Machine Learning Based Model For Design and Optimization of Rectangular Patch Antenna at 3.5 GHz for 5G Applications

Machine Learning Based Model For Design and Optimization of Rectangular Patch Antenna at 3.5 GHz for 5G Applications | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Machine Learning Based Model For Design and Optimization of Rectangular Patch Antenna at 3.5 GHz for 5G Applications Aremu Olaosebikan Akanni, Sheu Akeem Lawal, Makinde Oluniyi. Samuel, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7720277/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper presents the design and simulation of a low-profile patch antenna for 5G applications at 3.5 GHz. The substrate material selected is Rogers RT/Duroid 5880 epoxy, which has a permittivity of 2.2. The proposed antenna layout is simulated using Computer simulation Technology (CST) Microwave Studio Suite. The simulation results indicate significant variations in S11, gain, directivity, efficiency, and bandwidth due to differences in the substrate’s relative permittivity and thickness. The designed antenna achieved a return loss of -43.85 dB, VSWR of 1.015, gain of 8.90 dBi, directivity of 10.5 dBi, bandwidth of 177 MHz, and efficiency of 85.7%. With a gain exceeding 5 dBi, the antenna is well-suited for communication applications. The paper also develops a mathematical model employing the Multivariate Polynomial algorithm to predict three antenna parameters: Return Loss, VSWR, and Bandwidth. A machine learning-based model has been developed using the Levenberg-Marquardt algorithm with a feed-forward back-propagation learning approach and applied to patch antenna design. It processes input data, including the dielectric constant (εr), substrate thickness (hs), and dominant-mode resonant frequency (fr), to predict the S11, VSWR, and Bandwidth. The model’s performance has been evaluated by comparing its results with simulated values obtained from CST Studio Suite. The machine learning predictions closely align with the simulation results. Additionally, the neural network-based estimation offers the advantage of rapid and simultaneous output computation. Bandwidth Machine learning Optimization Return loss VSWR Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction Microstrip antennas are widely used in mobile communication applications that require multiband and/or wideband frequency operations, high power gain, and omnidirectional radiation patterns. Consequently, designing printed antennas to accommodate multiple operational services is a challenging task. This necessitates highly accurate calculations of various design parameters for microstrip patch antennas. The dimensions of a rectangular microstrip patch antenna play a crucial role in determining its performance and effectiveness. In this study, microstrip line feeding is chosen as the preferred method for delivering input power to the antenna. Precisely calculating the patch dimensions is especially critical when the antenna size is significantly compact. Several studies have been published on the calculation of microstrip antenna patch dimensions [ 1 , 2 ]. However, these studies exhibit significant deviations in the computed patch dimensions when compared to theoretical and simulation results. It is possible to use the machine learning models for optimization that is both effective and accurate, and these models can be generated within the range of training [ 3 – 4 ]. Exploiting the feed forward network permitted for the successful completion of the task of estimating the S11, VSWR, and bandwidth of the flexible antenna. Machine learning have recently gained attention as a fast and flexible tools for modeling, simulations and optimization of microstrip patch antenna. Recently CAD approach based on neural networks has been introduced in the microwave community for modeling of passive and active microwave component. Several research papers [ 5 – 10 ] indicates how Machine learning can be used efficiently to calculate different design and performance parameters of microstrip antennas. Nevertheless, the literature shows that only three layer MLPFFBP has been preferred to prove the utility of ANN in the area of microstrip antenna design. This study therefore, explores the potential of machine learning techniques to determine the length (L) and width (W) of a microstrip patch antenna on a ground plane, considering a substrate thickness (h) and dielectric constant (εr). 2. Design and Data generation This section outlines the materials and methodology used in the antenna design process. A Rogers’s substrate was selected for this design. Initially, CST software was utilized for modeling the antenna, with the substrate material being one of the key design requirements. The antenna’s performance will then be evaluated by comparing the results with previously published findings. If the obtained results are unsatisfactory, optimization will be carried out until the desired performance is achieved. The design process begins with estimating the antenna’s length (LP), width (WP), and the ground plane dimensions (Lg and Wg). These calculations were performed using equations (1–9) to achieve the required values [ 11 , 12 ]. Following this, modifications are applied to refine the design and obtain the most optimal results. Subsequently, modifications are applied to achieve the most optimal results. The material properties used in the antenna design include a substrate thickness of 0.27 mm, a dielectric constant ( \(\:{\epsilon\:}_{r}\) ) of 2.2, a loss tangent (tan δ) of 0.0009, and a patch thickness of 0.037 mm. Step 1: Calculation of the Patch width (Wp) In this study, the detailed procedure and design equations [ 13 – 15 ] for the proposed single element Microstrip patch antenna designed at the operating frequency 3.5 GHz for 5G applications using Rogers RO3000 substrate are as follows \(\:{f}_{r}=3.5\times\:{10}^{9}\:Hz\) , \(\:c=3\times\:{10}^{8}\:{ms}^{-1},\:\) \(\:h=1.6\:mm\:\:and\:{\epsilon\:}_{r}=2.2\) The patch width (Wp) of the antenna is computed based on (1) as: $$\:{W}_{P}=\frac{c}{2{f}_{r}\sqrt{\frac{{\epsilon\:}_{r}+1}{2}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)\:$$ Step 2: Design of effective dielectric constant, \(\:{ϵ}_{eff}\) The effective dielectric constant \(\:{\varvec{ϵ}}_{\varvec{e}\varvec{f}\varvec{f}}\) introduced to account for the fringing and the wave propagation in the line. \(\:{\varvec{ϵ}}_{\varvec{e}\varvec{f}\varvec{f}}\) is obtained from (2) $$\:{ϵ}_{eff}=\frac{{ϵ}_{r}+1}{2}+\frac{{ϵ}_{r}-1}{2}{\left(1+12\frac{h}{W}\right)}^{-0.5}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(2\right)$$ Step 3: Design of Effective and actual length of the patch. The effective length of the patch is calculated from (3): $$\:{L}_{eff}=\frac{c}{2{f}_{r}\sqrt{{\epsilon\:}_{eff}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(3\right)$$ The length extension (∆𝐿) is subtracted from the length of the patch with actual length of the patch unchanged. The length extension is considered due to fringing field as seen in (4) while the actual length of the patch is obtained from [ 9 ] using (5) $$\:\varDelta\:L=0.412h\frac{\left({ϵ}_{eff}+0.3\right)\left(\frac{W}{h}+0.264\right)}{\left({ϵ}_{eff}-0.258\right)\left(\frac{W}{h}+0.8\right)}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(4\right)$$ $$\:{L}_{P}={L}_{eff}-2\varDelta\:L\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(5\right)$$ Step 4: . Design of ground plane dimensions (Lg and Wg) The length and width of the ground is computed using (6) and (7) respectively as: $$\:{L}_{g}=6h+{L}_{P}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(6\right)$$ $$\:{W}_{g}=6h+{W}_{P}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(7\right)$$ Step 5: .Feedline Designed For a 50 Ω microstrip feedline, the characteristic impedance equation is used as, $$\:{Z}_{0}=\frac{60}{\sqrt{{\epsilon\:}_{eff}}}\:ln\:\left(\frac{8h}{{W}_{f}}+\frac{{W}_{f}}{4h}\right)\:\:\:\:\:\:\:=3.33\:mm\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(8\right)$$ The feedline length is chosen based on practical layout constraints and should be long enough to allow proper impedance matching. The expression is given by, $$\:{L}_{f}=\frac{{\lambda\:}_{g}}{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(9\right)$$ where the guided wavelength is given as, \(\:{\lambda\:}_{g}=\raisebox{1ex}{${\lambda\:}_{0}$}\!\left/\:\!\raisebox{-1ex}{$\sqrt{{\epsilon\:}_{eff}}$}\right.\) For 3.5 GHz frequency and \(\:c=3\times\:{10}^{8}\:{ms}^{-1}\) , free space wavelength \(\:{\lambda\:}_{0}=85.7\:mm\) . Hence, the value of \(\:{\lambda\:}_{g}\) and \(\:{L}_{f}\) are determined. Step 6. Computation of Antenna efficiency \(\:\left(\eta\:\right)\) and reflection coefficient \(\:\left({\Gamma\:}\right)\) Antenna efficiency \(\:\left(\eta\:\right)\) is determined by converting the gain (G) and directivity (D) obtained after simulation into linear scale, the relationship is as shown in Eq. (10) $$\:\eta\:=\frac{{10}^{\left(\raisebox{1ex}{${G}_{dBi}$}\!\left/\:\!\raisebox{-1ex}{$10$}\right.\right)}}{{10}^{\left(\raisebox{1ex}{${D}_{dBi}$}\!\left/\:\!\raisebox{-1ex}{$10$}\right.\right)}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(10\right)$$ The Reflection Coefficient (Γ) quantifies how much of the incident power is reflected due to impedance mismatch. It is related to S11 by, $$\:{\Gamma\:}={10}^{\raisebox{1ex}{$S11$}\!\left/\:\!\raisebox{-1ex}{$20$}\right.\:\:\:\:}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(11\right)$$ The reflected power \(\:\left({P}_{r}\right)\) presents the percentage of incident power that is reflected due to impedance mismatch, it is computed using the expression given by (12) $$\:{P}_{r}={{\Gamma\:}}^{2}\times\:{P}_{iincident}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(12\right)$$ The power radiated by the antenna can be estimated by subtracting the power reflected from antenna input power, equivalently as shown in (13) $$\:{P}_{radiated}=\left(1-{\left|{\Gamma\:}\right|}^{2}\right)\:.{P}_{input}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(13\right)$$ where \(\:{\Gamma\:}\) is the linear reflection coefficient given by, $$\:{\Gamma\:}={10}^{\raisebox{1ex}{$RL$}\!\left/\:\!\raisebox{-1ex}{$20$}\right.