# The Generalized Skin Depth for Polarized Porous Media Based on the Cole-Cole Model
## 3 Generalized Skin Depth Characteristic Analysis Based on the Cole-Cole Model
### Effect of the Polarization Parameters on the Generalized Skin Depth
#### 3.2.2 Effect of the Chargeability on the Generalized Skin Depth  
The chargeability is one of the most important parameters of a polarized medium based on the Cole-Cole model, which can well reflect the charging and discharging ability of the medium. To calculate the effect of chargeability on the generalized skin depth of polarized media, we chose the frequency range 10\({}^{-1}\)-10\({}^{4}\) Hz, frequency dependence c \(=\)0.5, characteristic time constant \(\tau\)\(=\) 1 s, and conductivity at infinite frequency \(\sigma_{\infty}\)\(=\) 0.3 S/m. The skin depth varied with frequency and chargeability, as shown in Figure 4. To quantitatively analyze the skin depth of generalized polarized media, we give the absolute error between the generalized skin depth and classical skin depth at different chargeability \(\delta_{\text{absolute error}}\)\(=\)\(|\delta_{\text{cole-cole}}-\delta_{\text{Classical}}|\) in Figure 5.  
Figure 3: The absolute errors of the generalized skin depth and classical skin depth under different conductivities ((**a**) and (**b**) represent the magnified maps of different frequency bands and red, purple, and blue represent the absolute error between the two when the conductivity is 0.1 S/m, 0.3 S/m, and 0.5 S/m, respectively). Figure 2: Generalized skin depth of polarized media at different conductivities (red, purple, and blue represent the \(\sigma_{\infty}\)\(=\) 0.1 S/m, \(\sigma_{\infty}\)\(=\) 0.3 S/m, and \(\sigma_{\infty}\)\(=\) 0.5 S/m generalized skin depths of the polarized media, respectively, and the black curve represents the classical skin depth). Figures 4 and 5 show that the generalized skin depth of polarized media and the classical skin depth decreased with increasing frequency. When other parameters remained unchanged, the generalized skin depth and absolute error increased with increasing chargeability. The effect of chargeability on generalized skin depth was mainly concentrated in the low-frequency band, approximately f \(<\) 10 Hz. When the frequency f \(<\) 10 Hz, the generalized skin depth was deeper than the classical skin depth, which also shows that the effective propagation distance of the EM wave was greater. When the frequency f \(>\) 10 Hz, with increasing frequency, the effect of chargeability on the generalized skin depth decreased, and the generalized skin depth under each chargeability was close to the classical skin depth. 2.3 Influence of the Frequency Dependence and Characteristic Time Constant on the Generalized Skin Depth  
In addition to the conductivity and chargeability, the generalized skin depth was also affected by the frequency dependence and characteristic time constant. Consider a frequency range of 10\({}^{-1}\)-10\({}^{4}\) Hz, a characteristic time constant \(\tau=1\) s, a conductivity at infinite frequency \(\sigma_{\infty}=0.3\) S/m, chargeability \(\eta=0.5\), and frequency dependence values of c =0.2, c =0.4, c =0.6, and c =0.8. The generalized skin depth under different frequency dependence values is shown in Figure 6.  
Figure 4: Skin depth of generalized polarized media at different chargeability values (black curve represents the traditional skin depth without polarization and red, blue, and green represent the generalized skin depth of polarized media with a chargeability of \(\eta=0.3\), \(\eta=0.5\), and \(\eta=0.7\), respectively). Figure 5: Absolute error between the generalized skin depth of polarized media and classical skin depth under different chargeability values ((**a**) and (**b**) represent magnified maps of different frequencies; red, blue, and green represent the absolute errors when the chargeability is 0.3, 0.5, and 0.7, respectively). As shown in Figure 6, the generalized skin depth decreased with increasing frequency dependence. The effect of the frequency dependence on the generalized skin depth was concentrated mainly in the low-frequency band at approximately f \(<\) 10 Hz.