Generating a starting-point solution

This help file contains only basic information; consult Sections 2 and 4 of tutorial if you wish to use all simulation and training capabilities of the code.

Normally, you will want to perform simulations for several values of the near-cathode voltage drop U. A solution for the first value should be generated separately. To this end, you need first to specify the near-cathode voltage U (in volts) which corresponds to the starting point desired. The code offers two procedures for generation of the starting-point solution. One of the procedures is automatic and is intended for finding solutions describing the diffuse mode of current transfer. The other procedure is manual and allows finding solutions describing both low- and high-voltage branches of the axially symmetric spot mode.

Generate starting-point solution automatically

In order to use the automatic procedure, you should specify the number Ns of steps in which the starting-point solution will be generated. Set U = 15V, Ns = 2, then press the button Generate starting-point solution automatically. The following information appears in the window:
s =  0.0000 (initial approximation)
           Tcen=3477       Tedge=3477
s =  0.5000
  1             3335             3475
  2             3358             3453
  3             3358             3453
  4             3358             3453
s =  1.0000
  1             3279             3428
  2             3282             3419
  3             3281             3419
  4             3281             3419
The starting-point solution is ready.
Here s is a transition parameter, s = 0 corresponding to the built-in initial approximation and s = 1 to a solution desired. You can see that the solution was successfully generated in 2 steps: there were 4 iterations with s = 0.5 and 4 iterations with s = 1. The printout includes the temperatures at the center and at the edge of the front surface of the cathode, Tcen and Tedge. Both temperatures ate equal to 3477K in the initial approximation. In the converged starting-point solution, Tcen equals 3281K and is somewhat lower than Tedge = 3419K; a feature which is typical for the diffuse mode at low currents and originates in the conditions for heat conduction cooling at the center being more favorable than those at the edge. Note that if Tcen exceeds Tedge in a converged starting-point solution, this is an indication that this solution describes not the diffuse mode of current transfer but rather a spot mode; see Section 4 of this tutorial.

Change Ns to 1 and press the button Generate starting-point solution automatically once again. You can see that you have obtained the same solution. Typically, lower values of Ns are suitable for thin cathodes, high ionization potential of plasma species and high work function. For example, if you try to perform the same procedure with Ns = 2 for a 1.6-mm radius cathode, the following information will appear on the screen:
s =  0.0000 (initial approximation)
           Tcen=3477       Tedge=3477
s =  0.5000
  1              632             5074
  2
The current iteration has been completed. The calculated temperature of some points of the front surface is below Tcol. The code terminated.
Here Tcol is the temperature of the base of the cathode, which is specified in the corresponding field in the tab Step 1. You can see that there were two iterations with s = 0.5. The temperature at the center of the front surface of the cathode after the first iteration was unusually low (632K). After the second iteration, the distortion of the surface temperature distribution became still stronger: there was at least one point on the front surface with a temperature below Tcol, so the code was interrupted. Setting Ns = 3 will fix the problem. Typically, suitable values of Ns are between 1 and 10, however values of Ns as high as 1000 may be needed in extreme cases.

The current-voltage characteristic U(I) of the diffuse discharge is U-shaped, i.e., falls at low currents, then passes through a minimum and starts rising. The above explains how to generate a starting-point solution for a low-current section of the diffuse mode, which is characterized by a falling current-voltage characteristic. In order to signal the code that you want to generate a starting-point solution for the high-current section, which is characterized by a rising current-voltage characteristic, it is sufficient to enter minus before Ns. However, the number of steps in which the starting-point solution is generated should be quite high in this case. For example, setting Ns = -50 for U = 15V results in the interruption of the code after the first iteration at s =  0.08 with diagnostics
The current iteration has been completed. The calculated temperature of some points of the lateral surface exceeds 6500K. The code terminated.
Setting Ns = -100 will fix the problem. Note that the number of steps in which the starting-point solution for the high-current section is generated may be decreased if U is as low as possible: for U = 12V, for example, Ns = -20 is fine.

If the lateral surface of the cathode is electrically and thermally insulated, the above-described (automatic) procedure generates a starting-point solution in one step (at which just one iteration is sufficient). Therefore, if parameter Insulated lateral surface is set equal to .t., then you should set Ns = 1 for the low-current branch of the diffuse mode or Ns = -1 for the high-current branch; the code will issue an error message and terminate if the specified value of Ns exceeds 1 or is below -1.

Generating starting-point solution from a manually-defined initial approximation

The code provides the possibility of specifying an initial approximation manually. This approximation is governed by three parameters: the temperatures at the center and at the edge of the front surface of the cathode, Tcen and Tedge, and a “spot radius” Rs. On the front surface of the cathode, the temperature varies between Tcen and Tedge exponentially in r, or, more precisely, the heat flux potential varies exponentially:

ψ = C1exp(-r/Rs) + C2,

where C1 and C2 are constants defined by the requirements

T = Tcen at r = 0,          T = Tedge at r = Ra

(here Ra is the cathode radius). On the lateral surface, the heat flux potential varies linearly in z between values corresponding to Tedge and Tcol.
 
The code provides the possibility of damping, which is in some cases critical for attaining convergence of iterations. Damping amounts to averaging the result of each iteration with the result of the previous one, with weights equal to (1-d) and, respectively, d, where d is an adjustable parameter, the so-called damper. d varies between 0 (no damping) and values close to 1 (heavy damping; only small changes of a solution between two successive iterations are allowed).