\:\:\:\:}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(14\right)$$ Table 1 shows the design specification of the proposed antenna while Table 2 shows the dimensions of the optimized antenna designed. Table 1 Design specifications of the proposed antenna. Parameter \(\:{\mathbf{W}}_{\varvec{p}}\) \(\:{\varvec{L}}_{\varvec{p}}\) \(\:{\mathbf{W}}_{\varvec{g}}\) \(\:{\varvec{L}}_{\varvec{g}}\) \(\:{\varvec{ϵ}}_{\varvec{r}}\) \(\:\varvec{h}\) \(\:\varvec{t}\) \(\:{\mathbf{W}}_{\mathbf{f}}\) \(\:{\varvec{L}}_{\varvec{f}}\) Dimensions (mm) 38.2 28.3 47.9 37.9 2.2 1.6 0.035 3.1 16.0 Table 2 Dimension of the Optimized antenna design Parameter Value Resonant frequency ( \(\:{\varvec{f}}_{\varvec{r}}\) ) 3.5 GHz Patch width ( \(\:{\mathbf{W}}_{\mathbf{p}}\) ) 26.2 mm Patch length ( \(\:{\mathbf{L}}_{\mathbf{p}}\) ) 18.7 mm Length of ground (Lg) 29.7 mm Width of ground (Wg) 35.8 mm Substrate height (h) 1.6 mm Dielectric constant \(\:\left({\varvec{ϵ}}_{\varvec{r}}\right)\) 2.20 Feedline Width ( \(\:{\varvec{W}}_{\varvec{f}}\) ) 2.27 mm Feedline length ( \(\:{\varvec{L}}_{\varvec{f}}\) ) 16.0 m 3. Optimization of Antenna Designed The optimization method will concentrate on two design parameters—substrate height (hs) and patch length (Lp)—selected based on the antenna designed in the previous section. Variations in these parameters significantly influence the antenna’s performance during optimization. Multiple simulations are conducted, testing various values to generate datasets for experimentation and mathematical modeling. After developing a suitable model, the parameters are fine-tuned using a constrained numerical method to achieve the desired performance level. The enhanced antenna performance is then validated through simulations in Computer Simulation Technology (CST) software, which utilizes an electromagnetic solver. Table 3 presents the variations in antenna performance along with detailed parameter values. 3.1 Machine Learning Back Propagation Technique In this study, the ANN model was trained using dielectric constant, substrate, thickness and cutoff frequencies as inputs, with patch length and width as outputs as shown in Fig. 1 . Back propagation shown in Fig. 2 was employed to minimize the error between predicted and actual resonant frequencies, ensuring that the forward and inverse models consistently yield accurate patch dimensions for a desired resonant frequency. 3.2 Treatment of Data and Development of Artificial Neural network A minimum-maximum normalization process was performed on the inputs and preprocessed path loss dataset obtained along the three itineraries using (16). This was done to prevent impulsive changes due to large variation in the datasets $$\:y=\frac{\left({y}_{max}-{y}_{min}\right)\times\:\left({x}_{in}-{x}_{min}\right)}{\left({x}_{max}-{x}_{min}\right)}+{y}_{min}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(15\right)$$ Since \(\:{y}_{min}=-1\) and \(\:{y}_{max}=+1\) , \(\:{x}_{R}\) is the original data, y is the result of normalization. Eq. (15) becomes, $$\:y=\frac{2\left({x}_{in}-{x}_{min}\right)}{\left({x}_{max}-{x}_{min}\right)}-1\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(16\right)$$ The normalized dataset was processed using a single-layer feed forward back propagation neural network trained with the Levenberg–Marquardt algorithm. The data were randomly partitioned into 70% for training, 15% for validation, and 15% for testing. All simulations were performed using MATLAB 2024a (MathWorks Inc.). The ANN architecture is illustrated in Figs. 1 and 2 . 3.3 Machine Learning Modeling A tansig activation function was employed in the hidden layer and a purelin function in the output layer, as described in Eq. (17). $$\:y=\:\sum\:_{J=1}^{m}\left\{Purelin\left[{LW}_{j,1}\left(\sum\:_{i=1}^{4}tansig\left({X}_{i}I{W}_{i,j}+{b}_{j}\right)\right)\right]+{b}_{o}\right\}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(17\right)$$ The network was trained using 1,000 epochs and a zero-error goal as stopping criteria. Following denormalization, the model’s performance was assessed by comparing actual and predicted values using correlation coefficients (R-values) for S11, VSWR, and bandwidth. As shown in Table 3 , the model demonstrated excellent predictive accuracy for S11 and VSWR, with slightly lower yet satisfactory performance for bandwidth. Table 3 R-values for trained data Antenna characteristics Training Validation Test All S11 0.9967 1.0000 1.0000 0.9667 VSWR 1.0000 1.0000 1.0000 0.9443 Bandwidth 1.0000 1.0000 1.0000 0.8995 3.4 Polynomial Regression Model A polynomial regression model for the two independent variables \(\:({x}_{1},\:{x}_{2})\) , and one dependent variable \(\:\left(y\right)\) is expressed as: $$\:y={\beta\:}_{0}+{\beta\:}_{1}{x}_{1}+{\beta\:}_{2}{x}_{2}+{\beta\:}_{3}{x}_{1}^{2}+{\beta\:}_{4}{x}_{2}^{2}+{\beta\:}_{5}{x}_{1}{x}_{2}+ϵ\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(18\right)$$ where \(\:{\beta\:}_{0}\) is the intercept, \(\:{\beta\:}_{1}\) and \(\:{\beta\:}_{2}\) are the linear coefficients, \(\:{\beta\:}_{3}\) and \(\:{\beta\:}_{4}\) are the quadratic coefficients, \(\:{\beta\:}_{5}\) is the interaction term coefficient, and \(\:ϵ\) is the error term. In this study, the second order polynomial with two independent variables \(\:{x}_{1}={f}_{r}\:and\:{x}_{2}={h}_{s}\) is given as, $$\:y={\beta\:}_{0}+{\beta\:}_{1}{f}_{r}+{\beta\:}_{2}{h}_{s}+{\beta\:}_{3}{f}_{r}^{2}+{\beta\:}_{4}{h}_{s}^{2}+{\beta\:}_{5}{f}_{r}{h}_{s}+ϵ\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(19\right)$$ The attained regression equation the return loss (S11) is: $$\:y=4032+(-1678.5*{f}_{r})+(-1222.4*{h}_{s})+235*{f}_{r}^{2}+\left(-11.6*{f}_{r}*{h}_{s}\right)+\left(872*{h}_{s}^{2}\right)\:\:\:\:\left(20\right)$$ The regression equation obtained for VSWR is: $$\:y=25.9+(-12.2*{f}_{r})+(-3.28*{h}_{s})+\left(1.72*{f}_{r}^{2}\right)+\left(-0.18*{f}_{r}*{h}_{s}\right)+\left(1.15*{h}_{s}^{2}\right)\:\:\:\:\left(21\right)$$ The regression equation obtained for Bandwidth is: $$\:y=-4808+\left(1865*{f}_{r}\right)+\left(2065*{h}_{s}\right)+(-307.3*{f}_{r}^{2})+\left(197.2*{f}_{r}*{h}_{s}\right)+(-832.4*{h}_{s}^{2}\:\:(22)$$ 4. Results and Discussion Error! Reference source not found. 3, 4 , and 5 present 3D plots illustrating the influence of resonant frequency (fr) and substrate height (hs) on S11, VSWR, and bandwidth. The results reveal that S11 exhibits a non-linear relationship with both fr and hs, with darker regions on the color map indicating improved impedance matching (lower S11 values). Figure 3 highlights that minor variations in hs significantly influence return loss. As shown in Fig. 4, VSWR also follows a non-linear trend with increasing fr, with measured values aligning well with the predicted surface. Regions with higher VSWR values indicate poorer impedance matching. Figure 5 illustrates the variation of bandwidth, which shows a mixed trend—expanding in certain regions and narrowing in others. The influence of fr on bandwidth is more pronounced than that of hs, suggesting frequency sensitivity in determining antenna performance. Table 4 compares the original and predicted values of the antenna performance metrics across thirteen samples. The results demonstrate a strong correlation between the actual and predicted data, confirming the reliability of the model and the effectiveness of the derived regression equations in estimating the target parameters. Table 5 presents the numerical relationship between substrate height (hs) and bandwidth based on test samples. While not strictly linear, the trend indicates that bandwidth generally increases with hs. It was noted that, at \(\:h=1.61\) mm, bandwidth starts at 145.13 MHz, then increases at different heights, reaching a maximum of 180.93 MHz a t h = 1.65 mm.. This behavior suggests that moderate increases in substrate height may enhance bandwidth due to improved impedance matching and reduced surface wave losses. Table 4 Comparison between original and predicted values of the performance metrics. \(\:{\varvec{f}}_{\varvec{r}}\) (GHz) \(\:{\varvec{L}}_{\varvec{g}}\) (mm) \(\:\varvec{\delta\:}\) \(\:{\varvec{h}}_{\varvec{s}}\) (mm) \(\:{\varvec{W}}_{\varvec{g}}\) (mm) \(\:{\varvec{W}}_{\varvec{P}}\) (mm) \(\:{\varvec{L}}_{\varvec{p}}\) (mm) \(\:{\varvec{\epsilon\:}}_{\varvec{r}}\) S11 (dB) VSWR Bandwidth (MHz) 3.351 29.7 0.009 1.61 35.8 26.2 20.11 2.2 -23.09 1.155 143 3.505 29.7 0.009 1.82 35.8 26.2 20.19 2.2 -30.79 1.062 153 3.852 29.7 0.009 1.72 35.8 26.2 20.21 2.2 -26.42 1.104 159 3.802 29.7 0.009 1.75 35.8 26.2 19.99 2.2 -28.05 1.084 153 3.755 29.7 0.009 1.63 35.8 26.2 20.04 2.2 -30.96 1.075 162 3.715 29.7 0/009 1.60 35.8 26.2 19.97 2.2 -32.73 1.052 168 3.670 29.7 0.009 1.62 35.8 26.2 20.13 2.2 -37.52 1.031 166 3.630 29.7 0.009 1.64 35.8 26.2 19.98 2.2 -39.39 1.022 169 3.505 29.7 0.009 1.69 35.8 26.2 20.12 2.2 -43.85 1.015 177 3.545 29.7 0.009 1.66 35.8 26.2 20.00 2.2 -34.28 1.046 173 3.465 29.7 0.009 1.61 35.8 26.2 19.97 2.2 -28.22 1.089 178 3.425 29.7 0.009 1.65 35.8 26.2 20.10 2.2 -26.31 1.103 181 3.388 29.7 0.009 1.60 35.8 26.2 20.21 2.2 -24.55 1.128 178 Table 5 Comparison of original and predicted antenna parameters F (GHz) \(\:{\text{h}}_{\text{s}}\:\left(\text{m}\text{m}\right)\) S11 original S11 predicted VSWR original VSWR predicted Bandwidth original Bandwidth predicted 3.351 1.61 -23.098 -22.01 1.155 1.085 143 145.13 3.505 1.82 -30.798 -30.798 1.062 1.094 153 153.79 3.852 1.72 -26.423 -26.423 1.104 1.104 159 161.33 3.802 1.75 -28.059 -28.059 1.084 1.084 153 159.39 3.755 1.63 -30.965 -30.965 1.075 1.075 162 167.06 3.715 1.60 -32.731 -34.560 1.052 1.052 168 166.79 3.670 1.62 -37.527 -37.527 1.031 1.031 166 168.15 3.630 1.64 -39.399 -37.772 1.022 1.022 169 170.30 3.505 1.69 -43.850 -46.980 1.015 1.015 177 175.76 3.545 1.66 -34.285 -34.285 1.046 1.046 173 176.07 3.465 1.61 -28.223 -28.223 1.089 1.038 178 176.29 3.425 1.65 -26.316 -26.316 1.103 1.103 181 180.93 3.388 1.60 -24.551 -24.551 1.128 1.134 178 179.65 4.1 Return loss (S11) Figure 6 illustrates the return loss (S11) values across various frequencies for the trained dataset, considering different substrate heights. The frequency range spans from 3.351 GHz to 3.852 GHz, with corresponding return loss values between − 22.02 dB and − 46.98 dB. The minimum return loss of − 46.98 dB is observed at 3.585 GHz, indicating optimal impedance matching at this frequency. Frequencies between 3.63 GHz and 3.67 GHz also exhibit high return loss values (below − 37 dB), suggesting a broad region of efficient antenna performance. Conversely, higher return loss values—such as − 22.02 dB at 3.351 GHz and − 24.55 dB at 3.388 GHz—indicate relatively poorer matching at those frequencies. Overall, the results confirm that the trained model accurately predicts the frequency range with optimal impedance characteristics, with peak performance centered around 3.585 GHz. Figure 7 presents the variation of the machine learning-predicted VSWR across the frequency range. The VSWR values remain consistently close to 1, indicating effective impedance matching throughout. The minimum VSWR of 1.015 occurs at 3.585 GHz, signifying optimal matching, while the maximum value of 1.134 at 3.388 GHz still falls within acceptable limits for efficient antenna operation. These results confirm that the trained model effectively maintains low reflection losses and stable VSWR across the target frequency band. 4.2 Gain and Directivity Figures 8 and 9 present the 3D gain and polar directivity patterns of the proposed antenna, respectively. The