# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 1 Introduction  
The manufacturing errors seriously restrict the performance of the products, and tolerance analysis plays a more and more important role for precision electromechanical product [1; 2; 3]. Based on the tolerance analysis, the effect of each tolerance variable on the product performance can be determined and then the tolerance can be optimized. State-of-the-art tolerance analysis methods can mainly be divided into two categories: the tolerance analysis methods without form error and the tolerance analysis methods considering form errors. (1) Tolerance analysis without form errors. Only translation and rotation errors are considered, while realistic form error (such as bending deformation) is ignored. For example, T-map method proposed by [PERSON] [4] maps all the possible errors of a feature to the hypothetical Euclidean volume point space, which focuses on the boundary of the Euclidean space and does not consider the specific form of a feature. [PERSON] [5] proposed the discrete method to model the geometrical tolerances, all the possible translation and rotation errors are sampled by the displacements of the ideal feature. [PERSON] [6] developed the pose decoupling method for the shaft-hole fit, and studied all the possible pose of the axis within the tolerance domain. These methods have achieved good results when dealing with traditional mechanical products, but encountered bottlenecks when dealing with the phased array antenna [7; 8]. Taking the spaceborne phased array antenna as an example as shown in Figure 1a. The whole array antenna comprises offour subarrays, the explosive view of the first subarray in the whole antenna is illustrated in Figure 1b. The performance of the antenna is determined by the realistic position (the amplitude and phase of each element is considered as ideal) of each element, details can be found in Appendix A. The position errors are mainly determined by the form errors of the reflection plate and the base plate, as a consequence, tolerance analysis without form errors cannot be applied on the antenna design. Many efficient methods such as the matrix model [9] and vector loop method [10] face the same problem. For the convenience of description, a linear array in the whole array antenna (shown in Figure 1a) was analyzed. The discrete horn element was simplified as a dot and the reflection plate was simplified as a curve, as illustrated in Figure 2a. If only the translation and rotation errors of the linear array (purple dash line in Figure 2a) are considered in tolerance analysis, then the maximum power density of the antenna will not change. However, when considering the form errors of the linear array (blue curve in Figure 2a), the maximum power density of the antenna will degenerate seriously. Figure 2b shows the relation between the maximum power density and the straightness considering form errors. The maximum power density depends not only on the size of the tolerance zone, it is also significantly  
Figure 1: (**a**) The CAD (computer aided design) model of the spaceborne phased array antenna. (**b**) The explosive view of the first subarray in the whole antenna. affected by the form error. Based on the engineering experience [3], the flatness of the reflection plate can be roughly estimated according to the wavelength of the antenna:  
\[F\in\left[\frac{1}{30}\lambda,\frac{1}{3}\lambda\right],F:\textit{flatness},\ \lambda:\textit{wavelength}\]  
For the X band spaceborne array antenna, the frequency of the electromagnetic wave is 10 GHz, and the wavelength is 30 mm. Supposing the flatness of the reflection plate is \(F=\lambda/30=1\) mm, then the tolerance zone \([-0.5,+0.5]\) is chosen in Figure 2b. Figure 2: **(a)** The linear array. The black square dot represents the ideal element, and the red circular dot represents the realistic element with form error. The blue curve represents the realistic linear array while the red dotted line represents the ideal array.