In order to study the role of contacts in the coherent transport of mesoscopic structure systems, a 2D-1D-2D model of nano-sized single-barrier system is considered, which includes a single-barrier structure conductor with two reservoirs. Basing on the scattering-matrix method and the Thomas-Fermi approximation, we have calculated the transmission probability and the distribution of internal potential when a dc voltage is applied to the system. The results show that: (1) the contacts can produce significant effects on the transmission probability; (2) the behaviors of conductance and the distribution of internal potential differ from the usual transport conductance give by the Kirchhoff's laws. Therefore we conclude that since contacts and the mesocopic systems are quantum coherent, the role of contacts is important for in-depth investigation of the transport in mesoscopic systems.