antenna demonstrates a gain of 8.91 dBi and a directivity of 10.4 dBi, indicating high efficiency and strong directional performance. With an efficiency of 85.7%, only 14.3% of the input power is lost due to factors like dielectric and conductor losses. The relatively small differences between gain and directivity suggests minimal energy loss, reflecting successful machine learning-based optimization. These characteristics make the antenna suitable for directional applications such as 5G base stations and point-to-point links. The performance summary of the machine learning-based antenna model is presented in Table 6 . The trained microstrip patch antenna, utilizing a Rogers substrate, exhibits outstanding electromagnetic performance. This is evidenced by a deep reflection coefficient (S11 = − 46.98 dB) and a near-ideal voltage standing wave ratio (VSWR) of 1.015, indicating excellent impedance matching and minimal signal reflection. The antenna achieves a bandwidth of 176 MHz, which is considerably wide for microstrip designs, thereby supporting stable frequency operation. Furthermore, the antenna demonstrates a gain of 8.9 dBi and a directivity of 10.4 dBi, confirming its strong radiative capability and directional beam characteristics. An overall radiation efficiency of 85.7% signifies low power losses, aligning with the typical 80–90% efficiency range reported for microstrip antennas [ 16 , 17 ]. These findings suggest that the proposed design is well-suited for high-performance wireless communication systems, including 5G base stations and radar applications. The results confirm that machine learning optimization has effectively enhanced key performance metrics, with potential for future improvements targeting higher gain and extended bandwidth to meet specific system requirements. Table 6 Summary of Simulation results Parameters Patch antenna with Rogers substrate S11 VSWR Bandwidth Gain Directivity Antenna efficiency Efficiency, \(\:\varvec{\eta\:}\) (Linear scale) Reflection coefficient ( \(\:\varvec{\Gamma\:}\) ) Reflected power \(\:\left({\mathbf{P}}_{\mathbf{r}}\right)\) Transmitted power \(\:\left({\mathbf{P}}_{\mathbf{T}}\right)\) -46.98 dBi 1.015 176 MHz 8.91 dBi 10.4 dBi 85.7% 71.2% 0.0142 2.02% 99.98% Table 7 presents the performance evaluation of the proposed microstrip patch antenna design in terms of return loss (S11), VSWR, and gain. The results demonstrate a significant improvement over previously reported designs. This conclusion is supported by detailed simulation data and comparative analysis, which collectively validate the effectiveness of the proposed approach. The findings confirm that the design methodology offers a practical and reliable solution for microstrip patch antenna development. Overall, the presented data serve as a valuable reference for the design and optimization of antennas in advanced wireless communication systems. Table 7 Comparison with similar works for 5G applications antenna. S/N Ref. Return loss (dBi) VSWR Gain (dB) 1. [ 13 ] -23.673 2.000 6.80 2 [ 14 ] -17.430 1.310 3.02 3 [ 15 ] -17.436 1.310 6.60 4 [ 16 ] -26.347 1.100 3.68 5 [ 17 ] -41.300 1.010 4.45 6 [ 18 ] -13.890 1.500 6.60 7 [ 19 ] -18.270 2.130 4.46 8 [ 20 ] -13.480 1.538 6.63 9 This work -46.984 1.015 8.91 5. Conclusion In this study, the microstrip patch antenna dimensions and parameters (return loss, gain, directivity, bandwidth, and VSWR) are determined with the help of CST studio to produce an effective machine leaning based model. The antenna design is modeled using the Levenberg Marquardt optimization method and feed-forward back propagation neural network at 3.5 GHz. The results obtained through the use of CST and machine learning based model are in conformity with one another in a manner that is acceptable. Moreover, the study incorporates training and test datasets alongside CST-generated data. The proposed antenna operates within a frequency range of 3.351 GHz to 3.852 GHz. The trained data at 3.5 GHz yielded a good results. Furthermore, a mathematical model based on the Multivariate Polynomial algorith m to predict three antenna parameters: Return Loss, VSWR, and Bandwidth was developed. Th e trained network reliably predicts antenna performance for new design cases, provided the input parameters—such as substrate height, resonant frequency, and dielectric constant—remain within the trained range. The proposed method improves design accuracy and significantly reduces development time, making it a practical tool for future wireless applications. Declarations Funding The authors received no funding for this research. Data availability “Data is provided within the manuscript files”. Ethics approval Ethics approval was not required for this study. Therefore, no ethics approval declaration is available Consent to publish Consent to publish was not required for this study. Therefore, no consent to publish declaration is available. Consent to participate Consent to participate was not required for this study. Therefore, no consent to participate declaration is available. Competing interests The authors declare no competing interests. References Mishra RK, Patnaik A. Neural Network-Based CAD Model for the Design of Square-Patch Antennas. IEEE Trans Antennas Propag. 2018;46(12):1890–1. 10.1109/8.743842 . Narayan JL, Krishna R, Reddy LP. (2019) Design of Microstrip Antenna Using Artificial Neural Networks, International Conference on Computational Intelligence and Multimedia Applications, Vol. 1, pp. 332–334. Kumar A, Kumar V, Singh. Radial Basis Function Neural Network for estimation of Bandwidth of Antenna. Int J Control Theory Application. 2017;10(9):901–6. Singh P, Kumar Singh V. Application of Multi-Layer Feed Forward Back Propagation Neural Network for Analysis & Modelling of Antenna. Int J Control Theory Application. 2017;10(9):895–900. Patnaik A, Mishra RK, Patra GK, and., Dash SK. (2017) An Artificial Neural Network Model for Effective Dielectric Constant of Microstripline, IEEE Transactions on Antennas Propagation, Vol. 45, No. 11, 2017, p. 1697. 10.1109/8.650084 Karaboga D, Giiney K, Sagıroglu S, Erler M. 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(2023) A Novel Method of Using Artificial Neural Networks to Calculate Input Impedance of Circular Microstrip Antenna, Antennas and Propagation Society International Symposium, Vol. 3, 2023, pp. 462–465. Ghazaoui Y, Alami AB, Ghzaoui MB, Das S, Barad D, Mohapatra S. (2020). Millimeter wave antenna with enhanced bandwidth for 5G wireless application, J. Instrum. 15 (2020). Abdulbari AA, Jawad MM, Hanoosh HO, Saare MA, Lashari SA, Sari SA, Ahmad Y, Khalill YM. (2021) Design compact microstrap patch antenna with T-shaped 5G application, Bull. Electr. Eng. Informatics 10 (2021) 2072–2078. Hossain Mollah MS, Faruk O, Hossain MS, Islam MT, Shafi ASM, Molla I. m. M. (2021) Design and Performance Improvement of Microstrip Patch Antenna Using Graphene Material for Communication Applications, 2021 IEEE 11th IEEE Symposium on Computer Applications & Industrial Electronics (ISCAIE), 2021, pp. 343–346. Rahman MA, Shaikat A, Iqbal IS, Hassan A. (2013) Microstrip patch antenna design and performance analysis for RFID applications at ISM band (2.45 GHz), 2nd International Conference on Advances in Electrical Engineering (ICAEE), 2013, pp. 305–308. } Paragya D, Hartono S. (2020). 3.5 GHz Rectangular Patch Microstrip Antenna With Defected Ground Structure for 5G. ELKOMIKA: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, &Amp Teknik Elektronika, vol. 8, no. 1, Institut Teknologi Nasional, Bandung, Jan. 2020, p. 31. Touko Tcheutou Stephane Borel, Priyadarshini R. U-Slotted Wideband Microstrip Patch Antenna for Ka Band and mmW 5G Applications, 14 July 2022, PREPRINT (Version 1). Rahman MA, Shaikat A, Iqbal IS, Hassan A. Microstrip patch antenna design and performance analysis for RFID applications at ISM band (2.45 GHz), 2013 2nd International Conference on Advances in Electrical Engineering (ICAEE), 2013, pp. 305–308. Paragya D, Hartono S. (2020). 3.5 GHz Rectangular Patch Microstrip Antenna With Defected Ground Structure for 5G. ELKOMIKA: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, &Amp Teknik Elektronika, vol. 8, no. 1, Institut Teknologi Nasional, Bandung, Jan. 2020, p. 31. Rafdzi MF, Mohamad SY, Ruslan AA, Malek NFA, Islam MR, Hashim AHA. Study for Microstrip Patch Antenna for 5G Networks, 2020 IEEE Student Conference on Research and Development (SCOReD), 2020, pp. 524–528. Hasan MM, Rahman Z, Shaikh R, Alam I, Islam MA, Alam MS. Design and Analysis of Elliptical Microstrip Patch Antenna at 3.5 GHz for 5G Applications, 2020 IEEE Region 10 Symposium (TENSYMP), 2020, pp. 981–984. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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5","display":"","copyAsset":false,"role":"figure","size":170206,"visible":true,"origin":"","legend":"\u003cp\u003e3D outlook of polynomial regression fit for bandwidth\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/060943cfb681f66b2f3b7648.png"},{"id":94060893,"identity":"ead90bfc-8235-4220-9861-9763474c3431","added_by":"auto","created_at":"2025-10-22 06:39:10","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":42218,"visible":true,"origin":"","legend":"\u003cp\u003eMagnitude of S-parameters based on different substrate height\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/64163b953dbb7c52f1d4ff06.png"},{"id":94060894,"identity":"880ed3d1-3194-405d-929c-f3e260028504","added_by":"auto","created_at":"2025-10-22 06:39:10","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":42263,"visible":true,"origin":"","legend":"\u003cp\u003eVSWR based on different substrate height\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/d5944f5384b0ec844c57d9c4.png"},{"id":94060898,"identity":"05f92ce5-17cf-4d73-b57d-57f32cfaf811","added_by":"auto","created_at":"2025-10-22 06:39:11","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":45863,"visible":true,"origin":"","legend":"\u003cp\u003e3D gain and radiation pattern of the proposed antenna\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/18904157dcb2860bba3db512.png"},{"id":94060900,"identity":"2e1ccb6a-3bb6-48f5-946d-d956b9d6354e","added_by":"auto","created_at":"2025-10-22 06:39:11","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":33292,"visible":true,"origin":"","legend":"\u003cp\u003eFar field Directivity and beam width of the proposed antenna.\u003c/p\u003e","description":"","filename":"image9.png","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/917aa08b68910147d54a3c8d.png"},{"id":98421266,"identity":"a1bbabe0-6a7c-4c38-91cb-207c0a2ddc9d","added_by":"auto","created_at":"2025-12-17 16:26:12","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1446368,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7720277/v1/dd755122-6ebb-403c-8c62-4a0e1ae34e27.