This document presents a study of a non-linear system for the regulation of the output voltage of a direct current converter of the quadratic reducer type, based on the static and dynamic equations of the system, taking into account its losses through The use of the equivalent series resistance, in its elements, and taking into account the conversion voltage ratio is equal to the square of a duty cycle, has good potential in DC-DC converter applications where a large voltage reduction is important, as in modern automobiles, employing a theoretical development scheme. To create this experimental design, firstly, the quadratic converter switched model is reviewed from its equations of state, the averaged model showing the non-linearity of the system, thereby constituting the linear model, of which later A linearization is applied to the system to obtain the transfer function of the quadratic Buck Converter to make a design from the most important construction variables such as Vg = 380V, Vo = 48V, Po = 500W, F = 100KHz with the purpose to evaluate the design parameters that lead to finding the passive and active components, by calculating capacitances and inductances, from voltage and ripple current of the quadratic Buck system in continuous conduction mode, to constitute a real model with losses, their simulations were carried out using MATLAB ® Simulink software, to ensure the performance of the system.