The operational instability of a cylinder/wire type superconductor subjected to an instantaneous linear thermal disturbance is investigated. The mechanism of loss and recovery of superconductivity is studied by calculating the instability parameter, Joule heating rate, and convective cooling rate based on a solution of the transient heat conduction equation with or without volumetric heating. The effects of operational current density and wire diameter on stability are investigated. A criterion is developed regarding the maximum current density for stable operation under thermal disturbance and the maximum current density for recovery after quenching. Data of NbTi superconductors are presented and the results are compared with tape/film type superconductors.