1 Introduction

The integrating ability of the MPA makes it more famous and constructive for appliances such as mobile, satellite or aerospace systems [1, 2]. Along with the striking characteristics like low-profile, light-weight, low-cost, and low-power, the realistic usage of MPA is restricted due to its narrow-band features, particularly in broad banded appliances [3, 4]. Improvement of MPA bandwidth is a key area of analysis for the past few decades. Various analyses are reported by means of diverse antenna geometry. Bandwidth improvements of patch antennas can be attained by a thicker substrate with lower permittivity; however radiation efficiency gets degraded because of the surface waves [5, 6, 7, 8].

Accordingly, MPAs are fed by the several techniques. These techniques are categorized into 2 types; “(a) contacting and (b) non-contacting”. As per the contact technique, the RF power is provided to radiation patch directly via a micro strip line [9, 10]. As per the non-contacting technique, EMF coupling is carried out for transferring the power among the radiating patch and micro strip line [11, 12]. The four well known feeding models are “the microstrip line, coaxial probe (both contacting schemes), aperture coupling and proximity coupling (both non-contacting schemes)”; however their patterns get vary with respect to frequency due to the false radiations. Moreover, they exhibit poor cross-polarization and back-radiation features [13, 14, 15].

Generally, the most renowned schemes for analysing MPA are the full wave model, transmit line and cavity models [16]. The full-wave scheme is very versatile, accurate and it treats the infinite and finite arrays, single elements, arbitrary elements, coupling, and stacked elements [17, 18]. The transmission line scheme is simpler and it offers better physical insights; however it is less precise. The cavity scheme offers good physical insight and is highly accurate; nevertheless it is complicated in nature.

The contributions of the work is given below.

  1. 1.

    Proposed a new algorithm termed as Elephant Herding Optimization with New Scaling Factor for fine-tuning the MPA parameters such as width, patch length, value of dielectric substrate, and substrate thickness for antenna gain enhancement.

The paper is arranged as: Section II portrays the review. Section III describe the modelling of MPA and section IV portrays the optimization of constraints of MPA design via Elephant Herding Optimization with New Scaling Factor. Section V and VI illustrates the outcomes and conclusion.

2 Literature Review

2.1 Related Works

In 2020, Dutta et al. [19] developed a RMPA with miniaturized and simple design with enhanced cross-polarization (XP) purity. The modelled scheme was highly flexible for tuning the antenna frequency for optimal performance. In addition, a comprehensive design was offered for estimating the resonance frequency of antenna and also for evaluating the respective resonant modes. Finally, the results obtained from the simulation have demonstrated the effectiveness of adopted concept.

In 2018, Qian et al. [20] have presented a new MPA design with narrow half-power beam-width and higher radiation gain loaded with MIS-MTM. Accordingly, a mathematical model was formulated for computing MPA in step by step procedure that aided in optimizing the constraints for optimal performance. This technique has also offered significant control for modelling the efficient transmitting antenna system in microwave appliances.

In 2020, Chandrashekar et al. [21] have presented a compact design of MPA for realizing the wideband features. Here, the annular metal-ring was deployed as superstrate and it was provided as inductive load to rectangular patch. In addition, the robust systematic method was exploited for choosing the size of ring and patch, in which they resonate closer to one another. This tactical model has ensured a wider harmonizing bandwidth with broader radiation. At last, the simulated results have revealed the enhancement of presented approach in terms of bandwidth.

In 2020, Rashmitha et al. [22] have designed and simulated a MPA that was well-suited for 5G communications. Here, the antenna worked at EHF ranges at 43.7 GHz. Accordingly, FR4 epoxy substrate was deployed in this work, and the antenna was examined for its radiation pattern, VSWR, return loss, distribution of current and gain. At last, the attained outcomes were confirmed and were examined for different appliances.

In 2019, Hocini et al. [23] have analyzed and designed the 5 terahertz MPA depending on “modified photonic band gap substrate” with a frequency of 0.5 to 0.8 THz. Consequently, the intention was to accomplish the optimal antenna features nearer to 0.65 THz that included appliances in communication and sensing fields. Moreover, the simulation was carried out on the basis of diverse substrates, together with aperiodic and periodic homogeneous photonic crystal substrates.