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Machine Learning Based Model For Design and Optimization of Rectangular Patch Antenna at 3.5 GHz for 5G Applications","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eMicrostrip antennas are widely used in mobile communication applications that require multiband and/or wideband frequency operations, high power gain, and omnidirectional radiation patterns. Consequently, designing printed antennas to accommodate multiple operational services is a challenging task. This necessitates highly accurate calculations of various design parameters for microstrip patch antennas. The dimensions of a rectangular microstrip patch antenna play a crucial role in determining its performance and effectiveness. In this study, microstrip line feeding is chosen as the preferred method for delivering input power to the antenna. Precisely calculating the patch dimensions is especially critical when the antenna size is significantly compact. Several studies have been published on the calculation of microstrip antenna patch dimensions [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. However, these studies exhibit significant deviations in the computed patch dimensions when compared to theoretical and simulation results. It is possible to use the machine learning models for optimization that is both effective and accurate, and these models can be generated within the range of training [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Exploiting the feed forward network permitted for the successful completion of the task of estimating the S11, VSWR, and bandwidth of the flexible antenna. Machine learning have recently gained attention as a fast and flexible tools for modeling, simulations and optimization of microstrip patch antenna. Recently CAD approach based on neural networks has been introduced in the microwave community for modeling of passive and active microwave component. Several research papers [\u003cspan additionalcitationids=\"CR6 CR7 CR8 CR9\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] indicates how Machine learning can be used efficiently to calculate different design and performance parameters of microstrip antennas. Nevertheless, the literature shows that only three layer MLPFFBP has been preferred to prove the utility of ANN in the area of microstrip antenna design. This study therefore, explores the potential of machine learning techniques to determine the length (L) and width (W) of a microstrip patch antenna on a ground plane, considering a substrate thickness (h) and dielectric constant (εr).\u003c/p\u003e"},{"header":"2. Design and Data generation","content":"\u003cp\u003eThis section outlines the materials and methodology used in the antenna design process. A Rogers\u0026rsquo;s substrate was selected for this design. Initially, CST software was utilized for modeling the antenna, with the substrate material being one of the key design requirements. The antenna\u0026rsquo;s performance will then be evaluated by comparing the results with previously published findings. If the obtained results are unsatisfactory, optimization will be carried out until the desired performance is achieved. The design process begins with estimating the antenna\u0026rsquo;s length (LP), width (WP), and the ground plane dimensions (Lg and Wg). These calculations were performed using equations (1\u0026ndash;9) to achieve the required values [\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e]. Following this, modifications are applied to refine the design and obtain the most optimal results. Subsequently, modifications are applied to achieve the most optimal results. The material properties used in the antenna design include a substrate thickness of 0.27 mm, a dielectric constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\epsilon\\:}_{r}\\)\u003c/span\u003e\u003c/span\u003e) of 2.2, a loss tangent (tan \u0026delta;) of 0.0009, and a patch thickness of 0.037 mm.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eStep 1:\u0026nbsp;\u003c/strong\u003eCalculation of the Patch width (Wp)\u003c/p\u003e\n\u003cp\u003eIn this study, the detailed procedure and design equations [\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e] for the proposed single element Microstrip patch antenna designed at the operating frequency 3.5 GHz for 5G applications using Rogers RO3000 substrate are as follows \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{r}=3.5\\times\\:{10}^{9}\\:Hz\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c=3\\times\\:{10}^{8}\\:{ms}^{-1},\\:\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:h=1.6\\:mm\\:\\:and\\:{\\epsilon\\:}_{r}=2.2\\)\u003c/span\u003e\u003c/span\u003e The patch width (Wp) of the antenna is computed based on (1) as:\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$$\\:{W}_{P}=\\frac{c}{2{f}_{r}\\sqrt{\\frac{{\\epsilon\\:}_{r}+1}{2}}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(1\\right)\\:$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eStep 2:\u0026nbsp;\u003c/strong\u003eDesign of effective dielectric constant,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{ϵ}_{eff}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eThe effective dielectric constant \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{ϵ}}_{\\varvec{e}\\varvec{f}\\varvec{f}}\\)\u003c/span\u003e\u003c/span\u003e introduced to account for the fringing and the wave propagation in the line. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{ϵ}}_{\\varvec{e}\\varvec{f}\\varvec{f}}\\)\u003c/span\u003e\u003c/span\u003e is obtained from (2)\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$\\:{ϵ}_{eff}=\\frac{{ϵ}_{r}+1}{2}+\\frac{{ϵ}_{r}-1}{2}{\\left(1+12\\frac{h}{W}\\right)}^{-0.5}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(2\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eStep 3:\u0026nbsp;\u003c/strong\u003eDesign of Effective and actual length of the patch.\u003c/p\u003e\n\u003cp\u003eThe effective length of the patch is calculated from (3):\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$\\:{L}_{eff}=\\frac{c}{2{f}_{r}\\sqrt{{\\epsilon\\:}_{eff}}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(3\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe length extension (∆𝐿) is subtracted from the length of the patch with actual length of the patch unchanged. The length extension is considered due to fringing field as seen in (4) while the actual length of the patch is obtained from [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e] using (5)\u003c/p\u003e\n\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equd\" class=\"mathdisplay\"\u003e$$\\:\\varDelta\\:L=0.412h\\frac{\\left({ϵ}_{eff}+0.3\\right)\\left(\\frac{W}{h}+0.264\\right)}{\\left({ϵ}_{eff}-0.258\\right)\\left(\\frac{W}{h}+0.8\\right)}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(4\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Eque\" class=\"mathdisplay\"\u003e$$\\:{L}_{P}={L}_{eff}-2\\varDelta\\:L\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(5\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eStep 4:\u003c/strong\u003e. Design of ground plane dimensions (Lg and Wg)\u003c/p\u003e\n\u003cp\u003eThe length and width of the ground is computed using (6) and (7) respectively as:\u003c/p\u003e\n\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equf\" class=\"mathdisplay\"\u003e$$\\:{L}_{g}=6h+{L}_{P}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(6\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equg\" class=\"mathdisplay\"\u003e$$\\:{W}_{g}=6h+{W}_{P}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(7\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eStep 5:\u003c/strong\u003e.Feedline Designed\u003c/p\u003e\n\u003cp\u003eFor a 50 Ω microstrip feedline, the characteristic impedance equation is used as,\u003c/p\u003e\n\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equh\" class=\"mathdisplay\"\u003e$$\\:{Z}_{0}=\\frac{60}{\\sqrt{{\\epsilon\\:}_{eff}}}\\:ln\\:\\left(\\frac{8h}{{W}_{f}}+\\frac{{W}_{f}}{4h}\\right)\\:\\:\\:\\:\\:\\:\\:=3.33\\:mm\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(8\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe feedline length is chosen based on practical layout constraints and should be long enough to allow proper impedance matching. The expression is given by,\u003c/p\u003e\n\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equi\" class=\"mathdisplay\"\u003e$$\\:{L}_{f}=\\frac{{\\lambda\\:}_{g}}{4}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(9\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere the guided wavelength is given as, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{g}=\\raisebox{1ex}{${\\lambda\\:}_{0}$}\\!\\left/\\:\\!\\raisebox{-1ex}{$\\sqrt{{\\epsilon\\:}_{eff}}$}\\right.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eFor 3.5 GHz frequency and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c=3\\times\\:{10}^{8}\\:{ms}^{-1}\\)\u003c/span\u003e\u003c/span\u003e, free space wavelength \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{0}=85.7\\:mm\\)\u003c/span\u003e\u003c/span\u003e. Hence, the value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{g}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{f}\\)\u003c/span\u003e\u003c/span\u003e are determined.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eStep 6.\u003c/strong\u003e Computation of Antenna efficiency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(\\eta\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e and reflection coefficient \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({\\Gamma\\:}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eAntenna efficiency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(\\eta\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e is determined by converting the gain (G) and directivity (D) obtained after simulation into linear scale, the relationship is as shown in Eq.\u0026nbsp;(10)\u003c/p\u003e\n\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equj\" class=\"mathdisplay\"\u003e$$\\:\\eta\\:=\\frac{{10}^{\\left(\\raisebox{1ex}{${G}_{dBi}$}\\!\\left/\\:\\!\\raisebox{-1ex}{$10$}\\right.\\right)}}{{10}^{\\left(\\raisebox{1ex}{${D}_{dBi}$}\\!\\left/\\:\\!\\raisebox{-1ex}{$10$}\\right.\\right)}}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(10\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Reflection Coefficient (\u0026Gamma;) quantifies how much of the incident power is reflected due to impedance mismatch. It is related to S11 by,\u003c/p\u003e\n\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equk\" class=\"mathdisplay\"\u003e$$\\:{\\Gamma\\:}={10}^{\\raisebox{1ex}{$S11$}\\!\\left/\\:\\!\\raisebox{-1ex}{$20$}\\right.