## 5 Results and Discussion
Based on these observations, the maximum communication distance for the equilateral triangular patch antenna was determined to be 16.5 cm, as its S\({}_{11}\) peak remains detectable at this range but becomes indistinguishable from background noise at 17 cm. In comparison, the rectangular and circular antennas exhibited maximum communication distances of 15 cm and 14.5 cm, respectively. Another observed phenomenon was the resonant frequency shift caused by the increasing distance between the antennas. The rectangular and circular patch antennas exhibited noticeable frequency shifts, highlighting their sensitivity to positional changes. In contrast, the equilateral triangular patch antenna maintained a relatively stable resonant frequency at approximately 2.42 GHz. This stability is attributed to its unique geometric and electromagnetic properties, which improve impedance matching and reduce susceptibility to distance-related variations. According to the results shown in Figure 9, the S\({}_{11}\) magnitude for all antenna shapes fluctuated within the initial temperature range of 25 to 150 \({}^{\circ}\)C and decreased steadily as the temperature rises from 200 to 500 \({}^{\circ}\)C. Among the designs, the circular patch antenna exhibited the lowest S\({}_{11}\) peak, as shown in Figure 9b. Additionally, its backscattering signal  
Figure 8: The observed S\({}_{11}\) of backscattering signals as the communication distance increases: (**a**) rectangular; (**b**) circular; (**c**) equilateral triangular. became unstable at elevated temperatures, deteriorating significantly as the temperature reached 500 \({}^{\circ}\)C. In contrast, the equilateral triangular patch antenna maintained the most stable S\({}_{11}\) magnitude across all temperatures, with its resonant frequency shifting from 2.4009 GHz (25 \({}^{\circ}\)C) to 2.3396 GHz (500 \({}^{\circ}\)C), as shown in Figure 9c. The application of third-order polynomial fitting to the frequency shift data as a function of increasing temperature demonstrates a high degree of accuracy, with an R\({}^{2}\) value ranging from 0.98 to 0.99, indicating a strong correlation between temperature variation and frequency response. Figure 9: The measured S\({}_{11}\) of backscattering signals and the fitted frequency shifts results: (**a**) rectangular; (**b**) circular; (**c**) equilateral triangular. According to the key performance results summarized in Table 2, the equilateral triangular patch antenna demonstrated superior performance, achieving a maximum communication distance of 16.5 cm and a sensitivity of 0.129 MHz/\({}^{\circ}\)C. Among the tested designs, it exhibited the highest simulated gain, resulting in the greatest communication range. Simulation results in Figure 5 showed that the equilateral triangular shape concentrated the electric field more strongly along its edges compared to the other designs. In contrast, the circular patch antenna, lacking sharp edges, displayed a more uniform but weaker electric field distribution. This behavior aligns with the established principle that electric fields tend to concentrate at sharp edges or regions of high curvature, where increased surface charge density generates a stronger electric field [24]. A stronger electric field enhances the sensor's ability to efficiently receive coupling energy from the interrogator antenna at its resonant frequency, leading to a more stable antenna backscattering signal that is reflected back to the interrogator antenna. While sensitivity variation among the designs was minimal due to the dominant influence of dielectric substrate properties, the choice of substrate material plays a critical role [12]. However, signal stability under temperature variations was significantly impacted by the antenna shape, with the equilateral triangular design proving to be the most reliable.