We investigate the creation of an arbitrary $kappa$-discontinuity of the Josephson phase in a long Nb-AlO_x-Nb Josephson junction (LJJ) using a pair of tiny current injectors, and study the formation of fractional vortices formed at this discontinuity. The current I_inj, flowing from one injector to the other, creates a phase discontinuity kappa ~ I_inj. The calibration of injectors is discussed in detail. The small but finite size of injectors leads to some deviations of the properties of such a 0-kappa-LJJ from the properties of a LJJ with an ideal kappa-discontinuity. These experimentally observed deviations in the dependence of the critical current on I_inj$ and magnetic field can be well reproduced by numerical simulation assuming a finite injector size. The physical origin of these deviations is discussed.
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