# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 2 Surrogate Model of the Electrical Performance and Hybrid Geometric Errors
### Hybrid Error Analysis  
Supposing the array antenna contains \(n\) elements in the \(x\) direction and \(m\) elements in the \(y\) direction. Then the hybrid position error of the elements is defined as Equation (1), as shown in Figure 3. Note that the \(x\) and \(y\) directional position error was ignored, because they can hardly affect the main beam and the first side lobe [15], which reflects the performance of the antenna, and details of this can be found in Appendix A. The influences of x and y directional position errors were ignored, and several researchers have also adopted this approximation [2,15,17].  
\[\Delta z_{ij}=\Delta z_{ij}^{d}+\Delta z_{ij}^{c}\,i=1,2,\cdots m,j=1,2,\cdots n \tag{1}\]  
Figure 3: The schematic diagram of hybrid position error of the subarray.  
The horns were bolted to the reflection plate, the position (especially the z directional position) of the horn elements is affected by the manufacture error and assembly deformation, as shown in Figure 4.  
The discrete errors \(\Delta z_{ij}^{d}\) are independent of each other, and they are assumed as a normally distributed variable.  
\[\Delta z_{ij}^{d}\sim N\left(0,\left(\sigma_{ij}^{d}\right)^{2}\right) \tag{2}\]  
For elements in the same subarray, the distribution parameters are the same naturally. Because the position tolerance T of elements in the subarray is the same, the standard deviation is defined as based on the 3-sigma theory:  
\[\sigma_{ij}^{d}=\frac{1}{6}T \tag{3}\]  
The continuous errors in the z direction are mutual dependent, they are assumed as multi-dimensional normally distributed variables, where the expectations equal zero and the covariance matrix \(\Theta\) is determined by the manufacturer.  
\[\begin{split}\left\{\Delta z_{1,1}^{c},\cdots\Delta z_{ij}^{c},\cdots\Delta z_{mm}^{c}\right\}\sim N_{m\times n}(\mu_{c},\Theta),\ \mu_{c}=\left[0,0,\cdots,0\right]\\ \Theta=\left[\begin{array}{ccc}\left(\sigma_{1,1}^{c}\right)^{2}&\cdots&cov \left(\Delta z_{1,1}^{c},\Delta z_{ij}^{c}\right)&\cdots&cov\left(\Delta z_{1, 1}^{c},\Delta z_{mm}^{c}\right)\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ cov\left(\Delta z_{ij}^{c},\Delta z_{1,1}^{c}\right)&\cdots&\left(\sigma_{ij} ^{c}\right)^{2}&\cdots&cov\left(\Delta z_{ij}^{c},\Delta z_{mm}^{c}\right)\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ cov\left(\Delta z_{mm}^{c},\Delta z_{1,1}^{c}\right)&\cdots&cov\left(\Delta z _{mm}^{c},\Delta z_{ij}^{c}\right)&\cdots&(\sigma_{mm}^{c})^{2}\end{array} \right]\\ \end{split} \tag{4}\]  
Figure 4: The z directional position error of the horn element.  
The covariance matrix is determined by the flatness of the reflection plate, which is estimated based on the measured data, and details can be found in our previous work [18].  
\[\Theta=\psi(F) \tag{5}\]  
Since the discrete error and continuous error are mutual independent, then the hybrid error \(\Delta z_{ij}\) is a normally distributed variable. The expectation and variance can be determined directly as follows:  
\[\begin{split}& E\Big{(}\Delta z_{ij}\Big{)}=E\Big{(}\Delta z_{ij}^{ d}\Big{)}+E\Big{(}\Delta z_{ij}^{c}\Big{)}=0\\ & D\Big{(}\Delta z_{ij}\Big{)}=D\Big{(}\Delta z_{ij}^{d}\Big{)}+D \Big{(}\Delta z_{ij}^{c}\Big{)}=\Big{(}\sigma_{ij}^{d}\Big{)}^{2}+\Big{(} \sigma_{ij}^{c}\Big{)}^{2}\\ & cov\Big{(}\Delta z_{ij},\Delta z_{pq}\Big{)}=cov\Big{(}\Delta z _{ij}^{c},\Delta z_{pq}^{c}\Big{)}\,ij\
eq pq\end{split} \tag{6}\]