In 2021, Susamay et al. [24] have developed a novel quantitative constraint that has identified the most important factor for suitable geometry election of MPA, which could be applied in lower XP appliances. The importance of NPS and PS geometries was examined based on FAR, a novel constraint that quantified the level of electric field irregularities in MPA. From the experimentation, it was found that the adopted model could ensure low XP radiation.

In 2019, Shalini et al. [25] have proposed a “microstrip fed graphene based dual-polarized patch antenna that included single graphene patch fed by dual ports oriented orthogonally”. This port has provided the horizontal and vertical radiations in its relevant port directions. Moreover, the patch has a frequency of 3.98 THz and it was mounted on silicon substrate. Finally, the simulations were carried out, and the supremacy of the presented scheme was validated in terms of peak gain, return loss and efficiency.

In 2019, Hossein et al. [26] established MPA model for terahertz appliances. The adopted antenna has obtained minimal cost as the substrate was FR-4 and it has included higher radiation efficiency. On deploying the fractal structure, the radiation length of present paths was raised and the antenna size was minimized. Finally, the results obtained from the simulation have revealed the efficacy of the modelled approach in terms of gain, efficiency and cost.

2.2 Review

Table 1 symbolizes the reviews on existing MPA design models. Cross-polarization [19] reveals improved gain and reduced polarization levels. However, there was degraded impedance matching. In addition, FEM exploited in [20] have offered high efficiency and increased gain efficiency, but it should consider more on radiation gain. Microstrip feeding technique [21] has effective control of Cross-polarization and attains efficient gain. However it needs consideration on radiation and band width properties. Trial and error method was deployed in [22] offered reduced impedance and high gain, but, there was implementation on real platforms. Finite integration technique [23] offers high radiation efficiency and reduced return loss. But, shock waves may degrade the performance of antenna. FAR [24] reduces cross-polarization and provides higher radiation efficiency. Yet, it requires consideration on varied types of feeding mechanisms. Dual-polarization based model [25] guarantees higher peak gain with minimal return loss. Nevertheless, it should consider more on polarization issues. FEM developed in [26] offers higher gain and minimized cost, but return loss should be focused more.

Table 1 Reviews on traditional MPA designs
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3 Modelling of Microstrip Patch Antenna

Figure 1 reveals the mechanism of radiation in MPA. Patch antennas usually include a ground plane and controlling patch. A dielectric media known as substrate is present at the centre of the design, which shows the exact value termed as dielectric constant. In general, patch sizes are said to be smaller than the ground and substrate, the. While designing a MPA, a resonance frequency and dielectric medium are chosen based on the size of MPA. In addition, the dielectric substrates with minimal dielectric constant are required for enhancing the antenna performances, wherein the efficiency is improved with higher radiation and bandwidths.

Fig. 1
figure 1

Radiation pattern for MPA design

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The noteworthy constraints while designing the MPA is depicted below.

Width computation Eq. (1) shows the width evaluation of MPA, wherein, \(r^{f}\) points out the resonant frequency, \(\varepsilon_{r}\) points out the value of dielectric substrate and \(L\) points out light speed. Equation (2) portrays the formulation for \(r^{f}\), wherein \(\varepsilon^{re}\) points out the effective refractive index and \(P\) points out patch length [27].

$$W = \frac{L}{{2r^{f} }}\sqrt {\frac{2}{{\varepsilon_{r} + 1}}}$$
(1)
$$r^{f} = \frac{L}{{2P\sqrt {\varepsilon^{re} } }}$$
(2)

Height assessment The height \(\left( H \right)\) of MPA is computed as given in Eq. (3), wherein \(T\) points out thickness of substrate [27].

$$H = \frac{T}{1000 \times 2.54}$$
(3)

ERI assessment The constraint,\(\varepsilon^{re}\) is evaluated as per the Eq. (4), wherein \(H\) points out the substrate’s height.

$$\varepsilon^{re} = \frac{{\varepsilon_{r} + 1}}{2} + \frac{{\varepsilon_{r} + 1}}{2}\left[ {1 + 12\frac{H}{W}} \right]^{{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} ,\frac{W}{H} > 1$$
(4)

