1. Applied Electromagnetic Waves
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes 22
Antenna Patterns

2. Infinitesimal Dipole
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 (e.g., see Chapter 7 of the Shen and Kong textbook).

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

4. Infinitesimal Dipole (cont.)
In the far field (r  ) we have:
Hence, we can identify

5. Infinitesimal Dipole (cont.)
The radiation pattern is shown below. -9 -3 -6
0 dB
30°
60°
120°
150°
45o
HPBW = 90o

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

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

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

9. Wire Antenna A center-fed wire antenna is shown below.
Feed
A good approximation to the current is:

10. Wire Antenna (cont.)
A sketch of the current is shown below for two cases.
Resonant dipole (l = 0 / 2, k0h =  / 2)
Short dipole (l <<0)
Use

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

12. Wire Antenna (cont.)
For an arbitrary length dipole wire antenna, we need to consider the radiation by each differential piece of the current.
Far-field observation point
Feed
Infinitesimal dipole:
Wire antenna:

13. Far-field observation point
Wire Antenna (cont.)
Far-field observation point
Feed

14. Far-field observation point
Wire Antenna (cont.)
Far-field observation point
Feed
Note:

15. Far-field observation point
Wire Antenna (cont.)
Far-field observation point
Feed
It can be shown that this approximation is accurate when

16. Far-field observation point
Wire Antenna (cont.)
Far-field observation point
Feed
Hence we have:

17. 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:
Note:
The term in front of the array factor is the far-field pattern of the unit-amplitude infinitesimal dipole.

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

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

20. Wire Antenna (cont.)
For a resonant half-wave dipole antenna: or

21. Wire Antenna (cont.) The directivity is:
The result (from numerical calculations) is:

22. Wire Antenna (cont.)
Results

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

24. Wire Antenna (cont.) Performing the  integral gives us
After simplifying, the result is then

25. For a resonant antenna (l  0 / 2), Xin = 0.
Wire Antenna (cont.)
The radiation resistance is defined from
Feed
Circuit Model
For a resonant antenna (l  0 / 2), Xin = 0.

26. The radiation resistance is now evaluated.
Wire Antenna (cont.)
The radiation resistance is now evaluated.
Using the previous formula for Prad, we have:
Resonant l0 / 2 Dipole:

27. The result can be extended to the case of a monopole antenna
Wire Antenna (cont.)
The result can be extended to the case of a monopole antenna
Feeding coax
(see the next slide)

28. This can be justified as shown.
Wire Antenna (cont.) + -
Monopole
This can be justified as shown. + -
Dipole
Virtual ground