Electromagnetic physics of the MPA: comprehensive review of applications, design, and analysis of microstrip patch antenas for 5G-technologies

Física eletromagnética da MPA: revisão abrangente das aplicações, projeto e análise de antenas de microfita para tecnologias 5G

Abstracts

This work presents a study focused on the electromagnetic evaluation of Microstrip Patch Antennas (MPAs) through electromagnetic simulation for 5G technology applications. The paper addresses two commonly used feeding structures – coaxial probe and transmission line. In our analysis, we observed the scattering and impedance matrices concerning feeding point variations. Additionally, a parametric study explored the impact of relative permittivity, length, width, and height on MPA electromagnetic performance for 5G frequency operation, understanding the physics of these antennas. The results revealed the following: varying the relative permittivity shifted the resonance frequencies and reduced gain; changing the patch length slightly shifted the resonance to lower frequencies without significant changes in the radiation pattern; adjusting the patch width maintained the resonance while increasing matching levels without significant radiation changes; and altering the substrate height shifted the resonance to lower frequencies.

Keywords Physics of the MPAs; Electromagnetic analysis and studies; Communication 5G technologies

Este trabalho apresenta um estudo focado na avaliação eletromagnética de Antenas Patch de Microfita (MPAs) por meio da simulação eletromagnética para aplicações de tecnologias 5G. O artigo aborda duas estruturas de alimentação comumente usadas – sonda coaxial e linha de transmissão. Em nossas análises observamoos as matrizes de espalhamento e impedância em relação às variações do ponto de alimentação. Além disso, um estudo paramétrico explorou o impacto da permissividade relativa, comprimento, largura e altura no desempenho eletromagnético da MPA para operação em frequências 5G, entendendo a física destas antenas. Os resultados revelaram o seguinte: a variação da permissividade relativa deslocou as frequências de ressonância e reduziu o ganho; a alteração do comprimento do patch deslocou ligeiramente a ressonância para frequências mais baixas sem mudanças significativas no padrão de radiação; o ajuste da largura do patch manteve a ressonância enquanto aumentava os níveis de correspondência sem alterações significativas na radiação; e a mudança na altura do substrato deslocou a ressonância para frequências mais baixas.

Palavras-chave: Física das MPAs; Análises e estudos eletromagnéticos; Comunicação em tecnologias 5G

1. Introduction

Microstrip Patch Antennas (MPAs) are planar in nature and represent a revolution in current communication systems by adapting them to printed circuits on chips, Systems-on-Chip (SoC) on what are called Integrated Circuits (IC) [¹]. Their planar characteristics allow them to be integrated into flat surfaces and facilitate design implementation when related to devices that require wireless communication, such as aircraft and commercial planes, automobile devices, smartphones, smart-TV circuits, among others [²].

In addition to these applications, they are easy to manufacture and have low cost, facilitating the mass production of commercial devices [³]. MPAs can take different shapes, such as rectangular (most common), circular, triangular, square, elliptical, trapezoidal [⁴], and several others, modifying radiation characteristics and impacting operational performance according to the project and analysis needs. MPAs are, by nature, resonant antennas, using methods such as cavity, transmission line, and coupling for their design. Their designs and performance can be modeled for bandwidth [⁵] in frequency and radiation pattern.

There are several notable advantages: their compact size and low profile make them easily integrable into compact devices such as mobile phones, RFID tags, GPS devices, and wireless communication systems, and Internet of Things (IoT) applications [⁶]; cost-effectiveness, owing to their relatively inexpensive manufacturing process involving simple design and printed circuit board technology; lightweight nature compared to other antenna types, making them suitable for weight-sensitive applications in aerospace and satellite communications [⁷, ⁸].

Their capability to operate across a broad frequency range demonstrates their versatility for diverse wireless communication applications like Wi-Fi, Bluetooth, and satellite communication [⁹]. Additionally, we highlight mobile device communications like fifth generation (5G) with microwave frequencies operating in the 5.8 GHz licensed spectrum, and other emerging technologies like beyond fifth generation (B5G) and sixth generation (6G) that will have extended spectrum for high frequencies. Additionally, their directional radiation patterns, when associated with MPAs, offer tailored patterns beneficial for point-to-point communication, effectively enhancing signal transmission and reception efficiency [¹⁰].

