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FRACTALCOMS Exploring the limits of Fractal Electrodynamics for

the future telecommunication technologies IST-2001-33055

Task 4.1 Final Report

Deliverable reference: D9

Contractual Date of Delivery to the EC: January 31, 2003

Author(s): José M. González, M. Barra, C. Collado, J. M. O’Callaghan, Jordi Romeu, Juan M. Rius, Eugenia Cabot, Michael Mattes, Juan R. Mosig, R. Gómez Martín, A. Rubio Bretones, M. Fernández Pantoja, F. García Ruiz , R. Godoy Rubio

Participant(s): UPC, EPFL, UGR

Workpackage: WP4

Security: Public

Nature: Deliverable

Version: 1.0 Date: December 30, 2003

Total number of pages: 110

Abstract:

Several fractal-shaped devices have been designed and simulated. Prototypes have been also fabricated for some of them. The new pre-fractal-shaped designs include antennas, resonators, filters, and antenna loads. This report gives a brief description of the more interesting ones. Special attention is paid to fractal-shaped resonators and filters fabricated with superconductor materials, since this is a very new and cutting-edge application of pre-fractal structures.

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TABLE OF CONTENTS 1 Summary................................................................................................................... 3

1.1 Superconductive resonators and filters............................................................. 3 1.2 Pre-fractal loading ............................................................................................ 6 1.3 Pre-Fractal Quasi self-complementary antennas .............................................. 8 1.4 The Y-Wired Sierpinski Monopole ................................................................ 11 1.5 3-D Pre-fractal tree ........................................................................................ 12 1.6 Design of small antennas using Genetic Algorithms (GA) ............................ 12 1.7 The H pre-fractal tree ..................................................................................... 13 1.8 Pre-fractal capillary devices ........................................................................... 15

2 Superconductive resonators and filters................................................................... 17 2.1 Fractal-shaped resonators ............................................................................... 17 2.2 Fractal-shaped filters ...................................................................................... 22

2.2.1 Quasi elliptical filters ............................................................................. 22 2.2.2 Chebychev filters.................................................................................... 25

2.3 Conclusions about the work on superconductor resonators and filters ......... 27 3 Pre-fractal loading .................................................................................................. 28 4 Pre-Fractal Quasi self-complementary antennas .................................................... 34

4.1 Self-complementary Koch-tie dipole.............................................................. 35 4.2 The Gosper Island........................................................................................... 37

5 The Y-Wired Sierpinski Monopole ........................................................................ 51 6 3-D Pre-fractal tree ................................................................................................ 55 7 Design of small antennas using Genetic Algorithms (GA) .................................... 58 References ...................................................................................................................... 60

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RELATED WP AND TASKS (FROM THE PROJECT DESCRIPTION) WP4. Fractal devices development.

Task 4.1: Design of fractal-shaped miniature devices. Objective: The advantages of fractal devices in the miniaturization is assessed by defining suitable geometries and analyzing them numerically. The following items will be considered:

a) Suggestion of usable fractal structures such as fractal wire antennas, fractal perimeter structures, space filling curves.

b) Numerical simulation of the resulting devices in order to assess the performance improvement over conventional ones.

Comparison of the new designs performance versus previously existing fractal miniature antennas (Koch).

1 SUMMARY

Several fractal-shaped devices have been designed, simulated and some fabricated. Here follows a description of the more interesting ones. Special attention is paid to fractal-shaped superconductive resonators and filters due to its first presentation in the frame of this project.

1.1 Superconductive resonators and filters One of the main conclusions of Task 1.1 is that most pre-fractal miniature antennas have a larger Q factor than conventional designs occupying the same enclosing sphere. While this is a major drawback for a communications antenna, the high Q is a very desirable feature for a microwave resonator. This makes pre-fractal structures excellent candidates to build miniature resonators for microwave filters. The use of pre-fractals in the miniaturization of planar microwave filters has been explored in this task. This type of filters have traditionally used half-wave resonant lines which required large substrate surfaces, especially if the frequency of operation is low. Classical miniaturization techniques have included several approaches of folding straight transmission lines. The work that we present in this report shows the initial steps taken to study the use of the Hilbert pattern to perform this folding, and the comparison with some non-fractal resonator geometries (the meander line). In recent years, the miniaturization trend of the planar filters has received a new and growing interest thanks to the discovery of high temperature superconductors (HTS). Indeed, while with traditional metallic microstrips, high current densities due to the miniaturization produce a very large amount of dissipative losses with the drastic and unacceptable worsening of filter performances, by using HTS films, thanks to their very low superficial resistance at microwave frequencies, it is possible to fabricate new compact planar resonators presenting however high (∼104) Q quality factors. So, nowadays, HTS filters seem to be the most adequate to satisfy the needs of the

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modern telecommunications systems, which, together a general criterion of maximum compactness of the microwave circuitry, require very low insertion losses levels and very steep skirts, thus reducing the interference problems coming out from adjacent bands signals. There are two main reasons for to build miniature resonators and filters using HTS materials:

• First, the substrate cost is high because it is difficult to make production HTS wafers larger than 2 or 3 inches (5 to 7,6 cm). As a result, the technical and scientific community working in HTS filters is very active in developing miniaturization techniques, and could potentially benefit from this work.

• The second reason for using HTS materials is a consequence of the miniaturization itself. Miniaturizing a planar resonator (while keeping its resonant frequency constant) tends to reduce its quality factor (Q), because the current density increases and this causes an extra metal loss. The use of HTS minimizes this effect, so higher miniaturization is possible without a significant degradation of the resonator performance.

Resonators We have focused on the Hilbert curve because it can pack the maximum length of a line in a given (square) area (see Fig. I). By simulating the performance of microstrip line resonators following this curve, we have found that they have a lower resonant frequency than a similar non-fractal geometry (meander line) occupying the same area and having the same line length. We have also found that the tolerance in variations in substrate thickness is lower for the Hilbert resonator.

The performance of the single Hilbert resonator in Fig. I(a), with a side of 3.58 mm and f0=2 GHz, has been tested at 77K showing a Q value about 30,000. The resonant frequency is about 30 MHz lower than the meander line of Fig. I(b). Both resonators occupy the same are and have the same line length. The per cent relative shift of the resonant frequency when the thickness substrate changes 5% is 1% for the Hilbert resonator and 1.5% for the meander line. Filters We also report on the minor modifications that are necessary in the Hilbert geometry to implement microwave filters. For example, the design of elliptic and quasi-elliptic

b)a)

Figure I: a) Hilbert resonator with k=4. b) Equivalent meander resonator.

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filters requires the implementation of inter-resonator couplings of opposite signs, and this is readily achieved if we can perform couplings that are dominated either by magnetic fields or by electric fields. To do this it is necessary that, at resonance, the electric and magnetic field maxima are located at the periphery of the resonator layout. The Hilbert microstrip resonator in Fig. I(a) does not achieve that: it has two separate electric field maxima, and the magnetic field maximum is close to the center of the layout. A slight modification of this layout (Fig. II) shows a resonator with a single electric field maximum (upper side of Fig. II) and a single magnetic field maximum (lower side of Fig. II), both at the periphery of the layout.

Different HTS pre-fractal filter configurations with quasi-elliptical and Chebychev responses have been designed, showing the flexibility of this type of resonator and its capability to obtain also very low couplings at relatively small distances. By using YBCO commercial 10 mm square films on MgO, one four pole filter quasi elliptical at f0 close to 2.45 GHz (Fig. III) and one Chebychev filter at f0=1.95 GHz have been fabricated and tested. The measured minimum insertion losses (0.1-0.2 dB) confirm the good trade off between quality factor and reduced dimensions. The filters performances appear without distortions until to Pin=10 dBm.

Figure III: Cascade-quadruplet quasi-elliptical filter: a) Geometry, b) Frequency

response simulated and measured in liquid nitrogen.

Figure II: Modified Hilbert resonator to build ellipctic filters.

a) b)

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1.2 Pre-fractal loading The low radiation resistance and high quality factors of pre-fractal designs suggests a very interesting application of pre-fractal technology: capacitive loads for monopole antennas. The comparisons among several iterations of a Hilbert pre-fractal used as top-loading of monopoles, with different ratios of pre-fractal size over the height of the monopole (Fig. IV), revealed that:

• For almost the same efficiencies and Q factors than a λ/4 monopole, smaller antennas can be fabricated (approx. 70% lower, according to simulations);

• Lower size reductions (in terms of k0a, being a the radius of the minimum sphere that encloses the monopole and its image on the ground plane, and k0 the wave number at self-resonance) than with a conventional top-loading with a circular plate (simulations show that the ratio k0a can not be lower that 0.4) can be attained using pre-fractal top-loading.

• Standard printed card board photo-etching technology can be used for fabricating the pre-fractal top-loaded monopoles.

Radiation efficiencies η and quality factors Q of pre-fractal top-loaded monopoles using first to third iteration Hilbert curves have been compared against some meander-line loaded monopoles (intuitively designed) (Fig. V). The comparison showed that better radiation performances (in terms of η and Q) are easily be achieved for the same electrical sizes (k0a) with the meander-line designs. The reason is that, the ohmic resistance and the stored energy in the surroundings of the antenna are higher in the Hilbert than in the meander loads. Besides, meander-line geometries allow additional degrees of freedom when designing the antennas.

Figure IV. Hilbert curves as top loads of monopoles compared with bannermonopoles. All the structures have the same overall height. The percentageindicates the relative height of the monopole occupied by the pre-fractal and the banner, respectively.

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Pre-fractal designs that include loops in their topology have been also tested as capacitive loads for monopole antennas (Fig. VI). Pre-fractals of this kind are the Delta-Wired Sierpinski (DWS) and the Y-Wired Sierpinski (YWS).

Not much difference in performance was observed when changing the topology (DWS or YWS) of the Sierpinski gasket nor the relative size of the pre-fractal load (when it is higher than 25% of the total height of the monopole). However, it is remarkable that the introduction of closed loop instead of bended wires in the geometry of the loads improve the radiation performance of the monopole (higher radiation efficiencies and lower Q factors). On the other hand, closed loop loads are unable to reduce the electrical size of the monopoles as much as the bended wire designs.

Figure V: Pre-fractal and meander-line loads used as top-loading of monopoles.

Percentages indicate the relative size of the load vs. the total height of the monopole.

Figure VI: Simulated pre-fractals used as top-loading of monopoles: Delta-Wired

Sierpinski of 3rd and 4th iteration and Y-Wired Sierpinski of 3rd iteration. Percentages indicate the relative size of the load versus the total height of the monopole.

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1.3 Pre-Fractal Quasi self-complementary antennas A planar metallic antenna is said to be self-complementary when the metal area and the open area have the same shape -but a rotation-, i.e. when they are congruent. In a strict sense, self-complementarity is only defined on infinite size antennas. According to Babinet’s principle, the input impedance of a self-complementary antenna has no frequency dependence and is equal to 188 Ω. The constancy of its radiation pattern is not ensured. A practical limitation in the frequency response of the input impedance of a self-complementary antenna comes from its whole size and the size of its terminals. The design of a self-complementary antenna with a pre-fractal profile is expected to provide a new family of antennas with combined performances. The frequency independent input impedance, typical of self-complementary antennas, and the miniaturization capability of pre-fractals. This combination of characteristics should be evidenced by the shift to lower frequency values of the frequency band where the input impedance is closer to 188 Ω when compared with a standard design of the same size. The self-complementary Koch-tie dipole The self-complementary Koch-Tie Dipole was built by mapping the Koch curve on the four sides of a bow-tie antenna (Fig. VII). The results of simulations show that the input impedance is approximately constant starting at a lower frequency than the conventional bow-tie antenna. The lower limit of the usable frequency band decreases with increasing number of iterations of the pre-fractal curve. However, the practical improvement of the usable frequency band is not significant.

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The Gosper island A quasi-self complementary pre-fractal antenna based in the Gosper Island (GI) has been investigated in order to evaluate its potentiality for designing wideband antennas. The GI pre-fractal curve is generated through an IFS of 7 affine linear transformations. A planar strip antenna is designed giving width to the pre-fractal curve. Fig. VIII shows the forth iteration of a Gosper island (GI-4) (left) and its complementary antenna (right). At first look they do not make any difference. A closer inspection reveals that they look self-complementary only in the central region of the pre-fractal (inside the green circle).

Although the Gosper Island pre-fractal is not strictly self-complementary, the quasi-self compementarity property of its surface and the existence of a large number of segments with different lengths make the GI pre-fractal a potential candidate for a dipole antenna with frequency independent input impedance or, at least, a multi-resonant antenna. Consequently, the input impedance response as a function of the feeding point position has been computed using the method of moments code FIESTA and the meshing software GiD. The unsymmetrical geometry of the GI pre-fractal dipole forces the search for the location of the antenna terminals. They should be located along the longest path on the antenna and in a position where the input impedance is constant and close to 188 Ω. According to the initial hypothesis of self-complementarity, the input impedance should be close to 188 Ω. The terminal position at which the dipole is well-matched at a wide frequency band has been determined by numerical simulations. The results show that there are bands where the dipole is matched to 188 Ω, but they are not as wideband as expected for a self-complementary dipole. Fore the GI-3 dipole, values of the matching coefficient to 188 Ω are lower than –10 dB for a frequency band of 5.5 to 7.9 GHz (35.8% fractional bandwidth) for the feeding point B, and from 6.4 to 8.6 GHz (29.3% fractional bandwidth) for the feeding point H.

Fig. VIII: Fourth iteration of a Gosper Island (GI-4) pre-fractal surfaces made with strips:complementary designs. The central surface of both designs (enclosed in thegreen circle) look self-complementary.

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Figure IX shows the current distribution on the surface of the GI-3 dipole fed at the terminal located at point H, for the operating frequencies 6.5, 7.5 and 8.5 GHz. These frequencies are in the band where the input impedance is well-matched to the expected 188 Ω for a self-complementary antenna. The same effect of current attenuation around the terminals of an spiral antenna seems to happen in the GI-3 dipole. This effect is supposed to be the main responsible for the input impedance near the 188 Ω, typical for a self-complementary antenna.

Figure X shows the 3-D radiation patterns at the same operating frequencies as in Fig. IX. The three patterns are similar, so apparently the radiation pattern does not change very much for operating frequencies inside the band adapted to 188 Ω.

Fig. X: Radiation patterns of the GI-3 at operating frequencies 6.5, 7.5 and 8.5 GHz (from left to right) for the GI-3 fed at point H.

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Fig. IX: Computed current distribution on the surface of a GI-3 fed at point H at operating frequencies 6.5, 7.5 and 8.5 GHz (from left to right).

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1.4 The Y-Wired Sierpinski Monopole The Y-Wired Sierpinski (YWS) monopole was introduced at Task 1.1 when comparing pre-fractal structures of the same fractal dimension and different topology. It has been observed that the 3rd iteration of the YWS (Fig. 11) succeeded in reaching a compromise in Q and η when compared with other wired Sierpinski designs.

More significantly, the Q factor, loss efficiency η and radiation pattern (Fig. 12) of the YWS-3 monopole are very similar to that of the λ/4 monopole, but the size is 68% the size of the λ/4 monopole, which means a 32% size reduction. On the other hand, the matching coefficient to 50 Ω of the YWS-3 pre-fractal is worse than that of a longer λ/4 monopole that resonates at the same frequency (-7 dB vs. –16.2 dB).

