## 3 Mathematical Model Development
### Motion Profile Analysis of the Characterization  
Developing mathematical models that combine mass and momentum conservation principles for energy conservation in movements through varying densities requires a comprehensive analysis of the trapezoidal movement pattern [33]. Such analysis is pivotal in understanding SOIC package trajectories, considering essential factors like speed, acceleration, and test time. It also requires a deterministic evaluation of dielectric fluids' physical and dynamic properties. This evaluation leads to an examination of hydrodynamic forces and physical dependencies. These factors determine how a SOIC package moves and positions itself in a dielectric fluid. The nozzle holds the package at its original position along the Z-axis, where the analysis begins. Figure 4 illustrates the complete time-speed trapezoidal motion pattern for the entire process divided in twenty stages represented by the Roman numbers (I-XX). The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**SOIC** & **Length** & **Width** & **Height** & **Equivalent** & **Area** & **Volume** & **Mass** & **Weight** \\
**Package** & **mm** & **mm** & **mm** & **mm** & **mm\({}^{2}\)** & **mm\({}^{3}\)** & **Kg** & **N** \\ \hline
**SOP-16** & 10.49 & 10.64 & 2.65 & 10.57 & 111.68 & 296.61 & \(1.36\times 10^{-4}\) & \(1.33\times 10^{-3}\) \\
**SOP-20** & 12.60 & 10.64 & 2.65 & 11.54 & 134.13 & 355.60 & \(4.99\times 10^{-4}\) & \(4.89\times 10^{-3}\) \\
**SOP-24** & 15.60 & 10.64 & 2.65 & 12.65 & 166.00 & 440.81 & \(6.35\times 10^{-4}\) & \(6.23\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SOIC packages’ physical and physic characteristics. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dielectric** & **Density** & **Viscosity at 25 \({}^{\circ}\)C** & **Surface Tension** \\
**Medium** & **Kg/m\({}^{3}\)** & **cSt** & **mN/m** \\ \hline
**Air** & 1.17 & 16.92 & N/A \\
**FR3** & 922.99 & 40.00 & 23.99 \\
**DTE-150** & 856.00 & 150.00 & 22.26 \\
**DPMS** & 967.99 & 350.00 & 21.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constants of the physical characteristics of dielectric fluids. detailed analysis examines the device's insertion and extraction displacements along the Z-axis, which the nozzle holds. The movement sequence involves first moving downwards through the air, then submerging into the dielectric fluid until reaching the upper surface of the zero-insertion force (ZIF) socket. The insertion process slows down and continues moving until the ZIF socket activates and stops it. Subsequently, the process allocates a specific time to release the SOIC package and then moves the Z-axis back to the ZIF socket's surface at a reduced speed. Once in position, the system increases the speed to the maximum allowed and then halts again upon reaching the home position. For extracting the SOIC package from the ZIF socket, the system replicates the same operational sequence in reverse order. A programmed speed ratio, tailored to the distance between each step, governs each travel time, featuring instant transitions with constant acceleration or deceleration. While inserting and removing the package are the most critical movements for the IC, technicians must remove the nozzle from the dielectric fluid to conduct high-voltage tests. Figure 5 showcases the trapezoidal motion pattern for either the insertion or extraction of the SOIC package selected from the twenty process stages represented by the Roman numbers. Figure 4: Trapezoid movement pattern: time-speed complete. Figure 5: Trapezoid movement pattern: time–speed. The insertion or extraction procedure requires a total time (\(t_{T}\)), which includes the sum of the downward travel time (\(\Delta t_{DEC}\)), the upward travel time (\(\Delta t_{ASC}\)), and a waiting period (\(t_{OUT}\)) to activate or deactivate the pneumatic system that releases or retains the SOIC package. Each downward or upward travel phase is segmented into periods with constant transitions, resulting in five distinct travel periods in each direction.