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Stability tests on channel cooled superconducting windings have revealed the existence of stationary normal zones as a possible response to transient disturbances. We have approached the phenomenon in analyzing the sets of solutions yielded by the time-independent heat equation of a conductor cooled with unsaturated superfluid helium. In good quantitative agreement with experiment our model confirms the existence of stable states with more or less extended normal zones in well defined regions of the I-B plane. It is concluded that stability of superconducting magnets should be evaluated in taking into account the absolute values given to the lengths of cooled and uncooled conductor sections rather than the average wetted conductor surface.