Normalized length assessment The MPA’s normalized length \(\left( {\Delta P} \right)\) is evaluated as per Eq. (5).

$$\frac{\Delta P}{H} = 0.412\frac{{\left( {\varepsilon^{re} + 0.3} \right)\left( {\frac{W}{H} + 0.264} \right)}}{{\left( {\varepsilon^{re} - 0.258} \right)\left( {\frac{W}{H} + 0.8} \right)}}$$
(5)

Length assessment The length of MPA indicated by \(\left( P \right)\) is evaluated as per Eq. (6).

$$P = \frac{L}{{2r^{f} \sqrt {\varepsilon^{re} } }} - 2\Delta P$$
(6)

CI calculation Eq. (7) demonstrates the CI of transition element, wherein,\(Y_{i}\) points out the patch impedance and is evaluated as per Eq. (8) [27].

$$Y_{I} = \sqrt {50 + Z_{i} }$$
(7)
$$Y_{i} = 90\frac{{\varepsilon_{r}^{2} }}{{\varepsilon_{r} - 1}}\left( \frac{P}{W} \right)^{2}$$
(8)

Gain computation The proportion of antenna intensity to the input power constitutes the MPA gain. Equation (9) defines the MPA gain, where \(I\) indicates the input antenna power and \(R_{I}\) points out radiation intensity [27].

$$M_{A} = 4\pi \left( {\frac{{R_{I} }}{I}} \right)$$
(9)

4 Optimized MPA Design via Improved Elephant Herding Optimization

4.1 Objective Model and Solution Encoding

The adopted scheme intends to increase the antenna gain as defined in Eq. (10). Accordingly, the design parameters like \(P\),\(W\), \(T\) and \(\varepsilon_{r}\) of MPA are optimally tuned by the EHO-NSF. The input solution to adopted algorithm is revealed in Fig. 2.

$$M_{A}^{ * } = \mathop {\arg \max }\limits_{{\left( {P,W,T,\varepsilon_{r} } \right)}} \left( {M_{A} } \right)$$
(10)
Fig. 2
figure 2

Solution encoding

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4.2 Proposed EHO-NSF Algorithm

The existing EHO model [28] shows better convergence and resolves varied optimization issues, nevertheless, to improve the searching quality, certain modifications are needed and this work introduces a modified version with new scaling factor. In general, self-improvement is proven to be promising in traditional optimization algorithms [29, 30, 31, 32, 33, 34, 35, 36]. The procedure of proposed EHO-NSF model is as follows: Elephants are societal creatures that survive in social groups including calves and females. The group involves various clans, and each clan is headed by a matriarch. Usually, female elephants live with the clans, whereas the male elephants leave the clans when they grow up. The following assumptions are considered in EHO-NSF.

  1. 1.

    The population includes several clans and every clan comprises of female and male elephants.

  2. 2.

    Some of the male elephants leaves the clan and live alone

  3. 3.

    Each clan is headed by a matriarch.

Clan-updating Operator As per the nature of the elephants, the elephants in a clan are lead by matriarch. Thus, the matriarch \(c\) has a major impact on the novel positions of all elephants. The elephant \(j\) in clan \(c\) is updated by Eq. (11), here \(Z_{c,j}\) and \(Z_{n,c,j}\) points out the old position and new position of elephant \(j\) in clan \(c\), correspondingly. \(Z_{best,c}\) is matriarch \(c\) that indicates the best elephant in clan \(c\). Conventionally, the scale factor \(\alpha\) lies between 0 and 1. As per the presented model, \(\alpha\) is computed as shown in Eq. (12), where \(K\) denotes the elitism parameter (that indicates how many best elephants to move from one generation to next generation) and \(Z_{best}\) refers to best fitness at each iteration.

$$Z_{n,c,j} = Z_{c,j} + \alpha .r.\left( {Z_{best,c} - Z_{c,j} } \right)$$
(11)
$$\alpha = \left( {{j \mathord{\left/ {\vphantom {j K}} \right. \kern-\nulldelimiterspace} K}} \right)*Z_{best}$$
(12)

However, the best elephant in each clan could not be updated as per Eq. (11). As per the developed model, it can be updated as per Eq. (13), where \(CL\) points out the chromosome length and \(ran\) points out the random integer.

