Grid performance models using Design of Experiments (DoE) methods

Transcription

1 Grid perforance odels using Design of Experients (DoE) ethods IEPC Presented at the 29 th International Electric Propulsion Conference, Princeton University, October 31 Noveber, 25 Michael Tartz *, Jan H. Peters, Horst Neuann Leibniz-Institute for Surface Modification, Leipzig, Gerany The Design of Experients (DoE) approach is used to build odels of bealet paraeters as the extracted bealet current, bealet divergence and the perveance liit in dependence on grid paraeters and plasa density eploying our well-approved bealet siulation code. Appropriate polynoial degrees for the input paraeters are deterined which result in a ean deviation of the odels fro additional siulated test data of less than %. The derived odels are obtained at a low coputational effort and are helpful in designing and optiizing ion source extraction systes. Noenclature t scr = screen grid thickness t acc = accelerator grid thickness d = grid hole diaeters (sae for both grids) l g = grid distance n, T e = plasa density, electron teperature p(x 1,x 2, ) = polynoial odel (e.g. bealet current, divergence, etc.) x 1, x 2, = independent paraeters in polynoial odels (e.g. grid paraeter as above) d 1, d 2, = degree of each paraeter x i c ijk = coefficients in polynoial (e.g. c ijk x di i x dj j x dk k ) w 1, w 2, = weights to calculate the coefficients c ikj I. Introduction Gridded ion thrusters are high-efficient electrostatic ion bea sources used for spacecraft propulsion 1. Successful issions proved their applicability for earth-orbit propulsion (e.g. NSSK) as well as for priary interplanetary propulsion 2,3,,5. Broad-bea ion sources, which basically use the sae technologies, find increasingly application in highdeanding ultra-precise surface processing technologies like soothing, polishing of high-precision optical eleents 7, structuring and thin fil deposition. Generally, in order to realize a technological application or a space flight ission a great deal of effort is invested to adapt the ion source to the particular requireents. Modelling of the ion bea generation opens a convenient way to study grid syste properties and proved to be a helpful tool in tailoring the ion bea perforance. At IOM a odu- * ichael.tartz@io-leipzig.de horst.neuann@io-leipzig.de 1 October 31 Noveber, 25

2 lar odelling of the broad-bea properties 9 and grid erosion and lifetie has been developed basing on the bealet trajectory code IGUN 11, and extensively validated. Figure 1 shows as exaple the changing bealet divergence with increasing plasa density and the occurrence of direct ipingeent when excessing the perveance liit. For any technological applications the ion bea current density on the target and its hoogeneity are the critical paraeters. On the bealet level the extracted bealet current and the angular distribution of ion directions are relevant. The latter is characterized by the bealet divergence defined as the half angle of the cone coprising 75% of the bealet ions. In propulsion applications also the lifetie of the grid is iportant which is deterined by the ipact of charge-exchange ions on the grids, by the inset of electron backstreaing and the perveance liit. The optiization of an ion extraction syste (e.g. towards low bea divergence or longer grid lifetie) is coplicated by the large nuber of paraeters (grid geoetry, voltages, plasa paraeters, ass flow properties), which influence the ion extraction and bea properties. Siulations usually yield the bealet properties for a given paraeter set only. In order to coe to a ore general grid design understanding, a very large nuber of siulations has to be perfored varying all of the relevant paraeters which leads to a vast quantity of data and reduces the clearness. Suarising this data in a odel would ake the changes of the bealet properties easier to understand. Figure 1: Bealets at increasing plasa density (upward) siulated with IGUN. In this paper we describe an approach to a general odel of the extracted bealet properties in dependence on the geoetrical grid and plasa paraeters and voltages using the Design of Experients (DoE) ethod. DoE is a systeatic approach to investigate the relationship between input paraeters (as listed above) affecting a process and the output of that process (e.g. bealet current or divergence) by designing a series of structured experients (one experient eans one siulation run here). The relation is approxiated by a polynoial with individual degrees of each paraeter. With DoE axiu inforation about the process can be obtained while iniizing the effort required. Software has been developed which allows to take as any paraeters into consideration as required. For each paraeter the polynoial degree can be defined separately. The software creates the experiental plan, calls the bealet siulations and builds up the odel autoatically. The precision can be checked by additional experients. The outcoe is a nuerical odel, which approxiates the interrelation between the bealet paraeter (e.g. divergence) and all relevant input paraeters. In this paper we give a short introduction into the DoE approach and the DoE-software. First results of the odels of bealet current and divergence in dependence on soe grid and plasa paraeters are presented. A odel giving the perveance liit in dependence on grid paraeters is presented. II. The Design of Experients (DoE) approach Siple Experiental designs and factorial plans where developed in the first half of the last century by Fisher and Box and Wilson 13. They offered a way to estiate the dependence of a easured result p on several paraeters x i fro a relatively sall nuber of experients and were ainly used to increase the efficiency of production processes. With the help of odern coputer technology, the principles of their work can be used to develop ore flexible odels whose coplexity is now ainly liited by the aount of available data. 2 October 31 Noveber, 25

