Linearization of AC power flow is crucial for simplifying the analysis and optimization of power systems, enabling more efficient and scalable computation for network operation and control. In this paper, a Neumann series expansion based linear bus injection model (NS-BIM) of three-phase AC power flow of active distribution network (ADN) is proposed. First, the full-nonlinear three-phase BIM is built based on AC power flow equations. Then, the mathematical derivation of the proposed NS-BIM is provided. Moreover, the approximation error analysis is given which guarantee the robustness of the NS-BIM. Finally, numerical case studies on different benchmark ADN systems are carried out to verify the accuracy of the NS-BIM in comparing with other representative linear BIMs.