Patterns of the radiation properties for Peano antennas

This paper studies metallic microstrip antennas with air as a substrate in the UHF band, patterned after space-filling, self-avoiding, and self-similar (FASS) Peano curves. Our novel study is based on context-free grammar and genetic programming as computing tools to unravel the role of geometry on both; the Voltage Standing Wave Ratio (VSWR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<2$$\end{document}<2) and frequency resonance patterns for Peano antennas. We use in our approach the numeric method of moments (MoM) implemented in Matlab 2021a to solve the corresponding Maxwell equations. Novel equations for the patterns of both features (resonance frequencies and frequencies such that the VSWR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<2$$\end{document}<2) are provided as functions of the characteristic length L. Antennas spanning a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\times L$$\end{document}L×L area (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\le 0.1\,m$$\end{document}L≤0.1m), feeding points set at seven places, and three widths of the metallic strip are introduced as instances of our approach. Finally, a Python 3.7 application is constructed to facilitate the extension and use of our results.

www.nature.com/scientificreports/ is 11.11% of Ŵ 2 ( where Ŵ is the reflection coefficient) at the antenna feeding point. This value is a sensible criterion for producing antennas suitable for use in several different technological fields 4 . The Peano microstrip antenna can be fabricated using photolithography and etching techniques on a dielectric substrate. Precision in the manufacturing process is essential to ensure that the patch geometry and antenna characteristics are consistent with the design. Figure 1 shows a Peano 2 antenna with a Teflon substrate. An antenna's substrate affects its efficiency and performance in size, bandwidth, and radiation efficiency. When designing an antenna, selecting a substrate with the appropriate characteristics is essential to achieve an optimal balance between size, weight, efficiency, and performance. In this work, the antenna's efficiency ( η ) is almost perfect when using air as the substrate. However, when other substrates are simulated, the antenna's performance may drop to 0.7 in some instances. Please note that this study does not cover radiation efficiency as there are numerous substrate and configuration options to consider.
An application in Python 3.7 is provided; it is available on GitHub to access the full extent of the results and examples reported here and produce additional ones that may be useful for the researchers' community 34 . This application works finding: (a) the resonance frequencies and b) frequencies such that VSWR<2) to produce Peano antennas spanning a square area smaller than 0.01m 2 , in the UHF band, with four (seven by symmetry) feeding point positions and three microstrip widths.
This paper is organized as follows. The following section introduces the model and gives an insight into the simulation details. Then, in Section 3, we present results, first (Subsection A) for the resonance frequency analysis and second for the VSWR< 2 (Subsection B) analysis. In Section 4 we summarize and conclude.

Numerical simulations
We choose metal to build the microstrip Peano antennas and air as the substrate to simplify the simulations and focus on the influence of the topology of the antenna, namely, the production rules iteration of the fractal Peano antenna, the feeding point positions, the spanned area by the geometry, and the microstrip Peano antenna width over the performance. Our simulations are conducted from 290 MHz to 3100 MHz; this frequency range contains the UHF frequency band and avoids discontinuities and other numerical problems in the UHF band boundaries. In addition, we use a 50 ohms resistance in the feeding line throughout this paper.
Here, the method of moments (MoM) 35 is applied to solve the Maxwell equations on the meshed metallic microstrip Peano antenna with air as substrate using the 2021a) MATLAB antennas toolkit 36 . We mesh the Peano antenna with square segments d × d , where d is set to 0.25 mm, 0.5 mm, and 1 mm; see Fig. 2. Figure 2 shows two feeding points; the first is a valid feeding point for the numerical computation since both (magenta) edges do not touch. The second is a not-valid feeding point since the (magenta) edges coincide in the euclidean (green) point, producing an electrical short circuit. The number of these segments is proportional to S; see Table 1. Please note that the number of corners in the Peano curve increases with the production rule i. Corners need a particular treatment in the MoM numeric; the feeding points cannot be placed there because the computing does not produce valid results.
The symbolic regressor GPlearn 37 identifies the underlying mathematical expressions in the data obtained from both simulations; resonance frequencies and the frequencies such that VSWR<2. At first, perform simulations for an iteration i of the production rules, size L, allowed feeding point, and microstrip width value. Next, we ran a batch of 1000 genetic algorithms on the data with randomly chosen configuration parameters, number of generations, and algorithm initial population. We report the resonance frequency and frequencies equations such that VSWR<2 as functions of the antenna size L that defines the spanned L × L area.  www.nature.com/scientificreports/ The Peano antenna simulation results, namely, the resistance, reactance, and VSWR simulations, produce a database saved for further analysis.
Our method constructs a Peano antenna geometry using an L-system geometry generator that sets an antenna width d, a feeding point, and an L for the spanned area L × L . Then we calculate the electromagnetic features for the antennas using MoM in MATLAB 2021a 36 . This procedure is repeated until generating a database of 996 antenna simulations. The MoM computing is the only part of the algorithm that uses MatLab; the remaining computing is homemade software coded in Python 3.7. The electromagnetic features produced by our simulations could be obtained by any other (commercial) software or computed directly from the Maxwell equations in Python 3.7. We use the Matlab 2021a antennas toolkit for expediency.
In the resonance frequency module of our method, we extract from the original basis (with 996 elements) one basis of the form (F, L) and apply genetic programming using GPlearn 37 to find the best and simplest equation L = f (F) to fit the data; we found this to be L = aF −1 + b . Then, we obtain the best a and b by the minimum least square. In the following module, from the original basis (with 996 elements), we get all the antenna frequencies such that VSWR < 2; and produce another (F, L) basis. Again, applying GPlearn 37 , we verify that L = aF −1 + b is the type of equation that fits best this new second database. Here a and b are obtained by minimum least square, as was done in the resonance frequency module 34 .   www.nature.com/scientificreports/ In the solver module, we implement two types of problems; a) set F to find the best L for any Peano_i antenna that resonates and has a VSWR < 2, and b) set L and find the best resonance frequency such that VSWR < 2. Finally, the remaining Peano_i properties are calculated 34 .

