# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 4 Design and Calibration of the Measurement Instrument  
In addition to the internal calibration of the instrument, the influence of the measurement device on the test results is also highly significant. The distinction between the rock and ore electrical measurement instruments and traditional impedance analyzers lies in the specific measurement objects. Enhancing the measurement accuracy of the rock sample's resistivity and providing a reasonable and practical sample holder design are the primary tasks in developing this instrument. In Figure 2, the dark box portion represents the device effects of the sample holder, where \(Z_{1}\) and \(Z_{4}\) denote the stray impedances between the measuring and receiving electrodes. In contrast, \(Z_{2}\) and \(Z_{3}\) represent the contact impedances between the acquisition channels and the measuring electrodes. These error terms may vary depending on the specific measurement device used. Referring to Figure 2, the actual current \(I_{\mathrm{x}}\) flowing through the sample is given by:  
\[I_{\mathrm{x}}\ =\ {I_{\mathrm{s}}}^{\prime}+I_{2}+I_{3}\ =\ {I_{\mathrm{s}}}{R_{ \mathrm{s}}}(\frac{1}{{Z_{\mathrm{s}}}}+\frac{2}{{Z_{\mathrm{c}}}}+\frac{{Z_{ 4}}}{{Z_{\mathrm{c}}}^{2}}+\frac{{Z_{4}}}{{Z_{\mathrm{c}}}{Z_{\mathrm{s}}}}) \tag{21}\]  
By substituting Equation (5) into Equation (21), we obtain the actual impedance value \(Z_{\mathrm{x}}\) of the sample, taking into account the device effects:  
\[Z_{\mathrm{x}}\ =\ \frac{{Z_{\mathrm{m}}}}{{R_{\mathrm{s}}}}\frac{{Z_{\mathrm{s}} }(1+{Z_{\mathrm{2}}}_{\mathrm{s}}\omega Ci)}{1+2Z_{\mathrm{s}}\omega Ci+{Z_{ \mathrm{4}}}\omega Ci-{Z_{\mathrm{4}}}{Z_{\mathrm{s}}}\omega^{2}C^{2}} \tag{22}\]  
To analyze the impact of the measurement device on the test results, we designed the circuit model shown in Figure 9a. With \(Z_{\mathrm{x}}=1\,\mathrm{k}\Omega\) and \(Z_{\mathrm{2}}=Z_{\mathrm{3}}=100\)\(\Omega\) held constant, we varied the values of \(Z_{1}\) and \(Z_{4}\). Figure 9b displays the phase test results of the instrument before calibration for different values of \(Z_{1}\) and \(Z_{4}\), compared with the calculated results from Equation (22). The results indicate that the calculated curve aligns with the measured results, and as the contact impedances \(Z_{1}\) and \(Z_{4}\) increase, the measurement errors gradually expand. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & **R/\(\Omega\)** & **C/pF** & \(R\) \\ \hline
1 & 1 k & 800 \\
2 & 10 k & 80 \\
3 & 100 k & 8 \\ \hline \end{tabular}
\end{table} 
Table 1: Resistance and capacitance values for the three resistor-capacitor models. Figure 9: (**a**) Diagram of the designed circuit model, and (**b**) Comparison of the phase testing results of this model (before calibration) with the calculated results obtained from Equation (22) upon varying the values of \(Z_{1}\) and \(Z_{4}\). In this study, we designed two measurement devices for different testing subjects. We chose the sample holder shown in Figure 10a for the measurement of rock and ore samples. We selected the device depicted in Figure 10b to test loose materials such as rock fragments and soil or for liquids. In both measurement setups, \(Z_{2}\) and \(Z_{3}\) represent the actual impedance of the electrodes, which is less than 1 \(\Omega\), thus \(Z_{2}\approx Z_{3}=Z_{2,3}\ll Z_{\text{in}}\) (where \(Z_{\text{in}}\) denotes the input impedance of the instrument's acquisition channel, which is approximately 17 G\(\Omega\) for this device). Additionally, \(Z_{4}Z_{s}\omega^{2}C^{2}\ll 1\). Combining this with the results from the different sampling resistance reverse connection methods in Equation (19), we can transform Equation (22) to:  
\[Z_{\text{x}}\ =\ \frac{Z_{\text{m1}}Z_{\text{m2}}Z_{\text{s}}}{{R_{\text{s}}}^{ 2}+Z_{4}Z_{\text{m2}}\omega Ci} \tag{23}\]  
The sample holder shown in Figure 10a can be divided into four parts: A for the power electrodes; B for the measurement electrodes; C for the fixture, and D for the sample holder container.