Figure XI: Three-iteration Y-Wired Sierpinski monopole (YWS-3).

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1.5 3-D Pre-fractal tree The performance as small antennas of several iterations of a 3D pre-fractal tree antenna (Fig. 13) has been analysed in the time domain. The dimensions of all the antennas are such that they fit in a half sphere. It has been observed that the 3D pre-fractal tree antenna behaves similarly to other pre-fractal antennas analysed in this project (Task 1.1): the resonant frequency and the radiation resistance of the antennas decrease as the number of IFS iterations increases.

1.6 Design of small antennas using Genetic Algorithms (GA) A multi-objective Genetic Algorithm (GA) in conjunction with the numerical electromagnetic code (NEC) has been applied to the optimisation of electrically small wire antennas seeking a compromise in terms of several parameters such as resonance frequency, bandwidth and efficiency. Figure XIV shows that the GA optimised designs perform better than pre-fractal antennas of the same electrical size in terms of loss efficiency and quality factor at the resonant frequency.

Figure XIV: Efficiency and quality factor of the GA optimized antenas (Pareto front) versus different pre-fractal configurations, as a function of the electrical size atresonance.

Fig XIII: 3-iteration pre-fractal 3-D tree

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1.7 The H pre-fractal tree Tree-shaped pre-fractals are attractive candidates to be used as antennas since the have many radiating elements of different sizes together with long wire packed into a small volume. The resulting antenna may have miniature size and broadband properties. The geometric characteristics and restrictions of filiform H-fractal trees have been studied (Fig. XV). The flat thick stemmed H-tree is considered afterwards (Fig. XVI).

Parameter η is defined as the ratio between the stem and the branches length, at a give iteration

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The conditions to have infinite wire length without overlapping and the size of the circumscribed rectangle have been studied for both the filiform and the thick stemmed versions. The “efficient” surface, or part of surface of the circumscribed rectangle that is actually occupied by the thick stemmed tree has also been derived. A transmission line model (Fig. XVII) has been proposed to compute the input reactance. Applying the branch model recursively, it has been observed that the electrical length of the equivalent transmission line, which determines the input reactance, converges to a limit for increasing number of iterations in the pre-fractal (Fig. XVIII). This is in full accordance with the theoretical findings in Workpackage 2: “Vector calculus on fractal domains”.

Fig XV: Filiform H-fractal tree with 2η = . Fig XVI: Thick stemmed H-tree, with η = 1.5 and

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Fig. XVII: Equivalent transmission line model for a branch of the tree terminated by an open circuit.

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1.8 Pre-fractal capillary devices The accurate prediction of the frequency response of a highly iteration pre-fractal structure is frequently a very consuming task in terms of computer resources. In practice, many objects called pre-fractal in the specialized literature should rather be considered as constructal objects as their generation should be understood as a building synthetic process going from an elementary small shape to a very complicated and large object, rather than an analytical process introducing complexity at smaller and smaller levels as the traditional IFS algorithms do. This is the case for fractal trees, capillars (Fig. XIX) and many other line structures and antennas.

An analysis method being able to give rapidly a first and reasonably accurate prediction of the frequency behavior of such highly iterated structures would be very useful and timesaving in the electromagnetic design of fractal-shaped devices. Following the above ideas, we start by presenting a transmission line model for the analysis of a family of fractal-shaped structures best represented by the fractal tree shape. First, the geometry of the structure is discussed, and some bounds are set to avoid overlapping of the different branches of the device. The structure is considered as a group of subnetworks, consisting each one on a set of transmission lines, connected in a combination of cascaded and parallel connections (Fig. XX). The subnetworks forming the global structure are related one another by an including or embedding property.

Figure XIX: Two-port capillar. The structure is build recursively from parallel-connection blocks (A, B, C).

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Figure XX: Transmission line model for the elementary block of the capillary structure. Hence, to analyze them we start with the inner and most basic structure, obtain its frequency response, and use this result as the seed that will be embedded in a higher level structure. The seed of the elementary block can take different values depending on the kind of connection between left and right side of the capillary loop: through, open circuit and short circuit. In this way, a recursive implementation of the transmission line equations can easily predict the responses associated to the different topologies of arboreal-like structures. Some prototypes, built in microstrip technology, have been measured to verify the validity of the method (Fig. XXI). The results are very encouraging, taking into account the simplicity of the transmission line model.

Figure XXI: Comparison of magnitude of S11 and S12 for a order-two square capillary (transmission line model, full wave model and measurements).

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2 SUPERCONDUCTIVE RESONATORS AND FILTERS

2.1 Fractal-shaped resonators It is well known that in the microstrip technology the simplest way to realize a resonator is to consider a straight line with open circuit ends. If the microstrip line has length L it resonates at the frequency at which L=λg/2, where λg is the wave length in the considered dielectric substrate. In the last decades, many planar filters based on this type of resonator have been studied and realized for many different applications. In them, the resonators have been coupled magnetically, putting close their long sides (backward and forward configurations) or electrically, by the field present at their edges (edge-coupled configuration) [Matthaei, 1980]. Despite the simplicity of this approach, for which a wide series of analytical expressions has been derived, it is evident that the use of this basic resonator does not optimize the occupied space. For this reason, in order to realize filters with more reduced dimensions, many other kinds of resonators have been investigated. Historically, the most common trend to miniaturize the microstrip resonators has been to propose opportunely folded rearrangements of the elementary straight line so that the first forms of these more compact structures are known in literature as “hairpin” resonators [Cristal, 1972]. In recent years, the miniaturization trend of the planar filters has received a new and growing interest thanks to the discovery of high temperature superconductors (HTS). Indeed, while with traditional metallic microstrips, high current densities due to the miniaturization produce a very large amount of dissipative losses with the drastic and unacceptable worsening of filter performances, by using HTS films, thanks to their very low superficial resistance at microwave frequencies, it is possible to fabricate new compact planar resonators presenting however high (∼104) Q quality factors. So, nowadays, HTS filters seem to be the most adequate to satisfy the needs of the modern telecommunications systems, which, together a general criterion of maximum compactness of the microwave circuitry, require very low insertion losses levels and very steep skirts, thus reducing the interference problems coming out from adjacent bands signals [Lancaster, 1997]. In this context, also owing to the development of new powerful electromagnetic simulators which make easy the analysis of complicated planar geometries, new highly compact HTS resonators have been recently proposed and successfully tested [Reppel, 2000] [Hong, 2000] [Huang, 2003] [Kwak, 2003] [Matthaei, 2003]. Despite their different shapes, all these structures are based on the miniaturizing principle to fold the elementary straight line resonator in very sophisticated ways in order to put it in an area as littler possible. Considering this scenario, we report the results, in terms of miniaturization and overall performances, obtained by the use of the Hilbert curves in the realization of HTS filters [O'Callaghan]. These fractal shapes seem to present a lower resonant frequency and a less dependence on the substrate thickness variations if compared to a classical meander resonator with same external side and microstrip width. In 1892, in a study about the existence of special curves which presented space filling capabilities and the property of being everywhere continuous, the German mathematician David Hilbert presented the sets of curves shown in fig.1 for the first four iterations. As evidenced by the presence of the background grid, the Hilbert curve with k=1 connects the centres of the four parts in which is divided the original square.

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For k=2, the same criterion can be applied dividing the square in 16 parts and connecting the centres in the same way. For the general kth iteration, we have 22k divisions and consequently the curve will be composed of (22k-1) segments, all with the same lengths. Another and probably more intuitive vision suggests that, at every stage, the geometry can be obtained by putting four copies of the previous iteration opportunely oriented and connected by additional short segments. For every curve, the total length L(k) exponentially grows with k and it can be written as

SkL k )12()( += (1) being S the length of the external dimension side [Anguera, 2003]. Furthermore, the fractal dimension D of the Hilbert curves, defined in terms of a multiple-copy algorithm, approaches the value 2 for very large values of k, indicating so that the curve tends to fill enterely the plane [Peitgen, 1992]. These simple geometrical considerations make clear the perspectives of resonators miniaturization by adopting this particular kind of curve which practically guarantees a reliable method to put a very long line in a very little region.

Recently and accordingly to what just mentioned, the miniaturization performances of the Hilbert curves in the fabrication of small antennas have been intensively investigated in many papers ([Vinoy, 2001], [Best, 2002], [Anguera, 2003], [González-Arbesú, 2003]). These studies have clearly shown that, increasing the iteration level k and keeping fixed the external side S, the resonant frequencies of a Hilbert antenna lower but contemporarily the radiation characteristics worsen with a rapid decreasing of the radiation resistance and the efficiency. From Hilbert microstrip resonators point of view, these last properties can suggest their good performances in lowering the packaging losses due to the radiated field. Moreover, analyzing the data reported in literature, it can be observed that in every k case the fundamental resonance frequency is however higher than the fundamental frequency of a corresponding λ/4 monopole with the same length. This phenomenon is due to the couplings between the different turns of the resonator which practically define an

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equivalent shorter path for the current. Obviously this effect gets stronger with k increasing, since a reduction of the interspacing among the turns takes place. In this way, for large values of k, these antennas show a saturation of their miniaturization power and, keeping fixed the external side S, the ratio f0(k+1)/f0(k) between the resonant frequencies of two next iterations, which ideally should be close to 0.5 considering the almost doubling of the length according to (1), tends to grow rapidly towards 1. This means, from a practical point of view, that only the first iterations (k ≤ 5 or 6) of a Hilbert resonator guarantee an effective miniaturization improvement. In the evaluation of Hilbert antennas performances, with a very good approximation, the transversal dimensions of the component wire can be considered negligible. On the contrary, for a Hilbert microstrip resonator, the microstrip width w plays a fundamental role, being the parameter which actually defines the trade off between miniaturization and obtainable quality factor Q. Indeed, trying to obtain high miniaturization levels, what in our case means adopting curves with high k, the value of w decreases considerably, with a consequent increasing of the dissipation losses and a quality factor lowering. In particular, named s the spatial gap among the resonator turns, the expression of the external side S as a function of k, can be written as:

( ) sswkS kk ⋅−++⋅= −− )12(342)( )2()2( (2) where s is the minimum spacing between strips. This formula, valid for k≥2, representing all the significant cases, can be derived from simple geometrical considerations mainly based on the condition that every k≥2 curve is formed by 22(k-2) elementary cells, which in their turn are k=2 Hilbert curves, connected by (22(k-2)-1) segments of length s. Looking at (2), it is so possible to conclude that keeping fixed S(k), the value of w almost behalves for two next iterations. In the first part of our investigation, different series of Hilbert resonators with k=3, 4, 5 have been designed. Values of S(k) from 3 mm to 10 mm have been considered and for every series the ratio w/s has been fixed to 1. The resonant frequencies of this structure have been evaluated by electromagnetic simulator IE3D, modeling the substrate with the characteristics of MgO (εr=9.6 and thickness h=0.508 mm), In fig. 2 we report for k=3, 4, 5, the obtained f0 as a function of S(k) and the corresponding microstrip widths. As reference points for the resonant frequencies obtainable in this range of dimensions, it can be observed that for S(k)=3 mm and k=3, f0 results to be approximately 3.6 GHz, while for S(k)=10 mm and k=5, it is 0.35 GHz.

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External Side S [mm] Fig.2 (a) Fundamental resonant frequencies of Hilbert resonators with k=3, 4, 5 and

(b) corresponding microstrip widths as a function of the external side S(k).

a) b)

- 20 -

In a second phase, our attention has been focused on the possible applications of these resonators at frequencies of interest for 3G wireless applications. In this context, again for k=3,4,5 and w/s =0.5,1,1.5, we have evidenced the values of external side S(k) for which in any case f0 was approximately 2 GHz. The detailed results of this analysis are summarized in Table 1. As shown, for every series w/s, passing from k=3 to k=5, S(k) almost behalves but w(k) is reduced of (8-10) times. From another point of view, increasing w/s and keeping fixed k, S(k) increases, thus making less miniaturized the resonator but contemporarily assuring larger values of the microstrip widths w.

In conclusion of this preliminary analysism the Hilbert resonator with k=4 and w/s=1 (fig. 3a) has seemed to be the best candidate to assure a good trade-off between miniaturization level and Q factor at f0=2 GHz. In this case, the external dimension S(k) is only 3.58 mm (0.06 λg, where λg at 2 GHz is the wavelength for a 50 Ω transmission line on MgO) and w is 115 microns.

The k=4 resonator was realized by using a 10 mm x10 mm MgO substrate, with 700 nm YBCO films on each side. For these commercial films by Theva, a critic temperature of 87 K is assured. The resonator performances were tested at T=77K in liquid nitrogen and a Q of about 30000 was measured. This value is in a very good agreement with the value predicted by Momentum software, while IE3D indicated a larger value close to 45000. Moreover it seems to be very similar to those reported in very recent papers for

wK (mm) SK (mm) w/s =0.5, k=3 0.2 4.6 w/s =0.5, k =4 0.069 3.3 w/s =0.5, k =5 0.024 2.35 w/s =1, k =3 0.33 4.95 w/s =1, k =4 0.115 3.58 w/s =1, k =5 0.041 2.58 w/s =1.5, k =3 0.4 5.09 w/s =1.5, k =4 0.142 3.72 w/s =1.5, k =5 0.051 2.71

Tab.1 Dimensions of resonators.

b)a)

Fig.3 a) Hilbert resonator with k=4. b) Equivalent meander resonator.

- 21 -

other compact resonators [Huang, 2003] [Matthaei, 2003]. A simulation by Momentum realized for k=5 and w/s=1 resonator has evidenced a Q factor of about 16000. In order to complete our analysis about the characteristics of the k=4 Hilbert resonator, its properties have been compared to those of an equivalent meander resonator (fig. 3b) with the same external side and the same microstrip width. The simulations by both IE3D and Momentum show that the meander resonator resonates at almost 30 MHz higher than Hilbert and this confirms the results contained in [Best, 2002], for the comparison between Hilbert and meander antennas. Geometrically, it can be easily demonstrated that a meander line (w=0) with same interspacing between turns of a Hilbert line of any k level, has the same length given by (1). However in a meander resonator, the coupling among the turns is higher due to their longer length, while in the fractal structure the size of the turns is reduced thanks to its particular shape. For this reason, the equivalent reduction of the path current is lower in the Hilbert structure. Another confirmation of the results of this kind of comparison can be given considering the dimensions of the meander (ziz-zag) treated in [Matthaei, 2003] which, on MgO at f0 very close to 2 GHz, are larger than those of our Hilbert resonator. The higher coupling between the turns of a meander resonator makes also its electromagnetic behaviour more dependent on the variations of physical parameters like the substrate thickness.

In fig. 4, the per cent relative shift of the resonant frequency for the resonators in fig. 3, as a function of the thickness substrate h, is reported. For an h change of ±5%, the shift range of the Hilbert is ±1% while for the meander is ±1.5%.

0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52 0.525 0.53-1.5

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Fig. 4 Comparison between the relative resonant frequency shift of the Hilbert resonator (continuous line) and the equivalent Meander resonator (dashed line) versus substrate thickness.

