# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 6 Conclusions  
A hybrid tolerance design method was proposed in this paper, and the position tolerance of the discrete elements and the flatness of the continuous reflection plate were optimized simultaneously. 1. The hybrid geometric errors of the array antenna were analyzed, and the surrogate model of the electrical performance and hybrid geometric errors were established based on the second-order Taylor expansion. 2. The expectation and the variance of the power pattern of the array antenna was derived, then the direct and explicit relationship between the performance and the hybrid tolerances was identified. 3. The proposed method was applied on an active phased-array antenna, and the hybrid tolerances were optimized finally. Numerical simulation results prove the effectiveness of the method. Conceptualization, Z.L. and G.S. ; methodology, G.S., Z.L. ; software, G.S. ; validation, G.S., C.Q. and Z.L. ; formal analysis, G.S. ; investigation, J.T. ; resources, J.T. ; data curation, C.Q. ; writing--original draft preparation, G.S. ; writing--review and editing, G.S. ; visualization, C.Q. ; supervision, Z.L. ; project administration, J.T. ; funding acquisition, J.T. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China, grant number 51490663, 51935009 and 51875517; Zhejiang Provincial Natural Science Foundation of China (grant No. LY20E050015). The authors declare no conflict of interest. The power pattern [13; 22] of the array antenna is defined as Equation (A1). \[ \begin{array}{l}P(\theta,\varphi)=F_{R}^{2}+F_{I}^{2}\\ F_{R}=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}\left|a_{ij}\right|cos\left\{ KC_{ij}\right\}\text{, }F_{I}=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}\left|a_{ij}\right|sin\left\{ KC_{ij}\right\}\end{array} \] (A1)  
Figure 13: The simulation results of the realistic gain of the antenna with the optimized tolerances. Only 100 sets of results are drawn in the figure for clarity. Only the array factor was considered and the effect of the horn antenna was ignored. where \(C_{ij}=u\cdot x_{ij}+v\cdot y_{ij}+w\cdot z_{ij},\ u=cosqsin\theta,v=sinqsin\theta,w= cos\theta\) and \(k\) is a constant related with the wavelength \(k=2\pi/\lambda\). \(a_{ij}\) is the amplitude, \((\theta,\varphi)\) is the spatial angle and \(\left(x_{ij},y_{ij},z_{ij}\right)\) is the position coordinate of each element, as shown in Figure A1. The first-order derivative of power pattern \(P\) to coordinate \(x\) is listed as Equation (A2). \[ \begin{array}{l}\frac{\partial P(\theta,\varphi)}{\partial x_{ij}}=\frac{ \partial F_{R}^{2}(\theta,\varphi)}{\partial x_{ij}}+\frac{\partial F_{I}^{2} (\theta,\varphi)}{\partial x_{ij}}\\ \frac{\partial F_{R}^{2}}{\partial x_{ij}}=-2k\big{|}a_{ij}\big{|}usin\big{(} KC_{ij}\big{)}\cdot\left[\sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\big{|}a_{pq} \big{|}cos\big{(}KC_{pq}\big{)}\right]\\ \frac{\partial F_{I}^{2}}{\partial x_{ij}}=2k\big{|}a_{ij}\big{|}ucos\big{(} KC_{ij}\big{)}\cdot\left[\sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\big{|}a_{pq} \big{|}sin\big{(}KC_{pq}\big{)}\right]\end{array} \] (A2)  
The second-order derivative of power pattern \(P\) to coordinate \(x\) is listed as Equation (A3). \[ \begin{array}{l}\frac{\partial^{2}P(\theta,\varphi)}{\partial x_{ij}^{2}}=2k ^{2}u^{2}\Bigg{(}a_{ij}^{2}cos\big{(}2 kC_{ij}\big{)}-\big{|}a_{ij}\Bigg{[}\\ \sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\big{|}a_{pq}\big{|}cos\big{(}KC_{pq }\big{)}\Bigg{]}cos\big{(}KC_{ij}\big{)}\\ \frac{\partial^{2}F_{I}^{2}}{\partial x_{ij}^{2}}=2k^{2}u^{2}\Bigg{(}a_{ij}^{ 2}cos\big{(}2 kC_{ij}\big{)}-\big{|}a_{ij}\Bigg{[}\\ \sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\big{|}a_{pq}\big{|}sin\big{(}kC_{pq }\big{)}\Bigg{]}sin\big{(}kC_{ij}\big{)}\\ p=1\quad q=1\\ p\
eq i\quad or\;q\
eq j\end{array} \Bigg{)}\] (A3)  
The first-order derivative of power pattern \(P\) to coordinate \(y\) is listed as Equation (A4). \[