1 Background

The advent of 5G and the emerging era of 6G networks hold the promise of connecting billions of devices, each with its own unique set of constraints and behaviors [1,2,3,4]. This upsurge in demand can be attributed to the convergence of various aspects of our daily lives toward smart systems and wireless connected devices [5,6,7,8,9]. From our homes to our workplaces, from transportation to healthcare, we find ourselves relying on an interconnected web of devices and systems that enhance our efficiency, convenience, and overall quality of life [10,11,12,13,14]. The proliferation of smart home technologies [15, 16], wearable devices [17, 18], and the Internet of Things (IoT) has accelerated this transformation, creating a landscape where wireless connectivity has become an indispensable part of our existence [19,20,21,22,23]. Undoubtedly, the emergence of wireless communications is closely linked to recent advancements in antenna design and manufacturing, which have been revolutionized by innovative design procedures involving artificial intelligence and deep learning methods [24,25,26,27,28,29]. These antennas are designed to operate across a wide range of frequency bands, while ensuring compatibility with the diverse environmental conditions found in various installations [30, 31]. Not to mention the cost factor, which also plays a crucial role [32].In addition to the RF bands commonly used for various traditional applications, the need for increased bandwidth has compelled 5G and 6G consortiums to explore additional bands, including millimeter wave, sub-6G, and terahertz bands. This is driven by the significant growth in bandwidth requirements [33,34,35]. The design of antennas operating in these frequency bands, while considering installation conditions, is directly linked to the choice of materials used for the antenna substrate and its patch, which represents a challenging research topic in this field [36,37,38]. Polymers, such as PMMA (polymethyl methacrylate), are considered as potential candidates for the design of future antennas, since they offer several advantages, including lightweight [38], flexibility [39, 40], and low-cost manufacturing capabilities [41]. PMMA polymer is widely used for plastic optical fibers (POFs) that are suitable for various applications like data communication and sensing in environments where traditional glass fibers may not be practical [42,43,44]. Additionally, PMMA-based fiber Bragg sensors are gaining prominence for their ability to measure strain, temperature, and other physical parameters with high sensitivity and reliability [45,46,47,48]. However, it is important to note that research and development in this field are still ongoing to optimize the properties of polymers and validate their use in specific antenna applications, especially within the aforesaid frequency bands [49,50,51,52]. Therefore, the current study focuses on investigating the impact of using PMMA as a substrate for the design of rectangular and circular patch antennas intended for applications in the sub-6 GHz band. Several polymers have been utilized in literature to address the challenges posed by next-generation networks in terms of gain, bandwidth efficiency, and ease of environmental installation. In [53], authors have performed a review on the involvement of the metamaterials for designing reconfigurable antennas for 5G and 6G networks. Authors in [54] have implemented RT/duroid 5880 laminates as a substrate for designing quad band antennas operating at mmwaves bands. The improvement of the electrical and radiation properties is realized by using a binary-coded genetic algorithm. In [55], authors have proposed a circularly polarized antenna for IoT energy constrained devices, measuring 15 × 35 mm2, serves as a versatile portable RF energy harvesting device, resonating from 2 to 10 GHz with band notches from 2.5 to 3.5 GHz, while a voltage double rectifier (VDR) with a 9.2 × 20 mm2 size converts RF to DC with a wideband of operation from 2 to 10 GHz, achieving a maximum DC output voltage of 0.94 V, 60% power conversion efficiency at -5 dBm input signal. Authors in [56] have studied the bending effect of polymer-based antennas intended for IoT devices, where numerous flexible materials were studied such as Polyimides (Pi), Liquid Crystal polymer (LCP), Polytetrafluoroethylene (PTFE), Polydimethylsiloxane (PDMS), Rogers RT/Duroid. Authors in [57] have demonstrated the use of chitosan polymer as substrate of ecofriendly antennas instead of plastic substrates. As chitosan is classified as sustainable material, the proposed antenna with 1 dBi realized gain, is proposed for healthcare IoT devices. In [58], authors have designed a natural rubber substrate based antenna with 80 mm telemetry range. The proposed antenna is intended for on-body devices. In addition, CNTs have been widely involved in various smart applications because of their excellent elctrolmechanical smart applications, such as THZ wave absobers, strain and piezoelectric sensors, smart wearables, and so on [59,60,61,62].