MPAs present a low profile and conformal design. This is because their low profile and conformal design make them suitable for applications requiring antennas to be mounted on irregular or curved surfaces, such as in vehicles, aircraft, or wearable devices, as seen in Table 1. MPAs are able to create designs based on their potential for array configurations. MPAs can be used in array configurations to achieve higher gain, increased directivity, and beam steering capabilities, enhancing their performance in certain applications. Additionally, they have good radiation efficiency; when designed properly, MPAs can exhibit high radiation efficiency, ensuring that a significant portion of the input power is radiated as electromagnetic waves [¹, ¹¹].

Table 1
Applications of MPAs [¹¹].

Application	Summary	Frequency range
Wireless communications, 5G, B5G and 6G	Some MPAs are used in wireless communication systems, Bluetooth devices, Wi-Fi routers, and base-station cellular and individual devices, as well as WLAN networks, since omnidirectional monopoles allow propagation in all directions, such as GSM, GPRS, etc. Other approaches, like fifth generation (5G), beyond fifth generation (B5G), and sixth generation communication devices, are already being developed and implemented in some countries.	2.4—5.8 GHz
Radar systems	It is employed in many radar systems with wide-range applications, including air traffic, military, control systems, and weather radars.	5.3—40 GHz (C- to Ka- Band)
Satellite Communication	Used in spacecraft and aerospace craft with ground stations, base stations, and on the top of fixed satellites in the L- and S-band.	1—4 GHz
Radio Frequency Identification (RFID)	When array antennas are used for identifying and tracking objects, bar codes, tagging codes, and others, some dipoles are allocated and integrated into readers within the RFID receiver to obtain information. In microwave frequencies, dipoles are employed in patch and microstrip antennas due to their small and compact design, which easily integrates into RFID systems.	2.4—5.8 GHz
IoT, Automotive and Industrial Automation	IoT (Internet of Things), automotive, and industrial automation use printed circuits (MPAs) as they require compact and efficient antennas. There is a wide range of IoT applications in mobile devices and sensors. Automotive antennas require monopoles for keyless entry systems, tire pressure monitoring systems (TPMS), Dedicated Short Range Communication (DSRC), and Vehicle-to-Vehicle (V2V) or Vehicle-to-Everything (V2X) communication. Industrial automation uses printed monopoles and patch antennas for remote monitoring and control of machines, such as in Bluetooth and Zigbee (ISM) applications.	IoT/V2V (0.7—1.9GHz); ISM and sub-GHz bands (2.4—5.8GHz)

2. Field Equations for a Typical Behavior of Radiation Patterns

Maxwell’s equations are solved using the resonant cavity method [¹¹], providing precise results and consistent approximations of electric and magnetic fields. In fact, Maxwell’s equations are resolved utilizing the potentials of A(x, y, z) and F(x, y, z) associated with the electric J_s(x, y, z) and magnetic M_s(x, y, z) surface current densities. Particularly, for the microstrip structure using the cavity method, we employ the potential A to solve the homogeneous Helmholtz equation and find stationary solutions in the form of (we have taken the electric field component in the x-direction, E_x):

(1) $$\begin{array}{lll}{E}_x\hfill & =\hfill & E_0\sin (k_{x}x)\cos (k_{y}y)\sin (k_{z}z)\hfill \\ E_y\hfill & =\hfill & E_0\sin (k_{x}x)\sin (k_{y}y)\cos (k_{z}z)\hfill \\ E_z\hfill & =\hfill & E_0\cos (k_{x}x)\cos (k_{y}y)\cos (k_{z}z)\hfill \end{array}$$

Where E₀ is the amplitude of electric field and the numberwaves are given at x, y, z-direction by terms:

(2) $$k_x=\frac{n\text{π}}{W}{k}_y=\frac{m\text{π}}{L}\text{and}{k}_z=\frac{p\text{π}}{h}$$

Where, n, m and p are integers number of null sinusoidal factors, and W, L, h are physicals parameters of antenna. In resonance operation mode, the numberwave can be write in terms of k_x, k_y and k_z, straightforwardly:

(3) $$\begin{array}{lll}{(k_r)}_{mnp}\hfill & =\hfill & \frac{2\text{π}}{{({\text{λ}}_r)}_{mnp}}=\sqrt{k_x^2+k_y^2+k_z^2}\hfill \\ {(k_r)}_{mnp}\hfill & =\hfill & \sqrt{{(\frac{m\text{π}}{L})}^2+{(\frac{n\text{π}}{W})}^2+{(\frac{p\text{π}}{h})}^2}\hfill \end{array}$$

And another figure of merit related to radiation parameters associated to far-field performance can be expressed from field equation E and H (approximated). However, it is not common to compute integrals of power densities associated with the far-field, as they are extremely difficult and often lack closed analytical solutions. Therefore, simulation software utilizes numerical methods such as finite element methods, method of moments, full-wave methods, among others and basic equations were presented in Table 2.

