# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 5 Results
The spectral response characteristics of the NaCl solution at 21 \({}^{\circ}\)C can be calculated using Equation (24) [38]:  
\[\varphi\ =\ -\arctan(\omega\varepsilon_{0}\varepsilon_{\mathrm{r}}\rho^{ \prime}) \tag{24}\]  
In Equation (24), \(\varepsilon_{0}\) = 8.854 \(\times\) 10\({}^{-12}\) F/m represents the vacuum permittivity, \(\varepsilon_{\mathrm{r}}\) = 79.7 is the relative permittivity of water at 21 \({}^{\circ}\)C, and \(\rho^{\prime}\) is the real part of the resistivity. The results indicate a pronounced downward trend in the high-frequency phase of the NaCl solution, with the decline accelerating as the resistivity increases. Below 10 kHz, the test results remain generally consistent with the theoretical curve, with an error controlled within 0.1 mrad. Above 10 kHz, the discrepancy between the two increases yet remains within 2 mrad, The decline in high-frequency calibration accuracy may be attributed to the following two factors: \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**R/\(\Omega\)**} & \multicolumn{3}{c}{**Measured Resistance/\(\Omega\)**} & \multicolumn{3}{c}{**Measured Capacitance/pF**} \\ \cline{3-8}  & & **CSUZA-** & **Error (\%)** & **Solartron** & **Error (\%)** & **CSUZA-** & **Solartron** \\  & & **FW08** & & **1260A** & & **FW08** & **1260A** \\ \hline
1 & 100 & 99.8 & 0.20 & 99.8 & 0.20 & 0.03 & 0.01 \\
2 & 1k & 1000.7 & 0.07 & 1000.4 & 0.04 & 0.42 & 0.44 \\
3 & 10k & 10,018.5 & 0.19 & 9998.7 & 0.01 & 0.44 & 0.47 \\
4 & 100k & 100,372.1 & 0.37 & 99,803.9 & 0.20 & 0.57 & 0.64 \\
5 & 1M & 1,005,816.4 & 0.58 & 989,711.0 & 1.03 & 0.48 & 0.70 \\
6 & 10M & 10,096,038.6 & 0.96 & 9,570,166.5 & 4.30 & 0.83 & 5.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of measurement results between the CSUZA-FW08 instrument and the Solartron 1260A impedance analyzer when measuring different resistances. Figure 13: Comparison of the test results between the CSUZA-FW08 instrument and the Solartron 1260A impedance analyzer when measuring the resistance of 1 k\(\Omega\)**(a)** and 1 M\(\Omega\)**(b)**. In Equation (23), the measurement results of the stray impedance \(Z_{1}\) and \(Z_{4}\) between the power supply electrode and the receiving electrode can influence the final calibration outcome. If we assume that the test solution is uniformly distributed within the measurement container, the relationship between the stray impedances \(Z_{1}\), \(Z_{4}\), and the impedance of the sample \(Z_{\rm x}\) can be roughly estimated based on the distance AM between the measuring electrode and the receiving electrode. However, for accurate results, precise measurements of \(Z_{1}\) and \(Z_{4}\) should be conducted using other testing methods. 2. During the calibration of complex resistivity data, we primarily analyze the distributed capacitance of the instrument's acquisition channels, the parasitic capacitance of the sampling resistor, and the device effects of the sample holder. In four-point measurements, electromagnetic coupling interference must also be considered, as this effect increases with the measurement frequency. Figure 14: Comparison of the complex resistivity testing results between the CSUZA-FW08 instrument and the Solartron 1260A impedance analyzer when the sample is lead-zinc ore.  
Figure 15: Comparison of the phase testing results of NaCl solutions with different resistivities against the calculated values. Below 100 kHz, the calibration method proposed in this study can meet the demands of high-precision testing. Since testing of rock and ore samples typically does not exceed 100 kHz, this calibration method remains highly applicable. Finally, we conducted tests on two rock samples using the proposed instrument (CSUZA-FW08) and calibration method, as shown in Figure 16. The two rock samples were collected from the Chanlinshan multimetallic mining area in Nanjing, Jiangsu Province, China, representing chalcopyrrite and diorite samples.