# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 5 Results  
The proposed hybrid tolerance design method was applied to a spaceborne active phased array antenna. The CAD model is shown in Figure 1. The array antenna is assembled from 4 subarrays, each subarray is composed of \(8\times 8=64\) horn elements. The operating wavelength is \(\lambda=30\) mm, and the element spacing is \(d=0.5\lambda=15\) mm. The initial phase of each element is set as zero and the amplitude of each element is set as one. All the elements are numbered as shown in Figure 8.  
Figure 8: The elements’ numbering. Figure 7: The interval of the power pattern of the \(8\times 8\) subarray with random geometric errors, the interval is predicted based on the Equation (33). The predicted interval in the main lobe area is precise and the predicted intervals in the side lobes are approximate results, because the x and y directional position errors of the antenna were ignored. Details can be found in Appendix A. The first-order derivation and second-order derivation in \((\theta,\varphi)=(0,0)\) were calculated according to Equations (A6) and (A7), since all the amplitudes are set as one, and then all the second order derivatives are the same. \[\frac{\partial P(0,0)}{\partial z_{ij}}=0,\frac{\partial^{2}P(\theta,\varphi)}{ \partial z_{ij}^{2}}=-5.5270\mathrm{e}+06 \tag{34}\]  
Due to the symmetry of the array antenna, all the tolerances of each subarray are the same:  
\[F_{1}=F_{2}=F_{3}=F_{4}=F;\;T_{1}=T_{2}=T_{3}=T_{4}=T \tag{35}\]  
The interval prediction (Equation (33)) was validated through 1000 sets of simulations on the whole array antenna. Simulation results show that the variation range of the power pattern based on Equations (17) and (30) was very accurate, and the experimental results showed that 996 results of 1000 total experiments satisfied the predicted fluctuation interval, as shown in Figure 9.  
The flatness of the reflection plate was discretized to 6 specific values, that is:  
\[F=\left\{\frac{1}{60}\lambda,\frac{1}{30}\lambda,\frac{1}{20}\lambda,\frac{1}{ 15}\lambda,\frac{1}{12}\lambda,\frac{1}{10}\lambda\right\} \tag{36}\]  
The covariance matrix \(\Theta\) of \(\left\{\Delta z_{11}^{c},\cdots\Delta z_{ij^{\prime}}^{c},\cdots\Delta z_{88} ^{c}\right\}\) for each flatness of reflection plate of the subarray was estimated. For example, the covariance matrix corresponding to \(F=0.1\lambda\) was shown in Figure 10. The diagonal elements represent the variance \(\left(\sigma_{ij}^{c}\right)^{2}\) and the rest represent the covariance \(cov\left(\Delta z_{ij^{\prime}}^{c},\Delta z_{pq}^{c}\right)\). Figure 9: Monte Carlo simulation results of the whole array antenna. Each green line represents the realistic power pattern with the random hybrid errors. Only 100 sets of results are drawn in the figure for the clarity. Only the array factor was considered and the effect of the horn antenna was ignored. Obviously, the values of the diagonal elements of the covariance matrix are much bigger than those of the rest, which indicates the correlation between continuous errors of adjacent assembly holes on the reflection plate. For example:  
Figure 8.0  
The 4 subarrays are independent of each other, and they have the same tolerances. Therefore, the covariance matrix \(\Sigma\) of the whole array can be determined easily based on the covariance matrix \(\Theta\) of the subarray plate. \[\Sigma=\left[ \begin{array}{cccc}\Theta&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\Theta&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\Theta&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\Theta\end{array} \right] \tag{38}\]  
where \(\Sigma\) is block matrix and \(\mathbf{0}\) represents the zero matrix.