Results
This Section presents the patterns in resonance frequencies and frequencies such that VSWR < 2 . First, we introduce novel equations that characterize the resonance frequency and frequencies such that VSWR<2 as leading rules to produce small Peano antennas (covering an area smaller than 0.01m 2 ) with adequate performance. We start by introducing the resonance frequency analysis for i = 1, 2, 3. Then, we study the frequencies where VSWR<2 for i = 1, 2, 3. This paper's most essential results are the equations obtained using GPlearn as a symbolic regressor for the resonance frequencies and frequencies such that VSWR<2 as functions of L; both equations are of the form L = aF −1 + b . Here a and b are parameters to be obtained using least-square fitting. F is the resonance frequency for the first case and the central frequency such that VSWR < 2 in the second case. We also provide the seconddegree polynomial coefficients that relate the so-called central frequency (the mean value) such that VSWR< 2 with the bandwidth around this as a percentage of the central frequency. F is always in MHz and L in meters. These curves are valid only in the UHF band; we should test other band frequencies in the near future.
Our simulations produce many numerical results that we illustrate in this paper. Since these simulation results plus additional numeric can be obtained through the application Peano_antennas 34 , we consider that the examples presented here and this application available in GitHub represent well all the possibilities of our approach. Therefore, in the following Subsections, we introduce some of our simulation results via examples.
Peano antennas: resonance frequency equations. We simulate the Peano antenna resonance frequencies, sweeping frequency from 290 MHz, with a frequency increment of 0.25 MHz (for i = 1, 2) and 1MHz (for i = 3), up to 3100 MHz. We obtain 11241 impedance data (for i = 1, 2) and 2811 impedance data (for i = 3). Then, we search for the zero crossings of the reactance; thus, we calculate the resonance frequencies. For the Peano antennas iteration 3, we perform calculations by a frequency step of 1MHz to keep a sensible balance between the computation time used for the simulations and the antenna complexity. We study the resonance frequency equations for Peano antennas generated by production rule iteration i = 1, 2, 3. Figure 6 shows how the electrical resonance of the antenna is calculated. The reactance property obtained in the feeding point is shown throughout the UHF band, and in two points, the resonance has a zero crossing. This figure shows an approach to these points, as seen in each value calculated before and after the crossing. For the calculation of the resonance, an equation is created between points 1 and 2 of the property and is resolved to obtain the most exact point of the zero crossing.
Peano_1. We show our results for a Peano antenna (production iteration rule equal to one); see Fig. 3a). In order to give an idea of the complexity of our simulations, we introduce our study with two examples.
Example 1 Let us consider L as 0.0225 m, the antenna width of 0.5mm, and all the possible feeding points, excluding corners, in total 195, throughout the antenna length S. The properties of these antennas in the UHF band can be seen in Fig. 4. Please, note that in Fig. 4a) we use the logarithm of the resistance and in Fig. 4d) the logarithm of the VSWR. Figure 4b) shows the antenna reactance; note that if this is zero, resonance occurs. The VSWR depends on the line's resistance (in this case, 50 ohms) and the resistance at the feeding point. It is clear from Fig. 4 that there are patterns in the electrical radiation properties of the Peano antennas that potentially can be described by equations; obtaining some of them is the goal of this paper. We reach this goal and present our results below. Figure 5 shows the calculated resonance frequencies within the UHF band for Peano_1 antennas, L < 0.10 m with feeding point placed at 0.125 (1/8) times the length S, and d=0.25 mm, d=0.5 mm, d=1.0 mm. Recall that the resonant frequencies depend on the feeding point's position, the geometry, and the microstrip antenna widths. Patterns, easily described by equations, appear in Fig. 5. We split this Figure into three cases, each for a particular d value: (a) 0.25 mm, (b) 0.5 mm (c) 1.0 mm. The resonance frequencies are reported for the UHF band (to be more precise, 300 MHz < F < 3000 MHz ) for different L ( L < 0.1 m) values, and the feeding point is placed at 0.125 times S. We apply GPlearn, dynamic programming, to obtain the possible functions to describe the curves shown in Fig. 5.

