Exploring a Directional Measurement Method of Three-Dimensional Electric Field Intensity in the Atmosphere

Abstract

The traditional process of measuring a three-dimensional (3D) electric field in the atmosphere may suffer from limited accuracy of the electric field component intensity due to multiple factors, such as the 3D electric field sensor’s pose change due to the sensor-carrier motion and the non-parallel positions between the 3D electric compass and field sensor’s measurement axes. Thus, this paper proposes a modified 3D electric field directional decomposition algorithm to overcome this problem. Specifically, the developed solution utilizes a parameter identification and fine-tuning scheme that solves the inaccurate measurement problem caused by the non-coaxial 3D positions between the electric field sensor and the compass. The measurement results highlight that the proposed method eliminates the instability of the electric field intensity measurement errors caused by the 3D electric field sensor changes because the measurement accuracy of the corrected electric field component intensity does not change with the sensor’s rotation angle. Our technique reduces the measurement error of the electric field intensity originating from the non-parallel position 3D electronic compass and field sensor measurement axes, further improving the measurement accuracy of the atmospheric field intensity.

1. Introduction

Atmospheric electric field intensity detection is of great significance, ensuring the safe launch of aircraft and rockets and the safe flight of helicopters. Therefore, three-dimensional (3D) electric field detection technology has essential applications in the aerospace industry and beyond [1,2]. In [3], a hybrid atmospheric electric field meter that was used to simultaneously measure the vertical and horizontal components of the atmospheric electric field on the ground was studied. An altitude correction method for atmospheric electric field data, which can measure the vertical component and polarity of the atmospheric electric field was proposed [4,5]. An aerial 3D electric field detection system comprising sensors, transmitters, receivers, and ground signal processing units was developed, which can effectively identify the aerial electric field vector signals [6]. A miniature foldable 3D electric field sensor based on a flexible substrate was proposed [7]. The sensor structure effectively eliminated the coupling effect, reduced the measurement error, and accurately measured the 3D electric field. Due to the sensor’s small size, this setup is deployable in small unoccupied aerial vehicles (UAVs) for onboard measurements. A new type of atmospheric electric field sensor based on a Micro-Electro-Mechanical system (MEMS) chip has also been proposed [8]. Through adoption of the sensing electrode of an axisymmetric structure, reducing the coupling interference between the axes is feasible. Furthermore, this structure’s simplicity and small size allow its deployment on UAVs. Ref. [9] proposed a 3D electric field measurement method utilizing a coplanar decoupling structure, where the sensing unit had a vibration structure based on MEMS technology. This setup has a simple structure and eliminates coupling between the electrodes, thus improving the measurement accuracy. In [10], the authors proposed a measurement scheme for digitally metering the atmospheric electric field based on FPGA. The digital signal processing circuit forms the core to accurately measure the electric field amplitude and assess its signal polarity.

Field mill-type 3D electric field sensors are commonly used instruments for measuring the electric field intensity; the stator, rotor structure, and the circuit model in field mill-type 3D electric field sensors were optimized in [11], the power consumption of the circuit system was reduced, and the the bandwidth improved. In [12], A low-power atmospheric electric field monitoring device was designed, powered by photovoltaic power generation and storage battery, and necessary data support for lightning monitoring and early warning were provided. The structure of the field mill-type 3D electric field sensors was improved in [13], a small three-dimensional electric field sensor was studied, and the electric field intensity was measured more accurately. In addition, an airborne electric field-measuring instrument was studied in [14]. This instrument was mainly used for high-altitude electric field measurement [15], but this measurement method does not consider the problem of insufficient measurement accuracy of the electric field intensity component caused by the movement of the carrying platform of the 3D electric field sensor in airborne mode.