The present in-depth study investigates the performance of rectangular and circular microstrip antennas utilizing PMMA substrate with varying thicknesses. In place of traditional copper, Carbon Nanotubes (CNTs) are employed for the conductive part and ground plane. Both PMMA-based antennas combined with CNTs demonstrate a compact size of 27.8 × 47.8 × 1.5 mm3 for the circular antenna and 22.8 × 39.5 × 1.5 mm3for the rectangular antenna.Remarkably, the realized gain exceeds 5 dBi for both antennas, exhibiting strong performance in both flat and bending scenarios across different substrate thicknesses.

2 Materials and methods

2.1 Circular antenna design procedure

The outlined procedure for designing the proposed antenna is comprehensive and detailed in this section. Initially, the design focuses on creating a circular patch antenna, utilizing carbon nanotubes (CNT) as the conductive material for both the circular-shaped patch and the ground plane. These components are printed on a polymethyl methacrylate (PMMA) dielectric substrate with a permittivity of of \(\varepsilon_{{\text{r}}} = 2.546\). The process commences with determining the antenna dimensions using Eqs. (1–4) [63] to achieve resonance at a frequency of \(f_{{\text{r}}} = 5.8\;{\text{ GHz}}\). The CNT radiating element and ground plane possess an electrical conductivity of \(3e^{5} \;{\text{S/m}}\) and a thermal conductivity of \(1000 \;{\text{W/K/m}}\). Additionally, they have a density of approximately \(0.81\;{\text{g/cm}}^{3}\), and they have a thickness of \(0.035\;{\text{ mm}}\). A visual representation of the basic circular antenna is provided in Fig. 1.

$$a = \frac{F}{{\left\{ {1 + \frac{2d}{{\pi \varepsilon_{{\text{r}}} F}}\left[ {\ln \left( {\frac{\pi F}{{2d}}} \right) + 1.7726} \right]} \right\}^{1/2} }}$$
(1)
$$F = \frac{{8.791\; \times \; 10^{9} }}{{f_{{\text{r}}} \cdot \sqrt {\varepsilon_{{\text{r}}} } }}$$
(2)
$$W_{{\text{S}}} = a + 6d$$
(3)
$$L_{{\text{S}}} = L_{{\text{f}}} + 2a + 6d$$
(4)

where c is the speed of ligth, \(f_{{\text{r}}}\) is the resonant frequency and \(\varepsilon_{{\text{r}}}\) is the relative permittivity of the substrate and \(d\) is its thickness where \(d \ll \lambda\). The antenna is fed by a microstrip line of width (Wf) of 3.1 mm and a length of \(\left( {L_{{\text{f}}} } \right)\).

Fig. 1
figure 1

Circular antenna basic geometry

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2.1.1 Return loss and L-shape impedance matching

The parameter S11, often referred to as the reflection coefficient or return loss, represents the amount of power reflected from the antenna. It is calculated using the Eq. (5):

$$S_{11} \left( {dB} \right) = - 20\log_{10} \left| \Gamma \right| = - 20 log_{10} \frac{{V_{ref} }}{{V_{inc} }}$$
(5)

\(\Gamma\): refers to the reflection coefficient, \(V_{{{\text{ref}}}} :\) the amount of the reflected voltage, \(V_{{{\text{inc}}}}\): the amount of the incident voltage.

The overall anetnna dimensions according to the above equations are 42.8 × 62.8 mm2 (0.82 \(\lambda_{r}\) × 1.21 \(\lambda_{r}\)).

The normalized S11 value, often expressed as a percentage or in dB, provides a clearer indication of the level of impedance matching. A lower normalized S11 value indicates better impedance matching and reduced reflection, ultimately leading to more efficient power transfer from the transmitter to the antenna. When S11 = 0 dB, it signifies that all the power is reflected from the antenna, indicating poor impedance matching. Conversely, lower magnitudes of S11 indicate better impedance matching and reduced reflection [63].

The antenna return loss is depicted in Fig. 2, revealing challenges in achieving satisfactory impedance matching at the desired frequency due to the presence of multiple other frequencies. To address this issue, the design implements the use of slots which are inserted on the radiating element in a second step. This intervention aims to enhance antenna impedance matching and simultaneously minimize antenna size.

