# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 3 Analysis and Calibration of the Instrument's Acquisition Circuit
### Reverse Connection of the Signal Source  
We compared the impedance measurement errors of the second circuit configuration, as shown in Figure 5. In this case, the sampling resistor is positioned on the output source side, which can be achieved by interchanging the connections of the signal source, referred to as the reverse connection of the signal source. Figure 4: Simplified acquisition circuit diagram with forward connection of signal source, with sampling resistor positioned at the terminal far from the signal source. After being generated by the signal source, the signal flows through the sample \(Z_{\rm x}\) and then to the sampling resistor \(R_{\rm s}\). Due to the presence of distributed capacitance in the acquisition channels, a leakage current \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\) is introduced, resulting in a discrepancy between the currents \(I_{\rm x}\) and \(I_{\rm s}\). In this circuit model, the current \(I_{x}\) is expressed as \(I_{x}=I_{s}-I_{2}-I_{3}\), and the voltage difference \(\Delta U_{mn}\) is equivalent to \(\Delta U_{mn}\). The measurement error in impedance is predominantly influenced by leakage currents associated with both the two-channel and three-channel configurations:  
\[Z_{x}\ =\ \frac{\Delta U_{mn}}{I_{x}}^{\prime}\ =\ \frac{\Delta U_{mn}}{I_{s}- \frac{2U_{mn}}{Z_{c}}}\ =\ \frac{Z_{m}}{1-2Z_{m}\omega Ci} \tag{12}\]  
Similarly, when the resistance of the measured sample is relatively low, given that \(2\,|\,Z_{m}\,|\,\omega C<0.1\), the impedance \(Z=\,|\,Z\,|\,\cos\varphi+\,|\,Z\,|\,\sin\varphi i\) can be approximated as \(Z\approx\,|\,Z\,|\,+\,|\,Z\,|\,\varphi i\) when the absolute value of the phase angle \(\,|\,\varphi\,|\,\) is small. Thus:  
\[Z_{x}\ =\ \frac{Z_{m}}{1-2Z_{m}\omega Ci}\approx\frac{|Z_{m}|e^{\varphi m \,i}}{e^{-2|Z_{m}|\omega Ci}}\ =\ |Z_{m}|e^{(\varphi_{m}+2|Z_{m}|\omega C)i} \tag{13}\]  
\[\Delta U_{mn}{}^{\prime}\approx\Delta U_{mn}+\frac{Z_{2,3}\Delta U_{mn}}{Z_{c}} \tag{14}\]  
At this stage, the measurement error primarily manifests in the phase, and its magnitude exhibits a similar pattern to the forward connection, albeit with an opposite sign. However, the phase error is independent of the sampling resistance and is solely related to sample impedance. When the impedance \(Z_{x}\) is low, we can utilize this approach to calculate the parasitic capacitance \(C\) of the internal channels of the instrument, which has been measured to be approximately 24 pF. However, as \(Z_{x}\) increases, \(Z_{m}\) also increases, leading to a larger measurement error with a more complex error pattern. Correction Methods Based on Forward and Reverse Connection Circuit Configurations of the Signal Source  
Based on the above two circuit configurations, the first method, although primarily dependent on the sampling resistance and therefore more convenient for calibration, poses challenges for high-impedance samples. For certain dense surrounding rocks, the sample resistance can exceed 10 M\(\Omega\), whereas the polarization effects remain minimal. In this scenario, employing a small sampling resistance for measurements will lead to increased errors in resistance calculations. Conversely, utilizing a large sampling resistance will cause the phase shift due to the components' parasitic capacitance at high frequencies, which will sig  
Figure 5: Simplified acquisition circuit diagram with reverse connection of signal source, with sampling resistor positioned at the front end near the signal source. After being generated by the signal source, the signal flows through the sampling resistor \(R_{s}\) and then to the sample \(Z_{x}\). Due to the presence of distributed capacitance in the acquisition channels, a leakage current \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\) is introduced, leading to a discrepancy between the currents \(I_{x}\) and \(I_{s}\). nificantly affect the phase measurement results of low-polarization samples.