We consider system of rod eutectic growth under directional solidification for the normal amplitude of the contact angles. The objective of this work is to obtain the uniformly valid asymptotic solutions for steady growth of rod eutectics using the asymptotic expansion method. For the contact angles being the normal order, system of rod eutectic growth will exist a boundary layer at the triple phase junction. According to different length scales, we divide the whole liquid region into three subregions. Solutions in every subregion are derived by asymptotic expansion method. Finally, all the asymptotic expansion solutions are matched in the intermediate regions and the composite solutions are obtained. Results show that the average undercooling in front of both phases are not equal. The dimensional average undercooling exhibits a minimum at the critical eutectic spacing. The critical spacing of SCN-DC is predicted and it is in agreement with the small-spacing stability limit.