# Simplified Optimization of the Magnetic Configuration of HEMP-Thrusters
## 4 Conclusions  
Optimization of the magnetic field geometry for a HEMP-thruster is very important for improved thruster operation. The use of an analytical Python package allows for simplified and fast optimization. The package was validated by comparison with the specific solution at the center of the cusp, where a particularly parallel magnetic field was derived at the origin. In experiments, two-stage thrusters were optimized by trial and error to reach a length-to-radius ratio of about 5 to 1. This means that the length-to-radius ratio for one stage is about 2.5 to 1, which is very close to the analytical optimization result of 2.45. The geometric relation can also be seen in Figure 6 and in [8,9] for comparison. A similar ratio was also found in the optimization procedure using genetic algorithms [7]. Further diagnosis is then numerically possible, motivated by physics. The length of the magnetic field lines was calculated numerically in a scan of the ring geometry, which is linked to the path lengths of the electrons. In addition, density weights from experiments were used for the field line analysis, and the mirror effect at the end of the cylindrical magnets was diagnosed. This is important for electron confinement. Confining the electrons between the cusps and minimizing radial losses to the walls improves ionization. The combination of different aspects of the optimization of the physics can be implemented very quickly and flexibly by defining a penalty function with different weightings for the different contributions. This allows numerical optimization of the magnetic field (including manufacturing constraints for the magnets) within minutes instead of days or weeks. Further combinations with power balance models or kinetic corrections can be made from this initial configuration, minimizing the risk of becoming stuck in local minima during the optimization process. An optimizing strategy for the magnetic field **B** is as follows:  
* Start from the optimal \(R\) and \(L\) of the analytical derivation. * Vary \(R\) and \(L\). \(R\) is expected to get a little smaller; see Figure 8. The magnetic mirror ratio becomes higher, and thus the confinement time. * \(L\) is expected to increase, which is the same statement as the previous point (cf. Figure 5). The intention is to avoid pushing the separatrix outward, which in turn may increase the exit angle of the ions. This is especially important for the exit stage. Figure 8: Same as Figure 7, but here, radius \(R\) is varied instead of the fixed length \(L=50\,\mathrm{mm}\). Since \(R\) changes, the three selected field lines are plotted for fixed ratios of \(r_{0}/R\). * The increase in \(L\) is limited by the reduction in the active volume \(\widehat{V}\) in Figure 5, which significantly reduces the electron density in these regions. In general, this simplified approach reduces the dimension of the remaining optimization problem. The calculation of magnetic fields is not only much faster but also has reduced error compared to common numerical approaches, which is particularly important for transport codes where artificial drifts and forces are minimized. Conceptualization, L.L. and R.S. ; methodology, L.L. and R.S. ; software, L.L. ; formal analysis, L.L. and R.S. ; writing--original draft preparation, L.L. ; writing--review and editing, R.S. ; visualization, L.L. All authors have read and agreed to the published version of the manuscript. This work is funded by the European Union as EU project HEMPT-NG2 [grant id 101004140]. The code for the numerical analysis can be found here: [[https://github.com/laleph/thruster_optimization](https://github.com/laleph/thruster_optimization)]([https://github.com/laleph/thruster_optimization](https://github.com/laleph/thruster_optimization)) (accessed on 20 February 2023). The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.