Fig. 2
figure 2

Basic circular antenna return loss

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The optimal locations for slot insertion are designated by analyzing the antenna’s current density at its resonant frequency, as illustrated in Fig. 3. Elevated current density is observed around both the feed line and the edges of the circular structure. Consequently, slots are strategically inserted at these regions of high current density to induce a corresponding change in the antenna's response, optimizing its performance accordingly.

Fig. 3
figure 3

Antenna current density at 5.8 GHz

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The design procedure for the proposed antenna is illustrated in Fig. 4. Initially, we introduce a horizontal slot, followed by the addition of a second vertical slot to optimize the antenna’s performance. Through several parametric studies to determine the dimensions of the slots, we achieve the desired results depicted in Fig. 5, showcasing an impressive return loss of − 19.91 dB at the resonant frequency with a bandwidth of 264 MHz. Consequently, the final design of the proposed antenna with the inverted L-slot configuration attains dimensions of 27.8 × 47.8− mm2, corresponding to 0.54λr *0.92λr.

Fig. 4
figure 4

The circular antenna design procedure

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Fig. 5
figure 5

Return loss vs frequencies for the different circular antennas in the design procedure

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The final antenna parameters values are summarized in Table 1.

Table 1 Circular antenna dimensions
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2.1.2 Realized gain and far field pattern for the circular antenna

The simulated antenna’s realized gain and far-field radiation pattern (for both E and H planes) at 5.8 GHz are presented in Fig. 6. As depicted, the radiation exhibits a predominantly omnidirectional pattern, which is suitable for receiving information signals from all directions. The antenna's gain at 5.8 GHz is approximately 4.76 dBi.

Fig. 6
figure 6

Circular antenna radiation pattern a E-plane and H-plane and b 3D realized gain at 5.8 GHz

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2.2 Rectangular antenna design

The proposed microstrip antenna is based on a rectangular shape, as depicted in Fig. 7. The antenna is designed on a PMMA substrate with a dielectric permittivity of \(\varepsilon_{{\text{r}}} = 2.546\), a thickness of \(d = 0.8\;{\text{ mm}}\), and dimensions of \(12\; \times \;18\; {\text{mm}}^{2}\). Both the CNT ground plane and the radiating element have a thickness of \(t = 0.035 \;{\text{mm}}\). The antenna dimensions are evaluated using Eqs. (6–10) [64].

$$W_{0} = \frac{c}{{2 \cdot f \cdot \sqrt {\frac{{\varepsilon_{{\text{r}}} + 1}}{2}} }}$$
(6)
$$L_{0} = L_{{{\text{eff}}}} - 2 \cdot \Delta L$$
(7)
$$\Delta L = 0.412 d \cdot \frac{{\left( {\varepsilon_{{{\text{reff}}}} + 0.3} \right) \cdot \left( {\frac{{w_{0} }}{d} + 0.246} \right)}}{{\left( {\varepsilon_{{{\text{reff}}}} - 0.258} \right) \cdot \left( {\frac{{w_{0} }}{d} + 0.8} \right)}}$$
(8)
$$L_{{{\text{eff}}}} = \frac{c}{{2 \cdot f \cdot \sqrt {\varepsilon_{{{\text{eff}}}} } }}$$
(9)
$$\varepsilon_{{{\text{ef}}f}} = \frac{{\varepsilon_{{\text{r}}} + 1}}{2} + \frac{{\varepsilon_{{\text{r}}} - 1}}{2} \cdot \left( {1 + 12 \cdot \frac{d}{{w_{0} }}} \right)^{ - 1/2}$$
(10)

where c is the speed of ligth, f is the resonant frequency (5.8 GHz) and \(\varepsilon_{{{\text{eff}}}}\) is the effective relative permittivity.

