In YBCO films doped with artificial pinning centers, such as BaZrO3 nanorods or BaCeO3 nanodots, the critical current density, J(c)(B vertical bar vertical bar c), is usually described with the form J(c) similar to B-alpha even though the shape of the J(c)(B)-curve does not really allow this. In the field region just above the low field plateau, the shape of the J(c)(B)-curve (in log-log scale) is rounded and not straight as in undoped films. The exponent alpha is found to decrease from 0.5 to around 0.2 in BaZrO3 doped films and to 0.4 in BaCeO3 doped films. The incompatibility with the curved data and the linear fit has lead to publication of alpha-parameters which are not comparable to each other due to different fitting limits. In this paper we show that it is better to use the Dew-Hughes pinning force F-p(B) = F-p0(B / Birr)(p)(1 - B / B-irr)(q) to describe the field dependence, where alpha approximate to 1 - p. We also show that the p and the roundness of the curve depend on the diameter of the pinning centers, but not, e.g., temperature or dopant concentration. This is shown from measurements of differently doped thin YBCO films and from large scale Ginzburg-Landau simulations. The result should have been expected since the diameters of the dopants are roughly the same size as the coherence length and it has been shown earlier that pinning centers much smaller than the coherence length lead to alpha = 0.5 and those much larger lead to alpha = 1.