## 2 Theory of Underwater Electric Field Communication
### Electric Field Communication Model  
As the electric field communication mechanism in weakly electric fish is complex, the fish emitting signals generate an electric field through a large array of cells on their bodies, and the receiving fish sense this electric field. They decode the electric signals using thousands of electric receptors distributed on their skin. Therefore, in order to develop a communication system suitable for underwater robots, the biological communication system of weakly electric fish is simplified into a model with a pair of transmitting electrodes and a pair of receiving electrodes, as shown in Figure 0(d). The electric field generated by weakly electric fish is essentially a low-frequency alternating electromagnetic wave. During the process of underwater electric field communication, the generation of this electric field can be simulated by emitting continuously varying currents through a pair of electric dipoles. Underwater channels are also good conductors, and the changing potential on the electric dipoles results in varying currents in the water, consequently generating alter  
Figure 1: The establishment of a biomimetic electrode model for weakly electric fish discharge and perception. nating electromagnetic fields. Therefore, the influence of both conduction current and displacement current needs to be considered. Specifically, conduction current is formed due to the movement of charges between atoms and is time-invariant. It follows [PERSON]'s law, and its expression is given by:  
\[J_{c}=\sigma E \tag{1}\]  
where \(\sigma\)--the conductivity S/m; \(E\)--the electric field strength V/m; \(J_{c}\)--the conduction current density A/m\({}^{2}\). The displacement current is a time-varying current generated by the motion of bound charges within atoms induced by an external field, and its density is expressed as:  
\[J_{d}=\varepsilon\frac{\partial E}{\partial t} \tag{2}\]  
where \(\varepsilon\)--the permittivity. Usually, at 20 \({}^{\circ}\)C, the permittivity of water is 80 times that of a vacuum (\(\varepsilon_{0}=8.85\times 10^{-12}\) F \(\cdot\) m\({}^{-1}\)). \[J_{d}=-\omega\varepsilon E_{0}\sin\omega t \tag{3}\]  
In the equation, \(E_{0}\)--amplitude of the alternating electric field; \(\omega\)--angular frequency of the alternating electromagnetic field. Likewise, the conduction current density \(J_{c}\) also has a similar form. \[J_{c}=\sigma E=\sigma E_{0}\cos\omega t \tag{4}\]  
As mentioned in the previous section, the electrical signal used by weakly electric fish for electric field communication is a quasi-static electric field. To meet the quasi-static field conditions, the field used for transmission should be primarily composed of a time-invariant conduction current. Therefore, the conduction current density and displacement current density for this alternating electromagnetic field should satisfy the following:  
\[\frac{|J_{d}|}{|J_{c}|}=\frac{\varepsilon}{\sigma}\omega<<1 \tag{5}\]  
Typically, in seawater, \(\sigma=3\)\(\sim 5\) Sm\({}^{-1}\), while in freshwater, \(\sigma=0.01\)\(\sim 0.1\) Sm\({}^{-1}\). In practical engineering applications, the prevailing condition for conduction current to be the primary component of the total current is \(\frac{|J_{d}|}{|J_{c}|}<0.1\). As per Equation (5), in a seawater environment, when the frequency of the alternating electric field is below 230 MHz, the time-varying displacement current can be neglected. However, in a freshwater environment, the frequency of the alternating electric field needs to be lowered to a few hundred kHz. When constructing an electric field communication model and designing a system, it is essential to consider the frequency of the alternating electric field.