In this paper, we propose a method to analytically solve some types of DC lumped-parameter equivalent circuit models for solar cells with S-shaped I-V characteristics measured under illumination. Based on the models proposed previously by other authors, we present the set of equations describing solar cell's terminal current and voltage, derive the analytical solutions of I-V characteristics, and give discussions about the effects from the model parameters on solar cells' I-V characteristics. The comparisons between the proposed solutions and the least square method results illustrate that the solution calculation scheme is not only both accurate and efficient, but also valid in the whole operation regime of solar cells especially for the S-shaped kink in the first quadrant. Finally, the solutions are validated by the reconstructed experimental data to demonstrate that they can be adopted in the practical applications of solar cells. As a result, the feature of the proposed solutions can decrease computation complexity, ease the extraction process of the solar cells' model fitting parameters, and increase simulation accuracy.