We present a model and a rigorous method to calculate the transmission coefficient of silicon micro-rings with generic waveguide cross-section including non-linear effects and self-heating. The method is applied to the design of MRRs in the SISCAP platform with high Q and reduced non-linearity. We demonstrate that the free carrier diffusion in rib waveguides and Shockely-Read-Hall recombination play a fundamental role in reducing the impact of non-linearities of the ring.