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1.2 Measurement of a Drifting Particle Distribution................ 2


E?ects of non-ideal biased grids on drifting particle distribution functions

by Je?rey Klenzing

A research proposal submitted to the faculty of the University of Texas a

t Dallas in partial fu?llment of the requirements for the degree of Doctor of Philosophy

September 2006

Contents

1 Introduction 1.1 1.2 1.3 A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of a Drifting Particle Distribution . . . . . . . . . . . . . . . . The Ram Wind Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5

2 Analysis Techniques 2.1 2.2 2.3 2.4 The Knudsen Analysis Technique . . . . . . . . . . . . . . . . . . . . . . . . Non-Uniform Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Need for an Improved Technique . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 10 10 11

3 Discrete Analysis 3.1 3.2 3.3 3.4 3.5 A Simple Construct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Idealized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . The General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15 16 17 18

1

3.6

Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4 Methodology 4.1 4.2 4.3 A Brief Description of ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . Veri?cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Research Approach and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 4.3.2 4.3.3 The ANSYS Code . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 23 23 26 26

The FORTRAN Simulator . . . . . . . . . . . . . . . . . . . . . . . . The Data-Fit Routine . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Summary and Conclusions

27

2

Abstract

The thesis proposal detailed here covers the study of the e?ects of non-ideal biased grids on drifting particle distribution functions. Instruments such as the Retarding Potential Analyzer measure the distribution of ?ux as a function of ion energy by use of these biased grids. A similar technique can be used to measure the ?ux distribution of neutral particles with the Ram Wind Sensor, a new instrument in development. In order to ?t collected data to physical parameters, the interaction of charged particles with the biased grid must be studied thoroughly. The program of study outlined in this proposal will discuss several shortcomings of non-ideal biased grids on data analysis techniques. In addition, I propose a new method of data analysis, which will take these shortcomings into account.

Chapter 1 Introduction

1.1

A Brief Overview

One of the fundamental instruments of space science is the Retarding Potential Analyzer (RPA). RPAs have successfully been ?own on satellites since the late 1950’s, including Sputnik [11], Atmospheric Explorer [6], Dynamics Explorer [7], and the DMSP family of satellites.

The function of the RPA is to measure the ion ?ux distribution in the ram direction as a function of energy, as outlined further in Section 1.2. At the heart of the RPA is a set of biased grids through which the particles of interest pass. The interaction between these charged particles and the grids is the focus of the study proposed here.

The retarding grid structure is also fundamental in the construction of a new instrument known as the Ram Wind Sensor (RWS), designed to measure the ?ux distribution of neutral atoms with constant angle of attack. This will be further outlined in the Section 1.3.

1

Over the years, various curve-?tting techniques have been developed to extract parameters such as density, temperature, drift velocity, and species composition from RPA data. One of these techniques will be described in Chapter 2, as well as several perturbations not considered in the ideal case.

A new analysis technique designed to consider these perturbations is outlined in Chapter 3. This technique uses a discrete analysis to simulate a set of collected currents as a function of voltage bias on the grid for a given incoming distribution of ions.

Chapter 4 outlines the methodology used to simulate the values of current. In addition, several practical considerations will be discussed in using this equation in a curve-?tting routine.

1.2

Measurement of a Drifting Particle Distribution

The RPA measurement technique consists of collecting ions above a speci?ed energy while excluding lower energy ions. This is achieved by passing the ions through a series of biased grids prior to measuring the ion current. Figure 1 shows a cartoon drawing of a normalized drifting Maxwellian distribution. The temperature has been arti?cially in?ated over that found in a naturally occurring ionosphere such that vdrif t = vthermal for the purpose of illustration. Both of these values have been set to a magnitude of 1 for simplicity. The graph is set up such that a negative velocity indicates ions moving toward the collector, and a positive velocity indicates those moving away from it. Note this implies that a certain fraction of ions in the distribution cannot be collected. 2

Figure 1: A sample density distribution. The shaded portion of the curve illustrates the ions collected for a particular bias voltage. Ions with a positive velocity relative to the collector are moving away from the collector and cannot be collected.

A speci?ed voltage is applied to one of the grids to select ions above a given energy. This grid is referred to as the RV (retarding voltage) grid. Ions with a kinetic energy less than the stopping energy speci?ed at the RV grid will not pass through the grid to be collected. The shaded portion of Figure 1 represents those ions moving fast enough to penetrate the RV grid for a given applied potential (0.32
2 mv0 ). e

Note that collected current decreases as

stopping potential increases, resulting in a current-voltage (IV) characteristic curve. The IV

3

curve shown in Figure 2 is derived from the sample distribution shown in Figure 1.

