# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## 6 Conclusions
\begin{array}{l}\frac{\partial P(\theta,\varphi)}{\partial y_{ij}}=\frac{ \partial F_{R}^{2}(\theta,\varphi)}{\partial y_{ij}}+\frac{\partial F_{I}^{2} (\theta,\varphi)}{\partial y_{ij}}\\ \frac{\partial F_{R}^{2}}{\partial y_{ij}}=-2k\big{|}a_{ij}\big{|}ysin\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 y_{ij}}=2k\big{|}a_{ij}\big{|}vcos\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} \] (A4)The second-order derivative of power pattern \(P\) to coordinate \(y\) is listed as Equation (A5). \[ \begin{array}{l}\frac{\partial^{2}P(\theta,\varphi)}{\partial y_{ij}^{2}}= \frac{\partial^{2}F_{R}^{2}(\theta,\varphi)}{\partial z_{ij}^{2}}+\frac{ \partial^{2}F_{I}^{2}(\theta,\varphi)}{\partial y_{ij}^{2}}\\ \frac{\partial^{2}F_{R}^{2}}{\partial y_{ij}^{2}}=-2k^{2}v^{2}\left\{a_{ij}^{2 }cos\left(2 kC_{ij}\right)+\left|a_{ij}\right|\left[\begin{array}{ccc}\sum &\sum\\ p=1&q=1\\ p\
eq i&or\ q\
eq j\end{array}\right]\left|a_{pq}\right|cos\left(kC_{pq}\right) \right\}\\ \frac{\partial^{2}F_{I}^{2}}{\partial y_{ij}^{2}}=2k^{2}v^{2}\left\{a_{ij}^{2 }cos\left(2 kC_{ij}\right)-\left|a_{ij}\right|\left[\begin{array}{ccc}\sum &\sum\\ p=1&q=1\\ p\
eq i&or\ q\
eq j\end{array}\right]\left|a_{pq}\right|sin\left(kC_{pq}\right) \right\}\\ \end{array} \] (A5)  
The first-order derivative of power pattern \(P\) to coordinate \(z\) is listed as Equation (A6). \[ \begin{array}{l}\frac{\partial P(\theta,\varphi)}{\partial z_{ij}}=\frac{ \partial^{2}F_{R}^{2}(\theta,\varphi)}{\partial z_{ij}}+\frac{\partial^{2}F_{I }^{2}(\theta,\varphi)}{\partial z_{ij}^{2}}\\ \frac{\partial^{2}F_{R}^{2}}{\partial z_{ij}}=-2k|a_{ij}|psi\left(kC_{ij} \right)\cdot\left[\sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\left|a_{pq} \right|cos\left(kC_{pq}\right)\right]\\ \frac{\partial^{2}F_{I}^{2}}{\partial z_{ij}}=2k|a_{ij}|wcos\left(kC_{ij} \right)\cdot\left[\sum\limits_{p=1}^{m}\sum\limits_{q=1}^{n}\left|a_{pq} \right|sin\left(kC_{pq}\right)\right]\end{array} \] (A6)  
The second-order derivative of power pattern \(P\) to coordinate \(z\) is listed as Equation (A7). \[ \begin{array}{l}\frac{\partial^{2}P(\theta,\varphi)}{\partial z_{ij}^{2}}= \frac{\partial^{2}F_{R}^{2}(\theta,\varphi)}{\partial z_{ij}^{2}}+\frac{ \partial^{2}F_{I}^{2}(\theta,\varphi)}{\partial z_{ij}^{2}}\\ \frac{\partial^{2}F_{R}^{2}}{\partial z_{ij}^{2}}=-2k^{2}w^{2}\left\{a_{ij}^{2 }cos\left(2 kC_{ij}\right)+\left|a_{ij}\right|\left[\begin{array}{ccc}\sum &\sum\\ p=1&q=1\\ p\
eq i&or\ q\
eq j\end{array}\right]\left|a_{pq}\right|cos\left(kC_{pq}\right) \right\}\\ \frac{\partial^{2}F_{I}^{2}}{\partial z_{ij}^{2}}=2k^{2}w^{2}\left\{a_{ij}^{2 }cos\left(2 kC_{ij}\right)-\left|a_{ij}\right|\left[\begin{array}{ccc}\sum &\sum\\ p=1&q=1\\ p\
eq i&or\ q\
eq j\end{array}\right]\left|a_{pq}\right|sin\left(kC_{pq}\right) \right\}sin\left(kC_{ij}\right)\right\}\\ \end{array} \] (A7)  
The power pattern of an array antenna is determined by 7 variables. The angle \(\theta,\varphi\), the wavelength \(\lambda\) and the amplitude \(a_{ij}\) of the elements are assumed constant. The variables considered in this paper are the position coordinates of the elements, which are affected by the hybrid errors. Due to the independence of all the variables, the rest of the second-order derivatives equal zero are as listed in Equation (A8). \[ \begin{array}{l}\frac{\partial^{2}P(\theta,\varphi)}{\partial x_{ij}\partial z _{pq}}=0,\frac{\partial^{2}P(\theta,\varphi)}{\partial x_{ij}\partial y_{pq}}= 0,\frac{\partial^{2}P(\theta,\varphi)}{\partial y_{ij}\partial z_{pq}}=0\\ \frac{\partial^{2}P(\theta,\varphi)}{\partial x_{ij}\partial x_{pq}}=0,\frac{ \partial^{2}P(\theta,\varphi)}{\partial y_{ij}\partial y_{pq}}=0,\frac{ \partial^{2}P(\theta,\varphi)}{\partial z_{ij}\partial z_{pq}}=0\\ pq\
eq ij,\ p=1,2,\cdots m,\ q=1,2,\cdots,n\\ i=1,2,\cdots m,\ j=1,2,\cdots,n\end{array}