Task 4.1: Design of fractal-shaped miniature devices

• Deliverables: D9 T4.1 Final Task Report
• Outputs: Miniature antennas and microwave devices designs to be used in tasks 4.3 and 4.4.

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.

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

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 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. 32). 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.

Figure 32: a) Hilbert resonator with k=4. b) Equivalent meander resonator

The performance of the single Hilbert resonator in Fig. 32(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. 32(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 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. 32(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. 33) shows a resonator with a single electric field maximum (upper side of Fig. 33) and a single magnetic field maximum (lower side of Fig. 33), both at the periphery of the layout.

Figure 33: Modified Hilbert resonator to build ellipctic filters.

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. 34) 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 P_in=10 dBm.

Figure 34: Cascade-quadruplet quasi-elliptical filter: a) Geometry, b) Frequency response simulated and measured in liquid nitrogen.

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. 35), 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.

Figure 35. Hilbert curves as top loads of monopoles compared with banner monopoles. All the structures have the same overall height. The percentage indicates the relative height of the monopole occupied by the pre-fractal and the banner, respectively.

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. 36). 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 36: 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.

Pre-fractal designs that include loops in their topology have been also tested as capacitive loads for monopole antennas (Fig. 37). Pre-fractals of this kind are the Delta-Wired Sierpinski (DWS) and the Y-Wired Sierpinski (YWS).

Figure 37: Simulated pre-fractals used as top-loading of monopoles: Delta-Wired Sierpinski of 3^rd and 4^th iteration and Y-Wired Sierpinski of 3^rd iteration. Percentages indicate the relative size of the load versus the total height of the monopole.

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.

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. 38). 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.

Figure 38: Bow-tie (left) and three-iteration Koch-tie (right) dipoles.

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. 39 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).

Fig. 39: Fourth iteration of a Gosper Island (GI-4) pre-fractal surfaces made with strips: complementary designs. The central surface of both designs (enclosed in the green circle) look self-complementary

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.

Figure 40 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.

Fig. 40: 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).

Figure 41 shows the 3-D radiation patterns at the same operating frequencies as in Fig. 40. 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. 41: 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.

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 3^rd iteration of the YWS (Fig. 42) succeeded in reaching a compromise in Q and η when compared with other wired Sierpinski designs.

Figure 42: Three-iteration Y-Wired Sierpinski monopole (YWS-3).

More significantly, the Q factor, loss efficiency η and radiation pattern (Fig. 43) 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).

Fig. 43: 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.

3-D Pre-fractal tree

The performance as small antennas of several iterations of a 3D pre-fractal tree antenna (Fig. 44) 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.

Fig 44: 3-iteration pre-fractal 3-D tree

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 45 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 45: Efficiency and quality factor of the GA optimized antenas (Pareto front) versus different pre-fractal configurations, as a function of the electrical size at resonance.

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 ?liform H-fractal trees have been studied (Fig. 46). The ?at thick stemmed H-tree is considered afterwards (Fig. 47).

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

and parameter μ is the ratio between the stem and the braches width:

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. 48) 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. 49). This is in full accordance with the theoretical findings in Workpackage 2: “Vector calculus on fractal domains”.

Fig. 48: Equivalent transmission line model for a branch of the tree terminated by an open circuit

Fig. 49: Electrical length of the equivalent transmission line for d₀=5w₀ and increasing number of pre-fractal iterations

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. 50) and many other line structures and antennas.

Figure 50: Two-port capillar. The structure is build recursively from parallel-connection blocks (A, B, C).

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. 51). The subnetworks forming the global structure are related one another by an including or embedding property.

Figure 51: 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. 52). The results are very encouraging, taking into account the simplicity of the transmission line model.

Figure 52: Comparison of magnitude of S₁₁ and S₁₂ for a order-two square capillary (transmission line model, full wave model and measurements).