Exploring Hannan Limitation for 3D Antenna Array

Hannan Limitation successfully links the directivity characteristics of 2D arrays with the aperture gain limit, providing the radiation efficiency upper limit for large 2D planar antenna arrays. This demonstrates the inevitable radiation efficiency degradation caused by mutual coupling effects between array elements. However, this limitation is derived based on the assumption of infinitely large 2D arrays, which means that it is not an accurate law for small-size arrays. In this paper, we extend this theory and propose an estimation formula for the radiation efficiency upper limit of finite-sized 2D arrays. Furthermore, we analyze a 3D array structure consisting of two parallel 2D arrays. Specifically, we provide evaluation formulas for the mutual coupling strengths for both infinite and finite size arrays and derive the fundamental efficiency limit of 3D arrays. Moreover, based on the established gain limit of antenna arrays with fixed aperture sizes, we derive the achievable gain limit of finite size 3D arrays. Besides the performance analyses, we also investigate the spatial radiation characteristics of the considered 3D array structure, offering a feasible region for 2D phase settings under a given energy attenuation threshold. Through simulations, we demonstrate the effectiveness of our proposed theories and gain advantages of 3D arrays for better spatial coverage under various scenarios.