Example 2
The best possible functions that fit these curves are L = aF −1 + b . The parameters a and b for all the curves shown in Fig. 5 are given in Table 2. In this Table, besides the corresponding parameters, a and b for each curve, the validity range Min(F) ≤ F ≤ Max(F) for each curve fitting and the root means square (RMS) are provided. Note that for each antenna width, there are ten resonance equations and curves; each curve is labeled by a number shown in Table 2 and Fig. 7.
Example 3 shows how the Peano_1 antenna produces three different equations of the type L = aF −1 + b , each with a validity frequency range and RMS, for a particular antenna width (0.25 mm, 0.5 mm, and 1.0 mm). The parameters a and b are obtained from Table 2; this is done for the feeding point 1/8 times S. From Table 2, the resonance curve labeled 0, for d=0. 25 Fig. 5 with green + symbols and in Fig. 7 with a blue ( • ) line.
In Table 3 we show the parameters of the resonance frequency curves L = aF −1 + b for the Peano_1 antennas in three different antenna widths d (0.25 mm, 0.5 mm, and 1 mm), with the feeding point placed at 0.3125 (or 5/16) times the antenna length S. Note that eight resonance curves exist instead of the ten resonance frequency curves if the feeding point is placed at 0.125 times S for the Peano_1 antenna simulation. Table 4 shows the parameters of the resonance frequency curves L = aF −1 + b for the Peano_1 antennas with three different antenna widths (0.25mm, 0.5mm, 1 mm), and feeding point placed at 0.4375 (or 7/16) times the length of the antenna S. In this case, there are ten resonance curves. The Peano_1 antenna parameters, feeding point located at 0.5 (or 1/2) times the length S are shown in Table 6. Only two resonance curves are possible www.nature.com/scientificreports/ for the antenna width of 1mm. Tables 2, 3, 4, and 6 provide the information needed to design, through their resonance frequencies, many Peano_1 antennas, in a similar way. Despite the Peano_1 antenna relative simplicity, we observe some intriguing patterns in the resonance frequency curves for antennas with feeding points placed at (1/8, 5/16, 7/16, 1/2, 9/16, 11/16, 7/8) times the length S, and the simulated d: 0.25mm, 0.5mm, and 1mm. The number of resonance curves and their parameter values vary for antennas with the same feeding point but with different microstrip Peano antenna widths or the other way around. Concerning these resonance curves, the Peano_1 antennas with feeding points placed at 1/8 times S are described in Fig. 5; there are similar curves for the three different antenna widths. However, it is also necessary to note that there are few curves for some antenna widths and feeding points location. Peano_2 and Peano_3. For higher iterations of the production rule i, in particular, i = 2,3, the radiation properties, particularly the resonance frequency of Peano antennas, can be obtained by using the Peano_antennas application available in the reference 34 . However, for i > 3 the application only provides the geometry of the Peano antenna so far. resonance curve 0 simulation data 0 resonance curve 1 simulation data 1 resonance curve 2 simulation data 2 resonance curve 3 simulation data 3 resonance curve 4 simulation data 4 resonance curve 5 simulation data 5 resonance curve 6 simulation data 6 resonance curve 7 simulation data 7 resonance curve 8 simulation data 8 resonance curve 9 simulation data 9  www.nature.com/scientificreports/   www.nature.com/scientificreports/   www.nature.com/scientificreports/ Peano antennas: Equations for the frequencies such that VSWR< 2 . We report the Peano antenna radiation features for a frequency range (also called bandwidth BW) such that VSWR< 2, which is a needed condition for a suitable antenna performance 4 . At first, the mean value of this frequency range is obtained and called the central frequency f. GPlearn shows that f can be related to L by equations of the form L = af −1 + b , with the parameters a and b from Table 5 for Peano_1 antennas with feeding point placed at 0.125 (or 1/8) and 0.3125 (or 5/16) times the length S. The bandwidth BW (or frequency range) is reported as a percentage of the central frequency f by using a second-degree polynomial fitting; namely by equations of the form BW=αf 2 + βf + γ , where the parameters α , β , and γ are taken from Table 7 for Peano_1 and feeding point placed at 0.125 (or 1/8) and 0.3125 (or 5/16) times the length S.
Peano_1. Figure 8 shows the simulated data for the frequency f and L grouped in curves, which correspond to equations of the form L = af −1 + b (obtained by GPlearn), for the feeding point placed at 0.125 (1/8) times the length S and three different antenna widths: 0.25 mm, 0.5 mm , and 1.0 mm. Table 5 shows the parameters of the equations L = af −1 + b that govern the curves in Fig. 8 for the antennas with feeding point placed at 1/8 and 5/16 times S. The numbers in these curves correspond to the labels in the Table. Here, we present the simulated values for three different antenna widths: 0.25 mm, 0.5 mm, and 1.0 mm.

