## 4 Discussion
### Identification of Circuit System Pattern
The electrical conductivity of the rock specimen, \(\rho^{\prime}\), is difficult to determine accurately, so the specific numerical value of \(R_{\sigma}\) is usually obtained through actual measurement. Under the excitation of \(U_{\sigma}(t)\), the model of the response voltage (\(U_{\sigma}(t)\)) of \(R_{\sigma}\) in the system shown in Figure 9 is as follows. \[U_{R\sigma}(t)=R_{\sigma}I_{\sigma}(t) \tag{11}\]  
(2) Identification of LCR series attenuating oscillation circuit  
(1) Mathematical model of LC series second-order circuit system  
The LCR components, representing the inductive impedance (\(Z_{L\sigma}\)), capacitive impedance (\(Z_{C\sigma}\)), and resistance (\(R_{\sigma}\)), are considered to be linked in series, as shown in Figure 9a. Based on [PERSON]'s voltage law, the complex frequency domain mathematical model of the second-order system \(H(t)\) comprised of LCR components in series is given by:  
\[U_{\sigma}(t)=I_{\sigma}(t)[R_{\sigma}+j\omega L_{\sigma}+1/(j\omega C_{ \sigma})] \tag{12}\]  
Converting Equation (12) to an equivalent model in the \(S\)-domain and simplifying it to obtain a mathematical model of the current in the \(S\)-domain, denoted as \(I_{\sigma}(S)\), then  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]0  
Solve Equation (13) and obtain its characteristic roots as  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]1  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]2According to the above expression, after inverse transformation, the current equation flowing through the impedance (\(Z_{\sigma}\)) is obtained, which is  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]3  
To determine whether the circuit in Figure 8 is a damped oscillation system, assuming \(\alpha=\frac{L_{\sigma}}{2R_{\sigma}}\) and \(\omega_{0}=\frac{1}{\sqrt{L_{\sigma}C_{\sigma}}}\), then  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]4  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]5  
(2) The attenuation and oscillation conditions of LC series circuit and its determination \(P_{1}\) and \(P_{2}\) are a pair of conjugated complex roots. When \(\alpha<\omega_{0}\) or \(R_{\sigma}<2(L_{\sigma}C_{\sigma})^{1/2}\) (i.e., \(R_{\sigma}\) is small), the high-quality factor (\(Q\)) \(L_{\sigma}C_{\sigma}\) circuit will satisfy \(Q=\omega_{0}/(2\pi)\), in which case, the circuit will undergo attenuated oscillation. The experimental results of the 8# granite specimen were used to verify the electrical decay oscillation system shown in \[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]6 When \(P=2\) kN, \(R_{\sigma}=131\) k\(\Omega\), \(f=18.99\) kHz, \(\omega_{0}=119.3\)\(\times\)\(10^{3}\) rad/s, and when \(C_{\sigma}=474\) pF, \(L_{\sigma}=148.2\) mH, so \(\alpha=R_{\sigma}/(2L_{\sigma})=441.97\)\(\times\)\(10^{3}\). It can be seen that \(\alpha>\omega_{0}\), which contradicts the condition that \(\alpha<\omega_{0}\), so the circuit system (\(H(t)\)) cannot oscillate. That is, the circuit system (\(H(t)\)) is not the damped oscillation system presented in Figure 8, and the assumption of the LCR series attenuation oscillation circuit model is not valid. (3) Identification of LCR parallel attenuation oscillation circuit  
(1) Mathematical model of LC parallel second-order circuit system  
The components corresponding to inductive reactance (\(Z_{L\sigma}\)), capacitive reactance (\(Z_{C\sigma}\)), and resistance (\(R_{\sigma}\)) are denoted as LCR. The connection between LCR elements is in parallel, as shown in Figure 9b. To analyze the attenuation oscillation phenomenon (as shown in Figure 8) that occurs in the LCR parallel circuit system, an electromagnetic radiation-generating inductor response model \(I_{L\sigma}(t)\) at zero input state in the system needs to be established. In accordance with [PERSON]'s current law, the current entering node \"1\" in Figure 9b is equivalent to the current exiting the node, and can be described by the following relationship.