This paper proposes a rigorous stability criterion for the 2 nd order digital phase locked loop (DPLL), with a charge pump phase frequency detector (CP-PFD) component. The stability boundary is determined using piecewise linear methods to model the non-linear nature of the CP-PFD component block. It calculates the control voltage, after a predetermined number of input reference signal sampling periods, to a small initial voltage offset. Using this piecewise linear model an exact closed form stability criterion is proposed for the second order system. The 2 nd order stability boundaries, as defined by the proposed technique, are compared to that of existing linear theory stability boundaries, and display a significant improvement.