## 4 Discussion
### Identification of Circuit System Pattern
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]7  
According to Equation (19), derive the characteristic equation of the second-order system, which is the impedance admittance. \[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]8  
Let \(1/Z_{\sigma}=Y\), \(1/R_{\sigma}=G\), \(1/Z_{C\sigma}=j\omega C_{\sigma}\), \(1/Z_{L\sigma}=(j\omega L_{\sigma})\), substitute these values into Equation (20), and then simplify it according to the parallel circuit conditions. \[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]9  
Convert Equation (21) to an equivalent model in the \(S\) domain. Then, the characteristic equation of the zero-input response in the \(S\) domain is:  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]0Solving Equation (22) yields the characteristic roots:  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]1  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]2  
If the damping coefficient of the circuit is \(\alpha=1/(2R_{\sigma}C_{\sigma})\) and the resonant frequency is \({\omega^{2}}_{0}=1/(L_{\sigma}C_{\sigma})\), then  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]3  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]4  
According to the characteristic roots (\(P_{1},P_{2}\)) of Equation (22), the solving equation for \(I_{L\sigma}(t)\) can be obtained through inverse transformation:  
\[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]5  
After undergoing time-domain and frequency-domain transformations, the current solution flowing through the impedance (\(Z_{L\sigma}\)) is obtained. \[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]6  
In Equation (28), \(A_{1}\) and \(A_{2}\), respectively, represent the amplitudes of the currents, where \(A_{1}=\frac{U_{C\sigma}(0_{-})}{L(P_{1}-P_{2})}\) and \(A_{2}=\frac{-U_{C\sigma}(0_{-})}{L(P_{1}-P_{2})}\). For specific rock materials that have not yet been deeply studied, the electromagnetic signals generated under compression are primarily manifested as direct current signals, signals with periodic variations, and signals with non-periodic variations. Regarding the Formula (28) that we have proposed, even when the rock material only generates a direct current signal, i.e., the frequency is zero, the sine term in the formula calculates to zero, yet the entire theoretical formula remains applicable. When the rock material generates a pulse signal under compression, we can decompose this signal into the fundamental wave and higher harmonics, and then substitute them into the theoretical formula for analysis, at which point the theoretical formula is still applicable. Furthermore, the theoretical analysis framework constructed in this study takes the example of a damped sinusoidal signal, and the derived theoretical formula naturally applies to such signals. Therefore, theoretically speaking, the results of our research can be extended to other types of geological materials. This indicates that our theoretical model has a wide applicability and can provide analysis and interpretation for the electromagnetic signals generated by different types of rock materials under compression. At present, most studies tend to explore rock mechanics problems using individual physical quantities such as resistivity, dielectric constant (permittivity), and magnetic permeability. In contrast, our constructed model (see Equation (28) for details) integrates these physical quantities into a comprehensive analytical framework. From the perspective of integrating multiple physical factors to construct a theoretical analysis model, our model provides a unified method of physical analysis. Looking from the angle of signal types, the model we have established is capable of fully adapting to the analysis needs of different types of signals, whether these are direct current signals, signals with periodic change patterns, or signals with non-periodic change patterns, all are applicable.