# Leakage Detection Method of Coal Mine Power Supply System Based on 3 AC
## 2 Leakage Protection Method Based on 3 AC Detection Circuit
### Mathematical Model Based on 3 AC Detection Method  
By collecting the ground current of '3 AC', the effective value and phase are calculated by the Fourier algorithm [15], and then the pattern recognition of the mathematical model of leakage resistance is established according to the variation law of the current phasor when single-phase, two-phase, and three-phase leakage faults occur. To optimize the system's performance, it is necessary to carefully select the parameters of the additional capacitors. Specifically, the capacitance values are determined such that \(C_{1}=2C_{2}=2C_{3}\), with the specific values set as \(C_{1}=2C_{2}=2C_{3}=0.2\)\(\mu\)F. The choice of 0.2 \(\mu\)F as the capacitance value is not only because it meets the basic functional requirements, but more importantly, it can operate reliably in a high-voltage environment of 1140 V, while taking into account safety, cost-effectiveness, and overall system performance optimization. This value is the optimal result obtained through simulation or experimentation and can meet the system's performance requirements under normal operating conditions. In order to simplify the theoretical analysis, it is usually assumed that the grid power supply, the insulation resistance, and distributed capacitors between the phase and the ground of all lines are symmetrically distributed, and the distributed capacitance of each phase to the ground is \(C\). When the three-phase insulation resistance decreases, the resistance values are represented by \(R_{\text{A}}\), \(R_{\text{B}}\), and \(R_{\text{C}}\), respectively. Figure 2: Principle diagram of power supply system based on 3 AC detection circuit. The node voltage method is used to analyze \(\dot{U}_{0}\) and \(\dot{I}_{C0}\), where \(\dot{I}_{C0}\) is a zero-sequence current artificially constructed by adding three asymmetric capacitors. Taking the earth as the reference node, the formula is as follows:  
\[\dot{U}_{0}=\frac{-\dot{U}_{A}(\frac{1}{R_{A}}+\mathrm{j}\omega C+\mathrm{j} \omega C_{1})-\dot{U}_{B}(\frac{1}{R_{B}}+\mathrm{j}\omega C+\mathrm{j}\omega C _{2})-\dot{U}_{C}(\frac{1}{R_{C}}+\mathrm{j}\omega C+\mathrm{j}\omega C_{3})}{[ \frac{1}{R_{A}}+\frac{1}{R_{B}}+\frac{1}{R_{C}}+\mathrm{j}\omega 3C+\mathrm{j} \omega(C_{1}+2C_{2})]} \tag{1}\]  
Expanding the denominator of Equation (1) and considering that the vector sum of the three-phase voltages is zero given the conditions \(C_{1}=2C_{2}=2C_{3}\) as set above, we can simplify the data to obtain Equation (2):  
\[\dot{U}_{0}=\frac{-\dot{U}_{A}\frac{1}{R_{A}}-\dot{U}_{B}\frac{1}{R_{B}}-\dot{ U}_{C}\frac{1}{R_{C}}-\mathrm{j}\omega C_{2}\dot{U}_{A}}{\frac{1}{R_{A}}+ \frac{1}{R_{B}}+\frac{1}{R_{C}}+\mathrm{j}\omega 3C+\mathrm{j}\omega 4C_{2}} \tag{2}\]  
In an ideal power system, the three-phase conductors are completely insulated from each other and from the ground. This means that no current will leak through the insulation materials to the ground or other conductors. Therefore, in this ideal scenario, the insulation resistance can be considered infinite. Although, in reality, any insulation material has a certain resistance value, under normal operating conditions, the insulation resistance is very high and can still be treated as infinite. Thus, \(1/R_{A}\), \(1/R_{B}\), and \(1/R_{C}\) can be approximated as zero. Consequently, Equation (2) can be further simplified as follows:  
\[\dot{U}_{0}=\frac{-\mathrm{j}\omega C_{2}\dot{U}_{A}}{\mathrm{j}\omega 3C+\mathrm{j}\omega 4C_{2}}\]  
Due to the addition of three-phase asymmetric capacitance, a zero-sequence current \(\dot{I}_{C0}\) will also be generated during the normal operation of the system.