Table 2
Principal parameters of a MPA [¹¹]

Effective dielectric constant, ∊reff	${\text{ϵ}}_{\text{reff}}=\frac{{\text{ϵ}}_r+1}{2}+\frac{{\text{ϵ}}_r-1}{2}{[1+12\frac{h}{W}]}^{-1/2},W/h>1.$
Normalized extension length, ΔL/h	$\text{Δ}L/h=0.412\frac{({\text{ϵ}}_{\text{reff}}+0.3)\left[\frac{W}{h}+0.264\right]}{({\text{ϵ}}_{\text{reff}}-0.258)\left[\frac{W}{h}+0.8\right]}$
Resonant frequency in fundamental mode, (fr)01	${(f_r)}_{01}=\frac{1}{2L\sqrt{{\text{ϵ}}_r{\mu }_0{\text{ϵ}}_0}}$
Slot conductance, G1	$G_1=\frac{W}{120{\text{λ}}_0}[1-\frac{{(k_{0}h)}^2}{24}],\frac{h}{{\text{λ}}_0}<\frac{1}{10}$
Slot conductance, B1	$B_1=\frac{W}{120{\text{λ}}_0}[1-0.636\ln \mathrm{}(k_{0}h)],\frac{h}{{\text{λ}}_0}<\frac{1}{10}$
Input slot resistance, Rin	$R_{in}=\frac{1}{2(G_1\pm G_{12})}\cos \left(\frac{\text{π}}{L}{y}_0\right)$
Resonance frequency in cavity method, (fr)mnp	${\left(f_r\right)}_{mnp}=\frac{1}{2\text{π}\sqrt{\text{ϵ}\mu }}\sqrt{{\left(\frac{m\text{π}}{L}\right)}^2+{\left(\frac{n\text{π}}{W}\right)}^2+{\left(\frac{p\text{π}}{h}\right)}^2}$
Characteristic impedance, Z0	$\frac{60}{\sqrt{{\text{ϵ}}_{\text{reff}}}}\mathrm{}\ln \mathrm{}(\frac{8h}{W}+\frac{W}{4h}),\frac{W}{h}\le 1\frac{120\text{π}}{\sqrt{{\text{ϵ}}_{\text{reff}}}\left[\frac{W}{h}+1.393+0.667\mathrm{}\ln \mathrm{}(\frac{W}{h}+1.444)\right]},\text{else}.$

3. Design and Geometry of the Microstrip Patch Antenna Proposed

In this section, we propose a simple and conventional antenna to analyze key performance metrics. Simulations are conducted based on foundational theory to assess the primary effects of physical parameters. This analysis is crucial, as it provides students and designers with an understanding of the principal parameters and how they influence antenna performance. In Fig. 1(a), the MPA consists of the following elements: a ground plane at z = 0 with thickness t = 18μm; at z = t, there is a substrate (dielectric with relative permittivity ϵ_r = 2.2 and loss tangent of tan δ = 0.0009 for Rogers 5880 material) with a height h = 1.575 mm, width W_sub = 40 mm, and length L_sub = 30 mm; at z = h+t, there is the patch resonator made of conductive copper material with dimensions of length l = 16.5 mm and width w = 20.5 mm (initially in the analyses) for a desired resonance frequency of 5.8 GHz (Wireless Communications).

Figure 1
(a) Physical parameters and dimensions of the MPA from the top isometric view (b) Bottom view of the MPA with details of the coaxial probe (c) The MPA proposed for microstrip transmission line feeding analysis.