- 22 -

2.2 Fractal-shaped filters Once experimentally tested the performances of the k=4 Hilbert resonator, many possible filters configurations realizable by it have been designed and analyzed. In the following, we report some of the investigated layouts with the corresponding responses, trying to put in evidence first of all the flexibility of this resonator in the realization of quasi elliptical and Chebychev responses. 2.2.1 Quasi elliptical filters As shown in fig. 5a, in order to realize couplings with different signs, what is the basic step for quasi elliptical responses, the Hilbert resonator has been provided with a capacitive load by lengthening the resonators terminations. For f0=1.95 GHz on MgO, the found dimensions of this resonator are 3.38 mm x 3.74 mm while the microstrip width is 110 µm.

Following the classical procedure described in [Matthaei, 1980] and [Hong, 2000], four pole quasi elliptical filters with different bandwidths (BW) have been succesfully designed and simulated. The couplings between the resonators have been obtained fixing opportunely the corresponding distances, while for the necessary Qext (Q external factor) an adequate geometrical parameter of the chosen feed-lines configuration has been defined. In particular, the feed lines can be coupled to the first and last resonators by a capacitive gap or by a direct connection (tapped line configuration). According to our results, the former, as a function of the gap, allows obtaining Qext larger than 200, suitable for BW lower than 0.5% f0. The latter gives, as a function of the tapping position, Qext lower than 20 and consequently make realizable BW larger than 3% f0 (about 60 MHz). By tapped feed lines, it is not possible to reach the intermediate values of Qext since the original Hilbert resonator centre point is not accessible. This last limitation has been simply overcome by rearranging the Hilbert resonator as depicted in fig. 5b. Practically, the orientation of the two of the four component elements with k=3, has been changed in order to supply the structure of a double axial symmetry which makes available the centre point. In this case, it is simple to show that increasing the distance between the tap-point and the centre, the realized Qext decreases [Wong, 1979]. At f0=1.95 GHz, this resonator is 3.15 mm wide and 3.9 mm high, with a microstrip width of 105 µm. By it, a four pole quasi elliptical filter in the classic quadruplet configuration (fig. 6a) has been designed with BW=15 MHz (0,77% f0). The necessary

a) b)

Fig. 5 a) Hilbert resonator with capacitive load; b) Hilbert resonator rearrangement with accessible centre point.

- 23 -

Qext =110 is obtained by tapping the 50 Ω feed lines and the overall dimensions are 10.2 mm x 7.5 mm. Fig 6b and 6c show respectively the in band and the large range simulated responses. It is significant to outline the absence of the second harmonic peak at 2f0, as a consequence of the capacitive load. Practically at 2f0, being the resonators long as one

wavelength, its ends present charges with the same sign and this reduces considerably the effect of the equivalent capacitance [Zhou, 2003]. Moreover, moving one of the tap points symmetrically respect to the centre, as reported in details in [Lee, 2000], a new filter response with the same bandwidth (same Qext), but with four zeroes near the pass band, improving the selectivity performance, can be obtained (fig. 6d). Indeed, in this new position the two feed lines produce a further double possible path for the signal which, owing to a destructive interference, causes the presence of other extra two zeroes. In order to verify experimentally the perspectives evidenced by the simulation work, a quasi elliptical filter with f0=2.45 GHz and BW=20 MHz has been fabricated by using a commercial 10 mm square double sided YBCO 700 nm thick film grown on 0.508 mm thick MgO substrate. In this case the basic resonator dimensions are 2.68 mm x 2.28 mm with a microstrip width of 90 µm. The overall dimensions of the filter are 7.9 mm x

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Fig. 6 a) Quasi elliptical four pole filter at f0=1.95 GHz; b) In-band

response; c) Large range response; and d) In-band response of the four zeroes filter.

- 24 -

6.2 mm. The filter has been tested in liquid nitrogen with Pin=0 dBm. In fig. 7, the measured performances obtained without the use of dielectric screws and a comparison with IE3D simulations are reported. A little discrepancy (not reported in fig. 7 where the simulated responses have been shifted) has been found between the simulated f0 (2.45 GHz) and the measured one (about 2.438 GHz). A slightly difference between the real and simulated permittivity can partially justify this occurrence. In particular, in the permittivity range (9.6-9.7), the simulated centre frequency is shifted down of about 12 MHz. Minimum insertion losses are about 0.2 dB, but the in band response seems to suffer a ripple distortion (maximum ripple 0.6 dB, minimum return loss in reflection 8 dB) due a little detuning condition. By the use of an equivalent lumped circuit, a difference of about 3 MHz between the central frequencies of the internal and external resonators has been estimated. The measured and simulated 3 dB bandwidths (fig. 7b) differ of about 1 MHz and also this can be attributed to the detuning distortion which makes less steep the upper frequency skirt and worsens the initial asymmetry of the response due to the unwanted couplings, not considered in the quasi elliptical model. Finally, the complete agreement for the large range responses with the confirmed absence of the second harmonic peak can be observed (fig. 7c).

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Fig. 7 a) Comparison between measured (continuous line) and simulated (dashed line) responses of the four pole quasi elliptical filters with about f0=2.45 Hz. In-band (b) and large range (c) responses.

a)

b) c)

- 25 -

2.2.2 Chebychev filters Similarly to the analysis realized for the quasi elliptical configuration, many four pole Chebychev filters in the frequency range of 2 GHz have been investigated by using the Hilbert resonators in fig. 5. The same considerations made in previous section, about the possibilities to utilize different feed lines configurations for the desired bandwidths, can be repeated. In order to provide typically obtained dimensions, it should be reported that by using the resonator of fig. 5a, a four pole filter with f0=1.95 GHz and BW=70 MHz resulted to be 14 mm wide and 3.6 mm high. In this case the particular shape of the resonators and the choice to put them in same orientation, so that the opposite flowing currents tend to cancel the field, weakening more and more the couplings (comb-line configuration [Matthaei, 2003]), allow obtaining the necessary couplings with distances lower than 0.2 mm. More in general, coupling coefficients of the order of 1·10-3, for very narrow bandwidths (<0.25% about 5 MHz) can be obtained with distances typically about 1 mm. However it is evident that, by these configurations, four pole filters, and even more increasing the resonators number, present a dimension much larger than the other. This last consideration has suggested us to realize a new rearrangement of the Hilbert resonator which can guarantee a better aspect ratio, resulting, for a four pole filter, in an almost square occupied area. Fig. 8 shows the new proposed layout of a four pole Chebychev filter.

In this case, considering as starting point a pure Hilbert resonator with k=4, we have changed the orientation and disposition of the component k=3 structures, rearranging them in vertical sequence. At f0=1.95 GHz, the obtained resonator is 1.83 mm wide and 7.91 mm high (w=120 µm). So a four pole filter with a bandwidth of 50 MHz has resulted to fit very well in a 10 mm square area with overall dimensions equal to 8.2 mm x 7.93 mm. As shown, every resonator is supplied with a final straight termination. The dimension and, in some cases, also the number of these terminations allows extending the range of obtainable Qext by a tapped line, making equivalently the tap point nearer to the resonator centre. The filter in fig. 8 has been fabricated by a doubled sided YBCO 300 nm thick film on MgO and tested at T=65K in a closed cycle cryogenic system. The measured response and a comparison with the simulations are presented in fig. 9.

Fig. 8 Four pole Chebychev layout.

- 26 -

For this filter, the measured f0 results to be about 1.9 GHz so that a discrepancy of about 50 MHz is found between simulation and measurement. Also the measured bandwidth (fig. 9b) resulted lower than expected (with a difference of about 10 MHz for 3 dB bandwidth). Probably, since this filter occupies almost all the superconducting area of the original 10 mm square film, factors as a not uniform value of the permittivity, the penetration depth effect, a possible non uniformity in the film thickness have influenced strongly the filter performance. Moreover, the simulations have not considered the effect of the reduced dimensions of the metallic box which certainly has limited also the out band rejection value (about 40 dB around f0) with a behaviour very similar to that described in [Huang, 2003]. However, the in band performances in terms of insertion losses (0.1 dB as minimum value) and of the ripple (0.2 dB as maximum value) are really very good, demonstrating again, beyond its flexibility, the optimum trade off between Q factor and miniaturization that this kind of resonator can offer. Finally, it is worth to note that the filter performances have been tested until to Pin=10 dBm and absolutely no responses distortion was observed.

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a)

b) c)

Fig. 9 a) Chebychev filter performances, comparison between

measurement at T=65K, (continuous line) and simulation (dashed line) a) in- band and c) large range details.

- 27 -

2.3 Conclusions about the work on superconductor resonators and filters In this work the miniaturization performances of a novel type of HTS microstrip resonator based on the Hilbert curves has been investigated. The fractal structure was analyzed considering different levels of iteration and putting in evidence the incidence of the different characteristic geometrical parameters on the obtainable miniaturization level. The performances in terms of the quality factor of a single Hilbert resonator with a side of 3.58 mm and f0=2 GHz, have been tested at 77K showing a Q value about 30000. Different filters configurations with quasi elliptical and Chebychev responses have been designed, showing the flexibility of this type of resonator and its capability to obtain also very low couplings at relatively small distances. By using YBCO commercial 10 mm square films on MgO, one four pole filter quasi elliptical at f0 close to 2.45 GHz and one Chebychev filter at f0=1.95 GHz have been fabricated and tested. The measured minimum insertion losses (0.1-0.2 dB) confirm the good trade off between quality factor and reduced dimensions. For the Chebychev filter, discrepancies between measured and simulated f0 and bandwidth have been evidenced, resulting object of future investigations. The filters performances appear without distortions until to Pin=10 dBm.

- 28 -

3 PRE-FRACTAL LOADING

While measuring Hilbert monopoles of high pre-fractal order at different heights from the ground plane, a high quality factor was observed. It suggested that Hilbert curves could be used as top loads useful for shorting monopoles. Several monopoles loaded with Hilbert pre-fractals from iteration 1 to 3 are depicted in Figure 10, and simulated results for efficiency and quality factor are shown in Figure 11. Dashed lines in Figure 11 join values of efficiency and quality factor at self-resonance for the same pre-fractal (H-i) when the relative size of the pre-fractal load to the height of the monopole is changed. All the analysed configurations have the same total height, 89.8 mm. The lowermost 2% of the monopole height is reserved to simulate the segment required for the welding of the feeding pin.

As observed in Figure 11, higher radiation efficiencies are obtained when small loads and long monopoles are used. Electrically smaller self-resonant monopoles are achieved by increasing the relative size of the Hilbert curve, but poorer values of radiation efficiency and quality factor are expected. Better values of quality factor are reached when increasing the iteration of the pre-fractal for almost the same radiation efficiency, but the improvement is not significant. Not negligible is the reduction in size achieved with the pre-fractal loading respective to the conventional λ/4 monopole (k0a=1.1 in front of k0a=1.5), for almost the same efficiency and quality factor. At these size reductions the same values are achieved with a banner monopole and a conventional circular top loading plate, but the later can not be simply manufactured with PCB photo-etching technology. In Figure 11 it is also shown that the conventional circular plate top loaded monopole (TLM) only achieves a

Figure 10. Hilbert curves as top loads of monopoles compared with bannermonopoles. All the structures have the same overall height. The percentageindicates the relative height of the monopole occupied by the pre-fractal and the banner, respectively.

- 29 -

maximum reduction of k0a closer to 0.4. At this point the loaded monopole has a really short feeding pin and a large plate, becoming more a patch than a small monopole. The banner monopole has a greater limitation in its size reduction, being k0a approximately 0.8 the minimum size for this kind of structure. In conclusion, loading a monopole with a pre-fractal seems useful for reducing the electrical size of the antenna. However, greater size reductions are achieved with an standard circular plate normal to the axis of the monopole, but this loading can not be fabricated using standard PCB technology.

-

Figure 11. Computed quality factor and efficiency at self-resonance of Hilbert loaded of monopoles compared with conventional Top Loaded Monopoles (TLM) (loaded with a circularplate), banner monopoles and a λ/4 monopole. All the monopoles have the same height.

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90

100

Electrical size at resonance, k0a

Rad

iatio

n ef

ficie

ncy,

η (%

)

Hilbert-1Hilbert-2Hilbert-3TLMBannerλ/4

0.6 0.8 1 1.2 1.496

98

100Increasing Size

of the Load

- 30 -

Radiation efficiencies η and quality factors Q of pre-fractal top-loaded monopoles using first to third iteration Hilbert curves have been compared against some meander-line loaded monopoles (intuitively designed). The comparison showed that better radiation performances (in terms of η and Q) are easily be achieved for the same electrical sizes (k0a) with the meander-line designs. Besides, meander-line geometries allow additional degrees of freedom when designing the antennas. Results for efficiency and Q factor are shown in figures 12 and 13, respectively. The computed geometries are presented in figure 14.

Simulations have been carried out for copper wire monopoles with the same height of 89.8 mm and a wire radius of 0.2 mm. All of these designs explored the potential of a bended wire, following a Hilbert pre-fractal curve, as top loads of a monopole. These fractally bended wires allowed a continuous decrease on the self-resonant frequency with the increasing iteration of the pre-fractal. However, an increase in the ohmic resistance and in the stored energy in the surroundings of the antenna was observed. These values were higher than the ones observed for other non-fractal designs.

Fig. 12: Radiation efficiency of Hilbert pre-fractal loading versus Meander-line loading

of monopoles. Dashed lines join families of pre-fractal loads and families of meander-line loads. Values are computed at self-resonance.

- 31 -

At the expense of lower reductions of size (k0a), higher efficiencies and lower Q factors have been observed when studying some other pre-fractal designs that include loops in their topology. Pre-fractals of this kind are the Delta-Wired Sierpinski (DWS) and the Y-Wired Sierpinski (YWS). Figures 15 and 16 show the behaviour of a DWS pre-fractal of 3rd and 4th iteration and a YWS of 3rd iteration when used as top-loading of a monopole. Several sizes of the pre-fractal load versus the height of the monopole have

Fig. 13: Quality factor of Hilbert pre-fractal loading versus meander-line loading of

monopoles. Dashed lines join families of pre-fractal loads and families of meander-line loads. Values are computed at self-resonance.

Fig. 14: Simulated pre-fractal and meander-line loads used as top-loading of

monopoles. Percentages indicate the relative size of the load vs. the total height of the monopole.

- 32 -

been tested and are displayed at figure 17. The monopoles were 89.8 mm height and they were simulated with copper wire of 0.2 mm radius.

Not much difference in performance was observed when changing the topology (DWS or YWS) of the Sierpinski gasket nor the relative size of the pre-fractal load (when it is higher than 25% of the total height of the monopole). Optimum values for η and Q factor for these designs are achieved when this relative size is somewhere in between 50% and 75% of the total height of the monopole. These values are very similar to those of a conventional λ/4 monopole (Q=7.4 and η=99%) but a significant reduction in the size (k0a) of the structure is noticed: the Sierpinski pre-fractal design is 60% the size of a λ/4 monopole (k0a is reduced from 1.51 to 0.91). Nevertheless, lower values are found with standard top-loading with a circular plate although this solution can not be fabricated with standard PCB photo-etching technology. We would remark that by changing the topology of the pre-fractal load, or in other words by using closed loop loads instead of bended wire loads, we are able to improve the radiation performance of the monopole (higher radiation efficiencies and lower Q factors). But with closed loop loads we are unable to reduce the electrical size of the monopoles as much as we could using bended wire designs.

