# Frequency Response Extension Method of MET Vector Hydrophone Based on Dynamic Feedback Network
## 2 Principles and Background
### Electrochemical Reaction Chamber  
The transfer function of the electrochemical hydrophone refers to the ratio of the Laplace transform of the output signal to the Laplace transform of the surface sound pressure signal of the hydrophone. The transfer function can be used to analyze the frequency characteristics and application range of the hydrophone. According to the mechanical structure and working principle of the electrochemical hydrophone, the working process of the hydrophone can be divided into two links: mechanical vibration and electrochemical charge transfer. The overall transfer function of the electrochemical hydrophone is the product of these two transfer functions. The mechanical vibration part and the electrochemical charge transfer part will be elaborated below. Molecular electron transfer (MET) technology is based on the sensing mechanism of redox reactions at the electrochemical interface. When a bias voltage is applied to the electrode surface, a current is generated. When acoustic pressure disturbances induce electrolyte flow, the resulting flow field motion alters the ion concentration gradient distribution near the electrode surface. By utilizing a differential electrode structure to detect sensitive variations in the cathodic current, an electrical signal linearly correlated with the velocity component of the acoustic pressure field can be extracted. This section details the operation scheme and principle of the MET transducer element and hydrophone (METH) [29]. The transfer function of METH consists of two subsystems: the mechanical part with the transfer function \(W_{\text{s,mech}}\) and the electrochemical part with the transfer function \(W_{\text{elch}}\). The mechanical subsystem represents the transfer of external pressure changes to the volume flow of the electrolyte, while the electrochemical part transfers the internal flow of the electrolyte to the signal current of METH. The volume flow \(q\) of the working fluid through the sensing element can be expressed as follows [30]:  
\[q=\frac{P_{\text{ex}}e^{i\omega t}-P_{\text{in}}-\frac{\rho dl}{s}}{R_{h}} \tag{1}\]  
where \(P_{\text{ex}}\) represents the amplitude of the external pressure that varies with frequency \(\omega\); \(P_{\text{in}}\) represents the pressure of the internal volume of the hydrophone; \(s\) represents the cross-sectional area of the hydrophone; \(l\) represents the length of the hydrophone; \(R_{h}\) represents the hydrodynamic resistance of the fluid through the sensing element. Figure 1: Schematic diagram of electrochemical hydrophone. If the internal pressure is expressed as \(P_{\text{in}}=\frac{\partial p}{\partial V}\Delta V=\frac{\partial p}{\partial V} \int q\,dt\), the complex amplitude of the harmonic solution of this equation can be expressed as  
\[q=\frac{P_{\text{ex}}}{R_{h}\Big{(}1+\frac{\partial p}{\partial V}\frac{1}{ \omega\omega K_{\text{s}}}+\frac{\rho l}{R_{\text{s}}}i\omega\Big{)}}, \tag{2}\]  
The mechanical part of the overall transfer function of METH is  
\[W_{\text{s\_mech}}=\bigg{|}\frac{q}{P_{\text{ex}}}\bigg{|}. \tag{3}\]  
Regarding the electrochemical part of the sensing process, its principle is the same as all types of molecular electronic devices [31], and the entire sensing mechanism can be explained by Figure 2. The sensing unit (composed of four anodes (A)--cathodes (K) pairs) is immersed in a liquid electrolyte. A small voltage (300 mV) is applied between the anode and cathode, and an oxidation reduction reaction occurs on the electrode. Excess current carriers are generated at the anode; at the cathode, these carriers recombine. As a result, a carrier concentration gradient is formed between each anode and cathode pair. When the fluid starts to flow due to external pressure changes, this concentration gradient changes. The transfer function of the electrochemical system is described by the Navier-Stokes equation, the incompressibility of the liquid, and convection diffusion, as described in [32].