Stable mode-locked operation in a simple normal-cavity-dispersion laser oscillator that consists of only Yb-doped fiber and saturable absorber is studied. The Yb-doped fiber design parameters: group velocity dispersion (GVD), nonlinearity coefficient, bandwidth (Yb-BW), length and gain are considered to be the controlling parameters of the laser cavity. The pulse characteristics such as the temporal width, spectrum and pulse energy as a function of these elements are reported here. A pulse spectrum transition from M-like to Pi-like and then to parabolic-like shape is observed with different values of the controlling parameters which are similar to that has been observed before in a solid-state laser. The stability limits in the domain of the Yb-BW and length are studied. The stability dependence on GVD, nonlinearity coefficient and gain of the Yb-doped fiber are elucidated. Moreover, the pulse instability dyn ics beyond stability limits are found to be similar to that reported before for similariton.