Fig. 7
figure 7

Rectangular antenna basic geometry

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2.2.1 Return loss and quarter-wavelength transformer for impedance matching

The rectangular-shaped patch is fed by a microstrip line with a characteristic impedance of 50Ω. In order to adapt the antenna to its feed, the width of this line is evaluated based on Eq. (10)

$$Z_{c} = \frac{120 \pi }{{\sqrt {\varepsilon_{{{\text{eff}}}} } \left[ {\frac{{w_{{\text{f}}} }}{d} + 1.393 + 0.667\ln \left( {\frac{{w_{{\text{f}}} }}{d} + 1.444} \right)} \right]}}$$
(11)

where \(w_{{\text{f }}}\) the feed line width and d represents the substrate thickness.

The antenna response in terms of return loss is presented in Fig. 8, revealing suboptimal impedance matching below − 10 dB at the desired frequency. To rectify this issue and enhance the antenna’s impedance matching, a quarter-wavelength transformer has been integrated, as depicted in Fig. 9. The final antenna geometry, showcased in Fig. 9, demonstrates improved impedance matching. Furthermore, Fig. 10 illustrates the antenna return loss, highlighting its enhanced matching performance. Detailed parameter dimensions for the final antenna design are provided in Table 2, measuring 22.8 × 39.5 mm², corresponding to 0.44 λrx by 0.76 λr.

Fig. 8
figure 8

Basic rectangular antenna return loss

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Fig. 9
figure 9

Final rectangular antenna geometry

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Fig. 10
figure 10

Return loss of the final rectangular antenna

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Table 2 Rectangular antenna parameters dimensions
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2.2.2 Realized gain and far field pattern for the rectangular antenna

The simulated antenna realized gain and far-field radiation pattern (for both E and H planes) at 5.8 GHz are presented in Fig. 11. As shown, the antenna exhibits a quasi-omnidirectional far field with a realized gain of 3.1 dBi at 5.8 GHz.

Fig. 11
figure 11

Rectangular antenna radiation pattern a E-plane and H-plane and b 3D realized gain at 5.8 GHz

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3 Results and discussion

3.1 Impact of the thickness ‘d’ for the return loss and the realized gain

Figures 12 and 13 depict the return losses of the circular and rectangular antennas, respectively, across various substrate thicknesses. This investigation endeavors to elucidate the impact of substrate thickness on antenna performance, encompassing factors such as impedance matching, gain, and radiation efficiency.

Fig. 12
figure 12

Circular antenna return losses for different substrate thicknesses

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Fig. 13
figure 13

Rectangular antenna return losses for different substrate thicknesses

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For the circular antenna, we observe that lower thicknesses result in a wider bandwidth compared to higher values. Conversely, we note an improvement in gain (Fig. 14) and efficiency values with increasing thickness. To strike a balance between impedance matching and gain, optimizing the dimensions of the slots is necessary to achieve a reasonable return loss. In our study of the circular antenna, we observe that slot 2 has a significant effect on enhancing antenna return loss around the \(5.8 \;{\text{GHz}}\) band (see Fig. 5), indicating the importance of studying the impact of its dimensions. It is evident from Fig. 15 that the antenna impedance matching is visibly enhanced, particularly for \(L_{2} = 40\;{\text{ mm}}\) and \(W_{2} = 3 \;{\text{mm}}\).For the rectangular antenna, we observe a slight shift in the resonance frequency as the substrate becomes thicker. As expected, the antenna gain improves with increasing substrate thickness, as illustrated in Fig. 16. Table 3 summarizes all the studied thicknesses.

Fig. 14
figure 14

Circular antenna simulated gain for different thicknesses a d = 0.4 mm, b d = 1 mm and c d = 1.5 mm

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Fig. 15
figure 15

Final circular antenna return loss

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Fig. 16
figure 16

Rectangular antenna simulated gain for different thicknesses a d = 0.4 mm, b d = 1 mm and c d = 1.5 mm

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Table 3 Antennas simulated gain and efficiencies for different substrate thicknesses
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3.2 Bending effect

The bending study primarily aims to investigate the response of flexible antennas in various scenarios and assess their ability to maintain good performance when implemented in practical applications. To evaluate the behavior of our antennas in such situations, we consider a vacuum cylinder to bend the antennas for different radii, thereby altering the degree of curvature. The dimensions of the antennas remain fixed as they are in a flat situation. Figures 17 and 18 depict the return losses of the circular and rectangular antennas, respectively.

