xmpl3d10, 10th 'example' data file for CPO3D
Space-charge limited planar diode, zero cathode temperature.
An infinite planar diode cannot be simulated and so it is necessary to add boundary electrodes in this simulation. The average current of the innermost rays is given after 3 iterations with an error of 0.7% (in a total computing time of less than 2 minutes, with a 33 MHz 486). The error becomes 0.1% in later iterations.
The answer is of course well known, but this cannot be a benchmark test because the beam is not infinite in the transverse direction.
(This is a 2-dimensional simulation and so of course can be solved with higher accuracy by CPO2D).
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
The boundary electrodes are cylinders of radius 5 that are given the theoretical potentials, proportional to z**(4/3). The accuracy of the solution is affected by the inability to reproduce the exact boundary conditions.
The cathode and anode are both discs of radius 5, and have voltages 0 and 100 respectively. 20 rays are used from the cathode.
As explained in the relevant note, the depth 'd' of the cathode region should be significantly greater than the mesh spacing 's'. In fact, for a quick calculation we have put 'd' = 1 and 's' = 1 (a larger 'd' would give a larger error in the total current and a smaller 's' would increase the computing time).
3 iterations are called, with a maximum total current density much higher than the theoretical value (because at this stage the program successfully limits the current). The fastest convergence seems to be obtained, in this example, with a damping factor of about 0.4 and an initial cathode current density of about 80% of the theoretical value.
The theoretical Childs Law current density is w*(V**1.5)/L**2, where w=.002334 mA/mm**2. In the present simulation, V=100, L=10, area=pi*25, and the number of reflection planes is 3, so the current before reflections should be 0.2291 mA.
The current given by this program is 0.1846 mA. The error would therefore seem to be 19% However, ignoring the rays nearest to the artificial boundaries (some of which go outside the boundary) and averaging over the 4 innermost rays gives a current equivalent to 0.2316, and hence an error of 1.1% (in a total computing time of less than 2 minutes, with a 33 MHz 486 PC). The errors become much smaller in later iterations.