Antenna in Plasma AIP Code Timothy W Chevalier

Antenna in Plasma (AIP) Code Timothy W. Chevalier Umran S. Inan Timothy F. Bell March 4, 2008

Stanford MURI Tasks Scientific Issues: q The sheath surrounding an electric dipole antenna operating in a plasma has a significant effect on the tuning properties. q Terminal impedance characteristics vary with applied voltage. § q Active tuning may be needed. Stanford has developed a general AIP code to determine sheath effects on radiation process. MURI Tasks: q q Validation of our AIP code by laboratory experiments using LAPD. § UCLA will provide time measurements of voltage, current and field patterns for dipole antennas to compare with Stanford model. § Locate sources of error in current model and identify means for improvement. Perform LAPD experiments on magnetic loop antennas. 1

Outline 1. Introduction 2. Cold Plasma Electromagnetic Model 3. Current Distribution and Impedance Results 4. Warm Plasma Electrostatic Model 5. Plasma Sheath Results 2

Coupling Regions Sheath Region (R ¿ ¸min ) § Near field § Reactive Energy (ES) § Highly nonlinear R ¼ Warm Plasma Region (R ¸) § Transition zone § Reactive/Radiated Energy (EM & ES) § Nonlinear effects still important À Cold Plasma (RRegion ¸) ES: Electrostatic EM: Electromagnetic § Far field § Radiated Energy (EM) § Linear environment 3

Modeling Methodology § Near field antenna characteristics § Electrically short dipole antennas § ES & EM approaches · ¸ (Poisson/Maxwell)-Vlasov Formulation ¢r F ¢r @f @t + (v )f + r m P v f =0 ~ +v F = q(E £ ~ B) (Lorentz Force) r ½ (¢ ~ E = ®P® ²o (Poisson) r£ ~ ~ = ~ d. E r £ H ¡ N J® + ²o dt ~ ~ = ¹ d. H E (Maxwell) o dt 4

Moments of Vlasov Equation F · ¸ ¢r @f F ¢r f (v) = + (v )f + r v m @t 8 > > F Z Z Z < Nth moment > m F(v)dv > > (v)dv ¡ F ¡ > : mv ¡ ¡ th Mn = m [v ¡ u] (v F ¡ u) ¡ ¡ u)d(v v m [v u] (v u)d(v u) ´ v ´ phase space velocity u average ¡ ´°ow velocity c = [v u] random velocity due to thermal motions 5

Fluid Representation of Plasma Fluid Momentsr ¢ =0 @t (nm) + r (nmu) ¢ ¡ £ mass density) =0 @t (nmu)r (nmuu + f. P) ¢ rnq (E + u£ B) +¢ g st (1 : momentum). . (u. P + Q) + P (u) + c P sym = 0 @t (P) + r¢ f ¢r £ ¡ r¢ (2 nd: pressure). . . 1 g sym = 0 @t (Q) + (v. Q + R) + Q (u) + c Q P (P) nm (3 rd: heat flux)…… (0 th: ´ Variables Fluid ´ Additional Variables n ´ number density u ´ average °ow velocity vector E ´ electric ¯eld vector B ´ magnetic ¯eld vector r-moment tensor u. P = tensor product c P ´ pressure tensor Q ´ heat °ux tensor R m´ mass q ´charge gyrofrequency vector 6

Outline 1. Introduction 2. Cold Plasma Electromagnetic Model 3. Current Distribution and Impedance Results 4. Warm Plasma Electrostatic Model 5. Plasma Sheath Results 7

Cold Plasma Fluid Approximation r¢ Fluid Description: (nmu) = 0 @ (nm) + r¢ ¡ £ = (nmuu +f. P) ¢ rnq (E + u£ B) @t (nmu)r +¢ g 0 (u. P + Q) + P (u) + c P sym = 0 @t (P) + f ¢r r¢ £ ¡ r¢ 1 g sym = 0 (v. Q + R) + Q (u) + c Q P (P) @t (Q) + nm t Closure =0 P = nk. T Assumption: ³ Generalized Ohms Law£ ~ q d. J® ~ + J~ ~ B + º® J~® = ® q® n® E ® o dt m® ´ 8

