# Inversion of a Thunderstorm Cloud Charging Model Based on a 3D Atmospheric Electric Field
## Materials and Methods
### Principles of Measurement and Calibration
\begin{array}{cc}-\sqrt{2}\pi \varepsilon E_{X(Y)}fS\cos\left(\frac{\pi}{4}-2\pi ft\right)&t\in(0,\,\frac{T} {2})\\ &\sqrt{2}\pi\varepsilon E_{X(Y)}fS\cos\left(\frac{\pi}{4}-2\pi ft\right)&t\in \left(\frac{T}{2},T\right)\end{array} \right. \tag{2}\]  
It could be seen that when the rotational frequency \(f\) of the motor was a constant value, the induced current generated by the \(X\), \(Y\) and \(Z\) electrodes was linear with the external electric field. Figure 1: A photograph (**left**) and schematic (**right**) of the three-dimensional atmospheric electric field sensor. Furthermore, the periods of the induced currents in the three directions were identical. The signal was calculated by measuring the induced voltage to determine the electric field intensity. The structure of the sensor causes distortion of the electric field near the sensing electrode, so the output voltage of the sensing electrode in each direction is not only related to the electric field in the direction perpendicular to the electrode, but is also weakly affected by the electric field in other directions. For the three-dimensional electric fields \(E_{x}\), \(E_{y}\), and \(E_{z}\), the measured induced voltage amplitudes of the three electrodes were \(U_{x}\), \(U_{y}\), and \(U_{z}\), respectively. When the electric field intensity in all three directions was 0, the induced voltages generated by the three electrodes were \(U_{x0}\), \(U_{y0}\), and \(U_{z0}\), respectively. Thus:  
\[\left| \begin{array}{c}U_{X}-U_{X0}\\ U_{Y}-U_{Y0}\\ U_{Z}-U_{Z0}\end{array} \right|=\left| \begin{array}{ccc}a_{0}&b_{0}&c_{0}\\ a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\end{array} \right|\left| \begin{array}{c}E_{X}\\ E_{Y}\\ E_{Z}\end{array} \right| \tag{3}\]  
where the matrix of \(a_{i}\), \(b_{i}\), and \(c_{i}\) represented calibration coefficients. To determine the calibration values, a DC voltage was applied to a parallel plate capacitor to generate standard electric fields, which were perpendicular to the \(X\), \(Y\), and \(Z\) electrodes, and the actual output voltage of the sensor was measured. Based on the measured voltage and voltage values, the calibration factor was calculated using Equation (3). The Relationship between a Three-Dimensional Atmospheric Electric Field and Thunderstorm Cloud Charge  
The three-dimensional atmospheric electric field meter installed in the field is shown in Figure 2a. To solve the relationship between the atmospheric electric field at the observation point and the thunderstorm cloud charge, an observation model (Figure 2b) was established. The \(X\)-\(Y\) plane was mounted on the ground; \(A\) (\(x,y,z\)) was an arbitrary observation point; \(B\) (\(x_{0},y_{0},z_{0}\)) was the equivalent charge point of thunderstorm cloud; and the charge of the thunderstorm cloud was \(q\). The horizontal azimuth of the thunderstorm cloud \(B\) relative to the observation point \(A\) was \(\alpha\), the elevation angle was \(\gamma\), and \(A^{\prime}\) and \(B^{\prime}\) were projections of the observation station \(A\) and the thunderstorm cloud \(B\) on the \(X\)-\(Y\) plane, respectively. According to the geometric relationship of Figure 2b, the horizontal azimuth angle \(\alpha\) and the elevation angle \(\gamma\) of the thunderstorm cloud satisfied:  
\[\alpha=\arctan\left|\frac{y-y_{0}}{x-x_{0}}\right| \tag{4}\]  
Figure 2: Observation experiment and model of the three-dimensional atmospheric electric field meter. (a) Installation of a three-dimensional atmospheric electric field meter; (b) observation model. \[\gamma=\arctan\frac{\left|z_{0}-z\right|}{\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{ 0}\right)^{2}}} \tag{5}\]  
The dielectric constants of region 0 above the surface \(X\)-\(Y\) and region 1 inside the earth were \(\varepsilon_{0}\) and \(\varepsilon_{1}\), respectively. The electric field of region 0 was generated by the thunderstorm cloud charge \(q\) and the induced charge of the surface. The induced charge was equivalent to the image charge \(q_{1}\).