In the above literature, the field mill-type 3D electric field sensor is the core of the measurement system, and the electric field component is measured by the sensing electrode in a local coordinate system established by the 3D electric field sensor. When measuring the atmospheric 3D electric field, the related sensor moves with the mobile platform, e.g., a drone. At this time, the position, attitude, and orientation of the sensor’s sensing electrode will continuously change. Thus, the electric field sensor cannot accurately measure the atmospheric 3D electric field along the geographic coordinate system’s axes. In order to solve the measurement problem accuracy, [16] developed a wireless directional measurement system that combined a 3D electric field sensor with a 3D electronic compass and a GPS to measure the components of the atmospheric electric field along the geographic coordinate axes. However, in the actual measurement process, the inevitable misalignment between the 3D electric field sensor and the electronic compass coordinate systems imposes a mismatch between the electric field measurement after conducting 3D electric field directional decomposition and the actual atmospheric electric field intensity.

In order to overcome the coaxial error per axis presented above, this paper proposes a modified directional decomposition model for the atmospheric 3D electric field that relies on parameter identification. The measurement results reveal that the suggested method can revise the atmospheric electric field’s directional decomposition model by solving a set of nonlinear equations and identifying the angles between the axis pairs of the 3D electric field sensor and the 3D electronic compass coordinate systems. After revision of the model, the measurement accuracy of the electric field components does not change with the sensor’s rotation angle, reducing the related electric field intensity measurement error due to the coaxial error presented above. Thus, our strategy further improves the measurement accuracy of the atmospheric electric field intensity.

2. Directional Decomposition Model of 3D Atmospheric Electric Field

The atmospheric electric field is a 3D field of vectors, with the 3D electric field intensity denoted as ${\overrightarrow{E}}_0$. Given that the three measurement axes of the 3D electric field are mutually perpendicular, the 3D electric field vectors in any measurement direction can be decomposed into three mutually perpendicular components, i.e., ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$, which determine the local coordinate system of a 3D electric field sensor. The components’ intensity changes with the position of the sensing electrode within the 3D electric field sensor.

A 3D electric compass is added to measure the intensity of the atmospheric electric field components. When the three axes of the 3D electric compass are parallel to the corresponding 3D electric field axes X, Y, and Z, the pitch α, roll β, and yaw θ represent the angles between the electric field components ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ and the geographic coordinate system axes can be measured using the 3D electric compass (Figure 1).

The directional decomposition model of the 3D electric field measures the intensity of the electric field components along the axes of the geographic coordinate system. For this measurement, first, the three orthogonal components ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ are decomposed in the local coordinate system of the electric field sensor. Then, along the pitch angle α, the three components in the geographic coordinate system, ${\overrightarrow{E}}_X^{\prime }$, ${\overrightarrow{E}}_Y^{\prime }$, ${\overrightarrow{E}}_Z^{\prime }$, are obtained:

$$\{\begin{array}{c}{\overrightarrow{E}}_X^{\prime }={\overrightarrow{E}}_X\\ {\overrightarrow{E}}_Y^{\prime }={\overrightarrow{E}}_Y\cos \alpha +{\overrightarrow{E}}_Z^{\prime }\sin \alpha \\ {\overrightarrow{E}}_Z^{\prime }={\overrightarrow{E}}_Z\cos \alpha -{\overrightarrow{E}}_Y^{\prime }\sin \alpha \end{array}$$

(1)

Next, the three components from the electric field sensor are decomposed along the roll angle β, obtaining the three components ${\overrightarrow{E}}_X^{″}$, ${\overrightarrow{E}}_Y^{″}$, and ${\overrightarrow{E}}_Z^{″}$ in the geographic coordinate system:

$$\{\begin{array}{c}{\overrightarrow{E}}_X^{″}={\overrightarrow{E}}_X^{\prime }\cos \beta -{\overrightarrow{E}}_Z^{\prime }\sin \beta \\ {\overrightarrow{E}}_Y^{″}={\overrightarrow{E}}_Y^{\prime }\\ {\overrightarrow{E}}_Z^{″}={\overrightarrow{E}}_Z^{\prime }\cos \beta +{\overrightarrow{E}}_X^{\prime }\sin \beta \end{array}$$

(2)