The feeding point specified by a coaxial probe is centered at (0, d_coax, h+h_Teflon+2t), where a cylindrical layer of Teflon insulator surrounds the inner copper conductor. The height Teflon is h_Teflon = 5 mm (arbitrary chosen). The ratio of the outer (r_ext = 1.675 mm for Teflon isolation) and inner (r_int = 0.500 mm for copper conductor) radii was previously calculated to provide an input impedance of Z₀ = 50 Ω, as shown in Fig. 1(b),

(4) $$\frac{r_{\text{ext}}}{r_{\text{int}}}=3.35\to Z_0\approx 50\mathrm{\Omega },$$

By the use of the equation (5):

(5) $$Z_0[\text{Ω}]=\frac{138\log \left(\frac{r_{\text{ext}}}{r_{\text{int}}}\right)}{\sqrt{{\epsilon }_r}},$$

Where, the cutoff frequency given by the relationship between external and internal radial distance of the conductor and insulator is (6):

(6) $$f_{\text{cutoff}}[\text{GHz}]=\frac{11.8}{\sqrt{{\epsilon }_r}\text{π}(r_{\text{ext}}+r_{\text{int}})}.$$

Fig. 1(c) the same MPA is presented with a different feeding mechanism using a matched microstrip transmission line for 50 Ω. The antenna’s dimensions remain unchanged, while the transmission line is parameterized by a width w_fed = 4.8 mm (ensure Z₀ ≈ 50Ω) and narrow slot openings w_slots = 1 mm (initially arbitrary chosen), with a length in the y direction given by a variable y₀ throughout the analyses (initially y₀ = 5 mm). Note that there is a coordinate system of Cartesian (x, y, z)- and spherical (θ, ϕ)-direction at the center of the MPA (refer to Fig. 1(a, c)).

4. Rectangular Patch Antenna with Coaxial Probe Feed

From the depicted project issues, we employed the Ansys Electronic 2023 software utilizing Finite Element Methods (FEM) to conduct various analyses for Rectangular patch antenna with coaxial probe feeding (first case).We have used d_coaxial = 3.2–7.0 mm at step 1.6 mm, such as we can see in the Fig. 2(a,c).

Figure 2
(a) Reflection coefficient (b) reflection resistance and (c) reflection reactance of the feeding point analysis at technical of the coaxial cable.

The variations in the feeding point for the coaxial probe resulted in an increase in the matching level by reducing the reflection coefficient |S₁₁| in dB, among the scenarios depicted for the most extreme cases like the purple and blue curves, showing a discernible trend indicated by the arrow. We observed that proper matching occurred for d_coaxial = 5 mm near the frequency of 5.75 GHz, |S₁₁| ≤ −10 dB, as we can observe in Fig. 2(a). In Fig. 2(b), the reflection resistance R₁₁ in Ω illustrates a decline in the relationship between peaks throughout the variation of the feeding point, promoting a matching for input resistance.

The appropriate scenario occurs at d_coaxial = 5 mm, where it is evident that a peak of approximately 200 Ω exists. Within the graphical illustration, a more detailed schematic is noticeable for the proposed coaxial probe, involving elements previously described in the preceding section.

In Fig. 2(c), the reactive reflection impedance X₁₁ in Ohms depicts the behavior relationship decreasing across the feeding point variation, displacing the frequency and promoting a match for input resistance. The appropriate scenario occurs at d_coaxial = 5 mm, where it is evident that there is an approximate peak-to-peak variation from −150 Ω to 50 Ω and a crossover X₁₁ ≈ 0 Ω at the resonance frequency f_r ≈ 5.5 GHz.

5. Parametric Analysis of Physical Dimensions

5.1. Permittivity analysis

In Fig. 3(a), the behavior of the reflection coefficient for variations (with a step of 0.2) illustrates a trend of shifting in the resonance frequency across the operating band. It is noteworthy that, besides the increase in resonance frequency due to the decrease in relative permittivity, the matching level between port 1 and the transmitted signal becomes more intense as the blue and orange curves already exceed |S₁₁| ≤ −10 dB. The same behavior is observed (Fig. 2(b) and (c)) for the reflection resistance and reactance (input impedance), with an increase from 150 Ω to 200 Ω for the peak resistance R₁₁ concerning the decrease in permittivity, as well as the shift in frequency for the center of the reactive reflection impedance X₁₁ crossing 0 Ω. In all cases, there is a slight rise in the X₁₁ curve centering at 50 Ω, causing a slight displacement in the operational aspect and signal performance.

