## 3 Results and Discussion
### Thermal Simulation Analysis of Packaging Defects
Additionally, the temperature in the bridged area is higher than that observed in the original air gap, thereby confirming that bridging may induce abnormal hot spots. In order to evaluate whether pad loss affects the temperature distribution, the temperature field of the package substrate is further analyzed. By integrating the heat dissipation path analysis with the substrate simulation results, several conditions are compared. Under normal conditions (Figure 7a), the temperature field exhibits a symmetrical diffusion pattern. When solder balls are missing (Figure 7b), a distinct hot spot with a temperature of 197 \({}^{\circ}\)C appears at the defect location in the upper-right corner, and a clear circular boundary corresponding to the pad is observed. Under the condition where both solder balls and solder pads are missing (Figure 7c), a similar hot spot emerges at the upper-right defect, with a slightly higher temperature of 197.22 \({}^{\circ}\)C. However, in this case, no clear circular boundary is observed, suggesting that the absence of the pad reduces thermal coupling in that region. In the solder ball bridging condition (Figure 7d), the temperature at the lower-left bridging site is notably lower than that of the surrounding solder balls, with a recorded temperature of 185.36 \({}^{\circ}\)C.  
Figure 6: Simulated temperature distributions on the PCB surface under various defect conditions: (**a**) normal package with symmetrical heat diffusion; (**b**) missing solder ball creating localized cold spots; (**c**) missing solder ball and pad showing similar thermal distortion; (**d**) bridging inducing lateral heat spreading without cold spots. This phenomenon corresponds to an abnormal cold spot, indicating that bridging may lead to localized thermal anomalies distinct from those caused by missing solder elements. Further transient analysis reveals that different types of defects have markedly different effects on the heating rate and temperature saturation time. As illustrated in Figure 8, a comparison between the normal package and the solder ball missing condition shows that the heating rate at the center point of the solder ball during the first 2 s reaches 41.9 \({}^{\circ}\)C/s under the missing condition, which is 47% higher than the 28.5 \({}^{\circ}\)C/s observed under normal conditions. Moreover, temperature saturation is achieved in just 1.4 s, significantly faster than the 37.5 s required under normal conditions, indicating a pronounced local heat load concentration effect. In contrast, under the solder ball bridging condition, the heating rate at the center of the bridged region during the first 2 s is 32.3 \({}^{\circ}\)C/s, representing a 13% increase compared to the normal case. The corresponding temperature saturation time is 17.0 s--faster than the normal condition but still slower than that observed in the solder ball missing condition. This suggests that although the lateral heat conduction path introduced by bridging accelerates the initial heat diffusion, the added thermal mass of the solder reduces the efficiency of longitudinal heat transfer. Notably, the heating rate and temperature saturation time in the solder ball missing condition and the combined solder ball and pad missing condition are almost completely overlapping. This is because both defects disrupt the vertical heat conduction path, forcing heat to diffuse laterally through the substrate, which has relatively low thermal conductivity. As a result, the heating rate is significantly reduced, and the thermal responses tend to converge.