Finally, the three components from the electric field sensor are decomposed along the yaw angle θ, obtaining ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, and ${\overrightarrow{E}}_Z^{‴}$ in the axes of the geographic coordinate system:

$$\{\begin{array}{c}{\overrightarrow{E}}_X^{‴}={\overrightarrow{E}}_X^{″}\cos \theta +{\overrightarrow{E}}_Y^{″}\sin \theta \\ {\overrightarrow{E}}_Y^{‴}={\overrightarrow{E}}_Y^{″}\cos \theta -{\overrightarrow{E}}_X^{″}\sin \theta \\ {\overrightarrow{E}}_Z^{‴}={\overrightarrow{E}}_Z^{″}\end{array}$$

(3)

From Equations (1)–(3), the three electric field components ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ can be transformed from their local sensor coordinate system to the newly decomposed yaw, pitch, and roll components, represented by ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, ${\overrightarrow{E}}_Z^{‴}$:

$$\left[\begin{array}{c}{\overrightarrow{E}}_X^{‴}\\ {\overrightarrow{E}}_Y^{‴}\\ {\overrightarrow{E}}_Z^{‴}\end{array}\right]=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \beta & 0& -\sin \beta \\ 0& 1& 0\\ \sin \beta & 0& \cos \beta \end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \alpha & \sin \alpha \\ 0& -\sin \alpha & \cos \alpha \end{array}\right]\left[\begin{array}{c}{\overrightarrow{E}}_X\\ {\overrightarrow{E}}_Y\\ {\overrightarrow{E}}_Z\end{array}\right]$$

(4)

The directional decomposition model of the 3D electric field along the axes of the geographic coordinate system is presented in Equation (5).

$$\{\begin{array}{c}{\overrightarrow{E}}_X^{‴}={\overrightarrow{E}}_X\cos \theta \cos \beta +{\overrightarrow{E}}_Y(\cos \alpha \sin \theta +\sin \beta \cos \theta \sin \alpha )+{\overrightarrow{E}}_Z(\sin \alpha \sin \theta -\sin \beta \cos \theta \cos \alpha )\\ {\overrightarrow{E}}_Y^{‴}=-{\overrightarrow{E}}_X\sin \theta \cos \beta +{\overrightarrow{E}}_Y(\cos \alpha \cos \theta -\sin \beta \sin \theta \sin \alpha )+{\overrightarrow{E}}_Z(\sin \alpha \cos \theta +\sin \beta \sin \theta \cos \alpha )\\ {\overrightarrow{E}}_Z^{‴}={\overrightarrow{E}}_X\sin \beta -{\overrightarrow{E}}_Y\cos \beta \sin \alpha +{\overrightarrow{E}}_Z\cos \beta \cos \alpha \end{array}$$

(5)

3. The 3D Electric Field Intensity Measurement System

The measurement system of a 3D electric field adopts a modular design approach, mainly comprising the GPS module, 3D electric compass, power supply, and WiFi module. During the measurement of the 3D electric field intensity, the current signals $i_X$, $i_Y$, and $i_Z$ output from the sensing electrodes of the 3D electric field sensor pass through the signal processing circuits, i.e., I–V transformation, amplification, and filtering circuits, and are converted into the voltage signals $u_X$, $u_Y$, and $u_Z$. Then, we decouple the intensity ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ of the electric field components, which are in the local coordinate system of the electric field sensor. The opto-electrical module measures the direction of the electric field components, while the position of the electric field is obtained via a GPS module, and the 3D electric compass measures the UAV carrier yaw θ, pitch α, and roll β. Then, through the directional decomposition model of the 3D atmospheric electric field, we obtain the intensity of the 3D atmospheric electric field components along the axes of the geographic coordinate system, ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, and ${\overrightarrow{E}}_Z^{‴}$. The corresponding measurement method and system are illustrated in Figure 2 and Figure 3, respectively.