Fig. 15 Radiation efficiencies of DWS-3, DWS-4 and YWS-3 top-loaded monopoles

compared with some Hilbert pre-fractal and meander line-loaded monopoles. Dashed lines join families of pre-fractal loads and families of meander-line loads. Values are computed at self-resonance.

- 33 -

The good matching between simulations and measurements in comparisons among meander-line monopoles, Hilbert pre-fractal monopoles, and several topologies of Sierpinski monopoles, was already assessed in previous tasks ([González, 2002]). According to this agreement, and from the point of view of radiation efficiency and quality factor, no experimental validation of the relative performance of these designs is required.

Fig. 16 Quality factors of DWS-3, DWS-4 and YWS-3 top-loaded monopoles compared

with some Hilbert pre-fractal and meander line-loaded monopoles. Dashed lines join families of pre-fractal loads and families of meander-line loads. Values are computed at self-resonance.

Fig. 17 Simulated pre-fractals used as top-loading of monopoles: Delta-Wired

Sierpinski of 3rd and 4th iteration and Y-Wired Sierpinski of 3rd iteration. Percentages indicate the relative size of the load versus the total height of the monopole.

- 34 -

4 PRE-FRACTAL QUASI SELF-COMPLEMENTARY ANTENNAS

Pre-fractal curves have been analyzed as candidates to design a new family of miniature antennas and even a new family of resonators and filters. In this section a self-complementary antenna based on the pre-fractal Kock curve and quasi-self complementary antenna based in the Gosper Island (GI) pre-fractal are investigated to evaluate its potential for designing wideband antennas. Self-complementary property is used for designing certain wideband antennas. For a planar metallic structure its complementary structure results from switching conductor and vacuum and viceversa. If both planar structures (the original and the complementary) are superposed, an infinite plane is obtained. An antenna is said to be self-complementary when this antenna and its complementary are the same shape, but a rotation. In a strict sense, self-complementarity is only defined on infinite size antennas. The input impedance for a complementary set of two antennas is related through the expression (derived from the Babinet’s principle):

4

2η=comp

aaZZ (1)

where Za and Za

comp are the input impedance of the antenna and its complementary, respectively, and η is the impedance of the vacuum. If the antenna is self-complementary, we get

Ω≈== 1882ηcomp

aa ZZ (2)

This is an interesting property because the input impedance of the antenna is well-known without any calculation and it is frequency independent. The limited size (the truncation) of the antenna stablishes the lower frequency where the input impedance is frequency independent. The upper frequency limitation comes from the non-zero size of the feeding point. The constancy of its radiation pattern is not ensured [Jasik, 1961, cap. 18]. The design of a self-complementary antenna with a pre-fractal profile is expected to provide a new family of antennas with combined performances. The frequency independent input impedance, typical of self-complementary antennas, and the miniaturization capability of pre-fractals. This combination of characteristics should be evidenced by the shift to lower frequency values of the frequency band where the input impedance is closer to 188 Ω when compared with a standard design of the same size.

- 35 -

4.1 Self-complementary Koch-tie dipole A typical self-complementary design is the bow-tie dipole of Figure 18. The concentration of currents in the borders of the bow-tie surface suggested that by replacing the straight borders of the antenna by a Koch curve: • The effective size of the truncated structure would be large due to the longer path

followed by the signal along the antenna borders. • The field radiated at the curve corners (see Deliverable D1 Task 1.1 Final Report)

would greatly reduce the amount of energy that reflects at the truncation of the structure.

If, additionally, the self-complementarity of the structure is maintained, the result would be a frequency independent behaviour of the input impedance starting at a lower frequency.

Self-complementary Koch-Tie Dipoles of four iterations have been designed and are displayed in Figure 19. Their input impedance behaviour was computed using FIESTA and are displayed in Figure 20. The expected frequency shift on the lower frequency range where the input impedance is constant is observed, but the improvement is too small to correspond with the increasing intricacy and contour length of the structure.

- 0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

Figure 18: An example of self-complementary dipole: the bow-tie.

- 36 -

- 0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

-0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

- 0.04-0.03

-0.02-0.01

00 .01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0 .02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

-0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

-0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

-0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

- 0.04-0.03

-0.02-0.01

00 .01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0 .02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

-0.04-0.03

-0.02-0.01

00.01

0.020.03

0.04 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

0.04

y

Figure 19: Four iterations of self-complementary Koch-tie dipoles. It is expected anenhanced band (in the lower range) where the input impedance is frequencyindependent.

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

f (GHz)

R (Ω

)

Resistance

KT-0KT-1KT-2KT-3KT-4

1 2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

f (GHz)

X (Ω

)

Reactance

KT-0KT-1KT-2KT-3KT-4

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

f (GHz)

R (Ω

)

Resistance

KT-0KT-1KT-2KT-3KT-4

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

f (GHz)

R (

f (GHz)

R (Ω

)

Resistance

KT-0KT-1KT-2KT-3KT-4

1 2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

f (GHz)

X (Ω

)

Reactance

KT-0KT-1KT-2KT-3KT-4

1 2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

1 2 3 4 5 6 7 8 9 10-200

-150

-100

-50

0

50

100

150

200

f (GHz)

X (Ω

)

Reactance

KT-0KT-1KT-2KT-3KT-4

f (GHz)

X (Ω

)

Reactance

KT-0KT-1KT-2KT-3KT-4

Figure 20: Input impedance of the bow-tie dipole and the first four iterations of self-complementary Koch-tie dipoles. The lower frequency of the frequency independentrange for the input impedance is slightly reduced.

- 37 -

4.2 The Gosper Island The design analyzed in this section is based on a Gosper Island pre-fractal curve generated through an IFS of 7 affine linear transformations [Mandelbrot, 1977, cap. 6]. Figure 21 shows the first 4 iterations of the algorithm. Blue lines are the GI pre-fractal iterations and red lines are the borders of the area where the pre-fractals are enclosed. These contours could be used to tile a whole plane.

Fig. 21: Blue line curves are the first four iterations of a Gosper Island pre-fractal. Red lines are the contours of the tile that enclose the pre-fractals.

Fig. 22: Fourth iteration of a Gosper Island pre-fractal surfaces made with strips: complementary designs. The central surface of both designs (enclosed in the green circle) look self-complementary.

- 38 -

A planar strip antenna is designed giving width to these curves. Its complementary can also be constructed. At first look they do not make any difference. They seem self-complementary in the central region of the pre-fractal, as figure 22 shows for a 4 iteration GI pre-fractal. Nevertheless, when a close inspection of both antennas is done (after a 120º rotation of one antenna and overlapping both designs), we should conclude that these designs are not self-complementary. Figure 23 shows the overlapped complementary designs for a 4 iteration GI pre-fractal and also for a GI-3 pre-fractal.

Although the Gosper Island pre-fractal is not strictly self-complementary, the quasi-self compementarity property of its surface and the existence of a large number of segments with different lengths make the GI pre-fractal a potential candidate for a dipole antenna with frequency independent input impedance or, at least, a multi-resonant antenna. Consequently, the input impedance response as a function of the feeding point position has been computed using the method of moments code FIESTA and the meshing software GiD. The GI pre-fractal dipole has been designed as a strip with a strip width equal to half the distance among two parallel segments in the pre-fractal curve. The borders of the curve have been terminated trying not to be en disagreement with the shape of the whole pre-fractal. The size of the dipole antenna has been scaled in a way that the longest path for the current flow into the pre-fractal were λ/2 at the stimated resonant frequency of 1 GHz. Once scaled, the GI-3 can be enclosed into a circle of 60.5 mm diameter. The unsymmetrical geometry of the GI pre-fractal dipole forces the search for the location of the antenna terminals. They should be located along the longest path on the antenna and in a position where the input impedance is constant and close to 188 Ω. Figure 24 displays the position of the tested terminal positions GI-3 pre-fractal. The

Fig. 23: Superposition of a 4 iteration Gosper Island pre-fractal complementary

surfaces (a) and a 3 iteration Gosper Island pre-fractal complementary surfaces (b). The areas where the two surfaces are not self-complementary are clearly shown. They result from overlapping the two complementary designs after a rotation of 120º of one antenna.

- 39 -

computations were carried out meshing the dipole surface with triangles not larger that λ/125 at 1 GHz.

The computed input impedances –actually, reactance and matching coefficient to 50 Ω- for these terminals in the frequency range of 0.2 to 4 GHz are displayed in figures 25 and 26. The computed impedances –also reactance and matching coefficient to 50 Ω- in the range 4 to 10 GHz are displayed in figures 27 and 28. Figures 29 and 30 show the matching coefficient to 188 Ω in the ranges 0.2 to 4 GHz, and 4 to 10 GHz, for the feeding point positions A to D and E to H, respectively. From the inspection of figures 25 to 28, no remarkable frequency band where the dipole is matched to 50 Ω is found. Only narrow gaps where the matching is best than -10 dB were attained. For instance, a 5.6 to 6.2 GHz band, corresponding to a 10.2% fractional bandwidth for the feeding point C; a 8.1 to 9.1 GHz (11.6%) for feeding point D; and for feeding point H a frequency band from 5.4 to 6.1 GHz corresponding to a 12.2% fractional bandwidth. The first resonance for the GI-3 dipole resulted to be 790 MHz (as observed from the reactance graphs) instead of the stimated 1 GHz. The cause of this shift is the high coupling among the segments of the pre-fractal.

Fig. 24: Positions where the input impedance of the GI-3 were computed.

- 40 -

Fig. 25: Computed input impedances for the feeding points A to D of a 3 iteration

Gosper Island pre-fractal. Reactances and matching coefficients are displayed in the left and right columns, repectively, for the frequency range of 0.2 to 4 GHz.

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

D

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

C

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

C

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

B

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

A

(A)

(B)

(C)

(D)

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)A

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

B

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

D

- 41 -

Fig. 26: Computed input impedances for the feeding points E to H of a 3 iteration

Gosper Island pre-fractal. Reactances and matching coefficients are displayed in the left and right columns, repectively, for the frequency range of 0.2 to 4 GHz.

(E)

(F)

(G)

(H)

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

H

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

H

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

G

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

G

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)

F

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

F

0.5 1 1.5 2 2.5 3 3.5 4-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

f (GHz)

Rea

ctan

ce (Ω

)E

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

E

- 42 -

Fig. 27: Computed input impedances for the feeding points A to D of a 3 iteration

Gosper Island pre-fractal. Reactances and matching coefficients are displayed in the left and right columns, repectively, for the frequency range of 4 to 10 GHz.

(A)

(B)

(C)

(D)

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

D

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

D

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

C

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

C

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

B

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

B

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)A

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

A

- 43 -

Fig. 28: Computed input impedances for the feeding points E to H of a 3 iteration

Gosper Island pre-fractal. Reactances and matching coefficients are displayed in the left and right columns, repectively, for the frequency range of 4 to 10 GHz.

(E)

(F)

(G)

(H)

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

H

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

H

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

G

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

G

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)

F

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

F

4 5 6 7 8 9 10-600

-400

-200

0

200

400

600

f (GHz)

Rea

ctan

ce ( Ω

)E

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

E

- 44 -

Fig. 29: Computed matching coefficients to 188 Ω for the feeding points A to D are

displayed in the left and right columns for the frequency ranges 0.2 to 4 GHz and 4 to 10 GHz, respectively.

(A)

(B)

(C)

(D)

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

D

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

D

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

C

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

C

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

B

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

B

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

A

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

A

- 45 -

Fig. 30: Computed matching coefficients to 188 Ω for the feeding points E to H are

displayed in the left and right columns for the frequency ranges 0.2 to 4 GHz and 4 to 10 GHz, respectively.

(E)

(F)

(G)

(H)

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

H

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

H

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

G

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

G

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

F

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

F

4 5 6 7 8 9 10-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

E

0.5 1 1.5 2 2.5 3 3.5 4-15

-10

-5

0

f (GHz)

Mat

chin

g Co

effic

ient

(dB

)

E

- 46 -

The current distribution on the dipole surface for the first resonance frequencies when the terminals are placed at position B (0.79 GHz, 1.54 GHz and 2.85 GHz) are shown in figure 31. Colorbar shows the current intensities in dB: in red highest currents and in blue the lowest ones. At 0.79 GHz the longest path through the dipole surface is where the current is concentrated. At 2.85 GHz the resonance is due to a current concentration in the opposite -to the feeding point- end of the dipole. According to the initial hypothesis of self-complementarity, the input impedance should be close to 188 Ω. For this input impedance a wide frequency band where the dipole is well-matched was expected at one of the terminal positions (to be determined by computation). Nevertheless, from figures 29 and 30 we see that for several feeding points (A, B, D, E and H) there are bands where the dipole is matched to 188 Ω but they are not as wideband as expected for a self-complementary dipole. Values of the matching coefficient to 188 Ω are lower than –10 dB for a frequency band of, for instance, 5.5 to 7.9 GHz (35.8% fractional bandwidth) for the feeding point B, or from 6.4 to 8.6 GHz (fractional bandwidth: 29.3%) for the feeding point H. Even slightly shifting the feeding point around H -trying to improve the matching to 188 Ω- the frequency band is not increased more than the 29.3% attained. Consequently, we should conclude that the quasi-self complementarity of the GI is translated into a significant frequency band where the input impedance is close to 188 Ω.

Figure 32 shows the current distribution on the surface of the GI-3 for the frequencies 6.5, 7.5 and 8.5 GHz for the terminal located at H. These frequencies are in the frequency band where the input impedance is well-matched to the expected 188 Ω for a self-complementary antenna. The same effect of current attenuation around the terminals of an spiral antenna seems to happen to these current distributions. This effect

J J Jf1=0.79 GHz f2=1.54 GHz f3=2.85 GHz

Fig. 31: Current distribution on the GI-3 surface for the first resonance frequencies when the terminals are at B. From left to right the resonance are: 0.79 GHz, 1.54GHz, and 2.85 GHz.

fl=6.5 GHz f0=7.5 GHz fh=8.5 GHz J J J

Fig. 32: Computed current distribution on the surface of a GI-3 feeded at H at frequencies 6.5, 7.5 and 8.5 GHz (from left to right).

- 47 -

is supposed to be the main responsible for the input impedance near the 188 Ω, typical for a self-complementary antenna.

Fig. 33: Computed radiation patterns at 6.5 GHz for the GI-3 feeded at H.

-30 -20 -10 0

30

-150

60

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90 -90

120

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180

0 Eθ , φ = 0

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120

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φ , φ = 0

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120

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0 Eθ, φ = 90

-30 -20 -10 0

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120

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0 E

φ, φ = 90

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120

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0 |E|, φ = 90

x

y

- 48 -

Fig. 34: Computed radiation patterns at 7.5 GHz for the GI-3 feeded at H.

-30 -20 -10 0

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E θ , φ = 0

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E φ , φ = 0

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0|E|, φ = 0

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Eθ, φ = 90

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Eφ, φ = 90

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0|E|, φ = 90

x

y

- 49 -

Fig. 35: Computed radiation patterns at 8.5 GHz for the GI-3 feeded at H.