3 A current ipleentation produces polynoials in N paraeters x 1,x 2,...,x N each to an individual degree d 1,d 2,...,d N d 1, d 2,, d N c n N n p x 1, x 2,, x N n, n n n 1, 2, N x 1,, n 2, N The results of the calculations are the coefficients c. The degree d n of each paraeter has to be chosen in advance. Given the nuber, degree and range of the paraeters, an experiental design can be created consisting of a list of experients to perfor (i.e. the points in the paraeter space, where easureents are to be taken) and a factorial plan to calculate the coefficients fro the easured experiental results. Usually, d n + 1 equally distributed values are chosen in the range of each paraeter x n and one experient is perfored for each possible cobination x 1,x 2,...,x N of these values in each paraeter. However, this choice is not andatory as long as still at least d N +1 different values are attained by each paraeter and additional easureents can be included into the design, often iproving the result. The factorial plan can be (and was in earlier applications of DoE) expressed in for of a table containing a colun for each ter (and therefore each coefficient) of the target function and a row for each experiental easureent. The entries of each row of the table are the weights w n of a weighted su by which the corresponding coefficient c can be calculated fro the easureents y(x 1,x 2,...,x N ) c n1, n 2,, n N w n y x 1, x 2,, x N N n w n x The appropriate choice of weights w n is indicated by assuing the easureents y(x 1,...,x N ) to be the values of a polynoial function of the actual for of the target-polynoial p(x 1,...,x N ). So that the weighted su takes the for c n1, n 2,, n N 1 W 1 w n n 1, n 2, d 1, d 2,, d N c n W n n, n d 1, d 2,, d N c n, n N 1, n 2, 1, n 2, N n, n N x N n, n N w n x where equality is attained if all but one of the suands turns zero. Hence the weights for each coefficient can be found as the solution of a syste of linear equations. A progra ipleenting the algoriths of DoE described above and interfacing the bealet siulation software has been developed to autoatically generate, execute and evaluate even ore coplex (> easureents on presently three paraeters) experiental designs. The target function was supposed to describe the dependence of ion bealet current and divergence on the plasa density, geoetrical grid paraeters and voltages. Coparison of the calculated odel with a set of randoly chosen easureents (i.e. bealet siulations) allows to evaluate the accuracy of the odel. III. Models of bealet properties In order to deterine the appropriate odelling degree of each input paraeter, 2-paraeter odels of the bealet current and divergence cobining each geoetrical grid paraeter with the plasa density are build. The larger odels coprising ore paraeters can be created with the estiated degrees assuing that there are no significant interrelations between the paraeters. In order to estiate the precision of each odel, for 5 additional easureent test points the deviation fro the respective odel results and the axiu and ean relative errors were calculated. For soe odels further coparison was done between odel-derived dependencies and siulated ones. Table 1 gives the standard grid paraeter values and the variation range within the odel. A. Two-paraeter odels of extracted bealet current The extracted bealet current is ainly a quadratic function of the hole diaeter d, therefore a degree of 2 was chosen. The current increases alost linearly with the plasa density n, however, in the low-density region (up to 3 October 31 Noveber, 25