4. Calibration Test of the 3D Electric Field Sensor

Before the measurement of the 3D electric field intensity, the sensor must be calibrated. Currently, a standard uniform electric field through the electric field box is generated, and then the induced voltage is measured by three orthogonal sensing electrodes in the 3D electric field sensor within a known uniform electric field. The magnitude of the induced voltage in different directions is measured to infer the inter-electrode coupling coefficient. The electric field calibration system mainly comprises a high-voltage DC power supply, parallel plates, and a host computer, as depicted in Figure 4.

A DC voltage is applied to the parallel plates through a high-voltage DC power supply, and the distance between the plates is adjusted to generate an adjustable and uniform electric field. The three sets of sensing electrodes in the 3D electric field sensor are perpendicular to the electric field. The electric field intensity can be adjusted by changing the DC power supply voltage, and then the voltage amplitudes generated by the three sets of sensing electrodes are measured and recorded. The voltage signal amplitude representing the electric field intensity is obtained according to Equation (6), revealing the relationship between the measured values and the sensitivity coefficient matrix.

$$U=KE$$

(6)

where $E$ is the electric field intensity component, $U$ is the voltage signal, and $K$ is a 3 × 3 sensitivity coefficient matrix, which is a constant matrix defined as

$$K=U{(E^{\mathrm{T}}E)}^{-1}{E}^{\mathrm{T}}$$

(7)

In the decoupling process of the 3D electric field sensor, the output voltage amplitude of the sensing electrode is the input variable, and the 3D electric field intensity component is the output variable, expressed as

$$E=CU$$

(8)

where C is the decoupling calibration matrix $C={(K^{\mathrm{T}}K)}^{-1}{K}^{\mathrm{T}}$ that adopts the least squares fitting method of multivariate functions. The calibration method for the data on the other two axes is similar (just changing the position of the sensing electrode of the electric field sensor). By substituting the sampled data into Equation (8), we obtain the decoupling calibration matrix:

$$C=\left[\begin{array}{ccc}36.3665& -10.019& -1.07461\\ -7.85491& 37.9304& -31.5856\\ -6.84204& -11.9843& 103.8711\end{array}\right]$$

(9)

The intensity of the 3D electric field components is

$$\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=C\left[\begin{array}{c}{U}_x\\ U_y\\ U_z\end{array}\right]=\left[\begin{array}{ccc}36.3665& -10.019& -1.07461\\ -7.85491& 37.9304& -31.5856\\ -6.84204& -11.9843& 103.8711\end{array}\right]\cdot \left[\begin{array}{c}{U}_x\\ U_y\\ U_z\end{array}\right]$$

(10)

where $U_X$, $U_Y$, and $U_Z$ are the amplitudes of the output voltages $u_X$, $u_Y$, and $u_Z$ from the sensing electrodes of the electric sensor, which is perpendicular to the X, Y, and Z axes in the 3D electric field. Additionally, ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ are the components along the orthogonal axes of X, Y, and Z, which constitute the local coordinate system of the electric field sensor.

5. Parameter Identification of Non-Coaxial Error of 3D Electric Field Sensor and Compass

Theoretically, when the 3D electric field intensity in space is constant, the theoretical intensity values of the electric field components along the geographical coordinate system axes are fixed and independent of the rotation angle of the 3D electric field sensor. Therefore, the electric field component intensity is obtained after the decomposition presented in Equation (5) and should remain constant.

However, the absolute coaxial alignment between the 3D electric field sensor and the 3D electronic compass cannot be achieved when installing the sensor measurement system. Indeed, there is always an angle between the three axes of two coordinate systems. Suppose the angles between the axes X, Y, and Z of the 3D electronic compass and the 3D electric field sensor are α₀, β₀, and θ₀, respectively. The traditional 3D electric field directional decomposition model is presented in Equation (5). After the correction, the intensity of the electric field along the three axes of the geographic coordinate system ${\overrightarrow{E}}_X^{(4)}$, ${\overrightarrow{E}}_Y^{(4)}$, and ${\overrightarrow{E}}_Z^{(4)}$ is