-30 -20 -10 0

30

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60

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120

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180

0 E

θ , φ = 0

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0 E

φ , φ = 0

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180

0 |E|, φ = 0

-30 -20 -10 0

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120

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0

Eθ, φ = 90

-30 -20 -10 0

30

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60

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120

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0

Eφ, φ = 90

-30 -20 -10 0

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90 -90

120

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150

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180

0|E|, φ = 90

x

y

- 50 -

Figures 33 to 35 show two main cuts of the radiation pattern for the frequencies 6.5, 7.5 and 8.5 GHz for the feeding point H. Figure 36 shows the three dimensional radiation patterns for the GI-3 also at frequencies 6.5, 7.5 and 8.5 GHz when feeded through terminal H. The attenuation of the current intensity around the feeding point suggests further research about a modified GI-3, and also about a 4 iteration Gosper Island pre-fractal (GI-4, see figure 32). For both designs the self-complementarity and the current attenuation around the feeding point are expected to be more remarkable.

Fig. 36: Radiation patterns at frequencies 6.5, 7.5 and 8.5 GHz (from left to right) for the GI-3 feeded at H.

Fig. 37: GI-4 pre-fractal dipole.

x

z

y

- 51 -

5 THE Y-WIRED SIERPINSKI MONOPOLE

From the results presented in deliverable D1 “Task 1.1 Final Report”, seccion titled “On the Influence of Fractal Dimension and Topology on Radiation Efficiency and Quality Factor of Self-Resonant Pre-fractal Wire Monopoles”, it was observed that a similar performance than a λ/4 monopole can be achieved with a pre-fractal design with a 70% smaller electrical size. This compromise among radiation performance and size is achieved with a Y-Wired Sierpinski Monopole of 3rd iteration (YWS-3), see Fig. 38.

Figure 39 shows the quality factor and the efficiency of the YWS-3 pre-fractal compared with the behaviour of the standard one-dimensional λ/4 monopole. The Q factor and loss efficiency η are very similar to that of the λ/4 monopole, but the size is much smaller.

DWS-1 YWS-1 SA-1 K1S-1

SA-5 K1S-5 YWS-5 DWS-5

Figure 38. Families of Sierpinski gasket pre-fractals. All of them have been generated using the same Iterative Function System but changing the initiator. The circle encloses the YWS-3 design that achieves similar performance than a λ/4 monopole.

- 52 -

Figure 39. Quality factor (left) and efficiency (right) for pre-fractal families (joined by dashed lines) of Sierpinski gaskets with different initiators. Circles points to the performance of the YWS-3. Its values are similar to a λ/4 monopole but with a 70% smaller electrical size. The performance –Q factor, radiation efficiency and main cuts of the radiation pattern- of the YWS-3 has been compared with an equal height λ/4 monopole (37.25 mm) and with a λ/4 monopole (62.25 mm height) with the same resonant frequency (1142 MHz). The wire radius of these simulated monopoles is 0.25 mm. Figures 40 and 41 show, respectively, the matching coefficient to 50 Ω and the input impedances (resistance and reactance) for these three designs. Figure 42 shows how the monopoles invest their resistance in radiation and in heating the wire.

Fig. 40: Matching coefficient to 50 Ω for a YWS-3 (red), a λ/4 monopole

resonant with the same resonance frequency than the pre-fractal (green), and a λ/4 monopole with the same height than the pre-fractal (blue).

- 53 -

From the figures we observe that the standard λ/4 monopole resonant at the same frequency than the YWS-3 monopole has higher radiation resistance and higher ohmic losses than the pre-fractal, both measured at resonance (1142 MHz). However, figure 43 assess that the Q factor and the radiation efficiency of the standard λ/4 monopole is maintained by the YWS-3, but with a remarkable reduction in its electrical size, k0a, of about 68%. As expected, for a monopole with the same size than the pre-fractal

Fig. 41: Input impedance (resistance on the left and reactance on the right) of a YWS-3 (red), a λ/4 monopole with the same resonance frequency than the pre-fractal (green), and a λ/4 monopole with the same height than the pre-fractal (blue).

Fig. 42: Radiation resistance (on the left) and ohmic losses (on the right) of a YWS-3 (red), a λ/4 monopole with the same resonance frequency than the pre-fractal (green), and a λ/4 monopole with the same height than the pre-fractal (blue).

- 54 -

resonance frequency, radiation resistance and ohmic resistance are higher, although the Q and efficiency figures stand around the values of their counterparts.

Figure 44 displays the computed main cuts for the radiation pattern of the YWS-3 and for the λ/4 monopole with the same resonance frequency. The cuts reveal that differences can be neglected. This similar behaviour is due to the small electrical size of the pre-fractal monopole. These computations assess that a YWS-3 performs like a λ/4 monopole resonant at the same frequency, in terms of Q factor, radiation efficiency and radiation patterns with a reduction in its electrical size of about a 68%. Nevertheless, the matching coefficient to 50 Ω of the YWS-3 pre-fractal is worse than a standard λ/4 monopole (-7 dB vs. –16.2 dB).

Fig. 43: Q factor (left) and radiation efficiency (right) of a YWS-3 (red) compared with a λ/4 monopole resonant at the same frequency than the pre-fractal (blue triangle), and a λ/4 monopole with the same height than the pre-fractal (blue circle).

0

180

30

210

60

240

90270

120

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330

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0 0

180

30

210

60

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90270

120

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-20

-10

0

λ/4 monopoleYWS-3 monopole

|E| , φ=0º |E| , φ=90º

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90270

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90270

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00

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00

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-10

0

-40

-30

-20

-10

0

λ/4 monopoleYWS-3 monopoleλ/4 monopoleYWS-3 monopole

|E| , φ=0º |E| , φ=90º

Fig. 44: Main cuts of the radiation pattern of a YWS-3 pre-fractal compared with a standard λ/4 monopole (dashed line) resonant at the same frequency.

- 55 -

6 3-D PRE-FRACTAL TREE

The performance as small antennas of several iterations of a 3D prefractal tree antenna has been analyzed. Figure 45 shows the geometries of the monopole antennas of order 1-3 that have been considered. The dimensions of all the antennas are such that they fit in a half sphere. The antennas are fed at its base with a Gaussian pulse and the feeding current is computed, in the time domain using the computer code DOTIG1.

Figure 46 presents the time history of the current at the feeding point for the first, second and third iterations and Figure 47 shows the input impedance of the three antennas. From figures 46 and 47 it is clear that the 3D prefractal tree antenna behaves similarly to other prefractal monopoles previously analyzed in this project, i.e, the resonance frequency of the antennas decreases as the number of iterations increases and their input resistance decreases.

-0.05

0

0.05

-0.05

0

0.050

0.02

0.04

0.06

0.08

0.1

-0.05

0

0.05

-0.05

0

0.050

0.02

0.04

0.06

0.08

0.1

-0.05

0

0.05

-0.05

0

0.050

0.02

0.04

0.06

0.08

0.1

It. 1 It. 2

It. 3

Figure 45: 3D pre-fractal tree monopole antenna: iterations 1, 2 and 3.

- 56 -

Figure 46: Current at the feeding point in the prefractal tree structures

0 0.5 1 1.5 2 2.5 3 3.5

x 10-8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time (s)

Cur

rent

(A)

0 0.5 1 1.5 2 2.5 3 3.5

x 10-8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time (s)

Cur

rent

(A)

0 0.5 1 1.5 2 2.5 3 3.5

x 10-8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time (s)

Cur

rent

(A)

(a) First iteration

(b) First iteration

(c) First iteration

- 57 -

Figure 47: Input impedance a) real part b) imaginary part of different iterations of the prefractal tree antenna. Comparison with NEC

2 4 6 8 10 12 14

x 108

-1500

-1000

-500

0

500

1000

Frequency (MHz)

Rea

ctan

ce ( Ω

)

It 1 - DOTIGIt 2 - DOTIGIt 3 - DOTIGIt 1 - NECIt 2 - NECIt 3 - NEC

2 4 6 8 10 12 14

x 108

0

500

1000

1500

2000

Frequency (MHz)

Res

ista

nce

( Ω)

- 58 -

7 DESIGN OF SMALL ANTENNAS USING GENETIC ALGORITHMS (GA)

Genetical algorithms have been used in task 1.2 “Fundamental limits of fractal miniature devices” in order to find what are the optimum antenna parameters that can be achieved with both pre-fractal and non pre-fractal configurations (see delierable D2 “Task 1.2 final report”). Although it is not the aim of task 1.2 to design antennas using genetical algorithms, but only to find what the optimum values of the antenna parameters are, some of the resulting antenna designs are interesting and deserve being mentioned here.

A multi-objective Genetic Algorithm (GA) tool has been applied in conjunction with the numerical electromagnetic code (NEC) to the optimization of wire pre-fractal Koch-like antennas in terms of bandwidth, efficiency and electrical size. Only the shape of the pre-fractal initiator is optimized, and the IFS recursive algorithm is used to obtain the final pre-fractal configuration (Fig. 48).

The performance of the genetically engineered pre-fractal antennas has been compared to that of other non-fractal wire geometries also designed by GA, like zig-zag and meander type antennas. Fig. 49 shows one example of each type, all of the having approximately the same wire length (10.22cm).

Their resonance frequencies, Q factor and loss efficiency have been calculated

with NEC and are given in Table I. Although all of them show similar bandwidth and efficiency, it is remarkable that the pre-fractal design have the poorest degree of miniaturization. This is due to the restrictions imposed by the pre-fractal structure: the

Figure 48: Geometry of the genetically optimized initiator and the 3rd iteration of the resulting pre-fractal. The parameters optimized by the GA are S1 , S2 , S3 , S4 , θ1 andθ2 .

Koch type Meander type Zigzag typeKoch type Meander type Zigzag type Figure 49: Three GA optimized designs that have approximately the same length (10.22cm ) and wire radius 0.1 mm.

- 59 -

geometry has to be generated by an IFS and therefore has less degrees of freedom that non pre-fractal designs like the meander and zig-zag examples.

Other numerical experiments and measurements were carried out using more pre-fractal wire antennas such as those with Peano, Hilbert or Sierpinsky arrow head type geometries (See the geometries in the Deliverable D17-WP0 T0+12). Similar results were obtained confirming that genetically optimized zigzag-like and meander-like antennas present a better performance that those of all the members of the families analysed. However, when several other pre-fractal families of Sierpinsky antennas (See the geometries in the Deliverable D17-WP0 T0+12) were considered, it was not always possible to find GA-designed zig-zag or menander type antennas with better characteristics than that of those pre-fractal monopoles. A common feature of all those Sierpinsky antennas was that they were formed by geometries which include closed loop shapes that obviously, were not possible to obtain with our GA designs as we had restricted ourselves to zig-zag and meander type Euclidean antennas. In order give more freedom to the GA code, the possibility of forming closed loops was subsequently allowed when looking for the best possible structure. The result was, as can be seen in Figure 50, that, as we expected, this new set of genetically optimised antennas presented a better performance than all the Fractal antennas including loops.

Figure 50: Pareto fronts corresponding to the GA designed antennas with closed loop shapes in part of their geometries. Comparison with several pre-fractal families.

Antenna Resonant

Frequency (MHz)

Quality Factor

Efficiency (%)

Koch 864.5 13.57 96.8 Meander 826.5 12.67 97.19 Zigzag 824 13.99 96.79

Table I: Parameters of the antennas in Fig. 16 computed by NEC.

- 60 -

REFERENCES

[Anguera, 2003] J. Anguera, C. Puente, E. Martinez and E. Rozan: “The Fractal

Hilbert Monopole: A Two-Dimensional Wire”, Microwave and Optical Technology Letters, vol. 36, issue 2, pp.102-104, Jan. 20, 2003.

[Best, 2002] S.R. Best: “A Comparison of The Performance Properties Of The Hilbert Curve Fractal And Meander Line Monopole Antennas”, Microwave and Optical Technology Letters, vol. 35, issue 4, pp. 258-262, Nov. 20, 2002.

[Cristal, 1972] E. G. Cristal and S. Frenkel: “Hairpin-line and hybrid hairpin-line/half-wave parallel-coupled-line filters“, IEEE Trans. on Microwave Theory and Techniques, vol. 20, no. 11, pp. 719-728, Nov. 1972.

[González, 2002] J. M. González and J. Romeu: Task 1.1 Final Report, Fractalcoms Project (IST 2001-33055), Deliverable D1, Dec. 17th, 2002.

[González-Arbesú, 2003]

J.M. González-Arbesú, S. Blanch, J. Romeu: “The Hilbert curve as a small self-resonant monopole from a practical point of view”, Microwave an Optical Technology Letters, vol. 39, issue 1, pp.45-49, Oct. 5, 2003.

[Hong, 2000] J. S. Hong, M.J. Lancaster, D. Jedamzik, R. B. Greed and J. C. Mage: “On The Performance of HTS Microstip Quasi Elliptic Function Filters For Mobile Communications Application”, IEEE Trans. On Microwave Theory and Techniques, vol. 48, no. 7, pp.1240-1246, July 2000.

[Hong, 2001] J. S. Hong and, M.J. Lancaster, Microstrip filters for RF/microwave applications, John Wiley & Sons, 2001.

[Huang, 2003] F. Huang: “Ultra-compact superconducting narrow band filters using single and twin spiral resonators“, IEEE Trans. On Microwave Theory and Techniques, vol. 51, no. 2, pp. 487-491, Feb. 2003.

[Jasik, 1961] H. Jasik (editor), Antenna Engineering Handbook, New York: McGraw-Hill, 1961.

[Kwak, 2003] J.S. Kwak, J.H. Lee, J.P. Hong, S.K. Han, W.S.Kim and K.R. Char: “Narrow Pass-band High – temperature Superconducting Filters Of Highly Compact Sizes for Personal Communication Service Applications”, IEEE Trans. On Applied Superconductivity, vol. 13, pp. 17-19, March 2003.

[Lancaster, 1997] M. J. Lancaster, Microwave passive device application of high temperature superconductors, Cambrige University Press, 1997.

- 61 -

[Lee, 2000] Sheng-Yuan Lee and Chih-Ming Tsai: “New cross-coupled filter

design using improved hairpin resonators”, IEEE Trans. On Microwave Theory and Techniques, vol. 48, no. 12, pp. 2482-2490, Dec. 2000.

[Mandelbrot, 1977]

B. Mandelbrot, The Fractal Geometry of Nature, New York: W.H. Freeman and Company, 1977.

[Matthaei, 1980] G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters Impedance- Matching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980.

[Mattaei, 2003] G.L Matthaei: “Narrow Band, Fixed Tuned And Tunable Bandpass Filters With Zig-Zag Hairpin Comb Resonators”, IEEE Trans. On Microwave Theory and Techniques, vol. 51, no. 4, pp 1214-1219, April 2003.

[O'Callaghan] J. M. O'Callaghan, C. Puente, N. Duffo, C. Collado, E. Rozan. Fractal and Space-Filling Transmission Lines, Resonators, Filters and Passive Network Elements, Patent-WO0154221. (Fractus S.A.)

[Peitgen, 1992] H.O. Peitgen, H. Jurgens and D. Saupe, Chaos and fractals: New frontiers of science, Springer-Verlag, New York, 1992.