Figure 2: The IV Curve corresponding to the sample distribution in Figure 1. The circle indicates the current collected for the applied RV in Figure 1. By comparing successive current measurements as a function of stopping potential, the relative ?ux of particles having energies between the speci?ed energy values can be measured.

?ψ ∝ I(j ) ? I(j +1)

(1.1)

where ψ represents the ?ux and I is the measured current. If a distribution function is assumed (in this case Maxwellian), then the parameters that uniquely determine the distri-

4

bution (such as drift velocity, temperature, and particle mass) can be obtained by ?tting IV curves like that in Figure 2. This process will be described in further detail in Chapter 2.

1.3

The Ram Wind Sensor

An exciting new instrument being developed by the University of Texas at Dallas is the Ram Wind Sensor (RWS). The ?rst space ?ight of this instrument will be on C/NOFS, the Communications/Navigation Outage Forecast System satellite. C/NOFS consists of a low inclination satellite in combination with a system of ground-based computational models. The RWS will be ?own along with a cross-track drift sensor. Together, these two sensors [8],[9] will be able to obtain the three-dimensional neutral wind velocity vector. The RWS determines the ram component of the wind by measuring the ?ux distribution in the ram direction. Figure 3a shows the interior of the engineering model of the RWS. Figure 3b shows the completed RWS delivered for integration onto the C/NOFS satellite.

(a) Engineering Model

(b) Flight Model

Figure 3: Images of the Ram Wind Sensor

The RWS instrument is mounted on the ram surface of a satellite to mitigate wake e?ects and to prevent interference to the neutral ?ow from obstacles such as boom systems. Figure 5

4 shows the conceptual block diagram of the RWS. The ionospheric particles enter the small knife-edge aperture from the left of the diagram. The beam of particles ?rst ?ows between two biased plates designed to collect ions and electrons, leaving only the neutral particles in the beam. Critical surfaces of the device are coated with black nickel [1] to suppress photons.

Figure 4: Conceptual Block Diagram of the RWS

A cross-track beam of energetic electrons is used to ionize a small fraction of the incoming neutral particles. The distribution of the resulting ions can then be analyzed with an RV grid, as described in the previous section. A channel electron multiplier (CEM) is used to detect the ions after they have passed through the RV grid. The sensitivity of the CEM is such that the small numbers of ionized neutrals at high altitudes can be detected.

6

Chapter 2 Analysis Techniques

2.1

The Knudsen Analysis Technique

The 1966 technique for analysis of drifting Maxwellian distributions of ions developed by Knudsen [10] is often used in conjunction with least-squares ?tting routines to ?t a distribution to a data set. First we consider that each ion species is in a thermal equilibrium subject to a drifting Maxwellian distribution over ram velocity (v⊥ ).

nj (v⊥ ) =

nj 0 2πkB T /mj

exp ?

mj (v⊥ ? vj 0 )2 2kB T

(2.1)

where nj 0 is the total density, mj is the ion mass, kB is Boltzmann’s constant, T is the ion temperature, and vj 0 is the ion drift in the rest frame of the instrument. We can simplify this expression by using a substitution for thermal velocity

αj =

2kB T mj 7

(2.2)

so that v⊥ ? vj 0 nj 0 nj (v⊥ ) = √ exp ?? αj π αj
?
2

? ?

(2.3)

We assume that all ions below the stopping energy are rejected. This energy value corresponds to a stopping velocity below which no ions will be collected.

vj ?stop =

2εstop mj

(2.4)

We can then eliminate the mass term by substituting αj .

vj ?stop = αj

2εstop kB T

(2.5)

If the potential of the grids is planar and normal to the velocity of the incoming particles and if the grids are considered to have negligible thickness, then the total number of each ion species collected per second will correspond to



Nj =

vj ?stop

χAvnj (v )dv

(2.6)

where χ is the total optical transparency of the grid stack and A is the area of the aperture. The collected current will be I=e
j

Nj

(2.7)

where e is the fundamental charge. We integrate over v for a given species, yielding

Nj = χA

exp(?κ2 nj vj 0 j) 1 + erf (κj ) + √ 2 πv/αj 8

(2.8)

where κj = vj 0 ? vj ?stop αj (2.9)

The total current collected will be

I (εstop ) = eχA
j

exp(?κ2 n j vj 0 j) 1 + erf (κj ) + √ 2 πv/αj

(2.10)

In principle, Equation 2.10 can be used in a least-square or a chi-square ?t routine to uniquely determine nj , v0j , and T , assuming that all species are at the same temperature. If the drift velocity is the same for all species, then

vj 0 = v0

(2.11)

for all j , and we only need to ?t for m + 2 parameters, where m represents the number of species under consideration.

