The problem of plasma-current build-up in a tokamak discharge is studied for the case of an expanding plasma column. A model is considered in which the thermal conductivity is pseudo-classical and a particle flux is assumed at the plasma boundary such that the density in the column remains constant. A class of analytical solutions for the evolution in time of the current and temperature profiles is found which reduces the system of equations to a system in only one variable. These solutions are numerically analysed and relations are found between several plasma and discharge parameters, as the expanding time, the value of q, the net energy load on the limiter, the poloidal β, the transformer flux, etc. The appropriate normalizations of these parameters and their relations allow the discussion of different characteristics of the discharge and of its optimization with respect to the presence of the skin effect and to the energy load on the limiter. Good profiles are obtained for an appropriate choice of parameters and of the time laws for limiter motion and current rise. A numerical code is also used to extend the solutions out of the particular class considered and to test the results.