[Reppel, 2000] M. Reppel and J.C. Mage: “Superconducting Microstrip Bandpass Filter on LaAlO3 With High Out-Of-Band Rejection”, IEEE Microwave and Guided Wave Letters, vol. 10, no. 5, pp. 180-182, May 2000.

[Vinoy, 2001] K. J. Vinoy, K.A. Jose, V.K. Varadan, and V.V. Varadan: “Hilbert Curve Fractal Antenna: A Small Reconant Antenna For VHF/UHF Applications”, Microwave and Optical Technology Letters, vol. 29, no. 4, pp. 215-219, May 20, 2001.

[Wong, 1979] J. S. Wong: “Microstrip Tapped-Line Filters Design”, IEEE Trans. On Microwave Theory and Techniques, vol. 27, no. 1, pp. 44-50, Jan. 1979.

[Zhou, 2003] J. Zhou, M. J. Lancaster and F. Huang: “Superconducting microstrip filters using compact resonators with double-spiral inductors and interdigital capacitors”, 2003, IEEE MTT- International Symp., pp. 1889-1892.

62

8 The H-fractal tree

8.1 Geometrical Properties

The present section discusses in a first step the geometric characteristics and restrictionsof filiform H-fractals and afterwards the flat thick stemmed H-tree ones.

8.1.1 H-tree skeletons

The H-fractal tree is formed by different H-shaped regions, see Figure 51. It consistsof a main trunk which subdivides at its end into two branches, which at their turnsubdivide again in two new branches, and so on. Therefore, summing up all individualline portions, the total geometrical length of the fractal is:

ltot =∞∑

i=0

2idi (1)

where di is the length of the i-th sub-branch. Assuming that in every iteration the newbranch length is reduced by a factor η (η > 1),

di =di−1

η(2)

then inserting (2) in (1) the total geometrical length of the tree can be expressed as

ltot = d0

∞∑

i=0

2i

ηi. (3)

From (3) it can be concluded that the length of the tree is finite for η > 2.Depending on the value of η the branches are overlapping. The limit value for η toavoid coincidence of the branches can be easily found. Let d0 be the main trunk of the

Figure 51: H-fractal tree.

63

d1

d3

d5

d7

d0

Figure 52: Representation to calculate the condition for infinitely thin branches not tooverlap.

tree, and let d1, d3, d5, d7 . . . be the horizontal branches of the tree (see first vertical andhorizontal black segments in Figure 52).Then the necessary condition in order to prevent branches from overlapping is

d1 >∞∑

i=1

d2i+1 = d3 + d5 + d7 . . . . (4)

Inserting (2) in (4) we obtain

d1 > d1

∞∑

i=1

1

η2i= d1

1

η2

1

1 − 1η2

. (5)

Working on expression (5)

1 >1

η2 − 1(6)

from which followsη >

√2. (7)

It has been shown that for η >√

2 the tree is self avoiding. Putting together conditions(3) and (7) (length limit and non-overlapping), we obtain that for

√2 < η < 2 the tree

has infinite length and non-overlapping branches. For η <√

2 branches are touchingone another and for η > 2 the length of the fractal is finite.Some examples of trees for different values of η are shown in Figure 53.

64

−8 −6 −4 −2 0 2 4 6 8−2

0

2

4

6

8

10

(a) η=1.3.

−6 −4 −2 0 2 4 60

1

2

3

4

5

6

7

8

(b) η=√

2.

−3 −2 −1 0 1 2 30

1

2

3

4

5

6

(c) η=2.

Figure 53: H-tree for different η values.

The circumscribed rectangle: The rectangle enclosing the H-tree can be easily com-puted, and its two sides l1 and l2 are (Figure 54)

l1 =∞∑

i=0

d0

η2i=

d0

1 − 1η2

(8)

l2 = 2∞∑

i=1

d0

η2i−1= 2

d0

η

1

1 − 1η2

(9)

With (8) and (9) we can easily compute the surface of the circumscribed rectangle as

Scr = l1 l2 =2d2

0η3

(η2 − 1)2. (10)

It is worthwhile to mention that the surface of the circumscribed rectangle diminishesonly with η and not with η2 (see Figure 55).

65

d1

d2

d4

d6

d0

l2

l1

d8

Figure 54: Schematic representation to calculate the circumscribed rectangle surface.

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

η

Scr2 d2

0

1η

Figure 55: Evolution of Scr2 d2

0

vs. η compared to 1η.

66

d1

d3

d5

d7

d0

. . .

w0

2

Figure 56: Representation of the right half of the tree to compute the condition forbranches of a thick tree not to overlap (I).

8.1.2 Thick H-tree

Previous conditions for the skeleton tree not to overlap are no longer valid if we introducesome width for the ramifications. It is obvious that the width must diminish as the treegrows, otherwise branches will glue in a few iterations. Let us suppose that in everyiteration new arms become narrower by a factor µ > 1. That is

wi =wi−1

µ, (11)

where wi is the width of the i-th arm. At first sight it seems logical to think that branch0 (the main trunk of the tree), as it is the thickest arm, will be the strongest restrictionfor the no longer negligible strip width. More precisely, looking at the black lines inFigure 56, in order to avoid overlapping

d1 >w0

2+ d1

∞∑

i=1

1

η2i=

w0

2+

d1

η2

1

1 − 1η2

. (12)

Inserting (2) in (12) yieldsd0

w0

>η

2

(η2 − 1

η2 − 2

)

. (13)

Together with (7) we can see that this condition is a lower limit for the ratio d0

w0(see

Figure 57).Equation (13) is easily generalized starting from a branch m and deriving the non over-lapping condition written in equation (12). Then it is obtained

67

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−3

−2

−1

0

1

2

3

4

5

η2−1

η2−2

η

Figure 57: Evolution of η2−1

η2−2vs. η.

d0

w0

>η

2

(η2 − 1

η2 − 2

)(η

µ

)m

,∀m ≥ 0. (14)

In order to accomplish equation (14) it is necessary that

η

µ≤ 1, (15)

otherwise the right hand side of equation (14) diverges as m → ∞.

We have obtained a limit for the ratio d0

w0for branches not to coincide. Nevertheless, one

can suspect that the factor in which the branches reduce in each iteration, µ, may causebranches to overlap if it is not big enough. The condition to avoid overlapping can beseen from Figure 58, and it is

wi+3

2+

wi−1

2+ di+2 < di. (16)

With (2) and (11) equation (16) can be rewritten

d0

w0

>η2

(

µ + 1µ3

)

2 (η2 − 1)

(η

µ

)i

. (17)

Again, in order to accomplish equation (17) it is necessary that

η

µ≤ 1, (18)

otherwise the right hand side of equation (17) diverges as i → ∞.

From expressions (14) and (17) we have to choose the most restrictive one to assure thetree being self avoiding.

68

di+2

di

wi−1

2

wi+3

2

Figure 58: Computing the condition for branches of a thick tree not to overlap (II).

0 5 10 15 20 25−0.5

0

0.5

1

1.5

2

2.5

3

Iteration#

d0

w0

Equation (14)

Equation (17)

Figure 59: Comparison of expressions for d0

w0to avoid overlapping in the thick fractal

tree.

69

Figure 60: Tree for η = 1.5, µ = 2, d0

w0= 4 < 3.75.

Figure 61: Tree for η = 1.5, µ = 2, d0

w0= 2 < 3.75 .

Both expressions are represented in Figure 59, where it can be seen how equation (14)offers the most restrictive condition. Thus, from now on this is the used expression.Once η and µ are fixed following (15), the only necessary condition for the tree com-ponents not to overlap is that the ratio between d0 and w0 accomplishes equation (14).It is obvious that these conditions to avoid overlapping in the case of the thick fractaltree are true provided that conditions to avoid coincidence in the filiform tree are alsofulfilled.For example, if we fix η = 1.5 (η >

√2 to avoid overlapping) and µ = 2, which

accomplishes equation (15), equation (16) forces us to take a ratio d0

w0> 3.75 in order

to avoid overlapping. Figure 60 shows a self-avoiding tree for η = 1.5, µ = 2 andd0

w0= 4. On the other hand, taking the same values for η and µ, but choosing d0

w0= 2

the overlapping tree shown in Figure 61 is obtained.

70

The ”efficient” surface : This concept is closely related to the circumscribed rectanglediscussed in the previous section. What we understand by ”efficient” surface is the partof the circumscribed rectangle surface that is actually occupied by the tree.

Ri−1

Ri

Ri ∩ Ri−1

wi−1

wi

Figure 62: Schematic representation of the 3 first iterations of a tree to compute theefficient surface.

The thick tree is constructed just giving some width to the segments of the skeletontree. Thus, all the segments are converted into rectangles. The intersection of eachsegment of the skeleton tree with the precedent one is one point, and in the same way,the intersection of each rectangle resulting from giving some width to the segments withthe precedent one is a surface. Thus, the effective surface of the fractal tree is thesum of the surfaces of all the rectangles minus the sum of all the intersections of allthe rectangles. In Figure 62 the skeleton tree is plotted in solid line, the rectangles indashed lines and the intersections between rectangles are shaded. From this figure canbe concluded that the value of the intersection surface of one rectangle, Ri, with theprecedent one, Ri−1, logically only depends on the width, and it is

Ri ∩ Ri−1 =wi

2

wi−1

2, i = 1, 2, 3 . . . , (19)

and the number of intersections is

N∩ i = 2i, i = 1, 2, 3 . . . . (20)

The efficient surface can be straightforward computed as

Seff =∞∑

i=0

2i d0

ηi

w0

µi−

∞∑

j=1

2j w0

2µj−1

w0

2µj(21)

Working in the expression we finally get

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)(22)

71

Instinctively we can affirm that the thicker the branches of the tree are, the bigger theefficient surface is. But it all is relative. Fixing a very long and thick initial segment doesnot assure the maximum efficient surface. To have the maximum efficient surface we haveto fix the minimum η, µ and d0

w0ratio that are allowed in order to avoid overlapping.

It is under these conditions that the maximum value of the efficient surface will bereached. We are going to calculate this value now, and consider the percentage of thecircumscribed rectangle surface filled by the maximum efficient surface.Inserting in (22) the value of w0 obtained taking the equality in (14), and for µ taking theequality in (15), that is, µ = η, we obtain the maximum value for the efficient surface,which is

Seffmax =2d2

0(η4 − 3η2 + 4)

η(η2 − 1). (23)

Now, for the lower bound of η, η =√

2, which ensures a self avoiding skeleton tree, butalso gives the lowest reduction rate in the length of the branches (thus, contributes tohave higher efficient surface), we obtain

Seffmax|η=√

2 =4d2

0√2. (24)

Computing the value of the circumscribed rectangle also for η =√

2 we get

Scr|η=√

2 = 4d20

√2. (25)

Thus the ratio between (24) and (25) is

Seffmax

Scr

∣∣∣∣η=

√2

=1

2. (26)

From (26) it can be concluded that fixing all the geometric parameters of the fractal treeto spread over the maximum possible surface, the 50% of the circumscribed rectanglesurface is covered.The evolution of ratio Seff

Scras a function of η, and for different values of µ is shown in

Figure 63.

72

1.5 2 2.5 3 3.5 40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

µ = η

µ = 1.1η

µ = 1.2η

µ = 2η

η

Seff

Scr

Figure 63: Evolution of Seff

Scrvs. η.

8.2 Transmission line analysis of a fractal Tree

A H-fractal tree is formed by a set of T-junctions. In particular, the H-tree has a centralbranch that bifurcates in two branches, and these two branches, bifurcate on their ownin two more branches, and so on. The equivalent transmission line representation of astage of the tree can be seen in Figure 64. This is a simple equivalent model in which the

w1

w2

w2

d2

d2

d1

(a)

YIN

Zc2, β2

Zc2, β2

Zc1, β1

YL

o.c.

o.c.

ZERO LENGTH

d1

d2

d2

x = d1x = d1x = 0 x

(b)

Figure 64: Equivalent transmission line model for a branch of the tree terminated by anopen circuit (o.c.).

effect of the T-junction is completely neglected and the lines are cascaded dependingon the ratio of the admittances. In an improved version the electrical length of theT-junction has to be also taken into account.

73

Computing the load admittance in Figure 64 we obtain

YL = 2jYc2T2, with T2 = tan θ2, θ2 = β2d2, (27)

with d2 the length of the line, β2 the propagation constant provided that the transmissionline has no losses and Yc2 the characteristic admittance of line number 2. Then, if wecalculate the value of the input admittance, we obtain

YIN = jYc1

2Yc2

Yc1T2 + T1

1 − 2Yc2

Yc1T2T1

, (28)

with T1 = tan θ1 = tan β1d1 and Yc1 the characteristic admittance of line segment 1.Expression (28) suggests to introduce an equivalent electrical angle θ∗2, given by

T ∗2 =

2Yc2

Yc1

T2 =2Yc2

Yc1

tan θ2, thus θ∗2 = arctan

(2Yc2

Yc1

tan θ2

)

. (29)

Hence

YIN = jYc1

T ∗2 + T1

1 − T ∗2 T1

= jYc1 tan(θ∗2 + θ1). (30)

The T-bifurcation behaves as a single line of characteristic impedance 1Yc1

and equivalentelectrical angle θ∗ = θ∗2 + θ1.Recursively for the i-th branch of the H-tree we find

θ∗i = arctan

(2Yci

Yci−1

tan(θ∗i+1 + θi)

)

. (31)

When the i-th branch corresponds to the last step of the tree ramification, θ∗i+1 = 0.If the tree is not H-shaped but each branch divides in n new branches, see Figure 65(a),the equivalent transmission line model in Figure 64 modifies to have n transmission linesin the right part, see Figure 65(b), and the expression for the equivalent electric anglecan be easily generalized

θ∗i = arctan

(nYci

Yci−1

tan(θ∗i+1 + θi)

)

. (32)

8.2.1 Asymptotic value of the equivalent angle

Particular case: In an H-fractal tree with Zci = 2 Zci−1, tan and arctan in expres-sion (31) cancel each other, and the equivalent electric angle is obtained by the simpleaddition of the electrical length of the branches as

θ∗i = θ∗i+1 + θi = θ∗i+2 + θi+1 + θi . . . (33)

with θi = βid0

ηi , η >√

2, thus

θ∗i = θ∗i+2 + βd0

ηi+1+ β

d0

ηi= θ∗i+3 + β

lspiral

︷ ︸︸ ︷(d0

ηi+2+

d0

ηi+1+

d0

ηi. . .

)

(34)

74

2

1

n

· · ·

(a)

YIN

Zc1, β1

Zc2, β2

Zc2, β2

Zc2, β2

Zc2, β2

YL2

o.c.

o.c.

o.c.

o.c.

d1

d2

d2

d2

d2

n

...

(b)

Figure 65: Equivalent transmission line model for a tree with n ramifications in eachiteration terminated by an open circuit (o.c.).

75

(a) η=1.5, µ=1.5. (b) η=1.5, µ=2.

Figure 66: H-tree for d0 = 3w0.

Expression (34) shows somehow that the structure is geometrically a tree, but in termsof electric length it is a spiral and the total length of the spiral is

lspiral = d0

∞∑

i=1

1

ηi=

d0

1 − 1η

, since1

η< 1. (35)

Therefore the H-fractal tree has infinite geometrical length for√

2 < η < 2, but finiteelectrical length.