In practice, these relatively simple assumptions do not fully describe the particle trajectories. Perturbations on the ideal case may arise from many factors, such as the non-uniformity of the potential due to the grid structure, the arrival angle of incoming ions, contaminants on the grid surface, and space charge e?ects. The work proposed here concentrates on the ?rst two perturbation sources, since these e?ects are likely to dominate in many spacecraft scenarios.

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2.2

Non-Uniform Potential

The geometry of a meshed grid system will set up a non-uniform potential in the grid plane, as described by Hanson et al. [6], with the applied potential appearing near the grid wires and a lower potential in the center of each hole. This allows some ions with energy just below the stopping energy (referred to hereafter as threshold ions) to pass through the grid depending on their relative position and trajectory with respect to the grid wires. This has the e?ect of increasing the collected current above that predicted by the formula above (Equation 2.10).

In practice, the leakage of ions can be reduced by using a double-grid system to reduce the lowest potential seen in the grid plane. A triple-grid system was recently proposed as a measure to further reduce leakage [4]. Further studies by Chao [2], [3] suggest using an depressed retarding potential in ?tting calculations, based on the average potential in the RV grid plane.

While recent simulation work has been done with ?at grid systems, in practice these grids are often woven, producing further perturbations in the grid-plane potential.

2.3

Angular Considerations

While it does reduce the total leakage, the double-grid system provides another distortion in the collected current. The previous calculations neglect the e?ects of any transverse velocity components that the incoming ions may have. These velocities may result from a thermal distribution as well as from a net drift of the ions parallel to the plane of the grid. 10

Figure 5 illustrates this problem in cartoon form. Depending on the relative grid orientations, a portion of the threshold ions may be de?ected due to the angular trajectory, thus reducing the expected leakage. Ions may be either de?ected by the second grid (Figure 5c) or translated along the grid plane (Figure 5d). Near the edge of the instrument, particles may be translated far enough away to miss the collector. Either of these e?ects will change the total leakage of the system, and therefore change the IV characteristic from which estimates of nj , v0j , and T are inferred.

(c)

(d)

Figure 5: Sample Ion Trajectories

2.4

Need for an Improved Technique

In order to improve our analysis of RPA data, as well as future RWS data, we must characterize the e?ects of the grid system on the ion distributions collected. While several simulations have been carried out in recent years [2] [4], these tests do not fully characterize the e?ects of real grids on particle trajectories and three-dimensional distributions.

I propose to address these concerns by using the ANSYS multiphysics simulator. By using an iterative technique, ANSYS can characterize transmission coe?cients for ions of speci?c ram energies and incidence angles. These values can then be incorporated into a new analysis technique (fully described in Chapter 3), which can be compared with Equation 2.10 to qualify and quantify changes in the ?tted parameters. Based on the preliminary results from 11

this study, alternative grid structures may be investigated to minimize leakage and lead to more accurate measurements.

12

Chapter 3 Discrete Analysis

3.1

A Simple Construct

The e?ects of non-ideal grids can be expressed as an energy and arrival angle dependent transmission coe?cient for particles encountering the grid. In order to better visualize the incorporation of these calculated transmission coe?cients, we ?rst consider a simpli?ed case for which there is only a ram velocity associated with the incoming ions. We de?ne a density distribution n as a function of the ram energy, or the kinetic energy due to the ram velocity
2 mv⊥ ), subject to the condition (1 2



N =
0

n(ε⊥ )dε⊥

(3.1)

where N is the density of particles moving toward the collector and n(ε⊥ ) is the density of particles as a function of ram energy. Note that N is not the total density of particles, as a percentage of the particles will be moving away from the collector.