Fig. 17
figure 17

Bending effect on circular antenna return loss a E-plane b H plane

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Fig. 18
figure 18

Bending effect on rectangular antenna return loss a E-plane b H plane

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The circular antenna is initially bent in the E plane and then in the H plane. We observe that the antenna’s impedance matching exhibits a slight shift to lower frequencies in the E plane. This shift is primarily caused by changes in the antenna’s effective length during the bending process, as well as alterations in the current distribution in this plane [64, 65]. However, the antenna maintains its good performance even with very small radii (R = 7 mm: severely bent) due to its large bandwidth, despite the frequency shift. Along the H plane, the antenna shows no resonant frequency offset, as the current path remains unaffected compared to the E plane. This is evident from Fig. 17b, where it maintains good matching until R = 10 mm.

In the E-plane bending for the rectangular antenna, the bending does not significantly affect the antenna response, while in the H-plane bending, it strongly impacts the antenna performance. Even with large radii (R = 19 mm), as specified in Fig. 18b, the antenna’s performance is notably affected. It is evident that the magnitude of the return loss increases when the antenna is bent, although the rectangular antenna experiences a significant increment even when slightly bent, compared to the circular antenna, which maintained the return loss under − 10 dB.

Tables 4 and 5 present the gain and efficiencies of the bent antennas, respectively, for a similar bending radius (R = 20 mm). We observe that both antennas’ gains are more affected in the E plane bending compared to the H plane bending. This can be explained by the fact that the current density is accumulated at the center of the antenna in the H plane, which increases the magnetic field, whereas in the E plane, the current is gathered at the antenna's side, leading to a decrease in its gain [66].

Table 4 Gain and efficiency performances of circular antenna at bending radius of R = 20 mm
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Table 5 Gain and efficiency performance of rectangularantenna in bent situations
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Furthermore, it is evident from Tables 4 and 5 that the rectangular antenna exhibits significant changes in its gain (in both E and H planes) compared to the flat situation. This behavior can be attributed to the edgy shape of the rectangular antenna.

This analysis shows the effectiveness of circular antennas which is less edgy and more stable comparing to rectangular antennas.

3.3 Comparative study between circular and rectangular design (PMMA-CNT)

We observe differences in the electrical and radiation performances between the circular and rectangular antenna structures, despite utilizing the same substrate and conductive materials characteristics. Notably, the circular antenna maintains a stable resonance frequency of approximately 5.8 GHz across varying substrate thicknesses, while also offering a wider bandwidth (5.51–6 GHz) compared to the rectangular antenna (5.47–5.68 GHz). Additionally, the circular antenna demonstrates superior gain performance, with nearly a 1 dBi difference compared to the rectangular antenna.

In the existing literature, there is a lack of examples combining PMMA substrate with CNT conductive materials. Therefore, we conducted a comparative analysis with antennas from recent related works, which were designed using different substrate and conductive materials. This comparison, detailed in Table 6, emphasizes factors such as compactness (antenna size), performance, and dielectric properties. Our findings indicate that while some reported antennas achieve good gains, they often do so at the expense of larger dimensions or lower efficiency. In contrast, our designs strike a suitable compromise between size, gain, and efficiency, highlighting their practical viability in real-world applications.

Table 6 Comparative study of different design antennas
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4 Conclusions

The present in-depth study investigates the performance of rectangular and circular microstrip antennas utilizing PMMA substrate polymer with varying thicknesses. In place of traditional copper, Carbon Nanotubes (CNTs) are employed for the conductive part and ground plane. Both PMMA-based antennas combined with CNTs demonstrate a compact size of 27.8 × 47.8 × 1.5 mm3 for the circular antenna and 22.8 × 39.5 × 1.5 mm3 for the rectangular antenna. Remarkably, the realized gain exceeds 5 dBi for both antennas, exhibiting strong performance in both flat and bending scenarios across different substrate thicknesses. The rectangular antenna achieves a bandwidth of approximately 200 MHz, while the circular microstrip antenna reaches an impressive 500 MHz bandwidth. These exceptional outcomes make the two microstrip antennas highly suitable for a wide range of emerging applications within the sub-6 GHz band, including but not limited to wireless communication systems and Internet of Things (IoT) devices.