## 4 Coordinate Transformation Model and Calibration System  
### Calibration Coordinate Transformation Model Based on Rotation Matrix  
To calibrate the 3D electric field sensor, it is essential to apply a known standard electric field to establish the relationship between the sensor's output voltage and the applied field. However, in calibration experiments, the standard electric field is typically defined in a global coordinate system (GCS) referenced to the calibration setup, while the relationship between the sensor's output voltage and the input field is established in the sensor's local coordinate system (LCS). Therefore, to accurately express the components of the input electric field in the sensor's coordinate frame, a coordinate transformation model must be established to convert the standard electric field from the GCS to the LCS.  
Assume that the three orthogonal axes of the GCS are denoted as \(OX\), \(OY\), and \(OZ\), while those of the LCS are denoted as \(OX^{\prime}\), \(OY^{\prime}\), and \(OZ^{\prime}\), as illustrated in Figure 3. Initially, the LCS is aligned with the GCS, meaning their coordinate axes coincide completely.  
During the calibration process, the sensor undergoes rotations around its local coordinate axes, causing the LCS to deviate from the GCS. Suppose the sensor rotates sequentially by angles \(\alpha\), \(\beta\), and \(\gamma\) about \(OZ^{\prime}\), \(OY^{\prime}\), and \(OX^{\prime}\) axes, respectively. The rotation directions follow the right-hand rule (i.e., counterclockwise rotation is considered positive when viewed from the positive semi-axis). Under these conditions, the standard electric field \(\mathbf{E}=\left[E_{x}\quad E_{y}\quad E_{z}\right]^{T}\) in the GCS is transformed into its components in the LCS, denoted as \(\mathbf{E^{\prime}}=\left[E^{\prime}{}_{x}\quad E^{\prime}{}_{y}\quad E^{ \prime}{}_{z}\right]^{T}\), through a rotation matrix \(\mathbf{R}\), which is expressed as  
\[\mathbf{E^{\prime}}=\mathbf{R}\mathbf{E} \tag{27}\]  
where \(\mathbf{R}\) is the rotation matrix that describes the coordinate transformation from the GCS to the LCS. The matrix representation of this transformation is given by  
\[\begin{bmatrix}E^{\prime}{}_{x}\\ E^{\prime}{}_{y}\\ E^{\prime}{}_{z}\end{bmatrix}=\begin{bmatrix}\cos\alpha\cos\beta&-\sin\alpha \cos\beta&\sin\beta\\ \sin\alpha\cos\gamma+\cos\alpha\sin\beta\sin\gamma&\cos\alpha\cos\gamma-\sin \alpha\sin\beta\sin\gamma&-\cos\beta\sin\gamma\\ \sin\alpha\sin\gamma-\cos\alpha\sin\beta\cos\gamma&\cos\alpha\sin\gamma+\sin \alpha\sin\beta\cos\gamma&\cos\beta\cos\gamma\end{bmatrix}\begin{bmatrix}E_{x} \\ E_{y}\\ E_{z}\end{bmatrix} \tag{28}\]  
This matrix describes the transformation of the standard electric field components from the GCS to their corresponding representations in the LCS after undergoing rotational changes. By utilizing this coordinate transformation model, the known standard electric field in the GCS can be accurately converted to the LCS, thereby establishing a precise mapping between the input electric field and the sensor's output voltage. This facilitates sensor calibration and decoupling computations.  
Figure 3: Coordinate system transformation model of the sensor.