13

We then consider the di?erential ?ux of particles through a grid as a function of this continuous density distribution function.

dψ = v⊥ n(ε⊥ )dε⊥

(3.2)

where v⊥ =

2ε⊥ . m

Note that the total ?ux toward the collector will be



Ψ=
0

v⊥ n(ε⊥ )dε⊥

(3.3)

For collected data, we must consider discrete measurements. We will consider M values of ε⊥ . Thus we rewrite Equation 3.2 as

?ψ(j ) = v⊥(j + 1 ) n(j ) ?ε⊥(j )
2

(3.4)

where ?ε⊥(j ) is the size of the energy bin associated with the density level of interest. Since the grid is designed to exclude particles below a given energy, we will de?ne the energy bin as lying between the energy of interest and the next energy, i.e., ?ψ(j ) is the ?ux of particles having ram energies between ε⊥(j ) and ε⊥(j +1) . We must use the v⊥ corresponding to ε⊥(j + 1 )
2

as the average velocity of particles in the energy bin of interest.

Now we de?ne a vector ψj , of which the components are the M values of ?ψ(j ) . We de?ne an additional vector Ii , consisting of N values, to represent the currents collected for N

14

values of retarding voltages applied. The current vector is found to be

?ij ψj Ii = eAT

(3.5)

? is an N × M transformation where e is the fundamental charge, A is the collection area, and T matrix that we call the Energy Flux Index (EFI). For this and all subsequent cases, a repeated index on a given side of the equation implies a summation over that index. The function of the transformation matrix is to specify what fraction of the ?ux in each energy bin will pass through the RV grid under each of the N applied voltages. Note that by this de?nition, the ?ij . transparency coe?cient χ is folded into the construction of T

?ij has an inverse, then we can solve for the If we consider only those conditions in which T ?ux vector. ψj = 1 ? ?1 T Ii eA ji (3.6)

In this case, we restrict ourselves to N values of ?ux (i.e., N = M ) for this solution to produce an exact value.

3.2

The Idealized Solution

First we look at the idealized solution of the above problem. In an ideal case, the grid itself has negligible thickness and the potential barrier it establishes is planar. There is no variation based on position or arrival angle, so either all of the ?ux is rejected or all of it is accepted. In this case, all components of the EFI must be either 0 or 1, scaled by the

15

transparency coe?cient. If our set of energy values corresponds to our set of RV values, then the EFI will look like
? ? ? ? ? ? T = χ? ? ? ?

1 0 0 . . .

1 1 0 . . .

1 1 1 . . .

··· ··· ··· ...

1 1 1 . . .

? ? ? ? ? ? ? ? ?

(3.7)

0 0 0 ··· 1

where χ is the optical transparency of the grid stack.

Likewise, the inverse of this matrix is reduced to
?

? ?1 T

? ? 1? = ? ? χ? ? ?

1 ?1 0 0 1 ?1 0 0 1 . . . . . . . . . 0 0 0

··· ··· ··· ...

0 0 0 . . .

? ? ? ? ? ? ? ? ?

(3.8)

··· 1

Note that by applying this inverted EFI matrix to the vector Ii , we end up with

?ψ(j ) =

1 I(j ) ? I(j +1) eχA

(3.9)

which is analogous to our de?nition of relative ?uxes described in Equation 1.1.

3.3

Perturbation Considerations

If we consider a realistic grid system where a small fraction of the ions near the stopping potential may leak through the grid, then the morphology of the EFI becomes slightly di?erent. The upper triangle of the EFI will be unchanged, but the values just below the diagonal will be replaced by fractional values.

16

? ? ? ? ? ? T =? ? ? ?

χ χ f12 χ f22 f23 . . . . . . 0 0

χ χ χ . . .

··· χ ? ··· χ ? ? ··· χ ? ? ? .. . . . . ? ? 0 ··· χ

?

(3.10)

The bottom right corner of the lower triangle will remain unchanged, as only particles near the stopping potential will leak through the grid. Note that by de?nition

fij ≤ χ

(3.11)

The relative magnitude of the fij terms e?ectively quantify the leakage of the non-ideal grid geometry under consideration.