$$Z_{n,c,j} = round\left( {1 + \left( {CL - 1} \right)} \right) \times ran$$
(13)

4.3 Separating Operator

The separating procedure, in which the male elephants depart their family group is modelled into separating operator. The separating operator is computed based on the worst fitness as shown in Eq. (14), where \(Z_{\min }\) and \(Z_{\max }\) are the minimum and maximum bounds of positions correspondingly. \(Z_{worst,c}\) indicate the worst elephant individual of clan \(c\) and \(r\) represents the random integer between 0 and 1. Algorithm 1 reveals the pseudo code of the presented EHO-NSF model.

$$Z_{worst,c} = Z_{\min } + r.\left( {Z_{\max } - Z_{\min } + 1} \right)$$
(14)
figure d

5 Results and Discussions

5.1 Simulation Set Up

The adopted MPA design was simulated in MATLAB and the results were achieved. Here, the population size was assigned as 10 and the iteration count was assigned as 100. In addition, the superiority of the presented antenna model was validated by comparing it with other existing schemes like AAD [37], WOAD [38], FAD [39], LAD [40], GAD [41], GWAD [42], PGWAD [43], PSAD [44], EHAD [28] and MP-LAD [27]. In addition, the analysis was carried out with respect to gain, directivity, cost, CI and efficiency.

5.2 Performance Analysis

The performance of the adopted EHO-NSFAD model over existing models like AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD is given in Fig. 3 for varied frequency levels that ranges from 77, 74, 72, 70, 68 and 65 GHz. Accordingly, the gain, directivity, CI, and efficiency are observed from Fig. 3(a) to (d) respectively. From the analysis, the presented model has attained better values for gain, directivity, CI, and efficiency when compared over the other existing models. Particularly, from Fig. 3(a), a higher gain of 17.2 dB has been attained by the EHO-NSFAD scheme, which is 8.14%, 5.23%, 23.26%, 23.26%, 5.23%, 12.79%, 8.14%, 23.26%, 5.23% and 1.16% superior to traditional AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD models at 77 GHz. Also, the directivity analysis and CI analysis for the proposed EHO-NSF model over conventional schemes is given in Fig. 3(b) and (c) for varied frequency levels. Moreover, from Fig. 3(c), the CI attained by the adopted scheme is 10%, 14.67%, 22%, 14.67%, 10%, 6.67%, 10%, 3.33%, 10% and 3.33% superior to traditional AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD models at a frequency of 70 GHz respectively. From the analysis, the adopted model is accomplished with higher directivity values, thus ensuring the superior performance of the work. Specifically, the directivity attained by the adopted scheme is 35.14%, 27.03%, 40.54%, 40.54%, 48.65%, 37.84%, 32.43%, 27.03%, 45.95% and 27.03% superior to traditional AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD schemes at a frequency of 75 GHz respectively. Consequently, the overall examination of the developed model over existing models for varied design constraints is shown in Table 2. Thus, from the evaluation, it is clear that the presented EHO-NSFAD model has obtained improved performance when compared to the state-of-art techniques.

Fig. 3
figure 3

Performance analysis of the developed scheme over traditional schemes in terms of (a) Gain (b) Directivity (c) CI (d) Efficiency

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Table 2 Overall analysis of developed model over traditional models
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5.3 Analysis on Radiation Pattern

The radiation patterns of the MPA design attained by the proposed EHO-NSFAD model over existing schemes are illustrated in Fig. 4. Consequently, the radiation patterns are observed for diverse frequencies. In fact, a higher radiation pattern ensures the better performance of the system. On observing the graphs, the presented scheme has accomplished bigger side lobes as well as main lobe than the existing schemes. Particularly from Fig. 4(a), the side lobes as well as main lobe of adopted scheme at 65 GHz has revealed a higher radiation pattern when computed over existing schemes. Similarly, from Fig. 3(b)–(f), the developed EHO-NSFAD model has obtained amplified side lobes as well as main lobe whose sizes are higher over other conventional schemes for varied frequencies.