4 Table 1: Geoetrical paraeters and voltages of the grid syste used in the odelling paraeter standard value odelling range screen grid thickness t scr accelerator grid thickness t acc 1. - grid hole diaeters (sae for both grids) d grid distance l g screen grid voltage V - accelerator grid voltage -3 V - operating gas argon - plasa density n (T e =3eV) - 1* * 11 c * 11 c -3 ) the siulated current is soewhat above the linear odel (cf. figure 2). This is traced back to the oveent of the plasa sheath with decreasing plasa density fro inside the screen grid hole towards the plasa, which leads to an increasing effective eitting sheath area 1. In order to describe this effect properly a higher degree was chosen for n. The iniu ean relative deviation of the additional test point currents fro the odel is achieved at degree 5 for n and aounts 1.1%. Figure 2 shows the good agreeent between experiental (i.e. siulated) and odeled bealet current obtained with a d 2 n 5 odel. The plot of the relative deviation between odeled and experiental bealet currents in dependence on plasa density and hole diaeter in figure shows that the largest deviation occurs at very low plasa densities and sall holes. However, depending on the ai of the odel a lower degree of n ight be sufficient where, as exaple with d 2 n 3 odel, odel and experiental values differ at a ean deviation of.7% and an axiu of 15% at lower plasa densities and sall hole diaeters. bealet current [µa] 2 Siulation d 2 n 2 odel d 2 n 5 odel plasa density 3.* 11 c * 11 c -3 * c -3 bealet current [µa] 2 siulation d 2 n 2 odel d 2 n 5 odel hole diaeter , 1,5 2, 2,5 3, hole diaeter d [] 2 3 plasa density [ c -3 ] Figure 2: Coparison of bealet currents derived fro odel d 2 n 5 with siulated values; right: dependence on plasa density, left: dependence on hole diaeter (all other paraeter as in table 1). An increased thickness of the screen grid t scr reduces the extracted bealet current approxiately according to 1/t scr. To describe this dependence with a polynoial properly a higher degree of t scr has to be chosen, a good agreeent was achieved with t scr. Again, at lower plasa densities the current shows the sae behavior as discussed above, therefore it sees obvious to apply the sae functional degree as above. Consequently, a t scr n 5 odel yielded a good agreeent with the siulated dependencies with a ean deviation of 2.1%. The largest deviations occur in the lower plasa density region. October 31 Noveber, 25

5 5 5 d 2 n 5 odel l(2) n(5) odel deviation odel-siulation [%] 3 deviation odel-siulation [%] 3 2 2,,15,2,25 hole diaeter [], plasa density [ c -3 ],5, grid separation [], plasa density [ c -3 ] Figure 3: Relative deviation between odeled and experiental bealet currents (projections over n and d shown), left: d-n-odel, right: l g -n-odel. Increasing the grid separation l g has a siilar effect on the current as the screen grid thickness. However, it was found that a l g 2 n 5 odel gives a good approxiation of the siulated dependencies (figure 3). The ean relative deviation aounts 1.1% at a axiu deviation of 11% found at low plasa densities and large grid separations (cf. right plot in figure ). 25 l g =1.1 l g =.5 2 n=2.5* 11 c -3 bealet current [µa] 2 15 l g =1. Siulation l(2) n(2) 5 l(2) n(3) l(2) n() l(2) n(5) 2 3 plasa density [ c -3 ] bealet current [µa] 15 5 n=1* 11 c -3 n=* c -3 Siulation l(2) n(2) l(2) n(3) l(2) n() l(2) n(5),,, 1, 1,2 1, 1, grid separation [] Figure : Coparison of bealet currents derived fro odel l g 2 n 5 with siulated values; right: dependence on plasa density, left: dependence on hole diaeter (all other paraeter as in table 1). B. Two-paraeter odels of bealet divergence The dependence of the bealet divergence on all of these paraeters is ore coplicated than that of the current because of the occurrence of a divergence iniu 15. Furtherore, inset of direct ipingeent on the accelerator grid affects the bealet divergence. The paraeter variation range was chosen so that direct ipingeent occurs only at the periphery and is therefore not particularly treated here. Figure copares the siulated dependence of the bealet divergence on the hole diaeter d and plasa density n with the respective odel results. The agreeent iproves with increasing degree of n. In order to describe the di- 5 October 31 Noveber, 25