\\:\\:\\:\\:}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(11\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe reflected power \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({P}_{r}\\right)\\)\u003c/span\u003e\u003c/span\u003e presents the percentage of incident power that is reflected due to impedance mismatch, it is computed using the expression given by (12)\u003c/p\u003e\n\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equl\" class=\"mathdisplay\"\u003e$$\\:{P}_{r}={{\\Gamma\\:}}^{2}\\times\\:{P}_{iincident}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(12\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe power radiated by the antenna can be estimated by subtracting the power reflected from antenna input power, equivalently as shown in (13)\u003c/p\u003e\n\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equm\" class=\"mathdisplay\"\u003e$$\\:{P}_{radiated}=\\left(1-{\\left|{\\Gamma\\:}\\right|}^{2}\\right)\\:.{P}_{input}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(13\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e is the linear reflection coefficient given by,\u003c/p\u003e\n\u003cdiv id=\"Equn\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equn\" class=\"mathdisplay\"\u003e$$\\:{\\Gamma\\:}={10}^{\\raisebox{1ex}{$RL$}\\!\\left/\\:\\!\\raisebox{-1ex}{$20$}\\right.\\:\\:\\:\\:}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(14\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e shows the design specification of the proposed antenna while Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e shows the dimensions of the optimized antenna designed.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eDesign specifications of the proposed antenna.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameter\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{W}}_{\\varvec{p}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{p}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{W}}_{\\varvec{g}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{g}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{ϵ}}_{\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{t}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{W}}_{\\mathbf{f}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{f}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDimensions (mm)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e38.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e28.3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e47.9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e37.9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.035\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e16.0\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eDimension of the Optimized antenna design\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameter\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eValue\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eResonant frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{f}}_{\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.5 GHz\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePatch width (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{W}}_{\\mathbf{p}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e26.2 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePatch length (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{L}}_{\\mathbf{p}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18.7 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eLength of ground (Lg)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e29.7 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eWidth of ground (Wg)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e35.8 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eSubstrate height (h)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.6 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDielectric constant \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({\\varvec{ϵ}}_{\\varvec{r}}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2.20\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eFeedline Width (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{W}}_{\\varvec{f}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2.27 mm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eFeedline length (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{f}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e16.0 m\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e"},{"header":"3. Optimization of Antenna Designed","content":"\u003cp\u003eThe optimization method will concentrate on two design parameters\u0026mdash;substrate height (hs) and patch length (Lp)\u0026mdash;selected based on the antenna designed in the previous section. Variations in these parameters significantly influence the antenna\u0026rsquo;s performance during optimization. Multiple simulations are conducted, testing various values to generate datasets for experimentation and mathematical modeling. After developing a suitable model, the parameters are fine-tuned using a constrained numerical method to achieve the desired performance level. The enhanced antenna performance is then validated through simulations in Computer Simulation Technology (CST) software, which utilizes an electromagnetic solver. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e presents the variations in antenna performance along with detailed parameter values.\u003c/p\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1 Machine Learning Back Propagation Technique\u003c/h2\u003e\n\u003cp\u003eIn this study, the ANN model was trained using dielectric constant, substrate, thickness and cutoff frequencies as inputs, with patch length and width as outputs as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. Back propagation shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e was employed to minimize the error between predicted and actual resonant frequencies, ensuring that the forward and inverse models consistently yield accurate patch dimensions for a desired resonant frequency.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2 Treatment of Data and Development of Artificial Neural network\u003c/h2\u003e\n\u003cp\u003eA minimum-maximum normalization process was performed on the inputs and preprocessed path loss dataset obtained along the three itineraries using (16). This was done to prevent impulsive changes due to large variation in the datasets\u003c/p\u003e\n\u003cdiv id=\"Equo\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equo\" class=\"mathdisplay\"\u003e$$\\:y=\\frac{\\left({y}_{max}-{y}_{min}\\right)\\times\\:\\left({x}_{in}-{x}_{min}\\right)}{\\left({x}_{max}-{x}_{min}\\right)}+{y}_{min}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(15\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSince \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{min}=-1\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{max}=+1\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{R}\\)\u003c/span\u003e\u003c/span\u003e is the original data, y is the result of normalization. Eq.\u0026nbsp;(15) becomes,\u003c/p\u003e\n\u003cdiv id=\"Equp\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equp\" class=\"mathdisplay\"\u003e$$\\:y=\\frac{2\\left({x}_{in}-{x}_{min}\\right)}{\\left({x}_{max}-{x}_{min}\\right)}-1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(16\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe normalized dataset was processed using a single-layer feed forward back propagation neural network trained with the Levenberg\u0026ndash;Marquardt algorithm. The data were randomly partitioned into 70% for training, 15% for validation, and 15% for testing. All simulations were performed using MATLAB 2024a (MathWorks Inc.). The ANN architecture is illustrated in Figs.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3 Machine Learning Modeling\u003c/h2\u003e\n\u003cp\u003eA tansig activation function was employed in the hidden layer and a purelin function in the output layer, as described in Eq.\u0026nbsp;(17).\u003c/p\u003e\n\u003cdiv id=\"Equq\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equq\" class=\"mathdisplay\"\u003e$$\\:y=\\:\\sum\\:_{J=1}^{m}\\left\\{Purelin\\left[{LW}_{j,1}\\left(\\sum\\:_{i=1}^{4}tansig\\left({X}_{i}I{W}_{i,j}+{b}_{j}\\right)\\right)\\right]+{b}_{o}\\right\\}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(17\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe network was trained using 1,000 epochs and a zero-error goal as stopping criteria. Following denormalization, the model\u0026rsquo;s performance was assessed by comparing actual and predicted values using correlation coefficients (R-values) for S11, VSWR, and bandwidth. As shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, the model demonstrated excellent predictive accuracy for S11 and VSWR, with slightly lower yet satisfactory performance for bandwidth.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eR-values for trained data\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAntenna characteristics\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTraining\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eValidation\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTest\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAll\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eS11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.9967\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.9667\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eVSWR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.9443\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBandwidth\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.0000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.8995\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n\u003ch2\u003e3.4 Polynomial Regression Model\u003c/h2\u003e\n\u003cp\u003eA polynomial regression model for the two independent variables \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({x}_{1},\\:{x}_{2})\\)\u003c/span\u003e\u003c/span\u003e, and one dependent variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(y\\right)\\)\u003c/span\u003e\u003c/span\u003e is expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equr\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equr\" class=\"mathdisplay\"\u003e$$\\:y={\\beta\\:}_{0}+{\\beta\\:}_{1}{x}_{1}+{\\beta\\:}_{2}{x}_{2}+{\\beta\\:}_{3}{x}_{1}^{2}+{\\beta\\:}_{4}{x}_{2}^{2}+{\\beta\\:}_{5}{x}_{1}{x}_{2}+ϵ\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(18\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{0}\\)\u003c/span\u003e\u003c/span\u003e is the intercept, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{1}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{2}\\)\u003c/span\u003e\u003c/span\u003e are the linear coefficients, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{3}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{4}\\)\u003c/span\u003e\u003c/span\u003e are the quadratic