A physics-based explicit mathematical model for the external voltage-dependent forward dark current in bulk heterojunction (BHJ) organic solar cells is developed by considering Shockley-Read-Hall (SRH) recombination and solving the continuity equations for both electrons and holes. An analytical model for the external voltage-dependent photocurrent in BHJ organic solar cells is also proposed by incorporating exponential photon absorption, dissociation efficiency of bound electron-hole pairs (EHPs), carrier trapping, and carrier drift and diffusion in the photon absorption layer. Modified Braun’s model is used to compute the electric field-dependent dissociation efficiency of the bound EHPs. The overall net current is calculated considering the actual solar spectrum. The mathematical models are verified by comparing the model calculations with various published experimental results. We analyze the effects of the contact properties, blend compositions, charge carrier transport properties (carrier mobility and lifetime), and cell design on the current-voltage characteristics. The power conversion efficiency of BHJ organic solar cells mostly depends on electron transport properties of the acceptor layer. The results of this paper indicate that improvement of charge carrier transport (both mobility and lifetime) and dissociation of bound EHPs in organic blend are critically important to increase the power conversion efficiency of the BHJ solar cells.