# A Hybrid Tolerance Design Method for the Active Phased-Array Antenna
## Appendix C  
Let \(P(\theta,\varphi)\) be the realistic power pattern of the antenna with hybrid geometric errors, assuming the efficiency does not change (efficiency factor \(c\)). The realistic gain of the antenna is [23]:  
\[G=c\cdot\frac{P(\theta,\varphi)_{max}}{P(\theta,\varphi)_{realistic}^{\alpha }}=c\cdot\frac{P(\theta,\varphi)_{max}}{\frac{1}{4\pi}\int_{\varphi=0}^{2\pi} \int_{\theta=0}^{\pi}P(\theta,\varphi)sin\theta d\theta d\varphi}\] (A14)  
where \(\ P(\theta,\varphi)_{max}\) denoting the maximum power density of the realistic antenna, and \(\ P(\theta,\varphi)_{realistic}^{\alpha\beta}\) denoting the average power density of the realistic antenna.  
The average power pattern can be calculated approximately based on Equation (A15).  
\[\begin{array}{lcl}P(\theta,\varphi)_{realistic}^{\alpha\beta}&\approx&\frac{ 1}{4\pi}\sum\limits_{N=1}^{N_{i}}\sum\limits_{M=1}^{M_{i}}P((N-1)\Delta\theta,(M-1)\Delta\varphi)sin((N-1)\Delta\theta)\cdot\Delta\theta\Delta\varphi\\ &=&P(\theta,\varphi)_{0}^{\alpha\beta}+\frac{1}{4\pi}\sum\limits_{N=1}^{N_{i}} \sum\limits_{M=1}^{M_{i}}\Delta P((N-1)\Delta\theta,(M-1)\Delta\varphi)sin((N- 1)\Delta\theta)\Delta\theta\Delta\varphi\\ &=&P(\theta,\varphi)_{0}^{\alpha\beta}+\sum\limits_{N=1}^{N_{i}}\sum\limits_{M= 1}^{M_{i}}H(N)\cdot\Delta P((N-1)\Delta\theta,(M-1)\Delta\varphi)\end{array}\] (A15)  
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