coefficients, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{5}\\)\u003c/span\u003e\u003c/span\u003e is the interaction term coefficient, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:ϵ\\)\u003c/span\u003e\u003c/span\u003e is the error term. In this study, the second order polynomial with two independent variables \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{1}={f}_{r}\\:and\\:{x}_{2}={h}_{s}\\)\u003c/span\u003e\u003c/span\u003e is given as,\u003c/p\u003e\n\u003cdiv id=\"Equs\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equs\" class=\"mathdisplay\"\u003e$$\\:y={\\beta\\:}_{0}+{\\beta\\:}_{1}{f}_{r}+{\\beta\\:}_{2}{h}_{s}+{\\beta\\:}_{3}{f}_{r}^{2}+{\\beta\\:}_{4}{h}_{s}^{2}+{\\beta\\:}_{5}{f}_{r}{h}_{s}+ϵ\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(19\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe attained regression equation the return loss (S11) is:\u003c/p\u003e\n\u003cdiv id=\"Equt\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equt\" class=\"mathdisplay\"\u003e$$\\:y=4032+(-1678.5*{f}_{r})+(-1222.4*{h}_{s})+235*{f}_{r}^{2}+\\left(-11.6*{f}_{r}*{h}_{s}\\right)+\\left(872*{h}_{s}^{2}\\right)\\:\\:\\:\\:\\left(20\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe regression equation obtained for VSWR is:\u003c/p\u003e\n\u003cdiv id=\"Equu\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equu\" class=\"mathdisplay\"\u003e$$\\:y=25.9+(-12.2*{f}_{r})+(-3.28*{h}_{s})+\\left(1.72*{f}_{r}^{2}\\right)+\\left(-0.18*{f}_{r}*{h}_{s}\\right)+\\left(1.15*{h}_{s}^{2}\\right)\\:\\:\\:\\:\\left(21\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe regression equation obtained for Bandwidth is:\u003c/p\u003e\n\u003cdiv id=\"Equv\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equv\" class=\"mathdisplay\"\u003e$$\\:y=-4808+\\left(1865*{f}_{r}\\right)+\\left(2065*{h}_{s}\\right)+(-307.3*{f}_{r}^{2})+\\left(197.2*{f}_{r}*{h}_{s}\\right)+(-832.4*{h}_{s}^{2}\\:\\:(22)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"4. Results and Discussion","content":"\u003cp\u003eError! Reference source not found.\u003cstrong\u003e3, 4\u003c/strong\u003e, and 5 present 3D plots illustrating the influence of resonant frequency (fr) and substrate height (hs) on S11, VSWR, and bandwidth. The results reveal that S11 exhibits a non-linear relationship with both fr and hs, with darker regions on the color map indicating improved impedance matching (lower S11 values). Figure\u0026nbsp;3 highlights that minor variations in hs significantly influence return loss. As shown in Fig.\u0026nbsp;4, VSWR also follows a non-linear trend with increasing fr, with measured values aligning well with the predicted surface. Regions with higher VSWR values indicate poorer impedance matching. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e illustrates the variation of bandwidth, which shows a mixed trend\u0026mdash;expanding in certain regions and narrowing in others. The influence of \u003cem\u003efr\u003c/em\u003e on bandwidth is more pronounced than that of hs, suggesting frequency sensitivity in determining antenna performance.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e compares the original and predicted values of the antenna performance metrics across thirteen samples. The results demonstrate a strong correlation between the actual and predicted data, confirming the reliability of the model and the effectiveness of the derived regression equations in estimating the target parameters.\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e presents the numerical relationship between substrate height (hs) and bandwidth based on test samples. While not strictly linear, the trend indicates that bandwidth generally increases with hs. It was noted that, at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:h=1.61\\)\u003c/span\u003e\u003c/span\u003e mm, bandwidth starts at 145.13 MHz, then increases at different heights, reaching a maximum of 180.93 MHz a\u003cstrong\u003et\u003c/strong\u003e h\u0026thinsp;=\u0026thinsp;1.65 mm.. This behavior suggests that moderate increases in substrate height may enhance bandwidth due to improved impedance matching and reduced surface wave losses.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab4\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eComparison between original and predicted values of the performance metrics.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{f}}_{\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e(GHz)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{g}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e(mm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{\\delta\\:}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{h}}_{\\varvec{s}}\\)\u003c/span\u003e\u003c/span\u003e (mm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{W}}_{\\varvec{g}}\\)\u003c/span\u003e\u003c/span\u003e (mm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{W}}_{\\varvec{P}}\\)\u003c/span\u003e\u003c/span\u003e (mm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{L}}_{\\varvec{p}}\\)\u003c/span\u003e\u003c/span\u003e (mm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{\\epsilon\\:}}_{\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eS11 (dB)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eVSWR\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBandwidth (MHz)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.351\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-23.09\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.155\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e143\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.505\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.82\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.19\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.79\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.062\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e153\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.852\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.21\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.42\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.104\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e159\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.802\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.99\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.05\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.084\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e153\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.755\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.63\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.04\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.96\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.075\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e162\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.715\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0/009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.97\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-32.73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.052\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e168\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.670\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.62\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.13\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-37.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.031\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e166\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.630\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.98\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-39.39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.022\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e169\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.505\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.69\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-43.85\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.015\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e177\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.545\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.00\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-34.28\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.046\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e173\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.465\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.97\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.22\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.089\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e178\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.425\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.103\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e181\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.388\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e29.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.009\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e26.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20.21\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-24.55\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.128\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e178\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eComparison of original and predicted antenna parameters\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eF (GHz)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{h}}_{\\text{s}}\\:\\left(\\text{m}\\text{m}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eS11 original\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eS11 predicted\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eVSWR original\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eVSWR predicted\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBandwidth original\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBandwidth predicted\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.351\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-23.098\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-22.