xmpl2d09, 9th 'example' data file for CPO2D
Space-charge limited planar diode, cylindrical symmetry.
An infinite planar diode cannot be simulated and so it is necessary to add boundary electrodes in this simulation. These are cylinders that are given the theoretical potentials, proportional to z**(4/3). Childs Law is used in the immediate vicinity of the cathode surface, assuming a zero cathode temperature (thermal emission can be treated with CPO3D). The average current of the innermost rays is given after 3 iterations with an error of 3.7% (in a total computing time of 48 sec, with a 66 MHz 486 laptop PC). This error decreases to 1.6% in later iterations.
Useable only with the full, space-charge version of CPO2D.
The exact answer is of course well known, but this cannot be a benchmark test because the beam is not infinite in the transverse direction.
Detailed description:
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 cathode and anode are both discs of radius 5 mm, and are separated by 10 mm and have voltages 0 and 100 respectively. 15 rays start from the cathode, and a 'type 2' subdivision is used to give a constant area for the subdivisions. The enclosing cylinders have a radius of 5 mm.
As explained elsewhere the depth 'd' of the cathode region should in principle be significantly greater than the mesh spacing 's'. In fact we have put 'd' = 1 and 's' = 0.3.
The theoretical Childs Law current density is w*(V**1.5)/L**2, where w=.0023340 mA/mm**2. In the present simulation, V=100, L=10, area=pi*25, so the current should be 1.8331 mA.
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 0.5 and an initial cathode current density of about 80% of the theoretical value.
The current given by this program is 1.5934 mA. The error would therefore seem to be 13% However the rays nearest to the artificial boundaries can be ignored because they cross the boundary (mostly due to the use of the 'mesh' method of ray integration, for speed, and the fact that some of the mesh points lie outside the boundary). Therefore averaging over the 5 innermost rays (which cover 1/3 of the cathode area) gives a current equivalent to 1.9002 mA, and hence an error of 3.7% (in a total computing time of 48 sec, with a 66 MHz 486 laptop PC). This error decreases to 1.6% in later iterations.