6 22 2 n = 5 * c -3 Siulation n = 2. * 11 c hole diaeter 1. Siulation hole diaeter d 2 n 2 odel d 2 n 3 odel d 2 n odel d 2 n 5 odel d 2 n odel d 2 n 2 odel d 2 n 3 odel d 2 n odel d 2 n 5 odel d 2 n odel 1, 1,5 2, 2,5 3, 1, 1,5 2, 2,5 3, hole diaeter [] hole diaeter [] plasa density [ c -3 ] plasa density [ c -3 ] Figure 5: Bealet divergence in dependence on hole diaeter (left) and plasa density (right) derived fro various d i n k odels in coparison with siulated values (all other paraeter as in table 1). vergence iniu properly (usually located in the low plasa density region) a degree of at least 5 for the plasa density is necessary (see botto of figure 2). An increase of the degree of d brings no further significant enhanceent. The ean deviation of the d 2 n 5 divergence odel is 3.9% and 3.7% for the d 2 n odel. The strongest deviation is found at lower plasa densities and larger hole diaeters. The dependence of the divergence on the grid separation is best described by a l g 3 n 5 odel resulting in a ean deviation of 2.%. The screen grid thickness is to be odelled as t scr 2 n 5 to yield a iniu ean deviation of 3.2% (see figure 5) n=5* c -3 1 n=1* 11 c -3 1 t scr =.2 1 t scr = ,2,,, 1, screen thickness [] 1,2,,, 1, screen thickness [] plasa density [ c -3 ] 2 3 plasa density [ c -3 ] 1 1 n=2* 11 c Siulation t(2) n(2) t(2) n(3) t(2) n() t(2) n(5) t(2) n() n=* 11 c t scr =1 Siulation t(2) n(2) t(2) n(3) t(2) n() t(2) n(5) t(2) n(),2,,, 1,,2,,, 1, 2 3 screen thickness [] screen thickness [] plasa density [ c -3 ] Figure : Bealet divergence in dependence on screen grid thickness (left) and plasa density (right) derived fro various t i n k odels in coparison with siulated values (all other paraeter as in table 1). Figure 7 shows the odel divergence plot in dependence on grid hole diaeter and plasa density illustrating the good agreeent with the siulated one. The ain deviations can be seen at low plasa densities and large grid holes as entioned above. Such plots are used to adapt the grid geoetry to application requireents, e.g. to low divergence, accounting for an inhoogeneous plasa density distribution. Whereas the left plot was obtained fro siulations, the right plot is calculated with the odel, which was obtained fro 1 siulations only. October 31 Noveber, 25

7 plasa density [ c -3 ] , 1,2 1, 1, 1, 2, 2,2 2, 2, 2, 3, hole diaeter () 1, 1,2 1, 1, 1, 2, 2,2 2, 2, 2, 3, hole diaeter [] Figure 7: The divergence in dependence on grid hole diaeter and plasa density, left: siulation, right: d 2 n 5 odel. C. Two-paraeter odel of the perveance liit The perveance liit gives the plasa density value (or the perveance value) where the direct ipingeent on the accelerator grid sets in. It is andatory in designing grid systes to stay below this liit even though an inhoogeneous plasa density distribution is taken into consideration. Figure shows the odel of the perveance liit (plasa density) in dependence on the screen grid thickness and the grid separation. As expected, the perveance liit decreases with growing grid distance and decreasing screen grid thickness. The ean deviation between the t 2 l 2 odel and additional test values is 1.7% with a axiu deviation of 5.2% occurring at larger grid distances. Figure 9 shows the perveance liit odel in dependence on the screen grid thickness and grid hole diaeters. With growing hole diaeters the perveance liit decreases. The ean deviation between the t 2 d 2 odel and additional test values is 1.3% with a axiu deviation of %.,,9 perveance liit (plasa density in c -3 ) 5E11 3E11 3, E11 25 screen grid thickness [c],7,,5,,3,2 E11 2E11,,,,,,1,1 deviation odel-siulation [%] ,2,,, screen grid thickness [c],,15,,5 grid distance [c] grid distance [c] Figure : Left: Model t 2 l 2 of the perveance liit in dependence on screen grid thickness and grid distance (all other paraeters as in table 1). Right: Relative Deviation between additional test values and odel. 7 October 31 Noveber, 25

8 ,,9 perveance liit (plasa density in c -3 ) 5 screen grid thickness [c] E11, E11,7 3E11 5E11,,5,,3,2,,,1,1,1,2,22,2,2,2,3 grid hole diaeter [c] deviation odel - siulation [%] 3 2,2,, screen grid thickness [c],,,,15,3,25,2 grid hole diaeter [c] Figure 9: Left: Model t 2 d 2 of the perveance liit in dependence on screen grid thickness and grid hole diaeter (all other paraeters as in table 1). Right: Relative Deviation between additional test values and odel. For a given axiu plasa density in the particular device, the grid design has to stay on the left side of the equidensity lines in the odel plots in figures and 9 in order to avoid direct ipingeent on the accelerator grid. D. Discussion The two-paraeter odels proved to describe the siulated dependencies very well and can be used for optiization of grid systes, e.g. to find the iniu divergence conditions or the conditions for a required divergence value. As copared to the direct creation of such dependencies by a very large nuber of single siulation runs the effort is significantly reduced by using the DoE approach. The here often-derived x 2 y 5 odel, as exaple, requires only 1 supporting points to be siulated notwithstanding the rather high-degree of the odel. If such odels have to be build for other paraeter sets (i.e. other values as in table 1) we assue that the derived odel structures will also be applicable in these cases and the optial degrees of the variables will not differ fro that obtained here. While the value of the odel sees to depend strongly on the degree of the paraeters, this choice currently depends either on prior knowledge or on the evaluation of the odels using a large nuber of further easureents. Algoriths to evaluate the quality of the odel by a sall nuber of additional easureents would be helpful. The inclusion of additional easureents into the experiental design ay iprove the accuracy of the odel. Heuristics to choose appropriate supporting points (e.g. near extree values) should be evaluated. Furtherore, the inclusion of non-polynoial target-functions adapted to the expected dependencies sees possible and could allow to reduce the nuber of required experients. This will probably overcoe the deviations occurring in the lower plasa density region where the bealet divergence inia are located and which is currently odeled at higher degrees (n 5 ) satisfactory. These topics will be addressed in the near future. IV. Conclusion Polynoial odels of the extracted bealet current, bealet divergence and perveance liit of a 2 grid extraction syste have been deterined in dependence on geoetrical grid paraeters like hole diaeters, screen grid thickness and grid distance and the plasa density using a validated bealet siulation code. The Design of Experients (DoE) approach gives an algorith to obtain the odels with a reduced coputational effort by using a few supporting values only. A dedicated software was developed to perfor this process. October 31 Noveber, 25

