## 2 Dual-Orthogonal Induction MEMS 3D Electric Field Sensor
### Basic Decoupling Method of Spatial 3D Electric Field Measurements  
A spatial electric field can be decomposed into three orthogonal components in any Cartesian coordinate system. During 3D electric field measurements, the output voltage of the sensor is affected by coupling interference among these components, which degrades measurement accuracy. Assuming that the sensor output noise follows a zero-mean Gaussian distribution, the relationship between the output voltage of the sensing chip \(i\) in the MEMS 3D electric field sensor and the vector electric field can be expressed as  
\[u_{i}-u_{i0}=k_{ix}E_{x}^{\prime}+k_{iy}E_{y}^{\prime}+k_{iz}E_{z}^{\prime}+n_{i} \tag{4}\]  
where \(u_{i0}\) is the output voltage of the sensing chip i in the absence of an external electric field, and \(u_{i}\) is its real-time output voltage. The terms \(E/x\), \(E/y\), and \(E/z\) represent the three orthogonal components of the measured electric field \(\mathbf{E}\) in the sensor coordinate system. The parameters \(k_{ix}\), \(k_{iy}\), and \(k_{iz}\) denote the sensitivity coefficients of sensing chip \(i\) to each electric field component, which depend on the sensor's structural design and the arrangement of the sensing chips. The term \(n_{i}\) represents the Gaussian noise of the sensor output signal, where \(n_{i}\sim N(0,\sigma_{i}^{2})\).  
For a MEMS 3D electric field sensor composed of multiple sensing chips, the relationship between the output voltages of all sensing chips and the electric field components can be expressed in matrix form as follows [20]:  
\[\begin{bmatrix}u_{1}\\ u_{2}\\ \vdots\\ u_{m}\end{bmatrix}-\begin{bmatrix}u_{10}\\ u_{2}0\\ \vdots\\ u_{m0}\end{bmatrix}=\begin{bmatrix}k_{1x}&k_{1y}&k_{1z}\\ k_{2x}&k_{2y}&k_{2z}\\ \vdots&\vdots&\vdots\\ k_{mx}&k_{my}&k_{mz}\end{bmatrix}\begin{bmatrix}E_{x}\\ E_{y}\\ E_{z}\end{bmatrix}+\begin{bmatrix}n_{1}\\ n_{2}\\ \vdots\\ n_{m}\end{bmatrix} \tag{5}\]  
where \(m\) represents the number of sensing chips (\(m\geq 3\)). The first term on the left-hand side is the real-time output voltage vector of the sensing chips, denoted as \(\mathbf{U}\), while the second term is the output voltage vector under zero-field conditions, denoted as \(\mathbf{U}_{0}\). On the right-hand side, the first term is the sensitivity coefficient matrix, denoted as \(\mathbf{K}\); the second term represents the 3D electric field components in the sensor coordinate system, denoted as \(\mathbf{E}^{\prime}\); the third term is the noise vector, denoted as \(\mathbf{N}\). Theoretically, the sensitivity coefficient matrix \(\mathbf{K}\) uniquely determines the mapping relationship between the output voltage and the measured electric field. Equation (5) can be rewritten in vector form as follows:  
\[\mathbf{U}-\mathbf{U}_{0}=\mathbf{K}\mathbf{E}^{\prime}+\mathbf{N} \tag{6}\]  
Figure 2: Working principle of the sensing structure.  
In practical measurements, the actual electric field components need to be decoupled from the output voltage of the 3D electric field sensor. By constructing a decoupled calibration matrix, the estimated value of the measured electric field \(\hat{\mathbf{E}}^{\prime}\) can be expressed as  
\[\hat{\mathbf{E}}^{\prime}=\mathbf{C}(\mathbf{U}-\mathbf{U}_{0}) \tag{7}\]  
where \(\mathbf{C}\) is the decoupled calibration matrix. By appropriately calculating the matrix \(\mathbf{C}\), the coupling interference among electric field components can be effectively compensated, thereby enabling high-precision 3D electric field measurements.