3.4

The General Solution

This tensor system of analysis can quickly be generalized to include angular dependence. We now de?ne ψjkl as a 3-rank tensor which describes the ?ux distribution as a function of ram energy (j ) and of angles in spherical coordinates (k, l). Again, we can characterize the transmission coe?cients as a function of all of these variables plus the applied retarding potential (i), yielding a 4-rank tensor EFI. Again, we sum over all repeated indices on the right-hand side. ?ijkl ψjkl Ii = eAT (3.12)

This becomes increasingly complicated should this equation be incorporated into a leastsquares ?tting routine. While this equation can be used to describe any ?ux distribution, we will limit ourselves to one speci?c case for the initial study. 17

3.5

A Proposed Solution

We ?rst assume is that there is no drift transverse to the ram direction and that the ?ux distribution is cylindrically symmetric about the ram velocity axis. The advantage of this setup is that the number of ?t parameters is reduced. This speci?c case can be described by

?ij ψj + T ? ψ Ii = eA T ijk jk

(3.13)

Note that the ?rst term in the brackets is identical to the simpler case outlined in Sections 3.1-3.3. This is the part of the distribution which has no transverse velocity. The second (primed) term in brackets describes the components which approach at a small angle. For a cylindrically symmetric drifting Maxwellian distribution, we can relate ψ to ψ by

ψjk = fk ψj

(3.14)

where fk is a proportionality constant related to the distribution for the k th incidence angle. Equation 3.9 can be rewritten as

?ij + T ?ijk fk ψj Ii = eA T

(3.15)

?ijk must be an average of the transparency coefIt should be mentioned that each value of T ?cients over the azimuthal angle b, as shown in Figure 6. This is done to enforce cylindrical symmetry across the system.

18

Figure 6: Azimuthal Angle b in the Grid Plane

The immediate bene?t of the form of Equation 3.15 is that it allows us to quickly judge the e?ects of angular dependency. In a case where there is no angular dependence, we see ?ij ) should be proportional to the angular EFI (T ? fk ), since both will that the ram EFI (T ijk act in the same way upon the ?ux vector. This can be characterized by

?ijk fk = ζ T ?ij T
k

(3.16)

where ζ is constant for all i, j .

?ijkl ) still must be calculated for this Note that all the components of the generalized EFI (T process. The assumptions are made to simplify the data-?tting process. These assumptions can easily be tested, as explained in the following chapter.

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3.6

Data Fitting

By using a least-squares ?tting routine, we can construct fk and ψj for a given drifting Maxwellian distribution and replace the Knudsen equation (2.10) with Equation 3.15 in standard ?tting routines. As a validation of this approach, we can simulate the case in ?ijkl takes on its ideal form and compare the results with Equation 2.10. which T

20

Chapter 4 Methodology

4.1

A Brief Description of ANSYS

The EFI can be constructed using the ANSYS Multiphysics simulation software. ANSYS is a Finite Element Analysis (FEA) tool used for the simulation of many physical phenomena, including structural, thermal, and electromagnetic analysis.

A three-dimensional geometry is de?ned, such as that seen in Figure 7e. Physical properties such as relative permeability and permittivity are de?ned and associated with given volumes, and boundary conditions are set according to user speci?cations.

The software then de?nes a three-dimensional adaptive mesh over the speci?ed geometry, as shown in Figure 7f. The shape and size of the cells is variable based on several user inputs and may be altered to optimize the calculations.

21

(e) Solid Elements in ANSYS

(f) All Elements Meshed

(g) Ion Trajectories

Figure 7: Some capabilities of ANSYS. These were made using a sample construction module. The user can then specify voltages to apply to certain volumes. In this case, the voltages are applied to the RV grid and the front and back ends of the 3D brick in Figure 7e. The software applies these voltages to the nodes of each cell in the volumes of interest and calculates the potential at each of the remaining nodes in the free space surrounding the grids. From this, ANSYS can also calculate the electric ?eld at each node.

Ion optics is another task easily handled by ANSYS. Once the electric ?eld is calculated for all space in the simulation, a charged particle can be placed anywhere within the volume according to user-de?ned initial conditions, including charge, mass, position and initial velocity. The trajectory path can then be calculated and displayed, as shown in Figure 7g.

4.2

Veri?cation

In order to verify the ANSYS method of solution, I propose to ?rst run the simulation using an idealized grid, as described in Section 2.1. The criteria for success is that the EFI constructed by the ANSYS routine will be identical to the ideal EFI (Equation 3.7).

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4.3

Research Approach and Goals

The research of this project can be carried out in three distinct steps. 1. Construct an RV grid system in ANSYS to build the EFI 2. Develop FORTRAN code to simulate IV curves using the EFI 3. Develop and test FORTRAN code for curve-?tting using the EFI The overall goals of this work are: A. Characterize the e?ects of a non-ideal grid on a drifting Maxwellian distribution of particles. (achieved by completion of Step 1) B. Quantify the errors that grid and arrival angle e?ects introduce into the measurement of nj , vj 0 , and T in standard curve-?tting analysis. (achieved by completion of Steps 2 and 3) C. Suggest improvements in grid designs and analysis techniques that will reduce these errors. (achieved by further iterations of Steps 1, 2, and 3)

4.3.1

The ANSYS Code

The ANSYS code in development is constructed as a modular system in order to simplify both the development process and future modi?cations to accommodate other grid geometries and incident particle distributions. The four modules described below can be seen in the ?ow chart in Figure 8.

