# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 5 Results
The highest tolerance cost was set as \(g_{0}=0.35\) for the spaceborne active phased-array antenna in Equation (32), then the feasible solutions of the hybrid tolerances were constrained in the area enclosed by the red line in Figure 11.  
The expectation of the \(\Delta P(0,0)\) for different position tolerance \(T\) and flatness \(F\) was calculated based on Equations (17) and (30), as shown in Figure 12a and Table 1. The standard deviation was calculated as shown in Table 2. Results shows that the power pattern is more sensitive to the flatness than the position tolerance. The flatness of the reflection plate plays a more important role in the tolerance design than the position tolerance. Figure 10: The covariance matrix \(\Theta\) of the reflection plate of the subarray (\(F=3\) mm). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Expectation (db)**} & \multicolumn{6}{c}{**Flatness of the Reflection Plate (mm)**} \\ \cline{2-7}  & \multicolumn{2}{c}{**0.5**} & **1.0** & **1.5** & **2.0** & **2.5** & **3.0** \\ \hline
0.50 & 0.023  
_Appl. Sci._**2020**, _10_, 1435  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Standard Deviation**} & \multicolumn{5}{c}{**Flatness of the Reflection Plate (mm)**} \\ \cline{2-7} \multicolumn{1}{c}{\(\sqrt{\text{D}(AP)}\) (db)**} & **0.5** & **1.0** & **1.5** & **2.0** & **2.5** & **3.0** \\ \hline \multirow{7}{*}{Position tolerance of the discrete elements (mm)} & 0.50 & 0.0030 & 0.0119 & 0.0268 & 0.0478 & 0.0727 & 0.1091 \\ \cline{2-7}  & 0.75 & 0.0031 & 0.0120 & 0.0268 & 0.0479 & 0.0728 & 0.1093 \\ \cline{2-7}  & 1.00 & 0.0033 & 0.0121 & 0.0270 & 0.0480 & 0.0730 & 0.1094 \\ \cline{2-7}  & 1.25 & 0.0036 & 0.0123 & 0.0272 & 0.0482 & 0.0732 & 0.1097 \\ \cline{2-7}  & 1.50 & 0.0040 & 0.0126 & 0.0274 & 0.0485 & 0.0735 & 0.1100 \\ \cline{2-7}  & 1.75 & 0.0046 & 0.0130 & 0.0277 & 0.0488 & 0.0738 & 0.1103 \\ \cline{2-7}  & 2.00 & 0.0053 & 0.0134 & **0.0281** & 0.0491 & 0.0741 & 0.1107 \\ \cline{2-7}  & 2.25 & 0.0062 & 0.0140 & 0.0285 & 0.0495 & 0.0746 & 0.1112 \\ \cline{2-7}  & 2.50 & 0.0072 & 0.0146 & 0.0290 & 0.0500 & 0.0 Finally, based on Equations (31) and (32), the position tolerance of the discrete elements and the flatness of the reflection plates were optimized, in the feasible area, the minimum degradation of the maximum power density locates on the point (\(T\), \(F\)), where the flatness is \(F\) = 1.5 mm, and the position tolerance is \(T\) = 2 mm, as shown in Figure 12b. The corresponding maximum power density degradation is 0.25 dB and the standard deviation is 0.03 dB. The performance of the antenna was optimized under the constraint of tolerance cost. For the optimized tolerances, the distribution parameters of the realistic gain were estimated by numerical simulation. In total, 1000 sets of samples of hybrid errors and corresponding gain were calculated, as shown in Figure 13. The expectation of the gain was 25.68 dB (ideal gain was 25.91 dB), and the standard deviation of the gain was 0.027 dB. Figure 12: (**a**). The expectation of the maximum power density degradation for different hybrid tolerances \(T\) and \(F\). (**b**). The constraint of the tolerance cost.