$$\{\begin{array}{c}\begin{array}{l}{\overrightarrow{E}}_X^{(4)}={\overrightarrow{E}}_X\cos (\theta +{\theta }_0)\cos (\beta +{\beta }_0)+{\overrightarrow{E}}_Y(\cos (\alpha +{\alpha }_0)\sin (\theta +{\theta }_0)+\sin (\beta +{\beta }_0)\cos (\theta +{\theta }_0)\sin (\alpha +{\alpha }_0))+\\ {\overrightarrow{E}}_Z(\sin (\alpha +{\alpha }_0)\sin (\theta +{\theta }_0)-\sin (\beta +{\beta }_0)\cos (\theta +{\theta }_0)\cos (\alpha +{\alpha }_0))\end{array}\\ \begin{array}{l}{\overrightarrow{E}}_Y^{(4)}=-{\overrightarrow{E}}_X\sin (\theta +{\theta }_0)\cos (\beta +{\beta }_0)+{\overrightarrow{E}}_Y(\cos (\alpha +{\alpha }_0)\cos (\theta +{\theta }_0)-\sin (\beta +{\beta }_0)\sin (\theta +{\theta }_0)\sin (\alpha +{\alpha }_0))+\\ {\overrightarrow{E}}_Z(\sin \alpha \cos (\theta +{\theta }_0)+\sin (\beta +{\beta }_0)\sin (\theta +{\theta }_0)\cos (\alpha +{\alpha }_0))\end{array}\\ {\overrightarrow{E}}_Z^{(4)}={\overrightarrow{E}}_X\sin (\beta +{\beta }_0)-{\overrightarrow{E}}_Y\cos (\beta +{\beta }_0)\sin (\alpha +{\alpha }_0)+{\overrightarrow{E}}_Z\cos (\beta +{\beta }_0)\cos (\alpha +{\alpha }_0)\end{array}$$

(11)

The solution to ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, and ${\overrightarrow{E}}_Z^{‴}$ in the traditional 3D electric field directional decomposition model is to obtain α, β, θ through the electric compass and $U_X$, $U_Y$, and $U_Z$ through the voltage output of the three sensing electrodes in the 3D electric field sensor. Then, through the decoupling calibration matrix (Equation (11)), the 3D electric field components ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, and ${\overrightarrow{E}}_Z$ can be calculated in the local coordinate system of the electric field sensor. The intensity components ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, and ${\overrightarrow{E}}_Z^{‴}$ of the 3D electric field along the three axes of the geographical coordinate system and under the uncorrected model can be obtained by

$$\{\begin{array}{c}\begin{array}{l}{\overrightarrow{E}}_X^{(0)}={\overrightarrow{E}}_X\cos (\theta +{\theta }_0)\cos (\beta +{\beta }_0)+{\overrightarrow{E}}_Y(\cos (\alpha +{\alpha }_0)\sin (\theta +{\theta }_0)+\sin (\beta +{\beta }_0)\cos (\theta +{\theta }_0)\sin (\alpha +{\alpha }_0))+\\ {\overrightarrow{E}}_Z(\sin (\alpha +{\alpha }_0)\sin (\theta +{\theta }_0)-\sin (\beta +{\beta }_0)\cos (\theta +{\theta }_0)\cos (\alpha +{\alpha }_0))\end{array}\\ \begin{array}{l}{\overrightarrow{E}}_Y^{(0)}=-{\overrightarrow{E}}_X\sin (\theta +{\theta }_0)\cos (\beta +{\beta }_0)+{\overrightarrow{E}}_Y(\cos (\alpha +{\alpha }_0)\cos (\theta +{\theta }_0)-\sin (\beta +{\beta }_0)\sin (\theta +{\theta }_0)\sin (\alpha +{\alpha }_0))+\\ {\overrightarrow{E}}_Z(\sin \alpha \cos (\theta +{\theta }_0)+\sin (\beta +{\beta }_0)\sin (\theta +{\theta }_0)\cos (\alpha +{\alpha }_0))\end{array}\\ {\overrightarrow{E}}_Z^{(0)}={\overrightarrow{E}}_X\sin (\beta +{\beta }_0)-{\overrightarrow{E}}_Y\cos (\beta +{\beta }_0)\sin (\alpha +{\alpha }_0)+{\overrightarrow{E}}_Z\cos (\beta +{\beta }_0)\cos (\alpha +{\alpha }_0)\end{array}$$

