We examine the existence condition of the quadratic soliton mode-locked degenerate optical parametric oscillator in the previously unexplored parameter space. We study the nature of the quadratic solitons and divide their dynamics into two distinctive branches, depending on the system parameters. Origin of the quadratic soliton perturbation is identified, and strategy to mitigate its detrimental effect is developed. Frequency comb with terahertz bandwidth and femtosecond pulse duration are attainable in an example periodically poled lithium niobate waveguide resonator. Design rules of the quadratic soliton mode-locking are summarized. The principle can be further extended to other material platforms, making it a competitive ultrashort pulse and broadband comb source architecture at the mid-infrared.