIIIIIn  00000 2,2,2,2,2 IIIIIIn  00000 5.0,,5.1,,5.0 IIIIIIn