(12)

The solution to ${\overrightarrow{E}}_X^{(4)}$, ${\overrightarrow{E}}_Y^{(4)}$, and ${\overrightarrow{E}}_Z^{(4)}$ in the revised 3D electric field directional decomposition model is to identify first the angles between the three corresponding axes of the 3D electric field sensor and 3D electric compass coordinate systems. Since α₀, β₀, and θ₀ are related to the measurement system installation, these are constant and do not change with the rotation of the measurement system in space. When modifying the model, α, β, θ, and ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, ${\overrightarrow{E}}_Z$ are all known. Hence, from a 3D electric field calibration system, we generate a known electric field, i.e., ${\overrightarrow{E}}_X^{(0)}$, ${\overrightarrow{E}}_Y^{(0)}$, and ${\overrightarrow{E}}_Z^{(0)}$, along the axes of the geographic coordinate system. Thus, a set of nonlinear equations is created (Equation (12)) and solved using the fsolve function based on the least squares method to obtain α₀, β₀, and θ₀. The latter angles are then input into Equation (12) to obtain the components of the 3D electric field intensity ${\overrightarrow{E}}_X^{(4)}$, ${\overrightarrow{E}}_Y^{(4)}$, ${\overrightarrow{E}}_Z^{(4)}$ under the revised model, along the axes of the geographical coordinate system.

6. Experiments and Outcome Analysis

In order to verify the accuracy of the revised 3D electric field directional decomposition model, we compare the traditional and revised 3D electric field directional decomposition models in terms of their outcomes, i.e., the intensity of the 3D electric field components along the three axes of the geographical coordinate system. The electric field box is placed horizontally to apply a uniform electric field along the negative direction that is geographically perpendicular to the ground with an intensity of 12 kV/m. The 3D electric field measurement system is placed on a test bench with a scaled compass, as illustrated in Figure 5. To minimize the interference of the rotating table with the measured field, the rotating table is fixed outside the electric field box. An insulating board from the rotating platform is connected to the electric field box, and the 3D electric field sensor measurement system is fixed on the insulating board and near the center of the electric field box. The rotating experimental platform with the 3D electric field measurement system is placed in a uniform electric field (Figure 4). Additionally, the platform body is adjusted so that three axes, X, Y, and Z, of the 3D electronic compass are parallel to the axes of the geographical coordinate system, ensuring that the sensing electrode of the 3D electric field sensor is perpendicular to the ground and parallel to the electric field direction.

The rotating table is adjusted to rotate around the X axis (in the plane of YOZ) clockwise for 90°, the angle between the rotating table and the plane of XOY is δ, and three voltage amplitudes $U_X$, $U_Y$, and $U_Z$ sensed by three sensing electrodes of the 3D electric field sensor are recorded, together with the angles of the electric compass in relation to the geographical coordinate system α, β, and θ, as shown in

Table 1. Voltage amplitude and angle measured by the electronic compass.

δ	${\mathit{U}}_{\mathit{X}}\left(\mathbf{V}\right)$	${\mathit{U}}_{\mathit{Y}}\left(\mathbf{V}\right)$	${\mathit{U}}_{\mathit{Z}}\left(\mathbf{V}\right)$	α	β	θ
0	1.26	0.08	0.08	−0.7	177.44	89.25
10	1.23	0.10	0.18	11.52	176.23	89.16
20	1.16	0.10	0.25	22.1	174.4	89.18
30	1.08	0.12	0.37	32.02	174.94	88.97
40	0.96	0.13	0.51	42.99	176.75	89.68
50	0.83	0.10	0.63	52.48	175.97	88.34
60	0.73	0.12	0.76	61.15	178.1	89.15
70	0.62	0.11	0.88	69.95	179.75	90.64
80	0.50	0.12	1.08	77.24	179.35	88.89
90	0.38	0.10	1.21	88.52	176.96	88.32

.