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.155\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.085\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e143\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e145.13\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.505\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.82\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.798\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.798\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.062\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.094\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e153\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e153.79\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.852\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.423\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.423\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.104\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.104\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e159\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e161.33\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.802\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.059\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.059\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.084\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.084\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e153\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e159.39\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.755\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.63\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.965\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-30.965\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.075\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.075\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e162\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e167.06\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.715\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-32.731\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-34.560\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.052\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.052\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e168\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e166.79\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.670\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.62\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-37.527\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-37.527\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.031\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.031\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e166\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e168.15\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.630\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-39.399\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-37.772\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.022\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.022\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e169\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e170.30\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.505\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.69\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-43.850\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-46.980\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.015\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.015\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e177\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e175.76\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.545\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-34.285\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-34.285\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.046\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.046\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e173\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e176.07\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.465\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.223\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-28.223\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.089\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.038\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e178\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e176.29\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.425\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.316\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.316\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.103\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.103\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e181\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e180.93\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3.388\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-24.551\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-24.551\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.128\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.134\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e178\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e179.65\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n\u003ch2\u003e4.1 Return loss (S11)\u003c/h2\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the return loss (S11) values across various frequencies for the trained dataset, considering different substrate heights. The frequency range spans from 3.351 GHz to 3.852 GHz, with corresponding return loss values between \u0026minus;\u0026thinsp;22.02 dB and \u0026minus;\u0026thinsp;46.98 dB. The minimum return loss of \u0026minus;\u0026thinsp;46.98 dB is observed at 3.585 GHz, indicating optimal impedance matching at this frequency. Frequencies between 3.63 GHz and 3.67 GHz also exhibit high return loss values (below \u0026minus;\u0026thinsp;37 dB), suggesting a broad region of efficient antenna performance. Conversely, higher return loss values\u0026mdash;such as \u0026minus;\u0026thinsp;22.02 dB at 3.351 GHz and \u0026minus;\u0026thinsp;24.55 dB at 3.388 GHz\u0026mdash;indicate relatively poorer matching at those frequencies. Overall, the results confirm that the trained model accurately predicts the frequency range with optimal impedance characteristics, with peak performance centered around 3.585 GHz.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e presents the variation of the machine learning-predicted VSWR across the frequency range. The VSWR values remain consistently close to 1, indicating effective impedance matching throughout. The minimum VSWR of 1.015 occurs at 3.585 GHz, signifying optimal matching, while the maximum value of 1.134 at 3.388 GHz still falls within acceptable limits for efficient antenna operation. These results confirm that the trained model effectively maintains low reflection losses and stable VSWR across the target frequency band.\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n\u003ch2\u003e4.2 Gain and Directivity\u003c/h2\u003e\n\u003cp\u003eFigures 8 and 9 present the 3D gain and polar directivity patterns of the proposed antenna, respectively. The antenna demonstrates a gain of 8.91 dBi and a directivity of 10.4 dBi, indicating high efficiency and strong directional performance. With an efficiency of 85.7%, only 14.3% of the input power is lost due to factors like dielectric and conductor losses. The relatively small differences between gain and directivity suggests minimal energy loss, reflecting successful machine learning-based optimization. These characteristics make the antenna suitable for directional applications such as 5G base stations and point-to-point links.\u003c/p\u003e\n\u003cp\u003eThe performance summary of the machine learning-based antenna model is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. The trained microstrip patch antenna, utilizing a Rogers substrate, exhibits outstanding electromagnetic performance. This is evidenced by a deep reflection coefficient (S11 = \u0026minus;\u0026thinsp;46.98 dB) and a near-ideal voltage standing wave ratio (VSWR) of 1.015, indicating excellent impedance matching and minimal signal reflection. The antenna achieves a bandwidth of 176 MHz, which is considerably wide for microstrip designs, thereby supporting stable frequency operation. Furthermore, the antenna demonstrates a gain of 8.9 dBi and a directivity of 10.4 dBi, confirming its strong radiative capability and directional beam characteristics. An overall radiation efficiency of 85.7% signifies low power losses, aligning with the typical 80\u0026ndash;90% efficiency range reported for microstrip antennas [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e17\u003c/span\u003e]. These findings suggest that the proposed design is well-suited for high-performance wireless communication systems, including 5G base stations and radar applications. The results confirm that machine learning optimization has effectively enhanced key performance metrics, with potential for future improvements targeting higher gain and extended bandwidth to meet specific system requirements.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab6\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSummary of Simulation results\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameters\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003ePatch antenna with Rogers substrate\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eS11\u003c/p\u003e\n\u003cp\u003eVSWR\u003c/p\u003e\n\u003cp\u003eBandwidth\u003c/p\u003e\n\u003cp\u003eGain\u003c/p\u003e\n\u003cp\u003eDirectivity\u003c/p\u003e\n\u003cp\u003eAntenna efficiency\u003c/p\u003e\n\u003cp\u003eEfficiency, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{\\eta\\:}\\)\u003c/span\u003e\u003c/span\u003e \u003cstrong\u003e(Linear scale)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReflection coefficient (\u003c/strong\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{\\Gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReflected power\u003c/strong\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({\\mathbf{P}}_{\\mathbf{r}}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTransmitted power\u003c/strong\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({\\mathbf{P}}_{\\mathbf{T}}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-46.98 dBi\u003c/p\u003e\n\u003cp\u003e1.015\u003c/p\u003e\n\u003cp\u003e176 MHz\u003c/p\u003e\n\u003cp\u003e8.91 dBi\u003c/p\u003e\n\u003cp\u003e10.4 dBi\u003c/p\u003e\n\u003cp\u003e85.7%\u003c/p\u003e\n\u003cp\u003e71.2%\u003c/p\u003e\n\u003cp\u003e0.0142\u003c/p\u003e\n\u003cp\u003e2.02%\u003c/p\u003e\n\u003cp\u003e99.98%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e presents the performance evaluation of the proposed microstrip patch antenna design in terms of return loss (S11), VSWR, and gain. The results demonstrate a significant improvement over previously reported designs. This conclusion is supported by detailed simulation data and comparative analysis, which collectively validate the effectiveness of the proposed approach. The findings confirm that the design methodology offers a practical and reliable solution for microstrip patch antenna development. Overall, the presented data serve as a valuable reference for the design and optimization of antennas in advanced wireless communication systems.