The large antenna arrays with hybrid analog and digital (HAD) architectures can provide a large aperture with low cost and hardware complexity, resulting in enhanced direction-of-arrival (DOA) estimation and reduced power consumption. This paper investigates the trade-off between DOA estimation and power consumption in large antenna arrays with HAD architectures. Particularly, the DOA estimation problem of fully-connected, sub-connected (SC), and switches-based (SE) hybrid architectures is formulated into a unified expression, with the compression matrix in a time-varying form. Based on this model, we derive a dynamic maximum likelihood (D-ML) estimator that is suitable for both HAD and conventional fully digital (FD) structures, and the closed-form expression of Cramér-Rao bound (CRB) to evaluate the performance limit of the D-ML estimator for different HAD structures. The theoretical CRB analysis in the single-source case reveals that, the SC structure has the ability to achieve approximately the same performance as the FD structures at DOAs around zero, but suffers from the inherent angle ambiguity because of the antenna grouping. In addition, we propose a dynamic SC (D-SC) structure that is proved to eliminate the angle ambiguity with time-varying phase shifters, and a switch optimization (SWO) algorithm to minimize the CRB of SE structures. Finally, we introduce a new metric, DOA efficiency, to measure the trade-off between the DOA estimation performance and power consumption of different structures. Simulation results verify our theoretical analysis and the superiority of the proposed D-SC structure and the SWO algorithm.