Theory of stability of self-sustaining dc discharges at inception with application to negative corona

The inception of self-sustaining dc discharges is analyzed in terms of the bifurcation theory. The existence of a nonphysical solution with negative ion and electron densities must be taken into account in order to identify the bifurcation type. The bifurcation is transcritical for positive and negative corona discharges and, in more general terms, it is expected to be transcritical for all discharge configurations except for the parallel-plate discharge, where the bifurcation is pitchfork. General trends of the bifurcation theory suggest that the corona discharges should be stable immediately after the inception. This conclusion is tested numerically for negative coronas in atmospheric-pressure air in coaxial-cylinder geometry. Two independent approaches have been used: (1) study of linear stability against infinitesimal perturbations with the use of an eigenvalue solver, and (2) following the time development of finite perturbations with the use of a time-dependent solver. The numerical results agree with each other and with the theory. In particular, it is shown that the negative corona is stable, i.e., pulseless, immediately after the ignition. The loss of stability occurs through growth of harmonic perturbations, which subsequently evolve into Trichel pulses, and this happens on the ascending branch of the current-voltage characteristic, contrary to the popular concept of negative differential resistance. Results of the work are of theoretical interest and offer further insights into physics of negative corona discharges.