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab7\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eComparison with similar works for 5G applications antenna.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eS/N\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eRef.\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eReturn loss (dBi)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eVSWR\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eGain (dB)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-23.673\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6.80\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-17.430\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.310\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3.02\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-17.436\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.310\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6.60\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-26.347\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.100\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3.68\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-41.300\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.010\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.45\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-13.890\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6.60\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-18.270\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.130\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.46\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-13.480\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.538\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6.63\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eThis work\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e\u003cstrong\u003e-46.984\u003c/strong\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e\u003cstrong\u003e1.015\u003c/strong\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e\u003cstrong\u003e8.91\u003c/strong\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eIn this study, the microstrip patch antenna dimensions and parameters (return loss, gain, directivity, bandwidth, and VSWR) are determined with the help of CST studio to produce an effective machine leaning based model. The antenna design is modeled using the Levenberg Marquardt optimization method and feed-forward back propagation neural network at 3.5 GHz. The results obtained through the use of CST and machine learning based model are in conformity with one another in a manner that is acceptable. Moreover, the study incorporates training and test datasets alongside CST-generated data. The proposed antenna operates within a frequency range of 3.351 GHz to 3.852 GHz. The trained data at 3.5 GHz yielded a good results. Furthermore, a mathematical model based on the Multivariate Polynomial algorith\u003cem\u003em\u003c/em\u003e to predict three antenna parameters: Return Loss, VSWR, and Bandwidth was developed. \u003cem\u003eTh\u003c/em\u003ee trained network reliably predicts antenna performance for new design cases, provided the input parameters\u0026mdash;such as substrate height, resonant frequency, and dielectric constant\u0026mdash;remain within the trained range. The proposed method improves design accuracy and significantly reduces development time, making it a practical tool for future wireless applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u0026nbsp;\u003c/h2\u003e\n\u003cp\u003eThe authors received no funding for this research.\u003c/p\u003e\n\u003ch2\u003eData availability\u0026nbsp;\u003c/h2\u003e\n\u003cp\u003e\u0026ldquo;Data is provided within the manuscript files\u0026rdquo;.\u0026nbsp;\u003c/p\u003e\n\u003ch2\u003eEthics approval\u0026nbsp;\u003c/h2\u003e\n\u003cp\u003eEthics approval was not required for this study. Therefore, no ethics approval declaration is available\u0026nbsp;\u003c/p\u003e\n\u003ch2\u003eConsent to publish\u003c/h2\u003e\n\u003cp\u003eConsent to publish was not required for this study. Therefore, no consent to publish declaration is available.\u003c/p\u003e\n\u003ch2\u003eConsent to participate\u003c/h2\u003e\n\u003cp\u003eConsent to participate was not required for this study. Therefore, no consent to participate declaration is available.\u003c/p\u003e\n\u003ch2\u003eCompeting interests\u003c/h2\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eMishra RK, Patnaik A. Neural Network-Based CAD Model for the Design of Square-Patch Antennas. IEEE Trans Antennas Propag. 2018;46(12):1890\u0026ndash;1. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1109/8.743842\u003c/span\u003e\u003cspan address=\"10.1109/8.743842\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNarayan JL, Krishna R, Reddy LP. (2019) Design of Microstrip Antenna Using Artificial Neural Networks, International Conference on Computational Intelligence and Multimedia Applications, Vol. 1, pp. 332\u0026ndash;334.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKumar A, Kumar V, Singh. Radial Basis Function Neural Network for estimation of Bandwidth of Antenna. Int J Control Theory Application. 2017;10(9):901\u0026ndash;6.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh P, Kumar Singh V. Application of Multi-Layer Feed Forward Back Propagation Neural Network for Analysis \u0026amp; Modelling of Antenna. Int J Control Theory Application. 2017;10(9):895\u0026ndash;900.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePatnaik A, Mishra RK, Patra GK, and., Dash SK. 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Instrum. 15 (2020).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eAbdulbari AA, Jawad MM, Hanoosh HO, Saare MA, Lashari SA, Sari SA, Ahmad Y, Khalill YM. (2021) Design compact microstrap patch antenna with T-shaped 5G application, Bull. Electr. Eng. Informatics 10 (2021) 2072\u0026ndash;2078.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHossain Mollah MS, Faruk O, Hossain MS, Islam MT, Shafi ASM, Molla I. m. M. (2021) Design and Performance Improvement of Microstrip Patch Antenna Using Graphene Material for Communication Applications, 2021 IEEE 11th IEEE Symposium on Computer Applications \u0026amp; Industrial Electronics (ISCAIE), 2021, pp. 343\u0026ndash;346.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRahman MA, Shaikat A, Iqbal IS, Hassan A. (2013) Microstrip patch antenna design and performance analysis for RFID applications at ISM band (2.45 GHz), 2nd International Conference on Advances in Electrical Engineering (ICAEE), 2013, pp. 305\u0026ndash;308.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003e} Paragya D, Hartono S. (2020). 3.5 GHz Rectangular Patch Microstrip Antenna With Defected Ground Structure for 5G. ELKOMIKA: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, \u0026amp;Amp Teknik Elektronika, vol. 8, no. 1, Institut Teknologi Nasional, Bandung, Jan. 2020, p. 31.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eTouko Tcheutou Stephane Borel, Priyadarshini R. U-Slotted Wideband Microstrip Patch Antenna for Ka Band and mmW 5G Applications, 14 July 2022, PREPRINT (Version 1).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRahman MA, Shaikat A, Iqbal IS, Hassan A. Microstrip patch antenna design and performance analysis for RFID applications at ISM band (2.45 GHz), 2013 2nd International Conference on Advances in Electrical Engineering (ICAEE), 2013, pp. 305\u0026ndash;308.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eParagya D, Hartono S. (2020). 3.5 GHz Rectangular Patch Microstrip Antenna With Defected Ground Structure for 5G. ELKOMIKA: Jurnal Teknik Energi Elektrik, Teknik Telekomunikasi, \u0026amp;Amp Teknik Elektronika, vol. 8, no. 1, Institut Teknologi Nasional, Bandung, Jan. 2020, p. 31.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRafdzi MF, Mohamad SY, Ruslan AA, Malek NFA, Islam MR, Hashim AHA. Study for Microstrip Patch Antenna for 5G Networks, 2020 IEEE Student Conference on Research and Development (SCOReD), 2020, pp. 524\u0026ndash;528.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHasan MM, Rahman Z, Shaikh R, Alam I, Islam MA, Alam MS. Design and Analysis of Elliptical Microstrip Patch Antenna at 3.5 GHz for 5G Applications, 2020 IEEE Region 10 Symposium (TENSYMP), 2020, pp. 981\u0026ndash;984.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Bandwidth, Machine learning, Optimization, Return loss, VSWR","lastPublishedDoi":"10.21203/rs.3.rs-7720277/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7720277/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents the design and simulation of a low-profile patch antenna for 5G applications at 3.5 GHz. The substrate material selected is Rogers RT/Duroid 5880 epoxy, which has a permittivity of 2.2. The proposed antenna layout is simulated using Computer simulation Technology (CST) Microwave Studio Suite. The simulation results indicate significant variations in S11, gain, directivity, efficiency, and bandwidth due to differences in the substrate\u0026rsquo;s relative permittivity and thickness. The designed antenna achieved a return loss of -43.85 dB, VSWR of 1.015, gain of 8.90 dBi, directivity of 10.5 dBi, bandwidth of 177 MHz, and efficiency of 85.7%. With a gain exceeding 5 dBi, the antenna is well-suited for communication applications. The paper also develops a mathematical model employing the Multivariate Polynomial algorithm to predict three antenna parameters: Return Loss, VSWR, and Bandwidth. A machine learning-based model has been developed using the Levenberg-Marquardt algorithm with a feed-forward back-propagation learning approach and applied to patch antenna design. It processes input data, including the dielectric constant (εr), substrate thickness (hs), and dominant-mode resonant frequency (fr), to predict the S11, VSWR, and Bandwidth. The model\u0026rsquo;s performance has been evaluated by comparing its results with simulated values obtained from CST Studio Suite. The machine learning predictions closely align with the simulation results. Additionally, the neural network-based estimation offers the advantage of rapid and simultaneous output computation.\u003c/p\u003e","manuscriptTitle":"Machine Learning Based Model For Design and Optimization of Rectangular Patch Antenna at 3.5 GHz for 5G Applications","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-22 06:39:06","doi":"10.21203/rs.3.rs-7720277/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"275e96ea-6c02-4424-a32b-b075e5762073","owner":[],"postedDate":"October 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-12-09T07:53:57+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-22 06:39:06","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7720277","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7720277","identity":"rs-7720277","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}