Antenna design
The proposed antenna dimension layout (\(w_s = 70, l_s = 60, w_p = 21, w = 5, a = 2, b = 21; \) Unit: mm).
The dimension layout of the proposed antenna is shown in Figure 1. The design features a hexagonal patch with rectangular slots cut from the center that extends to one end of the patch. Antennas are supplied at 50\(\omega\) Microstrip feedline. The flexible antenna is designed on a 1.8 mm thick PDMS board. The bottom layer of the PDMS board is completely covered with a ground plane, ensuring effective operation of the microstrip feed antenna. The substrate material, PDMS, was modeled in simulation using material properties obtained fromtwenty two. Specifically, the relative dielectric constant (\(\varepsilon _r \)) PDMS is considered 2.7 and the loss is tangential (\(tan~ \delta \)) was considered to be 0.022. These values were implemented in CST microwave studio to accurately represent the dielectric behavior of the PDMS substrate under antenna operating conditions. The dimensions of the hexagonal antenna were determined using the formula outlined in27. Slots introduced at the center of the patch promote excitation of higher-order modes28. When operating in higher order mode, the antenna helps maintain stable performance during mechanical deformation29. The surface current distribution of the proposed antenna at an operating frequency of 3.6 GHz is shown in Figure 2.
Surface current distribution of the proposed antenna at 3.6 GHz.
Electrical equivalent circuit model (a) Accounting for patch slot cutsb)( simplified version of )(a).
Equivalent circuit model of the proposed antenna
Figure 3(a) shows an equivalent circuit model for the proposed flexible antenna. component \(r_1, l_1, \) and \(c_1 \) It represents the resistance, inductance, and capacitance of the patch through which the primary current flows. Additionally, another current flows along the rectangular slot cut on the patch, as shown in Figure 2. \(l_2 \) and \(c_2 \). All these values are calculated based on the equation provided in30. Figure 3(b) shows a simplified circuit model for the proposed antenna. Table 1 shows the lump parameters calculated along with optimized values obtained through tuning in the Path Wave Advanced Design System 2024 (ADS) software (https:Keysight.com/zen/products/software/pathwave-design-software/puthwave-advanced-design-system.html). handle \(s_ {11} \) The curves are shown in Figure 4 compared to the simulated results in EM. The inconsistency between the EM simulated and initial circuit responses is attributed to centralized parameters calculated using the analytical formulas of rectangular patch geometry, but the proposed design features hexagonal patches.
Comparison of em simulations and circuit simulated responses of the proposed model.
Effect of mechanical bending on patch antennas.
To investigate the effect of bending, the performance of hexagonal patch antennas was analyzed in its underlying (planar) morphology and under cylindrical deformation with radii of curvature of r = 20 mm, 30 mm, and 40 mm, as shown in Figure 5. \(s_ {11} \) The curve shown in Figure 6(a) shows a consistent upshift in the resonant frequency as the bending radius decreases, but impedance matching is largely unaffected. Depth of \(s_ {11} \) The dip shown in Figure 6(a) confirms that the antenna continues to maintain a good match with 50. \(\omega\) Supply line despite curvature. For outward bending of the proposed antenna, the bending is applied and the current maximum value (as seen in Figure 2) is separated on either side of the bend. \(s_ {11} \) The plot shown in Figure 6(b) shows a minimal deviation from the existing state, further confirming the negligible effect of bending at the current null, thereby highlighting that higher-order mode excitation in the proposed flexible antenna design ensures stable performance under mechanical bending. As a result, the analysis emphasizes inward bending, making the impact on antenna performance more evident.
Different configurations ofa) Inside, and (b) Outward bending of the proposed antennas performed using Computer Simulation Technology (CST) 2021 software. (https://www.3ds.com/products/simulia/cst-studio-suite)
Comparison of simulations \(s_ {11} \) The proposed antenna for different configurations (a) inner bend, and (b) Outer bend.
