This paper proposes a more accurate characterization for the global stability of positive linear systems subject to saturation in the actuators. Exploring the description of saturation by means of the ramp function, that is positive in the non-negative octant, and the copositivity of quadratic forms, sufficient conditions for the global stability are given in terms of linear matrix inequalities for continuous- and discrete-time systems. Numerical comparisons with other techniques from the literature based on section conditions are made to illustrate the effectiveness of the new proposed conditions.