Antenna in Plasma (AIP) Code

Antenna in Plasma (AIP) Code Timothy W. ChevalierUmran S. InanTimothy F. BellMarch 4, 2008

Stanford MURI Tasks Scientific Issues: • The sheath surrounding an electric dipole antenna operating in a plasma has a significant effect on the tuning properties. • Terminal impedance characteristics vary with applied voltage. • Active tuning may be needed. • Stanford has developed a general AIP code to determine sheath effects on radiation process. MURI Tasks: • 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.

Outline • Introduction • Cold Plasma Electromagnetic Model • Current Distribution and Impedance Results • Warm Plasma Electrostatic Model • Plasma Sheath Results

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

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

Moments of Vlasov Equation · ¸ @ f F 8 ( ) d F v v m ( ) ( ) f f F r r + + v v ¢ ¢ > = r v > h l i @ t t ´ v p a s e s p a c e v e o c y > ( ) m d Z Z Z F < v v v m h t n M = ° l i t [ ] [ ] ( ) ( ) ´ u a v e r a g e o w v e o c y d F ¡ ¡ ¡ ¡ v u v u v u v u m > > v [ ] > d l d h l i i t t t t ¡ [ ] [ ] [ ] ( ) ( ) d ´ F c v u r a n o m v e o c y u e o e r m a m o o n s : = ¡ ¡ ¡ ¡ ¡ v u v u v u v u v u m Nth moment

Fluid Representation of Plasma ( ) ( ) @ r 0 + ¢ u n m n m = t l b d ¯ l d E i i t t t ´ ´ n n u m e e e c r r c e n s e y v e c o r ( ) ( ) ( ) @ P E B r 0 + + ¡ + £ u ¢ u u u n m n m n q = t ° ¯ l l d B i i t t t t ´ ´ u a v e m r a a g g e n e o w c v e e o c v e y c v o e r c o r ( ) ( ) f ( ) g s y m @ P P Q P P r r 0 + + + + £ ¢ u ¢ u = t c P t 1 ´ ´ m p r e m s s a u s r s e e n s o r ( ) ( ) f ( ) ( ) g s y m @ Q Q R Q Q P P r r r 0 + + + + £ ¡ ¢ v ¢ u ¢ = t c h h ° Q t t n m ´ ´ q e c a a r g u e x e n s o r f R t t t ´ ´ r - m g o y m r e o n r e q e u n e s n o c r y v e c o r c d P t t u e n s o r p r o u c = Fluid Moments (0th: mass density) (1st: momentum).. (2nd:pressure)...... (3rd:heat flux)…… Additional Variables Fluid Variables

Cold Plasma Fluid Approximation ~ ( ) ( ) d J @ r 0 + ³ ´ q ¢ u n m n m = ~ ~ ~ ~ t k P T 0 ® ® J E J B n + = = + £ º q n = ® ® ® ® ® o d t ( ) ( ) ( ) @ r P E B 0 + + ¡ m + £ u ¢ u u u n m n m n q = ® t ( ) ( ) f ( ) g s y m @ P r P Q P r P 0 + + + + £ ¢ u ¢ u = t c 1 ( ) ( ) f ( ) ( ) g s y m @ Q r Q R Q r Q P r P 0 + + + + £ ¡ ¢ v ¢ u ¢ = t c n m Fluid Description: Closure Assumption: Generalized Ohms Law

Finite Difference Time and Frequency Domain Techniques (FDTD/FDFD) ~ X ~ ~ ~ d E r H E E j X £ + ~ ~ ¾ ² ! = r H J £ + ® o ² = ® o d t N N ~ ~ r E H j £ ¡ ~ ¹ ! = d H o ~ Frequency Domain (FDFD) r E £ ¡ ¹ = o d t 1 ¡ ~ 2 ( ) I j d J ¡ ¾ ² ! ! ³ ´ = q ~ ~ ~ ~ ® o ® ® p J E J B + + £ º q n = ® ® ® ® ® o d 0 1 t m ¡ ¡ º ! ! ® b b z y ¡ ¡ ! º ! = b b @ A z x ¡ ¡ ! ! º b b y x Time Domain (FDTD) FDTD Method: • Time domain solution of Maxwell’s equations. • Wide spread use in EM community Computational Mesh: Solves: Ax=B