The voltage amplitudes are measured, while the intensity of the electric field components ${\overrightarrow{E}}_X$, ${\overrightarrow{E}}_Y$, ${\overrightarrow{E}}_Z$ in the local coordinate system of the 3D electric field sensor is obtained from the decoupling model (Equation (10)). The intensity of the 3D electric field components ${\overrightarrow{E}}_X^{‴}$, ${\overrightarrow{E}}_Y^{‴}$, and ${\overrightarrow{E}}_Z^{‴}$ in the geographical coordinate system is obtained by the traditional 3D directional decomposition model. According to the positions of the electric field box and the 3D electric field measurement system, the theoretic electric field intensity components are ${\overrightarrow{E}}_X^{(0)}=0$, ${\overrightarrow{E}}_Y^{(0)}=0$, and ${\overrightarrow{E}}_Z^{(0)}=$ 12 kV/m, respectively. The angles between the three axes of the 3D electric field sensor and the 3D electric compass are calculated using the fsolve function of Equation (12) and the least squares method, obtaining α₀ = −0.02199, β₀ = 0.03625, θ₀ = 0.01902 (unit is radians). Using the angles in the revised model presented in Equation (11) allows us to calculate the 3D electric field intensity components ${\overrightarrow{E}}_X^{(4)}$, ${\overrightarrow{E}}_Y^{(4)}$, and ${\overrightarrow{E}}_Z^{(4)}$ in the geographical coordinate system. The intensity of the electric field components is depicted in Figure 6.

Figure 6a highlights that in the geographic East–West direction, and since this direction is always perpendicular to the direction of the applied electric field, the electric field intensity is always 0. Moreover, the measured electric field intensity component is essentially generated by the coupling effect along the geographic East–West direction of the 3D electric field sensor (X). Therefore, the theoretical electric field intensity is always 0. The electric field intensity obtained by the revised 3D electric field directional decomposition model is smaller than the one calculated by the traditional model. Figure 6b,c reveals that since the sensor rotates vertically in the YOZ plane, the pitch angle gradually increases. In the local coordinate system of the 3D electric field sensor, the electric field intensity measured by the sensing electrode perpendicular to the Y axis of the sensor gradually increases. Accordingly, the electric field intensity measured by the sensing electrode perpendicular to the Z-axis of the sensor gradually decreases, i.e., the measured intensity of the 3D electric field component is related to the sensor pose. In the geographic coordinate system, the theoretical value of the electric field intensity component in the Y direction (geographic North–South direction) is 0 and in the Z direction (geographical North–South direction) is 12 kV/m. After calculating the revised 3D electric field directional decomposition model, we obtain the electric field intensity under the geographic coordinate system, which is closer to the theoretical value of the true electric field intensity, 0 and 12 kV/m, than the traditional model.

Figure 7 illustrates the errors between the electric field component intensity in the geographical coordinate system using the traditional and the revised decomposition models against the theoretical values. Since the electric field component intensities (geographical North–South and East–West) are 0, the error is expressed as a citation error.

Both of these electric field component intensities are erroneously different from the theoretical values while having the same error change trend. However, the error produced by the revised model is generally smaller than that obtained using the traditional model.

Table 2. The relative error between the calculated value of the electric field and the theoretical value obtained by two decoupling calibration methods.

