# Frequency Response Extension Method of MET Vector Hydrophone Based on Dynamic Feedback Network
## 3 Materials and Methods
### Calculation of Compensation Method for Low-Frequency Cutoff Frequency  
The key to the adaptive compensation of the low-frequency cutoff frequency is to calculate feedback parameters in the feedback system \(W_{\text{fb}}\). First, based on the test results of the excitation coil self-test, analyze the open-loop characteristics of the system. The excitation coil self-test can measure the frequency response \(H\) of the system. In the open-loop state, the transfer function of the system is given by Equation (8), so \(W_{\text{s}}\) can be expressed as Equation (12). \[W_{\text{s}}=-\frac{v_{\text{o}}}{v_{\text{l2}}}\cdot\frac{1}{W_{\text{a}}W_{ \text{f}}} \tag{12}\]  
\(W_{\text{f}}\) is a known circuit system, so the frequency response of \(W_{\text{s}}\) can be calculated using \(H\), \(W_{\text{f}}\), and \(W_{\text{a}}\). Then, \(W_{\text{s}}\) is fitted to obtain its transfer function. As shown in Figure 7, the transfer function of the preamplifier circuit \(W_{\text{f}}\) can be obtained through circuit analysis, as shown in Equation (13). \[W_{\text{f}}=\frac{C_{1}R_{2}s}{C_{1}R_{1}s+1}\cdot\frac{C_{2}R_{3}s+1}{C_{2}R_{ 2}s+C_{2}R_{3}s+1} \tag{13}\]  
where \(C_{1}\), \(C_{2}\), \(R_{1}\), \(R_{2}\), \(R_{3}\) are circuit parameters. In the experiment of this paper, the actual values used are \(C_{1}=22\)\(\upmu\)F, \(C_{2}=0.01\)\(\upmu\)F, \(R_{1}=100\) k\(\Omega\), \(R_{2}=3\) M\(\Omega\), and \(R_{3}=200\) k\(\Omega\). Next, fit \(W_{\text{s}}\). According to the analysis of the data results, the transfer function of \(W_{\text{s}}\) in the current research frequency band is mainly \(W_{\text{s,mesh}}\). The expression of \(W_{\text{s}}\) used in this paper is shown in Equation (6), and for the convenience of subsequent fitting calculations, it is equivalently rewritten as the fitting expression (14). \[W_{\text{s}}=\frac{As}{s^{2}+Cs+B} \tag{14}\]  
where \(A\), \(B\), and \(C\) are fitting coefficients. \[H_{0}=\frac{W_{\text{s}}W_{\text{f}}}{1+W_{\text{s}}W_{\text{f}}W_{\text{a}}W_ {\text{fb}0}} \tag{15}\]  
where \(W_{\text{fb}0}\) is the feedback system calculated based on the expected closed-loop response. The difficulty of this method lies in the construction of the expected frequency response. The closed-loop transfer function contains high-order terms, and it is difficult to construct directly. In addition, the constructed expected frequency response must meet the feasibility of signal processing. To solve this problem, this study adopts a piecewise approximation method to construct the expected frequency response [28]. The expected closed-loop response of the system is represented by \(H_{0}\). The frequency response of \(H_{0}\) can be divided into the passband and the attenuation section. Here, only the low-frequency attenuation section is considered, and the high-frequency part is regarded as an ideal passband without attenuation. By taking the limits of 0 Hz and infinity, the approximate expressions of the passband and attenuation section of the expected closed-loop response can be obtained, as shown in Equation (16). Here, \(H_{\text{0l}}\) and \(H_{\text{0h}}\) represent the attenuation section and passband of the expected closed-loop response, respectively. \[\left\{ \begin{aligned} \lim_{s\to 0}H_{0}&=H_{\text{0l}}\\ \lim_{s\to\infty}H_{0}&=H_{\text{0h}}\end{aligned} \right. \tag{16}\]  
Figure 7: Schematic diagram of the preamplifier circuit. First, take the limit of the open-loop transfer function at 0 Hz as the approximate expression of the open-loop attenuation section. In the attenuation section, the limit of \(W_{\text{s}}W_{\text{f}}\) at 0 Hz can be used to approximate the open-loop transfer function. \[