# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 2 Measurement System
### System Error Analysis  
In the measurement system, the factors that most often affect the instrument's accuracy are the distributed capacitance in the acquisition channels and the device effects caused by the sample holder. We have illustrated a simplified circuit model, as shown in Figure 2. Let \(\Delta U_{\text{mn}}=U_{\text{m}}-U_{\text{n}}\) represent the voltage measured across the sample, and \(\Delta U_{\text{mn}}{}^{\prime}=U_{\text{m}}{}^{\prime}\)\(-U_{\text{n}}{}^{\prime}\) represent the actual voltage across the sample. \(I_{\text{x}}\) and \(I_{\text{s}}\) denote the current passing through the sample and the sampling resistor \(R_{\text{s}}\), respectively. At low frequencies (<100 Hz), these effects can often be minimized through careful design of the acquisition circuit and optimization of the measurement apparatus. Under such conditions, according to [PERSON]'s law, the impedance of the sample is given by:  
\[Z_{\text{x}}\;=\;|Z_{\text{x}}|e^{\varphi_{\text{x}}i}\;=\;\frac{\Delta U_{ \text{mn}}{}^{\prime}}{I_{\text{x}}}\;=\;\frac{\Delta U_{\text{mn}}}{I_{\text{ s}}}\;=\;|Z_{\text{m}}|e^{\varphi_{\text{m}}i}\;=\;Z_{\text{m}} \tag{1}\]  
In Equation (1), \(Z_{\text{x}}\) represents the actual impedance of the measured sample, with \(|Z_{\text{x}}|\) and \(\varphi_{\text{x}}\) corresponding to its impedance magnitude and phase angle, respectively. Similarly, \(Z_{\text{m}}\) represents the impedance value calculated by the instrument, with \(|Z_{\text{m}}|\) and \(\varphi_{\text{m}}\) corresponding to its impedance magnitude and phase angle. However, as the measurement frequency increases, the errors introduced by the parasitic capacitance of the internal electronic components, the contact impedance of the measuring device, and the electromagnetic coupling effects between electrodes significantly intensify, making their impact on the measurement accuracy of the instrument non-negligible. This is particularly true for soil or non-mineralized rock samples, with a weak self-polarization effect. To accurately characterize the polarization characteristics of such samples, it is necessary to analyze the error response of the measurement system. In Figure 2, the dark box area represents the device effects of the sample holder. \(Z_{1}\) and \(Z_{4}\) indicate the parasitic impedance between the power supply electrodes and the receiving electrodes, while \(Z_{2}\) and \(Z_{3}\) denote the contact impedance between the acquisition channel and the receiving electrode. Additionally, the circuit model includes the internal acquisition circuit of the instrument, where \(R_{0}\) represents the internal resistance of the signal source, and \(R_{\text{s}}\) denotes the sampling resistance. For the sake of analysis, we utilize a  
Figure 2: Simplified circuit model of the complex resistivity measurement system. After being generated by the signal source, the signal flows through the sample (\(Z_{\text{x}}\)) and the sampling resistor (\(R_{\text{s}}\)) and is subsequently processed by a differential amplifier to yield the output signal (\(\Delta U_{\text{mn}}\), \(U_{\text{s}}\)). However, the measurement accuracy is influenced by contact impedances (\(Z_{1}\), \(Z_{2}\), \(Z_{3}\), \(Z_{4}\)) introduced by the measuring instruments, as well as leakage currents caused by the distributed capacitance (C) of the acquisition channels. capacitor \(C\) to simulate the distributed capacitance arising from leakage currents on the four acquisition channels, typically on the order of picoFarads (pF), with impedance defined as \(Z_{\rm c}=1/(i\omega C)\), where \(\omega\) is the angular frequency, and \(i\) is the imaginary unit. In the circuit model diagram, \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\) represent the magnitude of the leakage currents flowing through the four channels; therefore, the actual current flowing through the sample is given by \(I_{\rm x}=I_{2}+I_{3}+I_{\rm s}\).