Example 4
Let us consider the curve labeled as zero in Fig. 8 and Table 5  We complete this Peano_1 antenna VSWR description by calculating the bandwidth BW around the central frequency where VSWR< 2. Namely, we search for the upper and lower bound of BW as a percentage of the central frequency; the percentage values are related to the center frequency using a second-degree polynomial fit BW = αf 2 + βf + γ . Please, find in Table 7 the coefficients of the resulting bandwidth equations α , β , γ calculated for antenna widths of 0.25mm, 0.5mm, and 1mm. Besides, these coefficients are given for feeding points placed at 0.125 (1/8) and 0.3125 (5/16) times the length S.

Example 5
For a central frequency of 1000 MHz, we can simulate the suitable Peanno_1 antennas with VSWR < 2 and the corresponding bandwidth around the central frequency. From Table 5 any of the three different microstrip Peano_1 antenna widths (0.25 mm, 0.5 mm, 1.0 mm) and two feeding points (1/8 and 5/16 times S), can be chosen; since 1000 MHz is, in each case, a valid frequency range. It is easy to see that we obtain a suitable design if the feeding point is placed at 1/8 times S with an antenna width of 0.25mm. In these conditions, the equation to determine the size L of the antenna is L = 39.3205f −1 + 60.3841 × 10 −5 ; the L value for 1000 MHz from this equation is 3.9924 × 10 −2 m , RMS equals to 2305.10 × 10 −7 . Table 7 shows the parameters of the polynomial that provide the bandwidth values around the central frequency of 1000MHz; thus, BW = −5.5556 × 10 −8 f 2 + 36.3870 × 10 −5 f + 0.8283 is obtained, with an RMS of 0.1138 that for 1000MHz gives BW = 1.1366% . To summarize, the characteristics of the proposed Peanno_1 antenna with a VSWR< 2 are: L=3.9924cm, d=0.25mm, feeding point supply at 1/8 times S, and BW =1.1366%.
Peano_2 and Peano_3. For higher iterations of the production rule i, in particular, i=2,3, the radiation properties, in particular, the frequencies such that VSWR<2 of Peano antennas, can be obtained by using the Peano_ antennas application available in the reference 34 . However, for i > 3 , the application only provides the geometry of the Peano antenna so far. For instance, Fig. 9 shows the Peano iteration 3 frequency resonances. The equations that describe these curves are in the Peano_antennas application; the blue dots in Fig. 9 are the data obtained from the computational simulations. Figure 10 shows the magnetic (H) and electric (E) fields of a Peano fractal antenna iteration 3 at a resonance frequency of (a) 476,4532 MHZ and (b) 2241,8604 MHz, where the antenna resonates and has a VSWR equal to 1,1102 simultaneously. For the calculation of resonance frequencies, we use the bisection method such that each new point around the zero crossing of the reactance is calculated in each step of the method.
The bisection method is based on the intermediate value theorem, which establishes that if a function is continuous in a closed interval [A, B] and takes opposite values at the ends, then the function has at least one zero in this interval. The bisection method works by dividing the interval [A, B] in half and determining which subinterval the function zero is; we repeat this process until a desired precision is reached or until the iterations are exhausted. A more complete and detailed simulation of the Fig. 10 antenna has been carried out; its directivity can be seen from 100 MHz to 15 GHz in 38 https:// youtu. be/ 62Zfw XYbX3w; and the E and H fields for the same frequency range in 39 https:// youtu. be/ YB153 9T0pCw. www.nature.com/scientificreports/   www.nature.com/scientificreports/ Table 6. Peano_1. Parameters for the resonance frequency equations, feeding point placed at 0.5 (or 1/2) times the length S, for three different antenna widths.  www.nature.com/scientificreports/   www.nature.com/scientificreports/