8.3 Examples

In this section several tree conformations are studied. The tree shapes are formed fixinga w0 and then taking different values for d0, η and µ.

8.3.1 d0 = 3w0

Once w0 and d0 are fixed we are going to study different values for η and µ.

1. η = 1.5, µ = 1.5. The tree corresponding to this combination of values for w0, d0,η and µ is depicted in Figure 66(a).

It can be observed that the tree is not a self avoiding one, because equation (14)is not accomplished. Indeed, from equation (14) we obtain that the ratio for thebranches not to overlap is d0

w0> 3.75. However, in our case d0

w0= 3, thus branches

are overlapping.

2. η = 1.5, µ = 2. The tree corresponding to these values for the parameters can beseen in Figure 66(b). Branches are overlapping again because of the same reasonas in the precedent case, that is, equation (14) is not fulfilled.

76

Figure 67: Tree for η = 2, µ = 2 (d0 = 3w0).

3. η = 2, µ = 2. When these values are taken as parameters the tree shown in Figure67 is obtained. Equation (14) is fulfilled this time, as for η = 2 it is necessarythat d0

w0> 1.5 for a self avoiding tree. However, as it can be seen in Figure

67, for η = 2 branches are becoming short very fast, and this results in a treenot spreading over the whole circumscribed rectangle, but, appart from the firstbranch, it concentrates in the upper part.

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

(η2 − 1)2=

18w20η

3

(η2 − 1)2= 16w2

0 (36)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 5.5w2

0, (37)

which represents 34.37% of the total surface of the circumscribed rectangle.

• Main path

We call main path to the addition of the length of one arm of every iteration,that is, d0(1 + η−1 + η−2 + η−3...),

L =3w0η

2

η2 − 1= 4w0 (38)

In this case the length of the main path is exactly the square root of thesurface of the circumscribed rectangle.

77

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

Iteration #

θ∗i [rad]

Figure 68: Equivalent angle as a function of the iteration number for a tree with η = 2,µ = 2 ( d0

w0= 3).

• Equivalent angle

The equivalent angle of the tree is computed from (31) and it is shown inFigure 68.

8.3.2 d0 = 5w0

1. η = 1.5, µ = 1.5.

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

((η2 − 1)2=

50w20η

3

(η2 − 1)2= 108w2

0 (39)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 42w2

0, (40)

which represents 38.88% of the total surface of the circumscribed rectangle.

• Main path

L =5w0η

2

η2 − 1= 9w0 (41)

2. η = 1.5, µ = 2.

• Circumscribed rectangle

78

(a) η=1.5, µ=1.5. (b) η=1.5, µ=2.

(c) η=2, µ=2.

Figure 69: H-tree for d0 = 5w0.

79

From (10):

Scr =2d2

0η3

(η2 − 1)2=

50w20η

3

(η2 − 1)2= 108w2

0 (42)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 14.5w2

0, (43)

which represents 13.42% of the total circumscribed rectangle surface.

• Main path

L =5w0η

2

η2 − 1= 9w0 (44)

3. η = 2, µ = 2.

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

(η2 − 1)2=

50w20η

3

(η2 − 1)2= 44.4w2

0 (45)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 9.5w2

0, (46)

which represents 21.39% of the total circumscribed rectangle surface.

• Main path

L =5w0η

2

η2 − 1= 6.6667w0 (47)

4. Equivalent angle

The equivalent angles of the trees with d0

w0= 5 are computed from (31) and they

are shown in Figure 70.

8.3.3 d0 = 10w0

1. η = 1.5, µ = 1.5

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

(η2 − 1)2=

200w20η

3

(η2 − 1)2= 432w2

0 (48)

80

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

η=2,µ=2

η=1.5,µ=1.5

η=1.5,µ=2

Iteration#

θ∗i [rad]

Figure 70: d0 = 5w0.

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 87w2

0, (49)

which represents 20.13% of the total circumscribed rectangle surface.

• Main path

L =10w0η

2

η2 − 1= 18w0 (50)

2. η = 1.5, µ = 2.

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

(η + 1)(η − 1)2=

200w20η

3

(η + 1)(η − 1)2= 432w2

0 (51)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 29.5w2

0, (52)

which represents 6.82% of the total circumscribed rectangle surface.

• Main path

L =10w0η

2

η2 − 1= 18w0 (53)

81

(a) η=1.5, µ=1.5. (b) η=1.5, µ=2.

(c) η=2, µ=2.

Figure 71: H-tree for d0 = 10w0.

82

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

η=2,µ=2

η=1.5,µ=1.5

η=1.5,µ=2

Iteration#

θ∗i [rad]

Figure 72: Equivalent angle for d0 = 10w0.

3. η = 2, µ = 2.

• Circumscribed rectangle

From (10):

Scr =2d2

0η3

(η + 1)(η − 1)2=

200w20η

3

(η + 1)(η − 1)2= 177.7778w2

0 (54)

• Efficient surface

From (22)

Seff =d0w0

1 − 2ηµ

− µw20

2(µ2 − 2)= 19.5w2

0, (55)

which represents 10.96% of the total circumscribed rectangle surface.

• Main path

L =10w0η

2

η2 − 1= 26.6667w0 (56)

• Equivalent angle

The equivalent angles of the trees with d0

w0= 10 are computed from (31) and

they are shown in Figure 72.

83

d0 = 3w0 d0 = 5w0 d0 = 10w0

η = 1.5, µ = 1.5 - - - - 108 42 38.88 9 432 87 20.13 18

η = 1.5, µ = 2 - - - - 108 14.5 13.42 9 432 29.5 6.82 18

η = 2, µ = 2 16 5.5 34.37 4 44.4 9.5 21.39 6.6667 177.778 19.5 10.96 26.667

Scr/w

2 0

Seff/w

2 0

Seff

Scr(%

)

Lm

ain/w

0

Scr/w

2 0

Seff/w

2 0

Seff

Scr(%

)

Lm

ain/w

0

Scr/w

2 0

Seff/w

2 0

Seff

Scr(%

)

Lm

ain/w

0

Table 1: Geometrical characteristics of trees built with different values of η, µ and d0

w0

parameters.

8.4 Conclusions

In Table 1 the geometrical characteristics of the precedent examples are shown. Twomain conclusions can be extracted.First, the bigger the ration d0

w0is, and hence the farther it is from the non-overlapping

limit for a thick H-tree, see (14), the smaller the ratio Seff

Scris. To cover the surface of

the circumscribed rectangle in a more efficient way it is better to choose the value ofthe ratio d0

w0near to the non-overlapping limit, because like this the tree is enclosed in

a smaller surface (d0 is smaller, and remember that Scr ∝ d20), where branches do not

touch each other, but they spread more uniformly over the surface. On the contrary forbig ratios d0

w0(see Figure 71), we have the impression that the circumscribed rectangle

is somehow ”empty”.Then, for all the ratios d0

w0there are different behaviours of geometrical properties de-

pending on the values of η and µ, the factors in which the length and the width of thebranches diminishe, respectively. When µ = η the ratio Seff

Scris bigger than for µ > η, for

which the branches reduce their width more rapidly than their length. Thus, η affectsthe length and thus the circumscribed rectangle and µ affects the width, and thereforethe efficient surface. From Figures 70 and 72, it can be concluded that η acts on theequivalent electric angle in the same way it affects the geometrical length, that is , thebigger η the smaller values for the equivalent electrical angle. For the same value ofη and different µ, only slight differences in the equivalent electrical angle are observedwhen d0

w0is small.

Studies on the asymptotic value of the equivalent electric angle show that the limit valueexists, but it is analytically studied only in the case Zci = 2Zci-1, when the expressionfor the electric angle looses its non linear behaviour.

84

(a) rhomb (b) square

Figure 73: Filiform trees in different configurations.

9 Pre-fractal capillary devices: Analysis and

design

9.1 Geometry

Consider the filiform fractal trees in Figure 73. They are formed by a maintrunk with n ramifications in each stage, in this case 2, and keeping differentrelative angles in the ramifications. In any case the tree is a one port open-ended structure. Now let us connect one of these trees to a left to rightflipped version of itself, as it can be seen Figure 74, to obtain a new 2-portstructure called here ”capillar”, to describe the effect of ”passing through”.

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

Figure 74: Capillar: two-port structure.

85

A

BC

Figure 75: Different subnetworks in a capillar.

9.2 Transmission line theory

In this section the analysis of a capillar from a circuit point of view willbe done. It is useful to consider the capillar as a two-port network formedby several two-port subnetworks, where the basic unit is the smallest of theloops in the capillar.The analysis of the global structure starts with the characterization of thebasic subnetwork, that is, the smallest loop, which in Figure 75 is namedsubnetwork A.A single loop can be characterized in terms of its scattering matrix as theparallel of two branches, see Figure 76.Each of the branches consists in the cascade connection of three scattering(S) matrices, corresponding to the S matrix of a transmission line, the Smatrix that will be called here ”seed” whose function will be explained inthe following lines, and yet another S matrix corresponding to the right sidetransmission line. The seed matrix can take different values depending on thekind of connection between left and right side of the capillar loop. Typicallythere are 4 possibilities: through, open circuit, short circuit, gap. Table 9.2shows the different values the matrix must take in these cases.The cascaded connection of the two two-port networks shown in Figure 77

86

Stxl

Stxl

Sseed

Sseed

Stxl

Stxl

Figure 76: S matrix representation of a loop.

Seed Scattering matrix

S =

(

0 11 0

)

S =

(

1 00 1

)

S =

(

−1 00 −1

)

Table 1: Different possible values for the S matrix of the initial seed.

87

S ′ S ′′ S ′

Schain

Figure 77: Cascade connection

using scattering parameters is done in the following way [1]:

Schain =

[

S ′

11 + κS ′

12S′

21S′′

11 κS ′

12S′′

12

κS ′

21S′′

21 S ′′

22 + κS ′′

12S′′

21S′

22

]

(1)

where

κ =1

1 − S ′

22S′′

11

.

Symbolically we must do the following for a branch:

Sbranch = chain(chain(S ′, S ′′), S ′). (2)

Once the resulting Sbranch matrix of the cascade connection is obtained, theparallel connection between the two branches must be done. From [2] weobtain the general expression for the parallel connection of two scatteringmatrices S1 and S2

Sparallel = (3I +S1)−1(S1−I)+4(I +S1)S2[(3I +S1)−(S1−I)S2]

−1(I +S1)(3)

The global S matrix of the subnetwork becomes the seed for next iteration.In the case the characteristic impedance of the lines of iterations i and i + 1are different, a S matrix representing the jump of impedances between thelines must be added on the left and right of the seed. If the lines from oneiteration to the following one are of the same impedance the S matrix for thejump impedance is transparent.

9.3 Design: choosing the characteristic impedances of

the lines

A printed microwave structure is derived from a filiform capillar fractal bygiving some width to the lines. Choosing the right width we can get as a

88

(a) Top view

1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

freq [GHz]

|S11

|,|S

12|

(b) |S11|,|S12|

Figure 78: Top view and scattering parameters of a capillar whose lines havecharacteristic impedances following the law Zci = 2Zci−1.

(a) Top view

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

freq [GHz]

|S11

|,|S

12|

(b) |S11|,|S12|

Figure 79: Top view and scattering parameters of a capillar whose lines havecharacteristic impedances following the law Zci = Zci−1.

frequency response that one corresponding to a transmission line, see Figure78.For a capillar in which every branch divides into two new branches, it isevident that we need to double the impedance in the two new branches to geta single transmission line behavior for the whole structure, that is, nothingis reflected (|S11|=0) and all the input signal is transmitted (|S12|=1). Anyother relation between the impedances of the branches will give us a differentfrequency response; the idea is to find some interesting behavior betweenthese combinations.In Figure 79 we can see the top view of a capillar whose lines accomplishthe impedance law Zci = Zci−1, and its scattering parameters. In the band

89

4 l2

l2Zc

2 Zc

2 Zc

l1

(a) Top view

4 l1

l2

Zc

Zc

2 Zc

l1

(b) Top view

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

freq [GHz]

|S11

|,|S

12|

(c) |S11|,|S12|

Figure 80: Top view and scattering parameters of a capillar whose lineshave characteristic impedances following the laws that allow (a) only thesmall loop resonates, (b) only the big loop resonates. In (c) the scatteringparameter of the small loop has its first resonance at 2.5GHz, while the bigloop has its fourth resonance at 2.5GHz.

centered in 2.5GHz an incipient filtering-like behavior is observed, and it canbe improved by diminishing the ripple and increasing the slope with somedifferent impedance law.

9.4 Design: resonance frequency

In the last section we have seen how depending on the impedance law thecapillar behaves like a single transmission line or like something different,desirably a filter. In this section the principles to decide in which frequencyband the filtering properties appear are discussed.For this two iteration capillar, in which we distinguish one small inner loopand one big outer loop, the filtering properties appear at the frequency in

90

which both of the loops resonate. In general, for a n iterations capillar, itwill be interesting to place the resonances of the smallest loop in a frequencymultiple of its immediate consecutive exterior loop. To do so, let us call fn

the fundamental frequency of the n loop, the smallest loop of a n iterationcapillar, and fn−1 the fundamental frequency of the n − 1 loop. Then, therelation that these two frequencies must accomplish is:

knfn = kn−1fn−1 (4)

where kn and kn−1 are proportional constants. Taking into account thelengths of the two loops

lsmall = 4ln (5)

lbig = 4(ln−1 + ln) (6)

and inserting (5) and (6) in (4) the following equation is obtained:

ln−1 =kn−1 − kn

kn

ln. (7)

In general, for the i − th loop, we obtain:

li =ki − ki+1

kn

ln, for i = 1, . . . , n − 1. (8)

This expression gives the constant of proportion ki−ki+1

kn

for the relation ofall the lengths in the capillar, li, with the length of the smallest and innestloop, ln.

9.5 Design: geometry restrictions

Let us dimension a two order capillar fractal in taking k1 = 2 and k2 = 1.This choice leads to the following result (from (8)):

l1 = l2, (9)

the capillar is overlapping (see Figure 81). To avoid overlapping there aresome geometric restrictions to impose in (8), because it has been checkedthat not all ki are possible.Intuitively looking at Figure 82 it can be stated that for overlapping to beavoided the following condition must be accomplished:

li > li+1 + li+2 + · · · + ln (10)

91

Figure 81: Top view of a capillar whose lines have lengths l1 = l2.

that is to say, the length li must be bigger than the addition of the length ofall the branches on its right (see Figure 82). Thus

li >

n∑

j=i

lj =n−1∑

j=i

ki − ki+1

kn

ln + ln. (11)

This leads to

li >ki

kn

ln, i = 1, . . . , n − 1. (12)

Equation 12 gives us the inferior limit for all the lengths in the non-overlappingcapillar as a constant times the length of the smallest loop.