The radiation patterns for both the planar and bent configurations of inward bending are shown in Figure 7. These results minimize distortion in the pattern shape, but a significant decrease in gain is observed as the bending radius decreases. A comprehensive comparison of performance parameters across all configurations is shown in Table 2.
Parametric analysis
A parametric analysis was performed to investigate the effect of slot dimensions on antenna performance. As shown in Figure 8(a), increasing the slot width increased the resonance frequency from 3.51 GHz to 3.61 GHz, resulting in a decrease in the quality factor. This shows that impedance matching is sensitive to variations in slot width, as is evident from Figure 8(a). Furthermore, increasing slot length effectively segments the patch and expands the current path, thereby reducing the resonance frequency from 3.7 GHz to 3.5 GHz, as seen in Figure 8(b). However, within the slot length range of 20 to 26 mm, frequency shifts are found to be minimal, suggesting areas where sensitivity to this parameter is reduced.
Simulated radiation patterns of flexible antennas proposed under different conditions of inward bending: (a) 20 mm bending radius (b) 30 mm bending radius (c) 40 mm bending radius, and (d) Ready-made condition. (Solid line: electronic surface, dashed line: H plane)
Parametric analysis of the proposed antenna for various values (a)Slot width aand (b) Slot length b.
Performance analysis of machine learning models
Due to the increased complexity of antenna geometry and the demand for rapid prototyping of flexible designs, traditional parametric sweep methods are often time consuming and computationally expensive. To address this, ML techniques provide a promising alternative for efficient design space exploration and optimization. In this study, four different ML algorithms were employed to establish a predictive model that could identify the optimal combination of geometric parameters that minimize the reflection coefficients (\(s_ {11} \)) At the desired operating frequency. These models were trained on datasets generated through full-wave electromagnetic simulations, which significantly reduced computational overhead to allow for rapid prediction and parameter tuning. Specifically, the parameters \(w_p \) It changed from 15 mm to 22.5 mm in 1.5 mm steps. Slot width a Range from 0.5 mm to 5 mm in 0.5 mm steps. and slot length b Combining 5-35 mm increments in 2 mm increments, we created a comprehensive dataset for training and evaluation of machine learning models. All simulations were performed on a Windows 10 64-bit system with an Intel Core i7 9700kf processor and 32 GB RAM using MATLAB R2024B.
ML workflow for predicting antenna parameters using tree-based models.
The workflow for a tree-based machine learning model is shown in Figure 9. During model training, the dataset was split into 80% in training and 20% in testing. The training dataset is used to train the underlying patterns, while the test dataset is used to evaluate generalized performance. Hyperparameter tuning was performed using grid search and cross-validation, optimizing parameters such as number of estimates, maximum tree depth, and learning rates that improve prediction performance. The model was trained to predict the minimum value \(s_ {11} \) Values based on geometric input parameters. It allows for rapid estimation of antenna performance under a variety of design conditions. Figure 10 shows a comparison of the actual and predicted values of the proposed antenna using four machine learning models: Random Forest, xgboost, catboost, and lightGBM. Points near the ideal fit line indicate high prediction accuracy, minimizing deviations between the predicted and actual outputs. All models were implemented in Python and scripts were run to manage dependencies and environments via the Anaconda prompt. The selection of these specific models for this study was derived by their proven validity in handling regression problems with structured tabular data with nonlinear relations. All of these models are based on decision tree ensemble known for its robustness, interpretability, and ability to model complex dependencies without the need for feature normalization.
Actual and predicted values for different ML algorithms: (a) Random Forest, (b)xgboost, (c) lightgbm, and (d)catboost.
In general, a low mean square error (MSE) value indicates an improved prediction accuracy and reflects the minimum deviation between the predicted and actual output. Table 3 summarizes the performance of each model on the test dataset. Among the models evaluated, the Random Forest (RF) model demonstrated the best performance, achieving an MSE of 0.08137. \(r^2 \) Score of 0.996455 and mean absolute error (MAE) of 0.061535. These results highlight the powerful predictive capabilities of RF models in capturing the underlying relationship between design parameters and antenna performance.
