# A New Calibration Method for Broadband Complex Resistivity Measurement System of Rocks and Ores
## 3 Analysis and Calibration of the Instrument's Acquisition Circuit
### Forward Connection of the Signal Source  
In the first circuit design model (Figure 4), the sampling resistor \(R_{\rm s}\) is positioned at the end of the circuit, which is a commonly adopted design scheme in impedance analyzers [27; 29].  
To analyze the measurement errors within the instrument independently, we employed a BNC tri-coaxial shielded cable for direct resistance measurement. In this configuration, the error terms related to the measurement device, \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\), can be neglected, leading to a simplified circuit diagram as shown in Figure 4. The measurement errors are primarily induced by the leakage currents \(I_{2}\) and \(I_{3}\) of the second and third acquisition channels, while the fourth acquisition channel is short-circuited. In this circuit model, the actual current \(I_{\rm x}\) flowing through the sample can be simplified from Equation (3) as follows:  
\[I_{\rm x}\ =\ I_{\rm s}+I_{2}+I_{3}\ =\ I_{\rm s}(1+2\,R_{\rm s}\omega Ci) \tag{7}\]  
Similarly, the actual voltage value across the sample, \(\Delta U_{\rm mn}=\Delta U_{\rm mn^{\prime}}\). The actual impedance of the measured resistor can be calculated as:  
\[Z_{\rm x}\ =\ \frac{\Delta U_{\rm mn}}{I_{\rm x}}^{\prime}\ =\ \frac{\Delta U_{\rm mn}}{I_{\rm s}(1+2R_{\rm s}\omega Ci)}\ =\ \frac{Z_{\rm m}}{1+2R_{\rm s}\omega Ci} \tag{8}\]  
Figure 3: Impedance measurement results of a 100 \(\Omega\) resistor under the influence of distributed capacitance in the acquisition channel.  
In Equation (8):  
\[1+2R_{\rm s}\omega Ci\ =\ \sqrt{1+\left(2R_{\rm s}\omega C\right)^{2}}e^{\tan^{-1} \left(2R_{\rm s}\omega C\right)} \tag{9}\]  
The operating frequency range of the complex resistivity measurement system is from 1 mHz to 100 kHz, with an angular frequency \(\omega\) = 2\(\pi\)_rf_ < 6.3 \(\times\) 10\({}^{5}\) rad/s. Due to the distributed capacitance \(C\) being in the picoFarad (pF) range, when the sampling resistor \(R_{\rm s}\) has a relatively low resistance (<1 k\(\Omega\)), 2\(R_{\rm s}\omega C\) < 0.1. Thus, Equation (9) can be simplified as follows:  
\[1+2R_{\rm s}\omega Ci\approx e^{2R_{\rm s}\omega Ci} \tag{10}\]  
The error between the actual impedance of the sample and the measured impedance is given by:  
\[Z_{\rm x}\ =\ \frac{Z_{\rm m}}{1+2R_{\rm s}\omega Ci}\approx\frac{|Z_{\rm m}|e^{ \phi_{\rm m}i}}{e^{2R_{\rm s}\omega Ci}}\ =\ |Z_{\rm m}|e^{(\phi_{\rm m}-2R_{\rm s}\omega C)i} \tag{11}\]  
From the above equation, it is evident that the impedance measurement error is primarily reflected in the phase and is solely related to the value of the sampling resistor \(R_{\rm s}\), independent of the intrinsic impedance characteristics of the measured sample. When the sampling resistor \(R_{\rm s}\) is small, the calibration method is simple and quick, making it a commonly adopted calibration scheme. However, as the value of the sampling resistor \(R_{\rm s}\) increases, the impedance measurement error will rapidly escalate, complicating the calibration process, and the influence of the parasitic capacitance of the sampling resistor will also intensify. Therefore, this circuit model limits the range of sampling resistor values, rendering it inadequate for the measurement needs of high-impedance samples.