Three models regard the arc as a flexible plasma column that is fixed at one end (the spot) but free to flex along its length (see Figure 3.9).
Perhaps the first model to examine arc motion with regard to the arc column was that of Robson and von Engel [27]. In this instance the arc column is assumed to be deflected in the Lorentzian direction and be strongly curved in the vicinity of the cathode. The force then acting upon the cathode spot is the sum of the applied field and the self field generated by the curved current cariying element shown on Figure 3.10. Retrograde motion occurs if the self field exceeds the applied field, this is feasible only if the spot dimensions are small enough to give high enough self fields, in the light of recent experimental work investigating the sub structure of the spot this would seem to be the case (see Section 2.3).
Straight axis
Path along current channel axis in the direction oF the electric current Disturbance Discharge channel »►- \ Retrograde "7 Centre oF curvature
This model explains some interesting features of spot motion. Robson motion of the arc is the tendency of the arc to drift away from the true direction of retrograde motion when the applied magnetic field is non-parallel to the cathode surface [26]. This is explained in terms of the existence of a current component flowing parallel to the cathode surface giving rise to a force perpendicular to the retrograde direction. The absence of retrograde motion when observing a purely thermionic arc is to be expected as the current densities are much smaller in these cases. But perhaps the most interesting observation is effect on the arc velocity of decreasing the electrode separation, this will tend to force the arc column on to the cathode therefore decreasing the radius of curvature of the arc and increasing the retrograde velocity. This is increase in arc velocity with decreasing electrode separation is observed to occur experimentally.
Anode
Retrograde
Curved current
Carrying element Amperian
Cathode
The model of Hong and Allen [25], is based upon that of Robson and von Engel described above. It uses electrodynamic theory to describe the arc column and views the resultant force upon the spot as the sum of forces in the retrograde and Lorentzian directions. The arc column bends in the Lorentzian direction under the influence of an applied magnetic field, this has two main effects; firstly to produce a force on the arc column in the Lorentzian direction and secondly to distort the current density across the spot. The distortion of the current density profile gives rise to a force in the retrograde direction, the greater the applied field the more severely the column is bent and the greater the magnitude of the retrograde force. A limit is reached at which the column will bend no more and further increases in the applied field serve only to increase the force in the Lorentzian direction. This happens until both forces are balanced and any further increase in field results in Lorentzian motion.
This model qualitatively explains a number of features of arc motion including the temperature dependence of arc velocity, the pressure dependence of arc velocity and the material and surface condition dependence of arc motion. Also discussed is the importance of the current density at the spot, using this model retrograde motion is only predicted with current densities of the order of 1012 Am*2, this value is at the upper end of experimentally observed magnitudes for current density.
Finally a model proposed by Schrade [24] calculates the force per unit length acting upon a current carrying channel due to self and applied magnetic fields and comes to several interesting conclusions. In the first instance, the resultant force upon a straight column is zero, all forces being balanced, however if there is a small disturbance of the column resulting in it being bent then the resultant force becomes non-zero. In the case that the resultant force is in the opposite direction to the disturbance then the channel is driven back to its original position and becomes stable again. Alternatively, if the force is in the direction of the perturbation then the column is driven down into contact with the cathode surface, heating the surface there and creating a new spot and extinguishing the old. In the case of an applied magnetic field the conditions for instability in the column
are more favourable in the retrograde direction and new spots are more likely to be created in this direction.
This model also qualitatively explains Robson drift [26] in terms of the direction of the force resultant from the inclined field. This force reinforces a perturbation in the column not in the true retrograde direction but at a slight angle to it, giving rise to motion in this direction.