1. Plasma modelling in RF simulations
Alexej Grudiev
3/06/2013
HG2013, ICTP Trieste, Italy

2. Ion source for CERN LINAC4

3. 2 MHz Antenna + “plasma”

4. Plasma model: dielectric or conductor
For a fixed frequency the second term describing the losses in dielectrics equivalent to the one describing the losses in conductors.
We will take a conductor to represent plasma with certain number of free electrons per volume: N, then the conductivity is expressed (Jeckson: Ch.7.5)
It is also convenient for comparison to introduce plasma frequency
Where γ0 is so-called damping term describing energy loss of free electrons moving between ions and neutrals. it is proportional to the collision frequency

5. Range of interest for conductivity
In solid Copper: N = 8x1028/m3; σ = 58 MS/m =>
γ0 = 4x1013/s >> ω in microwave range
ωp = 6x1016/s >> γ0
Let’s also can assume that in our case ωp >> γ0 >> ω = 4πx106/s
The lowest value for conductivity: σ/ω ≈ ε0 => ωp ≈ ω ≈ γ0 : σ > [S/m]
For lower conductivity the lossy part: σ/ω is much smaller than ε0
The highest value of interest for conductivity is set by skin-effect:
δ = (2/σωµ0)1/2 > 1mm => σ < 105 [S/m]
For higher conductivity EM fields cannot be solved in “plasma”

6. E-field: antenna + “plasma” (σ = 0 S/m)

7. E-field: antenna + “plasma” (σ = 1e-4 S/m)

8. E-field: antenna + “plasma” (σ = 1e-3 S/m)

9. E-field: antenna + “plasma” (σ = 1e-2 S/m)

10. E-field: antenna + “plasma” (σ = 1e-1 S/m)

11. E-field: antenna + “plasma” (σ = 1e-0 S/m)

12. E-field: antenna + “plasma” (σ = 1e+1 S/m)

13. E-field: antenna + “plasma” (σ = 1e+2 S/m)

14. E-field: antenna + “plasma” (σ = 1e+3 S/m)

15. E-field: antenna + “plasma” (σ = 2e+3 S/m)

16. E-field: antenna + “plasma” (σ = 3e+3 S/m)

17. E-field: antenna + “plasma” (σ = 1e+4 S/m)

18. E-field: antenna + “plasma”
Along the coil axis
Across the coil axis
Higher field in the “extraction” region

19. H-field: antenna + “plasma”
For high sigma when skin depth is very small there are some numerical issues for magnetic field calculation

20. RF circuit (M.M. Paoluzzi, et. al., NIBS2010)

21. Measured parameters of the RF system. (M.Paoluzzi et.al. NIBS 2010)
IS-01* (simulated by AG)
68/60/26 6
(3.0)
(0.26)
2.3
0.14
1.5 nF
5.5 nF
~100=2pi*2*3.2/0.4
~9=100*0.4/( )
50Ω ?
* Private communication, Mauro Paoluzzi, (2013).

22. Input impedance of antenna + “plasma”
Imaginary part almost constant
It get reduced for high σ due to skin-effect in “plasma”
Real part stay constant at the level of Rant for low σ < 1 S/m
It is significantly high for σ > 10 S/m
It peaks at σ = 1000 S/m when skin depth is about the “plasma” radius

23. RLC parameters of antenna + “plasma”
Model used by Mauro
RANT
LANT
RPlasma
LPlasma
ZPlasma = ZAnt+Plasma(σ>0) - ZAnt(σ=0)
ZAnt = RAnt +jω0LAnt
ZPlasma = RPlasma +jω0LPlasma
IS-01 coil + “plasma” (simulated)
RAnt = Ω
LAnt = 2.65 µH
H- source (measured)
SPL PG (measured)
Mauro Paoluzzi
Plasma resistance and inductance calculated following Mauro’s equivalent circuit for a given plasma shape in bare antenna agree rather well with the measurement results for H- source.

24. Input impedance for different “plasma” shapes
Plasma R20mm
Plasma R24mm
The same “plasma” radius – the same Re{Zin} (red curves)
Larger “plasma” radius – larger Re{Zin}, larger Im{Zin} reduction
Plasma R28mm

25. RLC parameters for different “plasma” shapes
IS-01 coil + “plasma” (simulated)
RAnt = Ω
LAnt = 2.65 µH
H- source (measured)
SPL PG (measured)
Mauro Paoluzzi
Plasma R20mm
Plasma R24mm
Plasma R24mm
Plasma R28mm
Plasma R20mm
Plasma R28mm
The larger is “plasma” radius the higher is Rplasma and –Lplasma

26. Input impedance for full 3D model
Plasma R24mm
Plasma R24mm
RAnt = Ω
LAnt = 2.65 µH
Plasma Full SC
RAnt = Ω
LAnt = 3.02 µH
For IS-01, Mauro measured:
RAnt = 0.4 Ω
LAnt = 3.2 µH
Rplasma = 4.3 Ω
Lplasma = µH
plasma
Plasma Full SC
Inverted polarity
RAnt = Ω
LAnt = 3.03 µH

27. E-field for full 3D model plasma conductivity: ~316 S/m
frame # σ [S/m]
plasma

28. Possible applications to breakdown studies
Plasma ignited by a discharge
Antenna couples RF power to plasma and is used to measure plasma impedance
Additional passive or/and active diagnostics via damping waveguides
RF output
RF input
Higher frequency signal for break down plasma diagnostics
plasma
Plasma ignited by the breakdown

29. Back up slides

30. Power loss in antenna and “plasma”
Power loss on the antenna surface: Pant shows very small variation with respect to the “plasma” conductivity
Power loss in the “plasma”: Ppl linearly rising from ~0 level at σ=1e-1 S/m up to maximum at σ=1e+3 S/m.
Reason: E-field distribution is constant and Ppl ~ ∫dV σE2.
Equivalent circuit: Rp~1/σ => at constant “plasma” voltage: Vpl, Ppl~V2σ
For σ>1e+3 S/m, Skin-depth becomes comparable to the “plasma” radius what reduce E-field in the “plasma” volume
Vpl goes dawn
Ppl goes down ~σ-1/3.5

31. Equivalent circuit of a transformer
Volume current distribution in plasma
IAnt
RAnt LP
36RPl LM
Ipl
VAnt
Vpl
RP = RAnt ; RC = ∞ ; XP+XM = ωLAnt
NP:NS = 6:1
XS = 0; RS = Rpl – plasma resistance

32. Input impedance of antenna + “plasma”
ω0LP = Im{Zin}= Ω
LP= 2.04 µH
RAnt = Ω
LAnt = 2.65 µH
Re{Zin}= Ω
σ=∞

33. Plasma impedance: different view
System behaviour versus plasma conductivity:
σ<1e-4: no effect of the plasma
1e-4<σ<1e-1: plasma act as a dielectric pushing Er,Ez outside
1e-1<σ<1e3: plasma heats up due to Eθ. Power ~ ∫dV σEθ2.
1e3<σ: plasma heating is gradually reduced down to 0 due to skin effect. Coupling inductance LM is gradually reduced from LM=LM0 -> LM=0