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

Current Distribution for 100 m Antenna in Freespace ¸ ¸ L ¿ L = · µ ¶ ¸ L 2 2 ¼ I I i § / s n z o ¸ 2 Current distribution on linear antenna Excitation frequency: 10 kHz

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

Simulation vs. Theory ~ R ( ) d l ( ) E f V ¢ f d e e Z = = i n ~ ( ) f I H ( ) d l H ¢ f d e e Previous Analytical Work Input Impedance Formula [Wang and Bell., 1969,1970] [Wang., 1970] [Bell et. al., 2006] L=2 L=3

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?

Warm Plasma Fluid Approximation ( ( ) ) ( ( ) ) @ @ r r 0 0 + + ¢ ¢ u u n n m m n n m m = = t t k P Q r T 0 ¢ n = = ( ( ) ) ( ( ) ) ( ( ) ) @ @ P P E E B B r r 0 0 + + + + ¡ ¡ + + £ £ u u ¢ ¢ u u u u u u n n m m n n m m n n q q = = t t ( ( ) ) ( ( ) ) f f ( ( ) ) g g s s y y m m @ @ P P P P Q Q P P P P r r r r 0 0 + + + + + + + + £ £ ¢ ¢ u u ¢ ¢ u u = = t t c c 1 1 ( ( ) ) ( ( ) ) f f ( ( ) ) ( ( ) ) g g s s y y m m @ @ Q Q Q Q R R Q Q Q Q P P P P r r r r r r 0 0 + + + + + + + + £ £ ¡ ¡ ¢ ¢ v v ¢ ¢ u u ¢ ¢ = = t t c c n n m m Isothermal Approximation (2-moments) Closure Assumption: Adiabatic Approximation (3-moments) Closure Assumption:

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

Warm Plasma Simulation Setup(2-D) m i k P Q r T 2 0 0 0 ¢ n = = = m e Computational Domain: Antenna Properties • Length: Infinite in z-direction • Diameter: 10 cm • Position: Equatorial Plane Plasma Properties • L=2: • N = 2e9 #/m3 • fpe = 400 kHz • fpi = 28 kHz • fce = 110 kHz • fci = 550 Hz • L=3: • N = 1e9 #/m3 • fpe = 284 kHz • fpi = 20 kHz • fce = 33 kHz • fci = 163 Hz Fluid closure relations: • Isothermal (2 - moments) • Adiabatic (3 - moments) Mass ratio:

Simulation of Infinite Line Source Plane of symmetry: Simulation Properties • 25 kHz sinusoid • f>fpi • No magnetic field

Simulation of Infinite Line Source Simulation Properties • 25 kHz sinusoid • f>fpi • No magnetic field Plane of symmetry:

IV Characteristics (Sinusoid) 15 kHz (f < fpi) 25 kHz (f > fpi) Non-magnetized Non-magnetized Magnetized Magnetized

IV Characteristics (Pulse) 15 kHz (f < fpi) 25 kHz (f > fpi) Non-magnetized Non-magnetized Magnetized Magnetized

Warm Plasma Simulation Setup(3-D) m i 2 0 0 = m e Computational Domain: Antenna Properties • Length: 20 m • Gap: 2 m • Diameter: 10 cm • Position: Equatorial Plane • Electron gun (removes charge) Plasma Properties • L=2: • N = 2e9 #/m3 • fpe = 400 kHz • fpi = 28 kHz • fce = 110 kHz • fci = 550 Hz • L=3: • N = 1e9 #/m3 • fpe = 284 kHz • fpi = 20 kHz • fce = 33 kHz • fci = 163 Hz Mass ratio: Adiabatic (full pressure tensor)

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

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

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

Circuit Diagrams Tuning Circuit Diagram of Sheath Impedance:

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.

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 10,000V. • 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 the determination of sheath characteristics!

Antenna in Plasma (AIP) Code