Damping of cathode iterations

As is well known, it is notoriously difficult to achieve stability in calculations involving a cathode.

There tends to be positive feed-back between the ray currents and the space-charges, which can cause uncontrolled oscillations in the currents in successive iterations. It can easily happen that in one iteration there is too much current from the cathode so that in the following iteration the excessive space-charge of the rays causes a large reduction in the field at the cathode and hence gives a current that is too small, and so on.

The user must therefore be very careful.

The CPO programs provide a set of different methods of damping (not yet published) which are controlled by the user through a single global damping factor. Adjustment of the damping factor is vital for avoiding oscillations and obtaining fast convergence.

(The damping works as follows.  At the end of each iteration the program calculates the resulting fields at the cathode segments, then calculates the currents that would come from the segments, then looks at the resulting total current and calculates the ratio R of that to the total current of the previous iteration, then reduces the magnitude of the ratio to 1+(R-1)*d, where d is the damping factor, and finally uses the new ratio to scale the changes in the segment currents, limiting the maximum change to a factor of 2.)

The damping factor must be in the range 0 to 0.98. A small value of the damping factor gives light damping, while the maximum value 0.98 reduces the proposed changes in certain quantities by a factor of 50, giving very heavy damping. A suitable value for the damping factor is usually in the range from 0.25 to 0.5, but sometimes a larger value is required. If the damping is too light then violent oscillations might occur in the iterations, while if it is too heavy the iterations will converge too slowly. For an ordinary simple harmonic oscillator such as a pendulum there is an optimum ‘critical’ damping that gives only one excursion before the oscillator is stopped in the minimum time. If in doubt, decrease the value in steps, and observe when the calculated current starts to oscillate too much. Ideally the current should change smoothly towards its final asymptotic value.

In principle it would be easier if the program could automatically adjust parameters such as the ‘damping factor’ or the ‘total current at each iteration’, but in practice simulations and conditions differ so much that this is not possible. So the user cannot avoid being a vital part of the feed-back loop.