23

The Construction Module (Mod C) speci?es the three-dimensional geometry of the RV grid and the surrounding volume under consideration. It also applies material properties to the speci?ed volumes, speci?es boundary conditions, and de?nes the mesh. While a sample Module C has been constructed to test the capabilities of ANSYS, the module speci?c to the geometry of interest is still under construction.

The Voltage Module (Mod V) applies a speci?ed voltage to RV grid and sets the boundary plates to ground. It then calculates the potential and electric ?eld everywhere in the 3D volume. Module V is currently complete and has been tested for sample geometries in test versions of Module C.

The Transmission Module (Mod T) speci?es the properties of individual ions, including velocity, attack angle, charge, and mass. The module then calculates a trajectory for for a single ion at a given position and repeats this process for many positions, as determined by a random number generator. This is repeated many times (> 10, 000) to simulate a large population passing through the grids, and the fraction of those ions transmitted is saved to ?le, along with the corresponding RV, ε⊥ , and angular information. Module T is currently under construction.

The Trajectory Path Module (Mod P) is an optional module that displays the ion paths as a function of position for a given ion energy / angle combination and applied RV. Module P saves these tracks to an image ?le for demonstration purposes. It is currently complete.

24

Module C

Functions Construct Geometry Mesh Volume

V

Apply Voltages Calculate Resultant Fields

T

Calculate Transmission Save EFI info to ?le

P

Display Particle Paths Save Path as JPG

Table 1: Functions of ANSYS Modules

Figure 8: Flowchart for the ANSYS Modules

25

4.3.2

The FORTRAN Simulator

The second step is to write a program to process the ANSYS output to produce a simulated IV curve for a given Maxwellian distribution. This code will utilize the EFI generated by the ANSYS code and the general equation described previously (Equation 3.12). The Maxwellian distribution for a given set of input parameters (n0j , v0j , α, b, T ) can be calculated for all components of the tensor ψjkl . Random noise of a speci?ed level may be added to better simulate real collected data. In addition, the curve can be decimated to simulate data collection with sample and hold Analog-to-Digital (ADC) Circuit.

4.3.3

The Data-Fit Routine

The third step is broken into two parts. The ?rst is to ?t this simulated IV curve with a traditional ?t routine; the second is to ?t the curve with the new technique outlined in Section 3.5. Errors on the ?tted parameters can then be quanti?ed for several sets of input parameters for both codes.

The validity of neglecting the cross-track velocity in the ?t can also be determined during this stage. Additional simulations can be generated for several cross-track drift velocity inputs, and the errors on the ?t parameters can be quanti?ed by repeating Step 3 for these cases.

26

Chapter 5 Summary and Conclusions
The program of study outlined in this proposal will extend the current state of knowledge by considering several shortcomings of retarding grid-type devices to a level not previously studied. By using the discrete analysis approach outlined in Chapter 3, we will obtain both a numerical analysis tool and a convenient method for visualizing the relative ?ux leakage of non-ideal grids. This method allows for ease in comparison of the e?ectiveness of di?erent grid geometries. By incorporating the ANSYS simulation tool into our analysis, we will be able to realistically quantify the e?ects of the grid on both the IV characteristics and the ?t parameters inferred from them. All of these approaches will be new areas of applied research, and should therefore be publishable in journals such as Reviews of Scienti?c Instruments or IEEE Transactions on Plasma Science. We anticipate at least two such publications: one covering the e?ects of various grid geometries on the EFI, and a second quantifying the changes in inferred parameters caused by neglecting the e?ects of non-ideal grids.

27

The approximate time frame for completion of this work is given in the table below.

Milestone Complete ANSYS Modules Complete Code Validation Compare EFI for Ideal and non-Ideal cases Publish Results Complete FORTRAN Analysis Codes Perform IV case studies to quantify e?ects on inferred parameters Publish Results Defend Thesis

Estimated Completion Fall 2006 Fall 2006 Spring 2007 Summer 2007 Fall 2007 Fall 2007 Spring 2008 Summer 2008

Table 2: Estimated Time Frame for Completion

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