A model for the steady-state operation of an emissive cathode is presented. The cathode, biased negative with respect to a cold anode, emits electrons thermionically and is embedded within a large magnetized-plasma column. The model provides formulas for the spatial shape of the global current system, the partition of potential across the plasma-sheath system, and the effective plasma resistance. The formation of a virtual cathode is explored, and an analytical expression for the critical operating conditions is derived. The model is further developed to include the self-consistent increase in plasma temperature which results from thermionic injection. In a companion paper [S. Jin et al., Phys. Plasmas 26, 022105 (2019)], results from transport experiments in the Large Plasma Device at the University of California Los Angeles are compared with this model, and excellent quantitative agreement is achieved.