This paper presents a detailed study of the high-order superdirectivity of circular sensor arrays, which is aimed at completing the authors' recently proposed analytical superdirectivity model. From the limit expression of the maximum directivity factor, it is shown that the circular arrays possess good potential for directivity improvement. It is found that the sensitivity function used as a robustness measurement can also be accurately decomposed into a series of closed-form sensitivity functions of eigenbeams, similar to the optimal beampattern and its corresponding directivity factor. Moreover, the performance of eigenbeams can be regarded as an indicator of error sensitivity, and the robustness constraint parameters can be estimated easily. Two specific approaches are proposed for obtaining robust superdirectivity on the basis of robustness analyses, and their performance is demonstrated experimentally.