Homework 3 Solution Set

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Arleen Ryan
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1 Homework 3 Solution Set Blake Chapter Problems points (1 per problem) 1. Calculate the length (not speciied in metric or English) o a practical hal-wae dipole or a requency o 150 MHz. A practical hal-wae dipole will be approximately 95% o a hal-waelength: 150MHz.0m m L ( 0.95) (0.95) 0. 95m 14.5 Note: The text uses the ormula L to ind antenna length. You'll get the same answer with ( MHz) this method, but it's better to stick with basic ormulas (like the waelength equation aboe) and apply ariations based on your understanding o the system. On most wire antennas, the 95% igure arises because the elocity o waes on the antenna is about 95% o the ree-space alue.. Calculate the eiciency o a dipole with a radiation resistance o 6 Ω and a total eedpoint resistance o 75 Ω. The eiciency o an antenna system can be ound by: η P P AD IN TOTAL 6Ω % 75Ω 3. Gien that a hal-wae dipole has a gain o.14 dbi, calculate the electric ield strength at a distance o 10 km in ree space in the direction o maximum radiation rom a hal-wae dipole that is ed, by means o lossless, matched line, by a 15 W transmitter. First, ind the EIP o the antenna:.14 /10 EIP PINGT (15W )(10 ) 4. 55W (Note conersion o decibels back to a power ratio the 10^(.14/10) actor does this). Now use the electric ield equation or ree space: E 30P d T (30)(4.55W ) 10km.7mV / m

2 4. eer to the plot in Figure.3 and ind the gain and beamwidth or the antenna shown. This igure is not well dimensioned, howeer, by inspection the gain appears to be approximately +5 dbi and the beamwidth (distance between 3 db points) appears to be about Calculate the EIP in dbw or a 5 W transmitter operating into a dipole with 90% eiciency. The gain o a 100% eicient dipole is +.15 dbi or 10^(.15/10) [1.64] W/W. Howeer, a lossy dipole has a gain o: G T 1.64 η W The radiated EIP is thereore: EIP PT GT ( 5W )(1.47W ) 36. 9W Howeer, the answer was requested in dbw units: P 36.9W dbw 10 log 10log dBW 1W 1W 6. Calculate the length (not speciied in metric or English) o a quarter-wae monopole antenna or a requency o 900 MHz. A quarter-wae antenna is just 95% o a quarter-waelength (hal o a hal-wae dipole): 0.333m 900MHz 0.333m L ( 0.95) (0.95) 0.079m 79mm Calculate the optimum length (not speciied in metric or English) o an automobile FM broadcast antenna, or operation at 100 MHz. The hidden assumption in this question is that the optimum antenna is a quarter-wae monopole. That may not necessarily be true, howeer, in terms o this assumption, the solution is the same as problem 6: 100MHz 3m 3m L ( 0.95) (0.95) 0.715m 71. 5cm 4 4

3 H. Draw a dimensioned sketch o a discone antenna that will coer the VHF range rom 30 to 300 MHz. The ollowing design parameters are deried rom the AL Antenna Book ( ): H hypotenuse; H /4 at the lowest requency o the unit. Θ 64 (total angle o the cone; each hal is 3 ) Disc diameter D TOP should be about 70% o cone bottom diameter D(TOP) 3 D() From these parameters, we can derie the relationships by using some trigonometry: sin Θ D 0.53 H D 1.06H D TOP opposite hypotenuse 0.70D D H Θ 3 (0.70)(1.06) H 0.74H To design the discone, let H be about a quarter-waelength at the lowest requency o interest, the apply the aboe relationships. At 30 MHz, the dimensions in the igure will be: H 4. 5m 4 30MHz D 1.06H (1.06)(.5m). 65m D TOP 0.74H (0.74)(.5m) 1. 5m

4 9. A helical antenna consists o 10 turns with a spacing o 10 cm and a diameter o 1.7 cm. a) Calculate the requency at which this antenna should operate. The circumerence o the turns is about one waelength, and the spacing is 1/4 waelength or this type o antenna. First, we must ind the waelength: πd π ( 1.7cm) 39. 9cm The spacing between the coils is 1/4 waelength, and our times this alue yields 40 cm. The requency will be approximately: 75MHz 39.9cm (Note: Actual resonant requency will probably slightly lower due to 95% elocity actor on conductor.) b) Calculate the gain in dbi at the requency ound in part (a). Equation. calculates the gain as a ratio: 15NS( πd) (15)(10turns)(10cm)( π 1.7cm) G 37.6W 3 3 (39.9cm) In dbi, this gain is expressed as: G dbi 10 log( G) 10log(37.6W ) dBi c) Calculate the beamwidth at the requency ound in part (a). The beamwidth can be ound by Equation.9: Θ 5 πd 5(39.9cm) NS π (1.7cm) 39.9cm (10T )(10cm) deg 3.3deg

5 10. Assuming the aperture o a pyramidal horn is square, how large does it hae to be to hae a gain o 1 dbi at 1 GHz? Equation. states that the gain o a rectangular horn antenna is: d E d G H Since our antenna's aperture is square, d d E d H and we can write: d G Soling or d, the aperture cross section, we get: G 1dbi /10 d Where G the required gain 10 63W and 5mm 1GHz G 63W d 5mm 7. 4mm

Half-Wave Dipole. Radiation Resistance. Antenna Efficiency

Half-Wave Dipole. Radiation Resistance. Antenna Efficiency Antennas Simple Antennas Isotropic radiator is the simplest antenna mathematically Radiates all the power supplied to it, equally in all directions Theoretical only, can t be built Useful as a reference: