Tulip tree leaf-shaped microstrip MIMO antenna utilizing the superformula

1. Introduction

Exploring nature as a giant laboratory offers valuable insights into our lives and fundamental needs. A primary need that stands out is “Communication,” essential for all living beings. As technology advances, the demand for wireless communication devices has surged, catering to applications like remote sensing, radar systems, and critical medical and military uses (Bia et al., 2013). Antenna structures play a crucial role in these systems. In recent decades, there has been remarkable interest in enhancing communication systems, particularly through innovative antenna designs. The push for compact, lightweight wireless devices with broad bandwidth is more important than ever. This trend emphasizes the need to miniaturize antenna systems, keeping pace with the shrinking size of communication devices.

Communication systems inspired by nature have been interesting topics for researchers. For example, a Yagi-Uda array antenna has been designed based on the morphology of dolphin teeth and jaws (Flint, 2012). Additionally, a broadband microstrip antenna has been successfully created using a design that resembles a cockroach antenna (Mahmoud and Elkamchouchi, 2011). There are also exciting innovations in antenna designs; a Quasi-Yagi helical antenna has been developed by modeling the structure of DNA origami (Shah and Lim, 2019). Furthermore, the shapes of plant leaves have proven to be intriguing for antenna designs. One example is a patch-like antenna inspired by the Opuntia ficus indica, designed as a compact monopole antenna (da Silva et al., 2021).

The concept of the microstrip antenna was introduced by Deschamps in 1953 and patented in 1955. Since the 1970s, its usage has significantly increased (G.A.Deschamps, 1953; Gutton and Baissinot, 1955). Microstrip patch antennas (MPAs) consist of three essential layers: a ground plane, a dielectric substrate, and a patch. They are employed in a wide range of applications, including wireless communication, space technology, the global positioning system (GPS), and the global system for mobile communications (GSM). However, their applications are limited by narrow bandwidth and low gain. To enhance the performance of MPAs, the selection of patch shapes, along with optimizations and bio-inspired algorithms, can effectively increase bandwidth, improve directivity, and reduce size. There are various bio-inspired designs for microstrip (Nobrega et al., 2019; Silva Júnior et al., 2016; Victor Rocha Xavier et al., 2019) and optical antennas (Alessandri et al., 2020; Kong et al., 2014; Pita et al., 2017) that feature optimized structures. Particularly, leaf shapes serve as a source of inspiration because their properties allow them to capture sunlight and convert it into chemical energy, similar to how an antenna receives electromagnetic waves (Cruz et al., 2017). The broad surface areas and symmetric structures of certain leaves enable MPAs to have suitable radiation patterns and effective surface currents (Silva et al., 2016).

To develop nontraditional patch shapes beyond conventional rectangular or circular designs, Johan Gielis introduced a geometric approach known as the superformula (Gielis, 2003). This technique allows for the creation of specific geometric shapes while also mimicking natural forms. Simeoni et al. applied the superformula in the design of an ultra-wideband (UWB) antenna, resulting in a circularly polarized radiated field (Simeoni et al., 2010). By leveraging the superformula, researchers crafted the electromagnetic properties of a lens antenna, forming a hybrid system based on geometric and physical optics. In a study conducted by Shaima Naser and Nihad Dib, a single-pole UWB antenna was designed and analyzed using the superformula, showcasing several advantages, including compactness, consistent high gain, and nearly constant group delay (Naser and Dib, 2016).

In recent years, several studies have advanced compact and high-isolation Multiple Input Multiple Output (MIMO) antennas for modern wireless systems (Ali et al., 2024; Din et al., 2023; Elabd et al., 2025; Ghasemi, 2025; Khan et al., 2023; Patel et al., 2024; Rai et al., 2025; Ravi and Kumar, 2022; Tiwari et al., 2024). A MIMO structure was created by placing two UWB antennas, designed with identical superformula configurations, on a single substrate measuring 35x52 mm². Both antennas operated in the frequency range of 3.1 GHz to 12 GHz, and simulation and measurement results revealed that the group delay remained almost constant (Naser and Dib, 2017). Amjad Omar et al. proposed a UWB patch antenna characterized by a sawtooth-like circular shape, functioning within the frequency range of 3.1 GHz to 10.6 GHz. This design achieved an approximate 30% reduction in the circular patch diameter, improved group delay performance, and high radiation efficiency (Omar et al., 2017). Additionally, a study conducted by Alexandre Jean René Serres and his team focused on bio-inspired microstrip antennas designed using the superformula, highlighting the suitability of fractal features for leaf-shaped antennas (Serres et al., 2017). Related studies have also demonstrated that bio-inspired optimization algorithms, such as GA and PSO, can effectively minimize antenna size and mitigate mutual coupling in MIMO systems, thereby enhancing channel throughput (Leal et al., 2023).

In this study, we proposed an MPA shaped like a tulip tree leaf, modeled using the superformula. We compared the performance of this antenna with a conventional rectangular patch antenna as well as other super-shaped antenna designs. The results indicate that our proposed antenna exhibits better |S11| (reflection coefficient) while maintaining a smaller size. We enhanced the antenna’s performance by introducing a special cut at the center, utilizing specific superformula parameters; we refer to this design as the hollowed patch antenna. This improvement led to effective radiation at the desired operating frequency, resulting in better antenna properties compared to conventional designs. Additionally, we conducted further investigations into the MIMO structures of the hollowed patch antenna, revealing exceptional results and antennas with impressive isolation performance.

2. Material and Methods

The initial step in the design process is to determine the antenna’s resonance frequency. Once established, the appropriate substrate material can be selected. In this study, the targeted operational frequency is set at 5.2 GHz to comply with WLAN technology standards. The selected substrate is fiberglass, which has a relative dielectric constant of 4.4 and a thickness of 1.6 mm.

To initiate the design of the antenna patch shape, we began by calculating a traditional rectangular patch. The method for determining the standard size of the rectangular patch was adapted from Ref. (Ranjan, 2018). In our efforts to minimize size and enhance antenna performance, we employed the superformula to derive patch shapes, moving beyond conventional designs.

Once the superformula parameters have been determined, the patch geometry was created in 2D using MATLAB. This 2D geometry is then converted into a 3D model with the AUTOCAD program, after which the design is imported into the CST program for simulation. Following the initial simulation of the designed antenna, a genetic algorithm is employed to refine the selected parameters and enhance the overall design. Alongside optimizing the patch shape, the ground plane is also fine-tuned to achieve the desired specifications. This optimization process is carried out until the best possible results are obtained. The finalized designs are then fabricated on a fiberglass substrate. The antenna parameters are measured, and the results are compared with the simulations.

Nature displays a remarkable diversity of shapes that often resemble deformed ellipses. It is compelling to consider how mathematics serves as a lens through which we can interpret the wonders of nature. Johan Gielis effectively encapsulated this concept through his innovative Superformula, represented as f(θ) and comprising six adjustable parameters shown in Eq. 1. (Gielis, 2003).

In here, parameter m dictates the number of fixed points, sectors, corners, or grooves in the shape and their corresponding spacing. The parameters a and b are associated with the maximum extents along the horizontal and vertical axes. The dimensions of the shape can be normalized to ensure that all points reside on or within the unit circle. Additionally, the parameters n₂ and n₃ influence the degree to which the shape conforms to the unit circle. Utilizing the superformula makes it possible to parameterize and accurately depict nearly all closed curves and ellipse-like shapes.

When deriving the Super-Formula parameters, we assume that the center of the shape is positioned at the point (c_x, c_y) on the xy-plane, as depicted in Fig. 1. We then identify the smallest rectangle that can fully enclose the shape. The parameter a represents half the length of this rectangle, while b denotes half its width. To assign the four parameters (m, n₁, n₂, n₃, we can employ forecasting and genetic algorithms. To ensure the predicted values are accurate, it is essential to calculate the deviation. The process for calculating this deviation is detailed in Equations 2 and 3.

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Fig. 1.

The determination of the superformula parameters.

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Here, ${\hat{r}}_i$ denotes the observed value, while ${\tilde{r}}_i$ denotes the predicted value. When the predicted values align perfectly with the actual values, S² will be equal to 0. The goal is to minimize S² during the parameter selection process. Certain supershapes, generated by adjusting the six parameters in the formula, may exhibit symmetrical properties that make them suitable for antenna design.

3. Results and Discussion

The microstrip antenna is engineered to operate at a frequency of 5.2 GHz, which aligns with WLAN technology standards. The substrate material selected is fiberglass, characterized by a relative dielectric constant of 4.4 and a thickness of 1.6 mm. The design employs a superformula to create patch shapes that not only reduce size but also enhance the performance of the antenna compared to traditional designs. Initially, leaf geometry comprising five sections was developed; however, the radiation efficiency was suboptimal and did not align with the desired antenna characteristics. Consequently, a patch shape inspired by the leaf of a tulip tree, consisting of four sections, as illustrated in Fig. 1, was created. The proposed design exhibits a patch structure shaped like a tulip tree leaf, which was determined using the formulas. After the first simulation of the designed antenna, the genetic algorithm was applied to the selected parameters to improve the design along with intentional guidance. These calculated values played a crucial role in selecting the superformula parameters, which were optimized to create a closed and symmetrical shape. The parameters for the design have been outlined in Table 1. The shape of the patch is also vital for the fabrication process; intricate branching and frequent indentations can lead to reduced production precision.

The key innovaiton is that the central portion of the patch (Fig 2(a)) is removed in the Hollowed design. Through iterative adjustments to the cavity size of the patch, we found that the optimal outcome is achieved with a half-sized cut-out at the center while maintaining overall dimensions of 11.0 mm × 11.0 mm, as illustrated in Fig. 2(b). After covering the aimed antenna properties with the hollowed design, we produced the antenna as seen in Fig. 3, measured its antenna properties, and compared them with simulation results.

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Fig. 2.

Exciting perspectives of the designs, (a) the intact design, (b) the innovative hollowed-out design from top.

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Fig. 3.

Hollowed microstrip antenna.

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The simulation results show that the innovative hollow design reaches a peak radiated frequency of 5.2 GHz and exhibits an S11 of approximately -52 dB at this frequency. The antenna operates within a frequency range of 4.45 GHz to 6.51 GHz, providing a bandwidth of 2.07 GHz, as shown in Fig. 4. The measurement results are represented in the same graph, showing that the produced antenna has maximum radiation at 5.2 GHz and S11 of about -40 dB at this frequency. The proposed antenna radiates from the 5.04 GHz to 5.48 GHz frequency range. Thus, the experimental and simulation results are entirely consistent, and the resonance frequency of the hollowed design satisfies the WLAN technology standards.

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Fig. 4.

S11 graph for the hollow tulip tree leaf-shaped microstrip antenna for both simulation and measurement.

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The radiation patterns of the tulip tree leaf-shaped antennas were drawn and compared using the CST program and the HFSS program to get consistency in the simulations, as shown in Fig. 5. In these graphs, the red and black curves show the radiation pattern in the H-plane, and the blue and green curves represent that of the E-plane. The E-plane radiation pattern resembles that of an omnidirectional antenna. It is seen that the maximum gain of the hollowed antenna is 2.95 dBi.

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Fig. 5.

Power Gain Diagrams; (a) 2D and (b) 3D power gains of the hollowed design

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The tulip tree leaf-shaped design and some super-shaped antenna designs available in the literature have been given in Table 2, compared to the conventional rectangular patch antenna. The proposed antenna presented here is smaller in patch sizes and substrate dimensions than other super-shaped antennas, and the size reduction is about 64% smaller than the conventional rectangular patch antenna operating at the same frequency. Regarding S11, the designed antenna also presents a clear improvement compared to the other super-shaped antennas, with around 40 dB. S11 of the hollowed design is 32% better than that of the traditional one. Although the proposed antenna exhibits a slightly reduced gain compared to conventional designs, this is an expected trade-off in miniaturized bio-inspired antennas, where compactness and improved isolation are prioritized over maximum gain.

As further investigation, we designed a MIMO structure for the proposed antenna. When designing MIMO antennas, interference is the most significant limiting factor. Interference is a condition in an electronic communication system that degrades signal quality or hinders proper operation. It can occur due to the collision or overlapping of one or more signals within the system. As a result of this collision, signal power may decrease, data rates may drop, the signal-to-noise ratio may increase, and even complete disconnection may occur. The source of interference can be other electronic devices, cabling, or electromagnetic fields, among other factors. Reducing interference is essential to achieve high-quality and reliable communication. Proper positioning of antennas plays a crucial role in reducing interference. Additionally, interference can be mitigated by using noise-rejecting filters. In the antenna design provided in Fig. 6, the distance between antenna elements is set to 2 mm.

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Fig. 6.

The MIMO design of the proposed antenna

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When analyzing the S11 graph illustrated in Fig. 7, it becomes clear that the designed MIMO antenna exhibits a reflection loss of -11.972 dB at a frequency of 5.21 GHz and a more substantial loss of -23.668 dB at 10.04 GHz. The implementation of a MIMO design plays a significant role in enhancing the performance characteristics of the antenna, particularly by amplifying the second harmonic response. This feature is crucial for applications that require precise resonance at multiple frequencies, as it allows the antenna to efficiently operate at both its fundamental frequency and the second harmonic frequency. This dual-band capability makes the antenna especially suitable for a variety of applications that demand reliable performance across multiple frequency bands, thereby expanding its versatility in telecommunications and wireless communication systems.

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Fig. 7.

S11 graph of the proposed MIMO antenna

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The ECC is a key metric in MIMO antenna systems, primarily responsible for assessing the correlation between the S-parameters among diverse antenna elements. Fig. 8 presents the ECC results obtained from the designed MIMO antenna. The results demonstrate that enhanced isolation between these antenna components has led to significantly low ECC. Especially in the radiation frequency bands, the ECC values are below 0.001. This sought-after feature helps to reduce mutual coupling and improve overall system performance.

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Fig. 8.

The ECC and the DG of the proposed MIMO antenna

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Diversity Gain (DG) provides a quantitative assessment of the enhancement in signal reliability achieved through diversity techniques, such as the utilization of multiple antennas to mitigate fading. Fig. 8 also illustrates the DG results associated with the designed MIMO antenna. The findings indicate that the DG reaches 9.992 dB and above at their designated radiation frequencies for a configuration involving two antennas, which closely resemble those of an ideal uncorrelated MIMO system.

To further evaluate the MIMO performance of the proposed antenna, the Total Active Reflection Coefficient (TARC) and the Mean Effective Gain (MEG) were calculated. As shown in Fig. 9, the TARC remains well below the acceptable threshold of -10 dB across the operating bands, reaching -39.31 dB at 5.2 GHz and -22.09 dB at 10.1 GHz. These values confirm that the antenna exhibits excellent impedance matching and minimal signal degradation under multiport excitation. In addition, the MEG of the antenna was obtained as -3.01 dB, which is close to the ideal value of -3 dB for a two-element MIMO system. This result indicates balanced power distribution between the antenna elements and verifies that the proposed design provides reliable diversity performance for MIMO applications.

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Fig. 9.

TARC of the proposed MIMO antenna with minima at 5.2 GHz (-39.31 dB) and 10.1 GHz (-22.09 dB)

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4. Conclusions

This study presents microstrip antenna designs intended for WLAN applications that utilize patch shapes inspired by nature, instead of traditional designs. Specifically, antenna designs resembling the shape of Tulip tree leaves were created using the superformula. The microstrip antennas, operating at a frequency of 5.2 GHz, were fabricated on a fiberglass substrate with a hollowed form. The voltage standing wave ratio (VSWR) and S11 were measured. The optimized super-shaped patch reduced patch dimensions by 64% compared to a rectangular patched microstrip antenna at the same frequency. Notably, S11 of the hollowed design was 32% better than that of the rectangular patch antenna. Experimental measurements and numerical computations conducted using CST and HFSS showed strong agreement, highlighting the technical potential of the proposed tulip tree-inspired, superformula-assisted microstrip antenna designs. The implementation of a MIMO design significantly enhanced the performance characteristics of the proposed antenna, particularly by amplifying the second harmonic response. This dual-band capability makes the antenna especially suitable for various applications that require dependable performance across multiple frequency bands, thus increasing its versatility in telecommunications and wireless communication systems. The ECC results obtained from the designed MIMO antenna indicate that improved isolation between its components has led to significantly low ECCs, especially 0.001 or below at the desired frequency ranges. This desirable feature contributed to reduced mutual coupling, enhancing the overall system performance. Additionally, the diversity gain results for the designed MIMO antenna show 9.992 dB and above at their designated radiation frequencies for a configuration involving two antennas, which closely resemble those of an ideal uncorrelated MIMO system.