Implementation and verification of a hybrid performance and impedance model of gridded radio-frequency ion thrusters

  • Chris VolkmarEmail author
  • Ubbo Ricklefs
Regular Article
Part of the following topical collections:
  1. Topical Issue: Physics of Ion Beam Sources


In this paper, we show the development steps for an iterative performance and impedance model of gridded radio-frequency (RF) ion thrusters. The input parameters are equivalent to those of the real propulsion system; i.e., coil current, propellant mass flow, and extraction grid voltages. Therefore, the model is easily validated and verified by experimental data and can furthermore be used to optimize the overall thruster performance. The model predicts volume-averaged plasma parameters such as electron temperature, conductivity, total pressure, and ionization fraction as well as thruster performance data like generated thrust, specific impulse, and mass and electrical efficiency. The above mentioned plasma parameters are obtained as functions of the discharge chamber’s geometry by using a charge balance equation that relates generated ions to ions lost at the chamber’s walls. The plasma related quantities influence the electromagnetic field penetration which is here evaluated by means of a diffusion equation for the vector potential. The vector potential is obtained by a 3D Finite-Difference-Method on a cubic and rectangular grid which, in principle, offers the opportunity to have arbitrary plasma chamber and coil geometries. An actual ion thruster geometry is evaluated in this study in favor of experimental verification of the numerically obtained data. The thruster’s coil generates highly asymmetric electromagnetic fields which motivates the use of a three-dimensional solver. A dissipation model based on Ohm’s law and Poynting’s theorem is used to determine the absorbed power within the discharge. To obtain a stable solution, the electromagnetically absorbed power is equated to the power lost due to elastic and inelastic collisions and electron wall flux. This whole process is iteratively repeated until the degree of ionization converges within a given threshold. To relate the stable discharge parameters to the thruster performance an extraction model based on a modified version of Child-Langmuir’s law is used. Within the model, the extracted ion beam current is calculated as a function of extraction aperture, applied extraction voltage, and plasma sheath size. A particle balance equation is then used to iteratively compute the total pressure combining neutral, electron, and ion pressures. After convergence, the plasma impedance, typically described by a transformer model, is transformed to its equivalent series representation as seen by the RF source at its terminals. Furthermore, thruster performance data are evaluated which gives rise to use this model for optimization purposes. It can thus be regarded as a toolbox for virtual prototyping.

