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Homework lecture course Antenna Engineering

RF&Microwave Research LaboratoryUniv.-Prof. Dr. M. Hein

Status: Summer term 2011

1. Basics 1: Describe the wave parameters of spherical waves and plane waves. Explain thedifferences. Where do plane waves occur in nature?

2. Basics 2: Review typical values for the dielectric permittivity and magnetic permeability ofmedia like vacuum, dry sand, sweet water, and sea water (maybe you find additional infor-mation on further media). Calculate the resulting values for the phase velocity and the waveimpedance of plane electromagnetic waves traveling through these media. Discuss the re-sults.

3. Wave propagation: Recapitulate the terms free-space propagation, reflection, and dif-fraction. What are typical conditions for these types of propagation to apply? What are therelevant properties of typical propagation media?

4. Path loss 1: Estimate typical values for the path loss of different radio systems (e.g., point-to-point transmission, satellite TV, WLAN). Make realistic assumptions on the distance be-tween transmitter and receiver, frequency of operation, and antenna gains.

5. Path loss 2: What would be the path loss for a monostatic radar for any one of the scenariiof problem No. 1, if the relevant radar cross-section was 0 dBsm? Discuss the differences!

6. Path loss 3: Prove that |1-exp(-jx)| = 2|sin(x/2)|; similarly: |1+exp(-jx)| = 2|cos(x/2)|. Canyou imagine where such terms occur?

7. Free-space propagation 1: Describe how a perfectly isotropic radiator would need to looklike. Explain why isotropic radiators do not exist in nature. Try to devise (in a thought experi-ment) a radiating element that comes close to an isotropic radiator. How would its radiationcharacteristics differ from that of an isotropic radiator?

8. Free-space propagation 2: On the transmit side of radio systems, the term EIRP is rele-vant. Explain this term and give examples of how to get numerical values. On the receiveside, the antenna quality factor G/T is important for the link budget. Find out typical values forEIRP and G/T of satellite links (e.g., TV) and estimate the corresponding path loss. Howshould the satellite and a ground terminal share the values of EIRP and G/T in the downlink(from satellite to terminal)? What are the advantages of "spot-beam"-antennas mounted on asatellite?

9. Maxwells equations 1: Consider a planar interface between two homogeneous but differ-ent media. Discuss the continuity conditions for electrical and magnetic fields at such an in-terface. As specific applications, derive the law of reflection and/or Snells law of refractionfrom these conditions.

10. Maxwells equations 2: Consider a planar radiating aperture. Show that the six Cartesiancomponents of the electrical and magnetic fields in the aperture plane (in the image domain)are fully determined by the tangential components. Which conclusions can you draw for thegeneration of a certain radiation pattern?

11. /2-dipole: Prove that the radiation pattern of a Hertzian dipole can be derived from theresults for the half-wavelength dipole. Analyse the radiation pattern of dipole antennas ofdifferent lengths (e.g. /2, , 3 /2). Compare the corresponding directivities, and the direc-tions of the main beams and nulls. Can you explain the differences?

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12. Dipole antenna: Let () be the radiation pattern (field magnitude) of a dipole antenna.Show that the directivity is then given by D( ) = 120/R rad []|()|2, with R rad the radiationresistance of the antenna.

13. Aperture antenna: Imagine a rectangular aperture with different amplitude taperings: rec-tangular, cosine, Gaussian. Derive by Fourier transformation the corresponding beamwidthsand sidelobe levels. Review further commonly used amplitude tapers from the literature.

14. Horn antenna: Investigate, and justify, a suitable rule-of-thumb to establish a relationshipbetween the geometry of a horn antenna and its directivity.

15. Patch antenna 1: Show that the law of induction yields for the magnetic field distributioninside a rectangular waveguide resonator (e.g., a dielectric patch resonator) of negligibleheight (h ):

0 z j

H z E=

In which direction do the surface currents flow? Which conclusions can you draw about themodes exisiting inside such a resonator?

16. Patch antenna 2: Think about what determines the polarisation of a patch antenna.Based on your results, how could one feed a linearly polarised patch antenna? Correspon-dingly, which geometrical feed configurations would you consider suitable to excite circularpolarisation?

17. Linear arrays 1: Estimate the beamwidth of a linear array for low elevation ("endfire", 90 o). Compare the result with that of the lecture for = 0 o ("broadfire") and discuss the dif-ference.

18. Linear arrays 2: Show that the angular scan range of a linear array, expressed byM=sin M, is limited by the normalised element separation to: d/ < 1/(1+ M).

19. Antenna matching 1: Consider first a L-section consisting of a series reactance X s and ashunt reactance X p, aimed at matching a real input resistance R 1 to a real output resistanceR2 > R 1. Show that |X s | = (R 1[R2-R1])1/2 and X sXp=-R 1R2. How large is the phase lag acrossthe L-section?

20. Antenna matching 2: Based on transmission line theory, show that the characteristic im-pedance and phase of a transmission line transformer, which matches a complex input im-pedance Z in = R in + jXin to a complex load impedance Z L = R L + jXL, is given, within the rangeof validity, by

2 2L in in L

0in L

R | Z | R | Z |Z

R R= and

0 L in

L in in L

Z (R R )tan

R X R X =

Discuss the different matching approaches of problems #19 and #20.

Optional problems

1. Linear arrays 3: Consider a (2x2) Butler matrix. Verify for the corresponding scatteringmatrix that the three key conditions (reciprocity, losslessness, orthogonality) hold. Sketch the

two resulting orthogonal beams and calculate the directions of the main beams for a givenelement spacing (e.g., half-wavelength).

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