Finite Difference Time and Frequency Domain Techniques (FDTD/FDFD) X Time r£ ~ = H r£ Domain~ (FDTD) d. E J~® + ²o dt FDTD Method: § N ¡ d. H ~ ³ ´ ~ E = ¹o dt £ d. J~® q® ~ ~ = q® n® E + J® B + º® J® o m® dt Computational Mesh: § Time domain solution of Maxwell’s equations. Wide spread use in EM community X £ r Frequency Domain (FDFD) ~ ~ ~ H = ¾ E + ² j! E r£ ~ E ¾® = = = ® o ¡N ~ ¹o j! H 0 ²o ! 2 (j!I @ p¡ º ¡!bz !by ¡ ¡ ) ¡ ¡!bz º !bx 1 ¡!by ¡!bx º 1 A Solves: Ax=B 9

Outline 1. Introduction 2. Cold Plasma Electromagnetic Model 3. Current Distribution and Impedance Results 4. Warm Plasma Electrostatic Model 5. Plasma Sheath Results

Cold Plasma Simulation Setup Computational Domain: Antenna Properties § § Length: 100 m Diameter: 20 cm Orientation: Perpendicular to Bo Position: Equatorial Plane 11

Current Distribution for 100 m Antenna in Freespace · µ ¶¸ Current distribution on linear antenna I / L= 2¼ Io sin ¸ ¸ 2 L§ z 2 L ¿ Excitation frequency: 10 k. Hz ¸ 12

Current Distributions for 100 m Antenna at L=2 Excitation frequency: f < f. LHR Excitation frequency: f > f. LHR 13

Simulation vs. Theory Previous Analytical Work [Wang and Bell. , 1969, 1970] [Wang. , 1970] [Bell et. al. , 2006] L=2 R ¢ Input Impedance. H Formula ~ dl) ( E V (f ) ¢ f eed = Zin = ~ dl) I(f ) ( H f eed L=3 14

Conclusions Based upon Cold Plasma Approximation § Current distribution is triangular for cases demonstrated. § This result supports triangular assumption made in early analytical work. § Input impedance does not vary significantly as a function of frequency § The same antenna can be used over a broad frequency range; self tuning property. § Early analytical work should provide accurate estimates of radiation pattern of dipole antennas in a magnetoplasma [Wang and Bell. , 1972]. § What about the Sheath? 15

Outline 1. Introduction 2. Cold Plasma Electromagnetic Model 3. Current Distribution and Impedance Results 4. Warm Plasma Electrostatic Model 5. Plasma Sheath Results

Warm Plasma Fluid Approximation ¢ Isothermalr. Approximation (2 -moments) =0 @t (nm) + r (nmu) ¢ ¡ £ @t (nmu)r +¢ (nmuu +f. P) ¢ rnq (E + u£ B) g =0 @t (P) + (u. P + Q) + P (u) + c P sym = 0 r¢ f ¢r £ ¡ r¢ 1 g sym = 0 @t (Q) + (v. Q + R) + Q (u) + c Q P (P) nm Closure Assumption: P = nk. T ¢ Adiabatic r Approximation (3 -moments) =0 @t (nm) + r (nmu) ¢ ¡ £ @t (nmu)r +¢ (nmuu +f. P) ¢ rnq (E + u£ B) g =0 @t (P) + (u. P + Q) + P (u) + c P sym = 0 r¢ f ¢r £ ¡ r¢ 1 g sym = 0 @t (Q) + (v. Q + R) + Q (u) + c Q P (P) nm r¢ Closure Q = 0 Assumption: 17

Electrostatic Approximation Nonlinear Equations ¸ Sheath region < Time domain approach min P Electrostatic approach is valid Poisson’s Equation r¢ ~ = E ® Constant Voltage ½® ²o § Removes EM time-stepping constraint § Avoids problems associated with PML Triangular current distribution L ¿ ¸ 18

Outline 1. Introduction 2. Cold Plasma Electromagnetic Model 3. Current Distribution and Impedance Results 4. Warm Plasma Electrostatic Model 5. Plasma Sheath Results

