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Branching and Spatial Structure

Another element in understanding fractal antennae is the concept of branching. Take the simple branching element shown in Fig. 9 where the current is divided between the two branching elements. We can compute the radiation field, for a propagating current I, as

 

where and is the variation from the single dipole, i.e. we recover the dipole as .

  
Figure 9: A simple branching situation in which we distribute the current among the branching elements.

Again, the analysis can be simplified in the limit for small , i.e. . Of course P(0) is the dipole contribution, and is the change in the radiated power density due to the line branching. The dipole has a maximum in the radiated power density , while the branching contribution goes as . The function f depends on the given parameters, but its maximum is of the order with clear regions in where it is positive. Therefore, the branching process can give rise to an increase in the radiated power density at certain position. Of course this increase is due to the increase in the path length. This effect will saturate as is increased passed one, since then the strongest contribution will come from the dipole radiator given by 2.

A interesting and manageable broadband antenna can be described in terms of the Weierstrass functions [Werner and Werner, 1995]. We take successive branching elements, as shown in Fig. 10a, where we distribute the current at each branching point so that the branching element keeps a fraction of the current. The n branching element is displaced by a factor with respect to the origin. If we concentrate only on the contribution from the last branching set, as shown in Fig. 10a, we can write the field as

where we have redefined and a . In the limit and N we obtain the Weierstrass function that is continuous but not differentiable, i.e. is a fractal, and furthermore, its dimension in the sense given by Eq. ( 1) is d. For the purpose of illustration we truncate the above sum to N=8. In Fig. 10b, we show the dependence of the field as a function of with a for . The parameters values are shown in the figure caption. Figure 10c shows the gain factor given by

as a function of the dimension . We chose this range since the fractal already has a dimension 1 in the perpendicular directions, i.e. D=1+d.

  
Figure 10: The branching process to produce a Weierstrass radiation pattern. (a) The brunching process with the branching length increasing as and the current decreasing as . (b) the radiation pattern with given perfect coherence. (c) The gain vs the dimension. It also contains the parameters used in all 3 figures. (d) Patial coherence for .

Note the increase in the gain as a function of dimension. In general, there is an optimal value of D that generates the highest power density and that does not necessarily has to be for D=2. In Fig. 10b all the elements from the antenna add up coherently at , hence providing perfect coherence. For a finite the propagation brings a different phase shift at each element. Figure 10d shows the effect for as a function of . Note that at no point there is perfect coherence, but there is clear partial coherence. The peak value of E is actually sensitive to .

Even though fractal antennae naturally lead to the concept of an increase in the peak radiated power, it also has a second important consequence due to branching. As we have seen in the case of the Wiertrauss function, fractal antennae naturally result in the generation of a spatial structure in the radiated power density. This interplay between the spatial structure and the increase in the peak radiated power are the essential ingredients of fractal antennae and why they are so important. A clear example can be illustrated in Fig. 10d where there are multiple relevant peaks of the radiated power in space.



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Next: Modeling Lightning as Up: Radiation and Simple Previous: Fractal Tortuous Walk