Figure 3
(a) Reflection coefficient (b) reflection resistance (c) reflection reactance (d) pattern radiation (in gain format) at E- and (e) H-plane for permittivity analysis at near resonance frequency (5.5–5.8 GHz).

In Fig. 3(d), the radiation pattern is shown for the E-plane (yz-plane). A slight broadening of the main lobe is observed with the decrease in relative permittivity ϵ_r = 2.8 to 1.6 near the frequency f_r = 5.5 GHz. An increase from 6 dB to 8 dB in gain was noted. In Fig. 3(e), the radiation pattern in the H-plane (xz-plane) is depicted. A slight widening of the main lobe is observed as the relative permittivity decreases from ϵ_r = 2.8 to 1.6 around the frequency f_r = 5.5 GHz. An increase in gain from 6 dB to 8 dB was observed, along with an elevation in the sidelobe level.

5.2. Length of patch resonator analysis

In Fig. 4(a), the reflection coefficient is described in terms of a frequency shift due to the decrease in the resonator length for variations between l = 17.25–16.25 mm.

Figure 4
(a) Reflection coefficient (b) reflection resistance (c) reflection reactance and (d) gain at band frequency in front of length variation of resonator patch.

Increasing the l dimension results in an increase in the resonance frequency, from approximately 5.25 GHz to 5.75 GHz, causing a shift in the operating band. Another significant point is the elevation in the level of the reflection coefficient |S₁₁|, observed in the parametric analysis trend. Fig. 4(b) demonstrates a similar trend in the resonance shift. However, the peaks of the reflection resistance remain constant during the variation effect. In Fig. 4(c), a slight increase above the crossover at 0 Ohms demonstrates a mild impedance mismatch.

This could be associated with the probe, physical characteristics of the patch, among other factors. Finally, in Fig. 4(d), analyzing the effect of the physical length of the patch, we studied the relationship between its increase and the frequency gain, noticing a 0.1 dB increment and a slight shift in the peak of maximum gain within the operating band.

Fig. 5(a,b) reveals that the patch length does not significantly influence the radiation pattern despite its variation. The H-plane exhibits a slightly noticeable increase in the side lobes. The maximum value observed is 7.5 dB for both planes.

Figure 5
Pattern radiation (in gain) (a) for H- and (b) E-plane in front of length variation.

5.3. Width of patch resonator analysis

In Fig. 6(a), the reflection coefficient is depicted in terms of a frequency shift resulting from the variation in the resonator width, ranging between w = 20.5–16.5 mm. Increasing the w dimension leads to an elevated reflection coefficient |S₁₁| ≥ −10 dB, facilitating impedance matching between port 1 and the transmitted signal at a fixed resonance frequency f_r = 5.5 GHz within the operating band.

Figure 6
(a) Reflection coefficient (b) reflection resistance (c) reflection reactance and (d) gain at band frequency in front of width variation of resonator patch.

Notably, a decrease in the level of the reflection coefficient |S₁₁| is observed in the parametric analysis trend. Fig. 6(b) demonstrates a similar trend in the fixed resonance frequency concerning the increased reflection resistance peak R₁₁.

However, the peaks of the reflection resistance have slightly increased from 150 Ω to 200 Ω during the variation effect. In Fig. 6(c), a slight deviation above the crossover at 0 Ω indicates a mild impedance mismatch (as previously related in the parametric analysis). This discrepancy could be associated with the probe, physical characteristics of the patch, among other factors.

Finally, in Fig. 6(d), upon analyzing the effect of the physical width of the patch, we explored the relationship between its increase and frequency gain, observing a 0.1 dB increment and a minor shift in the peak of maximum gain within the operating band. The maximum gain observed corresponds to the green color (at w = 19.5 mm).

Figure 7(a,b) demonstrates that the variation in patch width does not notably impact the radiation pattern across both planes. However, a slight increase in the side lobes is observed in the H-plane, with the maximum observed value reaching approximately 7.5 dB for each plane.

Figure 7
Pattern radiation (in gain) for (a) H- and (b) E-plane in front of width variation.