Direction of Electric Field Component	Traditional Directional Decomposition Model		Modified Directional Decomposition Model
	Maximum Error	Mean Error	Maximum Error	Mean Error
X	13.6%	9.1%	9.4%	5.5%
Y	16.8%	11.7%	9.3%	7.0%
Z	9.3%	5.3%	7.9%	4.5%

reports the relative error between the electric field components of both directional decomposition models compared to the actual values. Moreover, Table 2 indicates that the revised 3D electric field directional decomposition model reduces the maximum errors of the electric field components in the geographic directions of East–West and North–South from 13.6% and 16.8% to 9.4% and 9.3% compared with the traditional model. Additionally, the average error decreases from 9.1% and 11.7% to 5.5% and 7.0%. In the loading direction of the electric field, i.e., the direction of the geographic plumb, the maximum error of the electric field component intensity decreases from 9.3% to 7.9%, and the average error decreases from 5.3% to 4.5%. The experimental results reveal that utilizing the revised 3D electric field directional decomposition model improves the measurement accuracy of the electric field component intensity. During the experiment, the electric field direction is along the Z axis. The intensity of the electric field component in the X and Y directions is formed by the coupling effect of the electric field intensity along the Z direction; because of the uncertainty of the coupling effect, the maximum and average errors of measurement in the X and Y directions are larger than those in the Z direction. At the same time, α₀ = −0.02199, β₀ = 0.03625, and θ₀ = 0.01902 were calculated by the proposed model, which showed that β₀ = 0.03625 is larger than both α₀ and θ₀ and there is a large coaxiality error between the geographic North–South direction of the electronic compass and the Y-direction of the electric field sensor, which is is due to the manufacturing process of the measurement system. After correction of the model proposed in this paper, the average error and maximum error of the Y component have the most obvious reduction effect.

7. Conclusions

In the traditional atmospheric 3D electric field measurement process, the measurement accuracy is affected by changes in the sensor’s pose due to the 3D electric field sensor’s and carrier’s motion. This imposes a misalignment between the coordinate systems of the 3D electronic compass and the 3D electric field sensor, resulting in inaccurate measurements. Thus, this paper proposes a revised 3D electric field directional decomposition algorithm appropriate for atmospheric measurements. Specifically, the developed scheme utilizes the spatial 3D electric field directional decomposition model and establishes three linear parameter equations to identify the angles α_0, β₀, and θ₀ between the 3D electric field sensor and the 3D electronic compass. Then, the fsolve function is used for parameter identification, which provides a 3D solution to the insufficient measurement accuracy of the electric field component intensity caused by the coaxial error of the 3D electric field sensor and the 3D electronic compass. The main contributions of this paper are as follows:

(1)

In the traditional 3D electric field directional measurement process, the 3D electric field sensor establishes a three-axis reference coordinate system, which is used to measure the electric field intensity component. However, it is impossible to accurately obtain the atmospheric electric field intensity in the geographical coordinate system. Hence, the proposed revised 3D atmospheric electric field directional decomposition model overcomes the problem of effectively measuring the atmospheric electric field intensity in the geographical coordinate system due to the 3D electric field sensor’s airborne motion.

(2)

Establishing a revised directional decomposition model of the 3D electric field in the atmosphere. The electric field component intensities obtained by the traditional 3D electric field decomposition model and the revised model in the geographical coordinate system are compared and analyzed. The experimental results highlight that, compared with the traditional 3D electric field decomposition model, when considering the non-coaxial error between the 3D electric field sensor and the 3D electronic compass, the maximum errors of the electric field component intensities in the geographical coordinate system (East–West and North–South directions) are reduced from 13.6% and 16.8% to 9.4% and 9.3%, respectively. The average errors are reduced from 16.8% and 11.7% to 5.5% and 4.5%, and the measured relative error of the electric field component along the geographic plumb direction is reduced from 5.3% to 4.5%. Additionally, the average relative error decreases from 9.3% to 7.9%. By adopting the revised model, we minimize the negative impacts on the measurement accuracy of the atmospheric electric field intensity caused by the angles between the axes of the 3D electric field sensor and the 3D electric compass. These angles are primarily incurred by the non-parallel installation of the corresponding devices. Hence, our developed method accurately measures the 3D electric field intensity in the geographical coordinate system.