Graphical abstract


Discharge Chamber Power Coupling Coil Wire Extraction Hole Screen Grid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    J.R. Brophy, Rev. Sci. Instrum. 73, 1071 (2002)CrossRefADSGoogle Scholar
  2. 2.
    Handbook of Space Technology, edited by W. Ley, K. Wittmann, W. Hallmann, 1st edn. (John Wiley & Sons, West Sussex, 2009)Google Scholar
  3. 3.
    H.W. Loeb, in Proceedings of the AIAA 7th Electric Propulsion Conference, Williamsburg, Virginia, USA, 1969Google Scholar
  4. 4.
    M.A. Lieberman, A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd edn. (John Wiley & Sons, Inc., New Jersey, 2005)Google Scholar
  5. 5.
    D.M. Goebel, I. Katz, Fundamentals of Electric Propulsion: Ion and Hall Thrusters (JPL Space Science and Technology Series), edited by J.H. Yuen, 1st edn. (John Wiley & Sons, Inc., New Jersey, 2008)Google Scholar
  6. 6.
    C. Bowick, J. Blyler, C. Ajluni, RF Circuit Design, 2nd edn. (Elsevier, Pennsylvania, 2007)Google Scholar
  7. 7.
    J.A. Hopwood, J. Microelectromech. Syst. 9, 309 (2000)CrossRefGoogle Scholar
  8. 8.
    P. Chabert, N. Braithwaite, Physics of Radio-Frequency Plasmas, 1st edn. (Cambridge University Press, New York, 2011)Google Scholar
  9. 9.
    V.A. Godyak, R.B. Piejak, B.M. Alexandrovich, Plasma Sources Sci. Technol. 3, 169 (1994)CrossRefADSGoogle Scholar
  10. 10.
    V. Vahedi, M.A. Lieberman, G. DiPeso, T.D. Rognlien, D. Hewett, J. Appl. Phys. 78, 1446 (1995)CrossRefADSGoogle Scholar
  11. 11.
    B. Keville et al., in Proceedings of the IFAC Workshop on Advanced Process Control for Semiconductor Manufacturing (Singapore, 2006), p. 56Google Scholar
  12. 12.
    C. Volkmar, T. Baruth, J. Simon, U. Ricklefs, R. Thueringer, in Proceedings of the 36th International Spring Seminar on Electronics Technology (Romania, 2013), p. 210Google Scholar
  13. 13.
    H.J. Leiter, R. Killinger, H. Bassner, J. Mueller, R. Kukies, in Proceedings of the 27th International Electric Propulsion Conference (California, USA, 2001)Google Scholar
  14. 14.
    H.J. Leiter, R. Killinger, H. Bassner, R. Kukies, J. Mueller, in Proceedings of the 38th Joint Propulsion Conference (Indiana, 2002)Google Scholar
  15. 15.
    H.J. Leiter, R. Killinger, H. Bassner, J. Mueller, R. Kukies, T. Froehlich, in Proceedings of the 28th International Electric Propulsion Conference (France, 2003)Google Scholar
  16. 16.
    H.J. Leiter, R. Killinger, M. Boss, M. Braeg, M. Gollor, S. Weis, D. Feili, M. Tartz, H. Neumann, D.M. Di Cara, in Proceedings of the AIAA 43rd Joint Propulsion Conference, Cincinnati (Ohio, 2007)Google Scholar
  17. 17.
    H.J. Leiter, H. Ellerbrock, M. Berger, M. Boss, D. Feili, B. Lotz, D.M. Di Cara, in Proceedings of the AIAA/ASME/SAE/ASEE 47th Joint Propulsion Conference & Exhibit, San Diego (California, 2011)Google Scholar
  18. 18.
    P. Chabert, J. Arancibia Monreal, J. Bredin, L. Popelier, A. Aanesland, Phys. Plasmas 19, 073512 (2012)CrossRefADSGoogle Scholar
  19. 19.
    D.M. Goebel, IEEE Trans. Plasma Sci. 36, 2111 (2008)CrossRefADSGoogle Scholar
  20. 20.
    J.T. Gudmundsson, M.A. Lieberman, Plasma Sources Sci. Technol. 6, 540 (1997)CrossRefADSGoogle Scholar
  21. 21.
    M.M. Tsay, M. Martinez-Sanchez, in Proceedings of the 30th International Electric Propulsion Conference (Italy, 2007)Google Scholar
  22. 22.
    C. Lee, M.A. Lieberman, J. Vac. Sci. Technol. A 13, 368 (1995)CrossRefADSGoogle Scholar
  23. 23.
    J.D. Jackson, Classical Electrodynamics, 3rd edn. (John Wiley & Sons, Inc., New Jersey, 1998)Google Scholar
  24. 24.
    M. Mitchner, C.H. Kruger, Partially Ionized Gases (Wiley Series in Plasma Physics), 1st edn. (John Wiley & Sons, Inc., New Jersey, 1973)Google Scholar
  25. 25.
    R.B. Piejak, V.A. Godyak, B.M. Alexandrovich, Plasma Sources Sci. Technol. 1, 179 (1992)CrossRefADSGoogle Scholar
  26. 26.
    F.F. Chen, Introduction to Plasma Physics and Controlled Fusion – Volume 1: Plasma Physics, 2nd edn. (Plenum Press, New York, 1984)Google Scholar
  27. 27.
    I.N. Bronshtein, K.A. Semendyayev, G. Musiol, H. Muehlig, Handbook of Mathematics, 5th edn. (Springer, New York, 2007)Google Scholar
  28. 28.
    M. Hayashi, J. Phys. D 16, 581 (1983)CrossRefADSGoogle Scholar
  29. 29.
    D.L. Book, NRL Plasma Formulary. Revision., No. NRL-PUB-177-4405, Naval Research Lab, Washington D.C., 1990Google Scholar
  30. 30.
    K. Miyamoto, Plasma Physics for Nuclear Fusion, 3rd edn. (The MIT Press, Massachusetts, 1989)Google Scholar
  31. 31.
    M.M. Turner, Phys. Rev. Lett. 71, 1844 (1993)CrossRefADSGoogle Scholar
  32. 32.
    E.S. Weibel, Phys. Fluids 10, 741 (1967)CrossRefADSGoogle Scholar
  33. 33.
    C. Volkmar, J. Simon, U. Ricklefs, J. Phys. Sci. Appl. 4, 262 (2014)Google Scholar
  34. 34.
    M.N.O. Sadiku, Numerical Techniques in Electromagnetics, 2nd edn. (CRC Press LLC, Florida, 2000)Google Scholar
  35. 35.
    P.-b. Zhou, Numerical Analysis of Electromagnetic Fields, 1st edn. (Springer, New York, 1993)Google Scholar
  36. 36.
    S. Yang, M.K. Gobbert, Appl. Math. Lett. 22, 325 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    A. Fridman, L.A. Kennedy, Plasma Physics and Engineering, 2nd edn. (CRC Press LLC, Florida, 2011)Google Scholar
  38. 38.
    T. Baruth, R. Thueringer, in Proceedings of the 33rd International Electric Propulsion Conference, Washington D.C., USA, 2013, Proceeding IEPC-2013-196Google Scholar
  39. 39.
    D.C. Rovang, P.J. Wilbur, in Proceedings of the AIAA/JSASS/DGLR 16th International Electric Propulsion Conference (Louisiana, 1982)Google Scholar
  40. 40.
    D.C. Rovang, P.J. Wilbur, J. Propul. Power 1, 172 (1985)CrossRefGoogle Scholar
  41. 41.
    M.M. Tsay, Master’s thesis, Massachusetts Institute of Technology, Cambridge, 2006Google Scholar
  42. 42.
    M. Martinez-Sanchez, Electrostatic Thrusters, in 16.522 – Space Propulsion, Lecture Notes, Massachusetts Institute of Technology (2004)Google Scholar
  43. 43.
    C. Volkmar, U. Ricklefs, in Proceedings of the Space Propulsion Conference (Cologne, Germany, 2014)Google Scholar
  44. 44.
    M.R. Ward, Electrical Engineering Science, 1st edn. (McGraw-Hill Education, New York, 1971)Google Scholar
  45. 45.
    P. Banks, Planet. Space Sci. 14, 1085 (1966)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Electrical EngineeringTHM University of Applied SciencesGiessenGermany

Personalised recommendations