Prof. D. R. Wilton

1. ECE 3317 Prof. D. R. Wilton Notes 22 Antennas and Radiation [Chapter 7]

2. Antenna Radiation + - Antenna Radiation We consider here the radiation from an arbitrary antenna. S z r y "far field" x The far-field radiation acts like a plane wave going in the radial direction.

3. Antenna Radiation (cont.) z S z H y S E H y x E x The far-field has the following form:

4. Antenna Radiation (cont.) The far-field Poynting vector is now calculated:

5. Antenna Radiation (cont.) Hence we have or Note: in the far field, the Poynting vector is purely real (no reactive power flow).

6. Radiation Pattern The far field always has the following form: In dB:

7. Radiation Pattern (cont.) 30° 30° 60° 60° 0 dB -10 dB -20 dB -30 dB 120° 120° 150° 150° The far-field pattern is usually shown vs. the angle  (for a fixed angle ) in polar coordinates.

8. Radiated Power The Poynting vector in the far field is The total power radiated is then given by Hence we have

9. Directivity The directivity of the antenna in the directions (, ) is defined as The directivity in a particular direction is the ratio of the power density radiated in that direction to the power density that would be radiated in that direction if the antenna were an isotropic radiator (radiates equally in all directions). In dB, Note: The directivity is sometimes referred to as the “directivity with respect to an isotropic radiator.”

10. Directivity (cont.) The directivity is now expressed in terms of the far field pattern. Hence we have Therefore,

11. Directivity (cont.) z +h feed y x -h Two Common Cases Short dipole wire antenna (l << 0): D = 1.5 Resonant half-wavelength dipole wire antenna (l = 0 / 2): D = 1.643

12. Beamwidth The beamwidth measures how narrow the beam is. (The narrower the beamwidth, the higher the directivity). HPBW = half-power beamwidth

13. Gain and Efficiency The radiationefficiency of an antenna is defined as Prad= power radiated by the antenna Pin= power input to the antenna The gain of an antenna in the directions (, ) is defined as In dB, we have The gain tells us how strong the radiated power density is in a certain direction, for a given amount of input power.

14. Infinitesimal Dipole z I l y x The infinitesimal dipole current element is shown below. The dipole moment (amplitude) is defined as Il. The infinitesimal dipole is the foundation for many practical wire antennas. From Maxwell’s equations we can calculate the fields radiated by this source (see chapter 7 of the textbook).

15. Infinitesimal Dipole (cont.) The exact fields of the infinitesimal dipole in spherical coordinates are

16. Infinitesimal Dipole (cont.) In the far field we have: Hence, we can identify

17. Infinitesimal Dipole (cont.) 30° 30° 60° 60° 0 dB -9 -6 -3 120° 120° 150° 150° The radiation pattern is shown below.

18. Infinitesimal Dipole (cont.) The directivity of the infinitesimal dipole is now calculated Hence

19. Infinitesimal Dipole (cont.) Evaluating the integrals, we have Hence, we have

20. Infinitesimal Dipole (cont.) 30° 30° 60° 60° 0 dB -9 -6 -3 120° 120° 150° 150° The far-field pattern is shown, with the directivity labeled at various points.

21. Wire Antenna z +h I(z) feed I0 y x -h A center-fed wire antenna is shown below. A good approximation to the current is:

22. Wire Antenna (cont.) +h +h l l -h -h A sketch of the current is shown below. resonant dipole (l = 0 / 2, k0h = /2) short dipole (l <<0 / 2)

23. Wire Antenna (cont.) +h l -h Short Dipole The average value of the current is I0/2. Infinitesimal dipole: short dipole (l <<0 / 2) Short dipole:

24. Wire Antenna (cont.) z R +h r dz' z' feed y x -h For an arbitrary length dipole wire antenna, we need to consider the phase radiated by each differential piece of the current. Far-field observation point I(z') Infinitesimal dipole: Wire antenna:

25. Wire Antenna (cont.) z R +h r dz' feed y x -h Far-field observation point

26. Wire Antenna (cont.) z R +h r dz' feed y x -h Far-field observation point  Note:

27. Wire Antenna (cont.) z R +h r dz' feed y x -h Far-field observation point

28. Wire Antenna (cont.) We define the array factor of the wire antenna: We then have the following result for the far-field pattern of the wire antenna:

29. Wire Antenna (cont.) Using our assumed approximate current function we have The result is (derivation omitted)

30. Wire Antenna (cont.) In summary, we have Thus, we have

31. Wire Antenna (cont.) For a resonant half-wave dipole antenna The directivity is

32. Wire Antenna (cont.) Results

33. Wire Antenna (cont.) Radiated Power: Simplify using

34. Wire Antenna (cont.) Performing the  integral gives us The result is then

35. Wire Antenna (cont.) z +h I(z) feed y x -h The radiation resistance is defined from Circuit Model Z0 Zin For a resonant antenna (l  0/2), Xin = 0.

36. Wire Antenna (cont.) The radiation resistance is now evaluated. This yields the result l0 /2 Dipole:

37. Wire Antenna (cont.) h Feeding coax The result can be extended to the case of a monopole antenna

38. Wire Antenna (cont.) Vmonopole Vdipole I0 I0 + + - - Virtual ground dipole This can be justified as shown below.

39. Receive Antenna l = 2h + Einc VTh - ZTh VTh + - The Thévenin equivalent circuit of a wire antenna being used as a receive antenna is shown below.