Conclusions
In this paper, we produce the geometry of the FASS antennas following the Peano curve. We set the iteration of the production rules equal to 1,2,3. Then, the MoM numerical method solves the Maxwell equations over the Peano_i metallic antenna (the Peano antenna produced by the iteration i) with air as a substrate. Let us choose i = 1, 2, 3, the microstrip antenna width d from the set (0.25 mm, 0.5 mm, 1.0 mm), L ≤.10m, the frequency range from 290 MHz to 3100 MHz (including the UHF band), and the feeding point from the set (1/8, 5/16, 7/16, 1/2, 9/16, 11/16, 7/8) times the length S. Then, our model delivers the main antenna electromagnetic properties directly, the patterns these data follow, and the corresponding governing equations.
Using genetic programming (GPlearn) as a novel tool, we find the best equations to fit the following Peano antenna radiation properties; (a) the resonance frequencies and (b) the central frequencies such that VSWR < 2, both as functions of L. For (b), we report the bandwidth BW around the central frequencies such that VSWR < 2 as a percentage of the central frequency. We conjecture that the existence of these curves is guaranteed for other iterations and parameters. Thus, we provide the systematic way and data needed to simulate efficient Peano antennas; our approach can be extended easily to include additional cases in the Peano_antennas application. The simulations provided in this paper give some insight into the technological applications of the patterns found in the Peano_i antenna resonance frequencies and VSWR<2. These calculations are presented within the UHF band; analysis of cases beyond this band is not in this paper's scope.
The simulations performed, plus the symbolic regressions (via GPlearn), produce the mathematical relation L = aF −1 + b that best describes the resonance frequency as functions of L and the VSWR<2 central frequency relation with L. We look for the crossing by zero for the first case and provide numerical methods for the best values of parameters a and b. For the second case, we look for all the values smaller than 2 (frequency range) and

L [m]
Curve 0 @ 1mm Data 0 @ 1mm Curve 1 @ 1mm Data 1 @ 1mm Curve 0 @ 0.5mm Data 0 @ 0.5mm Curve 1 @ 0.5mm Data 1 @ 0.5mm Curve 0 @ 0.25mm Data 0 @ 0.25mm Curve 1 @ 0.25mm Data 1 @ 0.25mm  www.nature.com/scientificreports/ again fit the L = af −1 + b through numerical methods and GPlearn. Here f is the central frequency within the frequency range; finally, we obtain the bandwidth around the central frequency as a percentage. For both cases, we optimize our numerics by the least possible RMS. We observe that the number of resonance frequency curves increases with increasing (production rules) iterations of the Peano antenna. Additionally, the change in antenna width, even by half a millimeter, favors the change in the number of resonance curves, which must also be considered for the design and applications of the antennas presented here.
From Fig. 4 and the corresponding Figures for Peanno_2 and Peanno_3 available in Peanno_antennas 34 , we can see the unmistakable patterns in the graphs feeding points vs. frequency for the; (a) logarithm of the resistance, (b) resonance, (c) return loss, and (d) logarithm of the VSWR. Furthermore, we observe how the graphs present patterns that follow increasingly complex curves suitable for being expressed by equations or relations in all these cases. Thus, we obtain equations for resonance frequencies and frequencies such that the VSWR is smaller than two from the data clustered in curves. In our case, we cluster the resonance frequency and frequencies such that VSWR < 2 in curves described by equations retrieved using GPlearn, as a successful first approach that supports our observations of patterns over these Figures; this is undoubtedly the best achievement reported in this paper. Besides, figures like Fig. 4 might be used as a hallmark of the Peano antenna for other purposes.
Analyzing the Peano antennas reported in this paper and produced in the Peano_antennas application, we conclude that the FASS geometry induces patterns in the radiation properties of the antennas suitable for being approached by equations whose parameters can be deduced from the geometry.