9.6 Iteration 2

From equation (8) and (12) we can conclude that fixing k2 = 1, the minimumvalue for k1 = 3, because k1 = 2 gives equal lengths for both loops, as it canbe seen in the precedent section, and this is not possible from the practicalpoint of view. In Figure 83(a) and Figure 83(b), the magnitude of scatteringparameters S11 and S22 for the big and small loop of the two iteration capillaris shown. The difference between taking k1 = 3 or k1 = 4 is observed. Whenk1 has an odd value the magnitude of S11 for the two loops coincide in onepole for the fundamental resonance of the small loop and the third resonanceof the big one, but they also coincide in a maximum of S11. So it canbe said somehow that when taking the odd value for k1 the coincidence isonce constructive and once destructive, while when taking the even value

92

l1

l2

l3

Figure 82: Schematic representation of the left part of a three order capillarand the respective lengths in each stage.

the coincidences are destructive and they are the resonant frequencies of thesmall loop.The resulting filtering band seems, although with improvable levels, more ev-ident when the coincidence is destructive, look Figure 84, where the responsefor an iteration 2 filter is plotted for the same relation of impedances in bothcases, Zc = [50, 50, 50]. The solid line corresponds to k1 = 4 and the dashedline to k1 = 3.We keep k1 = 4 and work with different values of Zc. Smaller values of Zc inthe last loop line give less ripple, better slope but less bandwidth, as can beseen in Figure 85.The top view and dimensions of these devices are shown in Figure 86. Thewidth of the lines is computed like in [3] for a substrate of ǫr = 1.07 andheight h = 1mm. The capillar can be built with different relative anglesbetween its branches, for example a square or rhomb or even in circularshapes. It is clear that the transmission line model only takes into accountthe length of the lines and not the relative position they have with respect tothe others, thus all these geometries give the same result from a transmissionline analysis point of view.

9.7 Higher order capillars

The same study is done for a capillar of order 3. With equation (8) and(12) we obtain the values of the constants ki: k1 = 8, k2 = 3, k3 = 1. Thelengths of the lines are: l1 = 5 ltot

k1, l2 = 2 ltot

k1, l3 = ltot

k1, with ltot the total

capillar length, from port 1 to port 2, without taking into account the short

93

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

freq [GHz]

|S11

|,|S

12|

k1=3

(a) k1=3

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

freq [GHz]

|S11

|,|S

12|

k1=4

(b) k1=4

Figure 83: Resonances of every single loop in a 2 iteration capillar fractal forbigger loop different resonant frequency factor.

94

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

Figure 84: Magnitude of S11 and S21 for two different constant k1, in dashedk1 = 3 and in solid k1 = 4, both for k2 = 1.

1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

Zc=[50 50 50]

Zc=50 50 33.3] (ρ=[−1/3 −1/2])

Zc=50 50 20] (ρ=[−1/3 −3/2])

Zc=[50 50 60]

(ρ=[−1/3 −1/4])

Figure 85: Magnitude of S11 and S21 for different Zc values.

95

(a) Zc = 50, 50, 50 (b) Zc = 50, 50, 33 (c) Zc = 50, 50, 20

(d) Zc = 50, 50, 50 (e) Zc = 50, 50, 33 (f) Zc = 50, 50, 20

Figure 86: Top view of iteration 2 filters.

lines for the connections. In Figure 87 we observe the response for a 3 ordercapillar for 2 different impedance relations. As for order 2, the filter getsbetter ripple and slope when the inner impedances are smaller. But it si notgood from the geometric point of view because the width of the line increaseswhen the characteristic impedance decreases and then the lines overlap (seeFigure 88).The top view and dimensions of these devices are shown in Figure 88. Thewidth of the lines is computed like in [3] for a substrate of ǫr = 1.07 andheight h = 1mm.For a capillar order 4 we obtain the following relation for ki with (8) and (12):k1 = 17, k2 = 8, k3 = 3, k4 = 1. The lengths of the lines are:9 ltot

k1, 5 ltot

k1, 2 ltot

k1, ltot

k1.

The top views corresponding to these dimensions can be seen in Figure 90.Conclusion: Controlling the level of the ripple by changing the relation be-tween the impedances in each stage of the capillar is feasible for low itera-tions, but it turns to be a complicate affair when handling with higher orderiterations. Ideally an optimization method will be needed for the success inreducing the ripple and increasing the slope in highly iterated capillars.

9.8 Comparison of fullwave responses

In this section we work with the capillar in its second iteration and keeping allthe impedances constant, which is not the filter giving the best response, as ithas been seen before, but for a matter of method comparison will provide the

96

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

Zc=[50 50 33 22 ]

Zc=[50 50 50 50 ]

Figure 87: Magnitude of S11 and S21 for 2 different Zc values for an order 3capillar.

(a) Zc = 50, 50, 50, 50 (b) Zc = 50, 50, 50, 50 (c) Zc = 50, 50, 33, 22

(d) Zc = 50, 50, 33, 22

Figure 88: Top view of iteration 3 filters.

97

9 9.5 10 10.5 11 11.5 12 12.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12| Zc=50 50 50 50 50

Figure 89: Magnitude of S11 and S21 for a 4 order capillar with an impedancelaw Zci = Zci−1.

(a) Zc = 50, 50, 50, 50 (b) Zc = 50, 50, 50, 50

Figure 90: Top view of iteration 4 filters.

98

(a) square (b) rhomb

1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

square

rhomb

(c) |S11|, |S12|

Figure 91: Comparison between the fullwave simulations of an order 2 squarecapillar and a rhomb one.

same information. As it has been said in precedent sections, the transmissionline model does not make differences between rhomboidal or square capillar,in which the relative positions of their lines is different respect the first one.On the contrary, from a fullwave model [4] a difference between the responsesof the two structures is expected. And so it is, as seen in Figure 91.The two capillars with different relative position of their lines are not the samestructure from the fullwave point of view. The most remarkable differencesare the frequency shift, the difference of level in the ripple and the peakappearing in the rhomboidal shape. The frequency shift is more noticeableat low frequencies; indeed, at high frequencies the slope of both structuresmatches well. The rhomboidal shape response has a spike appearing thatcannot be observed in the square one. This phenomenon might be a numericalerror, but it cannot be assured without contrasting the simulation with ameasurement. This point of the discussion will be retaken later on in thereport.

99

1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

fullwave

tx line model

Figure 92: |S11| and |S12| corresponding to an order two square capillar,obtained with the transmission line model (solid line) and fullwave model(dashed line).

9.9 Transmission line model vs. fullwave

Square The comparison between scattering parameters magnitude of anorder 2 square capillar is to be done. In Figure 9.9 we can see the magnitudesof S11 and S12 corresponding to the transmission line model and the fullwavesimulation for a square capillar. There is a shift between the two responses,but this shift seems to be constant in all the band, the central frequency isalso shifted. The level of ripple is constant for the transmission line model,but not for the fullwave model, where the ripple is more important for higherfrequencies. Otherwise the shape is the same. Even though the simplicityof the tx line model, results are quite accurate compared with the fullwavesimulation, the principal difference being a shift in frequency.

Rhomb The same comparison is now to be done with the rhomboidal cap-illar. In Figure 9.9 we can see the responses. The frequency shift is notconstant in all the band, the three poles of S11 have different shift, moreconcretely the difference in central frequencies is smaller. The shift is moreimportant at lower and at higher frequencies.

9.10 Different seed

The smallest loops of the capillar can be modified by changing the initialseed, as it has been explained in precedent sections. In this section the effectof changing the through connection by an open circuit is studied. The seed

100

1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

fullwave

tx line model

Figure 93: |S11| and |S12| corresponding to an order two rhomb capillar,obtained with the transmission line model (solid line) and fullwave model(dashed line).

could be changed in one, two, three or four branches, see Figure 94, where theseeds to be changed appear with a mark. The case 94(d) is not interestingbecause opening all the branches obviously the signal will not arrive to thesecond port. Cases 94(b) and 94(c) do not give either very good results,since only one branch is left connected. On the contrary, when there is onlyone opened branch, like in Figure 94(c), the response is ameliorated from thethrough seed case. In Figure 95 the magnitude of scattering parameters S11

and S12 is presented. In dashed line the values corresponding to a throughseed, and in solid line the corresponding to an open circuit seed. Although theband is decreased, the ripple level and the slope become importantly better.There are two clear transmission zeros and the slope becomes abrupt.In the same way, applying a different seed to a 3 order filter we obtaintransmission zeros and better level of ripple, see Figure 96.

Transmission line model vs. fullwave

In the same way as for the through seed capillar the transmission line responseand the fullwave response are going to be compared. For the transmissionline model the open circuit is simulated by introducing its scattering matrixinstead of the scattering matrix of a through connection. But for the fullwavemodel we have to make the layout of an open circuit, which is not so evident.What has been done here is to open the loop and put the horizontal branchesin a vertical position, keeping the same line length, see Figure 97.The results for both simulations are shown in Figure 9.10. It can be ob-

101

(a) One o.c. (b) Two o.c.

(c) Three o.c. (d) Four o.c.

Figure 94: Schematic representation of the point that must be replaced byan open circuit to simulate the capillar with different seed.

1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

connected

1oc

Figure 95: Magnitude of S11 and S21 for Zc = [50, 50, 50] for a direct con-nected capillar (dotted line), or a perturbed one (solid line).

102

4.4 4.6 4.8 5 5.2 5.4 5.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

connected

2 oc

1 oc

Figure 96: Magnitude of S11 and S21 for Zc = [50, 50, 50] for a direct con-nected capillar (dotted line), or a perturbed one (solid and dashed line).

Figure 97: Order two capillar with open circuit seed in one of its branches.

103

2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

tx line model

fullwave

Figure 98: |S11| and |S12| corresponding to an order two square capillar withopen circuit seed in one of its branches, obtained with the transmission linemodel (solid line) and fullwave model (dashed line).

served that the two transmission zeros are conserved, whereas the maximumin transmission is not constant in all the band, because there is a spikeappearing, and one of the poles in reflection is lost because of the spike.Whether the spike is a numerical effect or not will be only discerned withthe comparison to the measurements, but it could also been the result of thecoupling between the two open branches, as the simulated open circuit is notperfect.

9.11 Prototypes and measurements

Some prototypes have been built and measured to check the theory exposedin the precedent sections.

Materials. All the prototypes have been printed on a layer of epoxy 0.1mmand hot glued to a 1mm height foam, ǫr = 1.07 and the whole over a groundplane. In Figure 99 the top views of the built prototypes are shown.

Measurements In order to avoid the imperfections in the transition be-tween the connector and the lines, the soldering of the connectors is avoidedby the use of the text fixture shown in Figure 100. It consists of 4 coax-ial connectors, that can be moved on a guide, or moved translationarilly bychanging the position of the platform they are on.

104

(a) Order 2 square. (b) Order 2 square with differentseed.

(c) Order 2 rhomb. (d) Order 5 square.

Figure 99: Prototypes.

Figure 100: Measurement of one of the prototypes.

105

To improve the quality of the measurement a TRL (trough reflection line)calibration could have been done. This way the effect of the transition be-tween the connector and the line is taken into account and compensated.

Results In Figure 101 the measured scattering parameters of a two ordersquare capillar are compared to those obtained with the transmission linemodel and the fullwave commercial software. The first thing to commentis that the global shape of the three graphics is mostly similar, except forthe losses observed in the measurements and not appearing in the other twomodels, because losses are not taken into account in the simulations. Inthe precedent sections when comparing transmission line model to the full-wave model the most significant difference between the two models was ashift in frequency: the fullwave model had a mismatching toward the higherfrequencies with respect to the transmission line model. It was expectedfor measurements to be nearer to the fullwave model than to the transmis-sion line model. Unexpectedly it turns to be the contrary, the transmissionline model gives a response more approximated to the measured frequen-cies. Particularly good is the matching at the lower resonance frequency,where the transmission line model coincides with the measurement, whereasthe fullwave model gives a far prediction of the measurement. Then, forhigher frequencies there is a shift appearing for both models with respect tothe measurements, but the shift is always smaller for the transmission linemodel. Respect to the ripple it must be mentioned that the fullwave modelpredicts better the level of the ripple than the transmission line model, surelybecause of the coupling between the lines, that is completely neglected in thetransmission line model.In Figure 102 the measured scattering parameters of a two order rhombcapillar are compared to those obtained with the transmission line model andthe fullwave commercial software. Contrarily to the square case, the globalshape is changing from one model to another and also from measurementsto the models. The spike appearing in the fullwave model analysis appearsalso in the measurements, although less emphasized in the measurements.So the spike is not a numerical problem, it is real and due to the couplingsexisting for this topology of capillar and not for a square one. Again as inthe square case the transmission line model gives a better approximation tothe measurements for low frequencies than the fullwave model but there isa frequency shift appearing, shift always more important for the fullwavemodel.In Figure 103 the measured scattering parameters of a two order squarecapillar with a open circuit as the seed of one of its branches are compared to

106

1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

fullwave

tx line model

MEASUREMENTS

Figure 101: Comparison of magnitude of S11 and S21 for a two order squarecapillar (transmission line model, fullwave model and measurements).

1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

fullwave

tx line model

MEASUREMENTS

Figure 102: Comparison of magnitude of S11 and S21 for a two order rhombcapillar (transmission line model, fullwave model and measurements).

107

1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq [GHz]

|S11

|,|S

12|

fullwave

tx line model

MEASUREMENTS

Figure 103: Comparison of magnitude of S11 and S21 for a two order squarecapillar with different seed (transmission line model, fullwave model andmeasurements).

those obtained with the transmission line model and the fullwave commercialsoftware. The spike predicted with the fullwave software is also appearingin the measures, but less strong. For this structure neither transmission linemodel nor fullwave model are giving a reasonable prediction of the measuredresponse, there is a frequency shift for all the frequencies, although again itis smaller for the transmission line model. Nevertheless, the models are moresimilar to each other than to the measurements.

108

9.12 Conclusions and future work

An analysis method for a kind of printed line fractal-shaped structure hasbeen proposed and tested by the building of some prototypes. Results areencouraging, taken into account the simplicity of the model, and are leadingto the improvement of the model by:

• Consider effective lengths in TL models and discontinuities compensa-tion. This leads to a more sophisticated, but still fully tractable, modelwhich allows easy fine tuning of the algorithms.

• Doing a more efficient fullwave analysis based on a backbone approachwhich should analyze with the same easiness the different topologiesfor a capillar (square, rhomb, circular).

109

References

[1] J.S. Hong, M.J. Lancaster: Microstrip filters for RF/Microwave appli-cations, Wiley, (2001)

[2] P. Bodharamik, L. Besser, R.W. Newcomb: Two Scattering Matrix Pro-grams for Active Circuit Analysis., IEEE Transactions on circuit theory,vol. 18, no 6, November 1971.

[3] H.A. Wheeler:Transmission line properties of parallel strips by a dielec-tric sheet, IEEE Transactions MTT-13, March 1965, pp. 172-185.

[4] ADS-Momentum, Agilent Technologies.

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DISCLAIMER The work associated with this report has been carried out in accordance with the highest technical standards and the FRACTALCOMS partners have endeavoured to achieve the degree of accuracy and reliability appropriate to the work in question. However since the partners have no control over the use to which the information contained within the report is to be put by any other party, any other such party shall be deemed to have satisfied itself as to the suitability and reliability of the information in relation to any particular use, purpose or application. Under no circumstances will any of the partners, their servants, employees or agents accept any liability whatsoever arising out of any error or inaccuracy contained in this report (or any further consolidation, summary, publication or dissemination of the information contained within this report) and/or the connected work and disclaim all liability for any loss, damage, expenses, claims or infringement of third party rights.

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