Fig. 4
figure 4

Radiation pattern of the developed scheme over traditional schemes for varied frequencies like a 65 GHz, b 68 GHz, c 70 GHz, d 72 GHz, e 74 GHz and f 77 GHz

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5.4 E-Plane Analysis

Usually, E-plane includes the capability to discover the orientation of the antenna radio waves. Accordingly, E-plane analysis is done by altering the frequencies at 77, 74, 72, 70, 68 and 65 GHz, where the DOA is assigned as 900. A proficient antenna design can be accomplished only if the side lobes and main lobe attain maximal dimensions. From Fig. 5(a), the dimensions of side lobes and the main lobe offered by the developed EHO-NSFAD scheme at the frequency of 65 GHz is higher than the existing AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD schemes. Moreover, at the frequency of 68 and 70 GHz, the sizes of side lobes as well as main lobe of presented EHO-NSFAD model are much bigger over the compared schemes. In the same way, the size of side lobes as well as main lobe of developed EHO-NSFAD model at diverse frequencies such as, 77, 74 and 72 GHz are found to expose higher radiation patterns than the conventional schemes.

Fig. 5
figure 5

E-plane analysis of developed scheme over traditional schemes for varied frequencies namely a 65 GHz, b 68 GHz, c 70 GHz, d 72 GHz, e 74 GHz and f 77 GHz

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5.5 H-Plane Analysis

The analysis on H-plane attained by the developed EHO-NSFAD model over existing models at varied frequencies such as, 77, 74, 72, 70, 68 and 65 GHz is shown in Fig. 6. For all the frequencies, the sizes of main lobe and the side lobes of presented antenna model are amplified than the existing models. Thereby, the improvement of the developed EHO-NSFAD scheme in terms of H-plane is proved from the outcomes.

Fig. 6
figure 6

H-plane analysis of developed scheme over traditional schemes for varied frequencies namely “a 65 GHz, b 68 GHz, c 70 GHz, d 72 GHz, e 74 GHz and f 77 GHz a 65 GHz, b 68 GHz, c 70 GHz, d 72 GHz, e 74 GHz and f 77 GHz”

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5.6 Convergence Analysis

The convergence analysis (cost) of adopted scheme (EHO-NSFAD) over the existing schemes is shown in Fig. 7 for varied frequencies such as, 77, 74, 72, 70, 68 and 65 GHz. Here, cost is evaluated as: (cost = 1/gain). Accordingly, the analysis is performed by varying the iterations from 0 to 100. On examining the outcomes, the cost values attained by the developed EHO-NSFAD model are found to be lower than the compared schemes for all frequencies. For instance, at 65 GHz, the proposed model at iteration 50 has attained a minimal cost value of 0.06, which is much lower when compared to other methods. That is, the adopted scheme is 6.67%, 11.67%,6.67%, 6.67%, 11.67%, 6.67%, 11.67%, 5.83%, 30% and 5% superior to traditional AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD schemes. Thereby, the superiority of presented EHO-NSFAD method was validated.

Fig. 7
figure 7

Cost of for developed scheme over traditional schemes for varied frequencies namely a 65 GHz, b 68 GHz, c 70 GHz, d 72 GHz, e 74 GHz and f 77 GHz

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The results are encouraging and they show better performance on the microstrip patch antenna. Further, based on the availability of the resources like hardware requirements this work will be further extended to experiment real-time.

6 Conclusion

This paper introduced a novel antenna design by integrating the EHO-NSFAD based optimization concept. Accordingly, the MPA parameters such as width, patch length, substrate thickness and dielectric substrate value were optimized by the presented EHO-NSFAD for attaining increased gain. Furthermore, analysis was carried out that proved the efficiency of the presented work. From the analysis, the adopted model was found to accomplish higher directivity values, thus ensuring the superior performance of the work. Specifically, the directivity attained by the adopted scheme was 35.14%, 27.03%, 40.54%, 40.54%, 48.65%, 37.84%, 32.43%, 27.03%, 45.95% and 27.03% superior to traditional AAD, WOAD, FAD, GAD, GWAD, PSAD, LAD, PGWAD, EHAD and MP-LAD schemes at a frequency of 75 GHz respectively. Thus, the effectiveness of the presented EHO-NSFAD method was proved via valuable analysis.