# Inversion of a Thunderstorm Cloud Charging Model Based on a 3D Atmospheric Electric Field
## Materials and Methods  
### Principles of Measurement and Calibration  
The physical and structural details of the three-dimensional electric field sensor we developed are shown in Figure 1. It consisted mainly of a rotating shield; an induction electrode perpendicular to the radial \(X\), radial \(Y\), and axial \(Z\) directions; a photoelectric switch; a synchronous blade; and a motor. The sensing electrodes \(X\) and \(Y\) were composed of two opposite rectangular metal sheets distributed on the side of the circular body. The sensing electrode \(Z\) was a metal disk that was equally divided into eight sectors. The three sensing electrodes were perpendicular to each other. When the motor drove the shield to rotate at a constant speed, the electrodes \(X\), \(Y\), \(Z\) and the atmospheric electric field were periodically shielded and contacted, and an induced current was generated on the electrodes. The magnitude of the induced current was proportional to the atmospheric electric field, and the small blades were continuously driven by the motor. The shield rotated synchronously, so that the photoelectric switch generated a square wave signal of the same frequency and the same phase, which was convenient for phase sensitive detection of the induced voltage. The traditional atmospheric electric field meter only had an electrode \(Z\) and a corresponding rotating shield on the bottom of the cylinder. Because the shield rotated periodically on the bottom surface, the \(Z\) electrode was periodically exposed to the vertical electric field, resulting in induced current. This kind of electric field meter could only measure the vertical component of the atmospheric electric field and was suitable for measuring the atmospheric electric field on sunny days, because the horizontal component of the atmospheric electric field was very weak on sunny days. The two pairs of parallel electrodes were added to the side of the cylinder and the shield was extended along the side. When the shield was rotated, the electrodes on both sides and the bottom of the cylinder were periodically exposed to the space electric field, and the induced current was generated at the same time. It was suitable for detecting horizontal and vertical electric fields in thunderstorms. The electrodes in all three directions were composed of two symmetrical metal pieces, which could generate differential induced currents and suppress common mode noise. The \(X\) and \(Y\) electrodes were each composed of two rectangular metal sheets distributed at opposite positions on the sides of the cylinder. The width of each metal piece (chord) corresponded to an arc length of \(1/8\) of the entire circumference. The \(Z\) electrode was composed of two metal sheets, \(A\) and \(B\) (blue fans \(A\) and brown fans \(B\) in Figure 1). The shielding cover rotation frequency was \(f\); the atmospheric dielectric constant was \(\varepsilon_{0}\); the electric field intensity in the three directions were \(E_{x}\), \(E_{y}\), and \(E_{z}\); and the induced charges were \(q_{x}(t)\), \(q_{y}(t)\), and \(q_{z}(t)\). The area of a single metal piece of the \(X\) and \(Y\) electrodes was \(S\). The outer radius of the \(Z\) electrode disk was \(r_{2}\), and the inner radius was \(r_{1}\). With the rotation of the shield, the three sensing electrodes were gradually exposed from time \(0\), and the electrodes were completely exposed at time \(\frac{T}{2}\). During the time \(\frac{T}{2}-T\), the three electrodes were gradually blocked by the shield. Then, the induced currents \(i_{x(y)}(t)\) and \(i_{z}(t)\), generated by one of the metal pieces of the \(X\) (\(Y\)) electrode and by the \(A\) portion of the \(Z\) electrode, respectively, were:  
\[i_{z}(t)=\frac{dq_{z}(t)}{dt}=\left\{ \begin{array}{cc}-4\pi\varepsilon_{0}E_ {z}f({r_{2}}^{2}-{r_{1}}^{2})&t\in(0,\,\frac{T}{2})\\ &4\pi\varepsilon_{0}E_{z}f\big{(}{r_{2}}^{2}-{r_{1}}^{2}\big{)}&t\in\left(\frac {T}{2},T\right)\end{array} \right. \tag{1}\]  
\[i_{x(y)}(t)=\frac{dq_{x(y)}(t)}{dt}=\left\{