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A generalized master mode-locking model is presented to characterize the pulse evolution in a ring cavity laser passively mode-locked by a series of waveplates and a polarizer, and the equation is referred to as the sinusoidal Ginzburg-Landau equation (SGLE). The SGLE gives a better description of the cavity dynamics by accounting explicitly for the full periodic transmission generated by the waveplates and polarizer. Numerical comparisons with the full dynamics show that the SGLE is able to capture the essential mode-locking behaviors including the multi-pulsing instability observed in the laser cavity and does not have the drawbacks of the conventional master mode-locking theory, and the results are applicable to both anomalous and normal dispersions. The SGLE model supports high energy pulses that are not predicted by the master mode-locking theory, thus providing a platform for optimizing the laser performance.