# 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
To accurately characterize the internal measurement errors of the instrument, it is essential to consider the parasitic capacitance of the sampling resistance concurrently. Consequently, the current \(I_{\rm x}\) flowing through the sample in the first circuit configuration (forward connection) should be modified according to Equation (7):  
\[I_{\rm x}\ =\ I_{\rm s}{}^{\prime}+I_{2}+I_{3}\ =\ I_{\rm s}{}^{\prime}+2I_{\rm s }R_{\rm s}\omega Ci\ =\ I_{\rm s}\left(\frac{I_{\rm s}{}^{\prime}}{I_{\rm s}}+2R_{\rm s}\omega Ci \right)\ =\ I_{\rm s}\left(\frac{R_{\rm s}}{Z_{\rm s}}+2R_{\rm s}\omega Ci\right) \tag{15}\]  
\(I_{\rm s}{}^{\prime}\) represents the actual current flowing through the sampling resistance \(Z_{\rm s}\) in Equation (15). Substituting the equation back into Equation (8) yields:  
\[Z_{\rm x}\ =\ \frac{\Delta U_{\rm mn}}{I_{\rm s}\left(\frac{R_{\rm s}}{Z_{\rm s }}+2R_{\rm s}\omega Ci\right)}\ =\ \frac{Z_{\rm m}}{R_{\rm s}}\frac{Z_{\rm s}}{1+2Z_{\rm s}\omega Ci} \tag{16}\]  
In the reverse connection configuration of the signal source, the current \(I_{\rm x}\) flowing through the actual sample is expressed as:  
\[I_{\rm x}\ =\ I_{\rm s}{}^{\prime}-I_{2}-I_{3}\ =\ I_{\rm s}{}^{\prime}-2I_{ \rm s}Z_{\rm m}\omega Ci\ =\ I_{\rm s}\left(\frac{I_{\rm s}{}^{\prime}}{I_{\rm s}}-2Z_{\rm m}\omega Ci \right)\ =\ I_{\rm s}\left(\frac{R_{\rm s}}{Z_{\rm s}}-2Z_{\rm m}\omega Ci\right) \tag{17}\]  
Equation (12) is transformed to:  
\[Z_{\rm x}\ =\ \frac{\Delta U_{\rm mn}}{I_{\rm s}\left(\frac{R_{\rm s}}{Z_{\rm s }}-2Z_{\rm m}\omega Ci\right)}\ =\ \frac{Z_{\rm m}}{\frac{R_{\rm s}}{Z_{\rm s}}-2Z_{\rm s}\omega Ci} \tag{18}\]  
To characterize the error between the actual impedance \(Z_{\rm s}\) of the sampling resistance and the calculated value \(R_{\rm s}\), we connect a component identical to the sampling resistance in the BNC measurement line as the test resistance, i.e., \(Z_{\rm x}=Z_{\rm s}\). Equation (18) can be transformed to:  
Figure 4:0  
To differentiate the measured values \(Z_{\rm m}\) in Equations (16) and (19), we denote the measurement in Equation (16) as \(Z_{\rm m1}\) and that in Equation (19) as \(Z_{\rm m2}\), representing the measurements obtained under the forward and reverse connection configurations of the signal source, respectively. By combining these two equations, we obtain:  
Figure 4:1  
In Equation (20), \(|Z_{\rm m1}|\), \(|Z_{\rm m2}|\), \(|Z_{\rm s}|\) and \(\varphi_{\rm m1}\), \(\varphi_{\rm m2}\), \(\varphi_{\rm s}\) represent the magnitudes and phases of the impedances \(Z_{\rm m1}\), \(Z_{\rm m2}\), and \(Z_{\rm s}\), respectively. From Equation (20), it is evident that the actual impedance of the sample is related to the measured values \(Z_{\rm m1}\) and \(Z_{\rm m2}\) under the forward and reverse connection configurations of the signal source, as well as the actual impedance \(Z_{\rm s}\) and calculated value Rs of the sampling resistance. The phase error is primarily influenced by \(\varphi_{\rm m1}\), \(\varphi_{\rm m2}\), and \(\varphi_{\rm s}\), where \(\varphi_{\rm m1}\) and \(\varphi_{\rm m2}\) are mainly determined by the parasitic capacitance distribution within the instrument. In contrast, \(\varphi_{\rm s}\) is affected by the parasitic capacitance of the components themselves. When the sampling resistance is low and the measured impedance \(Z_{\rm x}=Z_{\rm s}\), we have \(|Z_{\rm m1}|\approx|Z_{\rm m2}|\approx|Z_{\rm s}|\approx R_{\rm s}\) and \(\varphi_{\rm s}\approx 0\), leading to \(\varphi_{\rm m1}=-\varphi_{\rm m2}\). This conclusion is consistent with Equations (11) and (14). Essentially, Equation (20) generalizes the error term by considering the parasitic capacitance of the sampling resistance, thus enhancing calibration accuracy. Additionally, the form of this error term is simple and facilitates calibration without the need for separate analysis of the internal channel distribution capacitance \(C\). The sampling resistance Rs does not need to be limited to low values.