## 4 Simulation Analysis and Measurement Verification
### Simulation Analysis and Verification of Mutual Inductance  
In this section, the transmission characteristics of the balise uplink FSK signal are used as an example to validate the mutual inductance of the circuit model in free space and water debris. The simulation is conducted using Comsol 5.6, with a magnetic permeability of \(4\pi\times 10^{-7}\) H/m. The experiment is carried out on a non-magnetic acrylic framework, as shown in Figure 6. The OAU and balise are both designed with dimensions of 390 mm \(\times\) 200 mm in length and width. A network analyzer is employed to measure the transmission parameters, with the frequency set to 4.234 MHz. Figure 5: Dual-port coupled network for underwater debris. To maintain consistency with practical engineering, both the simulation and experimental processes keep the balise stationary, while the relative position changes are achieved by moving the OAU. They are made of copper and have identical dimensions, with a length and width of 2a = 2c = 390 mm and 2b = 2d = 200 mm. When the OAU and the balise are in different installation scenarios, their relative vertical distance h varies accordingly. As the OAU moves along the Z-axis with a step size of 30 mm, and the vertical distance h between the geometric centers of the OAU and the balise increases from 220 mm to 460 mm, the mutual inductance variation curves for free space and underwater debris are shown in Figure 7.  
In Figure 7, the mutual inductance between the OAU and the balise gradually decreases as the vertical distance increases in free space and water debris. Due to the coupling magnetic flux between the two antennas diminishing with increased distance, the mutual inductance is reduced. In water debris, as the electrical conductivity \(\sigma\) increases, the mutual inductance between the OAU and the balise decreases. The discrepancy between the experimentally determined mutual inductance measurements and the values obtained through theoretical calculations and computer simulations is minimal. The calculated results are generally consistent with simulation and experimental findings. Figure 8 shows the mutual inductance variation curves corresponding to the X-axis horizontal offsets. The vertical distances between the OAU and the balise are selected as 220 mm, 340 mm, and 460 mm. When the vertical distance \(h\) is 220 mm, horizontal offsets occur in 25 mm increments, with the offset \(\Delta x\) increasing from 0 mm to 250 mm. When \(h\) is 340 mm, horizontal offsets occur in 30 mm increments, with \(\Delta x\) increasing from 0 mm to 300 mm. When \(h\) is 460 mm, horizontal offsets occur in 35 mm increments, with \(\Delta x\) increasing from 0 mm to 350 mm. Under all three height conditions, the mutual inductance decreases as the horizontal offset \(\Delta x\) increases. This is primarily due to the reduction in the overlapping area between the OAU and the balise as \(\Delta x\) increases, leading to the magnetic flux between the antennas being reduced. As the conductivity \(\sigma\) increases, the mutual inductance between the two antennas decreases. As shown in the figure, the deviation  
Figure 6: Measurement experiment system of mutual inductance; (**a**) Free space; (**b**) Water debris. Figure 7: Mutual inductance variation curve versus vertical displacement; (**a**) Free space (air); (**b**) Water debris (water). between the measured mutual inductance values and those obtained through theoretical calculations and simulations varies within a narrow range, demonstrating excellent overall consistency. Notably, as the central positions of the two reference loops gradually move apart, the relative deviation gradually increases. This is primarily due to the decrease in mutual inductance with distance. However, considering the operational range of the balise, the model's results remain valid for our purposes. Figure 9 shows the mutual inductance variation curves corresponding to the Y-axis horizontal offsets. The vertical distances between the OAU and the balise are selected as 220 mm, 340 mm, and 460 mm.