Warm Plasma Simulation Setup (2 -D) Computational Domain: Antenna Properties § § § Length: Infinite in z-direction Diameter: 10 cm Position: Equatorial Plane Plasma Properties Fluid closure relations: § Isothermal - moments) P =(2 nk. T r¢ § Adiabatic Q (3 -=moments) 0 L=2: L=3: § § § § § N = 2 e 9 #/m 3 fpe = 400 k. Hz fpi = 28 k. Hz fce = 110 k. Hz fci = 550 Hz N = 1 e 9 #/m 3 fpe = 284 k. Hz fpi = 20 k. Hz fce = 33 k. Hz fci = 163 Hz mi = 200 Mass ratio: me 20

Simulation of Infinite Line Source Simulation Properties § Plane of symmetry: 25 k. Hz sinusoid § f>fpi § No magnetic field 21

Simulation of Infinite Line Source Simulation Properties § 25 k. Hz sinusoid § f>fpi § No magnetic field Plane of symmetry: 22

Simulation of Infinite Line Source Simulation Properties § 25 k. Hz sinusoid § f>fpi § No magnetic field Plane of symmetry: 23

IV Characteristics (Sinusoid) Non-magnetized Magnetized 25 k. Hz (f > fpi) Magnetized 15 k. Hz (f < fpi) 24

IV Characteristics (Pulse) Non-magnetized Magnetized 25 k. Hz (f > fpi) Magnetized 15 k. Hz (f < fpi) 25

Warm Plasma Simulation Setup (3 -D) Computational Domain: Antenna Properties § § § Length: 20 m Gap: 2 m Diameter: 10 cm Position: Equatorial Plane Electron gun (removes charge) Plasma Properties L=2: L=3: § § § § § N = 2 e 9 #/m 3 fpe = 400 k. Hz fpi = 28 k. Hz fce = 110 k. Hz fci = 550 Hzm i Mass ratio: me N = 1 e 9 #/m 3 fpe = 284 k. Hz fpi = 20 k. Hz fce = 33 k. Hz fci = 163 Hz = 200 Adiabatic (full pressure tensor) 26

Simulation of 20 m Dipole at L=3 Orthographic Projection Potential and Density Variation Current-Voltage Gap Current 27

Simulation of 20 m Dipole at L=3 with 20 cm Gap Orthographic Projection Potential and Density Variation Current-Voltage Gap Current 28

Simulation of 20 m Dipole at L=3 without Electron Gun Orthographic Projection Potential and Density Variation Current-Voltage Gap Current 29

Circuit Diagrams Diagram of Sheath Impedance: Tuning Circuit 30

Conclusions Based upon Sheath Calculations § Sheath structure is periodic with both sinusoid and pulse waveform excitation. § Sheath is a quasi-steady state structure. § Proton densities vary significantly throughout sheath region and contribute to current collection. § Commonly used assumption of immobile protons within sheath region for frequencies above and below proton plasma frequency is not necessarily accurate. § Most notable in case of floating antenna. 31

Validity of Fluid Code for Sheath Region § Ma and Schunk [1992], Thiemann et al. [1992]: Compared PIC and 2 -moment fluid codes with diagonal pressure tensors surrounding spherical electrodes stepped to 10, 000 V. § Noisy PIC simulations agreed with results of fluid code with addition of more particles § Under-sampled distribution functions in PIC code are inherently noisy. § Plasma ringing and double layer formation was captured in both fluid and PIC simulations. § Very good qualitative agreement § Borovsky [1988], Calder and Laframboise[1990], Calder et al. [1993]: PIC simulations of spherical electrodes stepped to very large potentials. § Calder and Laframboise [1990], noted ringing effects could be driven to large amplitude by ion-electron two steam instability which a fluid code can capture. § No presence of electron-electron two-stream instability in any of the PIC simulations § Landau damping is negligible since the phase velocity of waves within the sheath region are generally different than thermal velocities. § No need to capture this effect in fluid code. § § Though particle trapping within sheath is possible (mainly slow moving ions), the relatively small number of trapped particles results a minimal deviation of the potential variation within the sheath. A fluid code can provide an accurate and more computationally efficient method for 32 the determination of sheath characteristics!