9 The resulting two-paraeter odels include the plasa density at a degree of 5 and the geoetrical paraeters at lower degrees. Lower-degree odels best describe the perveance liit. The odels agreed very well with the siulated dependencies at a ean relative deviation of less than %. The work will be continued to include ore paraeters in the odels aking the ore generally. Such ultiparaeter odels are useful in designing grid systes. They help to coe to a quick understanding what happens if one changes a grid paraeter. Soe topics to iprove the accuracy and effectivity of the odel were addressed which will be studied in future work. References 1 P. J. Wilbur, V. K. Rawlin, J. R. Beattie, J. Propulsion Power 1 (199) 7. 2 J. E. Polk, R. Y. Kakuda, J. R. Anderson, J. R. Brophy, V. K. Rawlin, M. J. Patterson, J. Sovey, J. Haley, Validation of the NSTAR ion propulsion syste on the Deep Space One ission - Overview and initial results, AIAA , 35 th Joint Propulsion Conf., Los Angeles, CA, June 2-2, R. Killinger, R. Kukies, M. Surauer, H. Gray, G. Saccoccia, Final Report on the ARTEMIS Salvage Mission Using Electric Propulsion, AIAA th Joint Propulsion Conf., Huntsville, AL, July 2-23, 23. J. Brophy, M. Marcucci, J. Gates, C. Garner, B. Nakazono, G. Ganapathi, Status of the Dawn Ion Propulsion Syste, AIAA-2-333, th Joint Propulsion Conf., Fort Lauderdale, FL, July 11-1, 2. 5 H. Kuninaka, Y. Shiizu, T. Yaada, I. Funaki, K. Nishiyaa, Flight Report During Two Years on HAYABUSA Explorer Propelled by Microwave Discharge Ion Engines, AIAA , 1 st Joint Propulsion Conf., Tucson, AZ, July -13, 25. B. Wolf, Handbook of ion sources, CRC Press, Boca Raton, A. Schindler, T. Hänsel, F. Frost, G. Böh, W. Frank, A. Nickel, T. Arnold, R. Schwabe, S. Gürtler, S. Görsch, B. Rauschenbach, Modern Methods of Highly Precise Figuring and Polishing, Glass Science and Technology 7 (Suppl. C) (25) 111. T. Chassé, H. Neuann, B. Ocker, M. Scherer, W. Frank, F. Frost, D. Hirsch, A. Schindler, M. Lorenz, G. Otto, M. Zeuner, B. Rauschenbach, Mo/Si-Multilayers for EUV-Lithography by Ion Bea Assisted Deposition, Vacuu 71 (23) 7. 9 M. Tartz, R. Deltschew, E. Hartann, H. Neuann, Effect of the plasa inhoogeneity on grid perforance, Surf. Coat. Technol. -1 (21) 3. M. Tartz, E. Hartann, H. Neuann, Evolution of extraction grid erosion with operation tie, Paper AIAA-2-377,. Joint Propulsion Conf., Ft. Lauderdale, R. Becker, W. B. Herrannsfeldt, Rev. Sci. Instru. 3 (1992) 275. R. A. Fisher, The Design of Experients, London Edinburgh, Oliver and Boyd G. E. P. Box, K. B. Wilson, On the experiental attainent of optiu conditions, J. Royal Statistical Soc., Ser. B13 (1951), M. Tartz, E. Hartann, F. Scholze, H. Neuann, Modelling of broad bea ion sources, Rev. Sci. Instr. 9 (199) T. S. Green, Rep. Progr. Phys. 37 (197) October 31 Noveber, 25