# Frequency Response Extension Method of MET Vector Hydrophone Based on Dynamic Feedback Network
## 2 Principles and Background
### Closed-Loop System Analysis of MET  
Figure 5 shows the block diagram of the electrochemical vector hydrophone system (i.e., closed-loop system) using the force balance feedback system, where \(W_{\text{a}}\) and \(W_{\text{fb}}\) are the transfer functions of the feedback coil and the feedback system, respectively. The feedback coil's electromechanical conversion process is characterized by the following transfer function:  
\[W_{\text{a}}=\frac{1}{K_{\text{a}}s} \tag{9}\]  
where \(K_{\text{a}}\) represents the electromechanical coupling coefficient, which is experimentally calibrated. This coefficient reflects the relationship between the input current and the generated force within the feedback coil. The integral relationship in Equation (9) results from the integration of acceleration into velocity in the force balance mechanism.  
Figure 4: Block diagram of the METH system without force balance feedback.  
Figure 3: Schematic of the digital force balance feedback METH; (**a**) The internal structure showing magnetic rubber and coil components. (**b**) The physical image of the device.  
The detailed expression of the total transfer function \(H\) of the system is shown in Equation (10), and the total transfer function of the system can be adjusted by adjusting \(W_{\text{fb}}\). The expected total transfer function after adjustment is denoted by \(H_{0}\), and the transfer function \(W_{\text{fb0}}\) of the feedback system required to achieve the expected value is determined by Equation (11).  
\[H=\frac{W_{\text{s}}W_{\text{f}}}{1+W_{\text{s}}W_{\text{f}}W_{\text{a}}W_{ \text{fb}}} \tag{10}\]  
\[W_{\text{fb0}}=\frac{\frac{1}{H_{0}}-\frac{1}{W_{\text{s}}W_{\text{f}}}}{W_{ \text{a}}} \tag{11}\]  
The circuit diagram of the force balance feedback system is shown in Figure 6. The system is designed digitally and includes a preamplifier, an input range converter, an AD converter, a DSP, two DA converters, a coil drive circuit, and a feedback coil.  
The preamplifier preprocesses the current signal generated by the electrochemical reaction chamber, which is converted by the AD converter and further processed by the DSP. In the DSP, the signal generates a feedback signal through the feedback system and DA1, and the feedback signal drives the coil to generate feedback force through the coil drive circuit, which is applied to the reaction chamber to form a closed-loop feedback loop of force balance feedback. The DSP generates an output signal through DA2 and a differential signal converter, which is provided to the subsequent data acquisition system. The frequency characteristics of the system can be tested by the internal test signal generated by the DSP, tested through the excitation coil. Moreover, in order to improve the accuracy  
Figure 5: Block diagram of the METH system with force balance feedback.  
Figure 6: Implementation of the digital force balance feedback system.  
of the internal test, this study also uses a sound pressure calibration tube to test through an external test method. The external test results will be used to provide a reference and correction for the internal test.