5.4. Height of patch resonator analysis

In Fig. 8(a), the reflection coefficient shows a frequency shift due to variations in the resonator height, ranging between h = 2.5–0.5 mm. An increase in h results in an elevated reflection coefficient |S₁₁| ≥ −10 dB, causing a shift in the resonance frequency. This variation aids in impedance matching between port 1 and the transmitted signal at a fixed resonance frequency f_r = 5.25–5.75 GHz within the operating band.

Figure 8
(a) Reflection coefficient (b) reflection resistance (c) reflection reactance and (d) gain at band frequency as a function of height variation in the substrate (dielectric) of the MPA.

A noticeable increase in the reflection coefficient |S₁₁|is observed in the parametric analysis. Fig. 8(b) shows a similar trend in the resonance frequency shift relative to the decreasing peak of the reflection resistance R₁₁. The peaks of the reflection resistance significantly decrease from 500 Ω to 100 Ω due to the height variation. In Fig. 8(c), the crossover at 0 Ω indicates improved impedance matching. Finally, in Fig. 8(d), the analysis of the impact of patch height h on frequency gain shows a 0.38 dB increase and a slight shift in the peak of maximum gain within the operating band. The maximum gain, highlighted in red, corresponds to h = 1.0 mm at f_r = 5.8 GHz, while the minimum gain is observed at h = 0.5 mm.

Fig. 9(a,b) illustrates that the physical variation in patch height does not significantly impact the radiation pattern across both planes. Nevertheless, a slight increase in the side lobes is observed in the H-plane, especially at the maximum height of 2.5 mm. Notably, the E-plane demonstrates an enlargement in the main lobe. The maximum observed value is approximately 7.5 dB for each plane.

Figure 9
Pattern radiation (in gain) (a) for the H-plane and (b) the E-plane as a function of height variation.

6. Impedance Matching Analysis

The parametric analyses revealed, from the scattering matrices and complex impedance, that the microstrip transmission line feeding resulted in a matching effect with the reflection coefficient |S₁₁| ≤ −10 dB for the evaluated scenarios in Fig. 10(a), with the length y₀ = 3–7 mm. In this context, the suitable value of y₀ for efficient matching was found at y₀ = 5 mm (orange curve), as demonstrated by |S₁₁| ≥ −10 dB. Observing Fig. 10(b), we notice that the peaks of reflection resistance shift within the operating band, ranging from resonance frequencies of 6.5 GHz to 5.5 GHz. This behavior is expected as the input resistance is proportional to the cosine factor:

Figure 10
Parametric analysis for impedance matching at the transmission line schematic: (a) reflection coefficient (b) input reflection resistance (c) input reflection reactance.

(7) $$R_{\text{in}}=R_{\text{in}}(y_0=0)\cos \left(\frac{\text{π}}{l}{y}_0\right)$$

It varies with the position of the input slots in the transmission line for patch feeding. This aspect is advantageous since the ideal position is observed between the edge (end) of the patch resonator and the central point, where the input resistance is null.

Finally, Fig. 10(c) exhibits a similar behavior, as previously described, for the reactance centers in the parametric study. These centers shift within the operating band, ranging from resonance frequencies of 6.5 GHz to 5.5 GHz, varying with the position of the input slots in the transmission line used to feed the patch antenna.

7. Conclusion

This work aimed to conduct a review of the main applications of MPAs, describing the primary parameters related to the construction and design of these antennas. Two antennas with common and distinct feeding structures were simulated: coaxial probe and transmission line. The analysis covered scattering and impedance matrices concerning variations in the feeding point. In this context, the approach also presented a parametric study of relative permittivity, length, width, and height, and their influence on the electromagnetic performance of the MPA. The studies revealed that:

• The relative permittivity shifts the resonance frequency and decreases gain in both planes with its increment.

• The physical length of the patch resonator slightly shifts the resonance frequency towards lower frequencies with its increase, without significantly altering the radiation pattern.

• The physical width maintains the resonance frequency but increases the matching level in the scattering and impedance matrices, without significantly altering the radiation pattern.

• The substrate height shifts the resonance frequency towards lower frequencies, increasing the matching level with its increment. It induces minor changes in the radiation pattern by broadening the E-plane and increasing side lobes in the H-plane.

Acknowledgments

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.

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Publication Dates

• Publication in this collection
  08 Nov 2024
• Date of issue
  2024

History

• Received
  05 Aug 2024
• Reviewed
  14 Sept 2024
• Accepted
  25 Sept 2024