• I wanted to get a sense of the baseline electrical field magnitude in different regions and to check the sensitivity to changes in the dielectric properties of surrounding structures.
• I could use the DXF import capability of the Mesh drawing editor to create a set of outlines for the complex turnings. After testing, I moved them directly to the MetaMesh script.
• I wanted benchmark values to check that the 3D solution was set up correctly.

Because 2D solutions run quickly, I incorporated the entire injector assembly in the mesh. Extending this solution volume to three dimensions would have resulted in a huge mesh and many hours of run time. I feel that if a run takes more than one hour, there is probably a better way to solve the problem.

Since the flaws were relatively localized, it would be possible to get a good idea of their effect by limiting the axial extent of the solution to a space near the insulator. In this case, I could define a fixed-potential region on the solution volume boundaries in -z and +z and assign a spatial variation using data from the EStat solution. The existing version of HiPhi supported the definition of potential variations from mathematical functions using the command:

POTENTIAL NoReg > f(x,y,z)

Here, f(x,y,z) is any algebraic function of the Cartesian coordinates. In order to use the feature, it would have been necessary to take a radial scan of potential and then to fit the results with a power series  in √[x^2 + y^2]. This approach seemed like it would be a lot of work, and I didn’t feel like doing it.

Instead, I took advantage of my unique position as deity of the HiPhi source code and added a new program feature. It is ideally suited to using a 2D solution as the basis of a 3D microscopic solution. Here is the corresponding script command

POTENTIAL NoReg TABLE [x,y,z,r] TabName

The string TabName is the name of a text file defining the potential variation along the specified direction (the variable r is interpreted as √[x^2+y^2]). It consists of a set of data lines:

r[n]    phi(r[n])

For the project I prepared tables for the downstream and upstream boundaries directly from EStat scans, using ConText to remove unwanted columns. The process took only a few minutes.

The figure below shows a test solution for the potential inside a grounded cylinder. The bottom boundary in z is grounded and the potential on the top follows a tabular variation in r (a partial cosine function). The top picture shows equipotential lines in the plane x = 0.0 and the bottom shows lines in a plane normal to z near the defined fixed-potential boundary.

I modified the HiPhi instruction manual, describing how to use the new command forms. While I was at it, I decided to ex[and the existing commands for relative dielectric constant, conductivity and space-charge density to handle table input.

For more information on HiPhi, please use this link: http://www.fieldp.com/hiphi.html.

With two independent time-varying electrode voltages, the spatial distribution of the field changes at each instant. Clearly, it would be impractical to solve for the three-dimensional electric field at each integration step of a particle orbit. Fortunately, there is a simple way to find the time-varying field when there are two independent voltages. We can understand the method with reference to the field solution illustrated in the first figure below. The simple three-electrode system has a ground plane at the bottom. Independent voltages may be applied to the electrode at the top left (V1) and right (V2). We can construct two normalized base solutions: solution S10 has V1 = 1.0 V and V2 = 0.0 V while solution S01 has V1 = 0.0 V and V2 = 1.0 V. In the absence of nonlinear materials, the set of all possible electrostatic solutions with arbitrary voltages V1 and V2 may be generated by a linear combination of the base functions:

S = V1*S10 + V2*S01.

To illustrate, the bottom solution in the first figure is the sum of the base functions. The same solution would follow from a direct HiPhi solution with V1 = V2 = 1.0 V.

The following procedure is used for modulated dual-voltage solutions in OmniTrak:

• Prepare two base electrostatic solutions with HiPhi using the same mesh. One or more reference electrodes in both solutions are grounded (φ = 0.0 V). In the first solution, one electrode (or set of electrodes) has V1 = 1.0 V and the other has V2 = 0.0 V. The voltages levels are reversed in the second solution.
• Load the first solution with the EFIELD3D command. The program reads the mesh characteristics as well as the potential values at all nodes, φ1(i,j,k). A normalization factor may be included in the command to set the amplitude V1 in solution S10.
• Load the second solution with the new EFIELD3D2 command. In this case, OmniTrak checks that the mesh is identical to the previous one and stores the potential values φ2(i,j,k). Again, a normalization factor may be used to set the amplitude V2 for S01.
• The command MODFUNC E is used to define a modulation function M1(t) for S10, while the command MODFUNC E2 sets the modulation function M2(t) for S01.

During orbit tracking, the program uses electrostatic potential values of the form

φ = M1(t)*S10 + M2(t)*S01,

as input to the electric field interpolation routine. Because much of the computational work involves identification of the element occupied by the particle, there is little change in the orbit tracking time penalty for a dual-voltage solution.

We can track a proton through the solution shown in the first figure to demonstrate the procedure. The Field section of the OmniTrak script includes the following commands:

EFIELD3D: ThreeElect01.HOU 100.0
MODFUNC(E) > sin(1.2566E7*$t)
EFIELD3D2: ThreeElect02.HOU 200.0
MODFUNC(E2) > sin(3.1416E7*$t)

The left-hand electrode has a sinusoidal variation at 2.0 MHz with amplitude 100 V. The peak electric field far from the gap is Ex = -2000 V/m. The voltage on the right-hand side varies at 5.0 MHz with amplitude 200 V (Ex = -4000.0 V/m). A 250 eV proton moving in z is injected near the left-hand boundary. The transit time over the 20.0 cm distance is about 1.0 μs. The commands

LISTON 1
LISTTYPE EField

specify an electric-field listing along the particle trajectory. The second figure shows the resulting electric field experienced by particle as a function of axial position.

For more information on OmniTrak, please use this link: http://www.fieldp.com/omnitrak.html.

I recently had an inquiry about how to find the self-consistent potential of a floating open region. Such a region consists solely of node assignments and has zero volume. In this case, it is not possible to assign special material properties. One resolution is to perform a series of calculations with different potentials assigned to the region and to look for the one with the lowest electric-field energy. This involves some extra work. In the next post, I will show how to automate the procedure using a Perl script.

The first picture below shows the geometry and equipotential lines of a test solution in cylindrical coordinates. The inner electrode has potential φ = 10,000 V and the outer boundary is grounded. There is a fine mesh grid at unknown potential in the intervening space. The goal is to find the grid potential determined by capacitive division for a rapidly-pulsed voltage. I made a series of 20 calculations with grid potentials from 2000 to 10,000 V. The top section of the first figure shows the distorted field distribution at 9200 V. I calculated the electrostatic field energy for each case. The second figure shows a plot of energy versus grid voltage. There is a minimum at 4000 V. The bottom section of the first figure shows the resulting smooth field distribution. Sample files for this example are available in the next post.

Please use these links for more information on EStat (http://www.fieldp.com/estat.html) and HiPhi (http://www.fieldp.com/hiphi.html).

I added entries to the index of the EStat manual and decided to write this post. The figure below shows a plot of |E| for a cylindrical example. The solution volume is bounded on the right and top by the shaped inner wall of a grounded vacuum chamber. A high-voltage rod electrode with spherical tip is supported by an insulator on the left-hand side. The goal is to find the variation of electric field over the electrode and the vacuum surface of the insulator.

The Region command in the Analysis menu of EStat generates the listing. We could also set up analysis script like the following:

INPUT FieldList.EOU
OUTPUT FieldList
* Electrode
REGION 2
* Insulator
REGION 3
ENDFILE

EStat performs several activities in response to the command, including volume integrals of field energy, surface integrals of normal electric field to infer induced charge and surface integrals of the Maxwell stress tensor to determine force if the region is surrounded by air or vacuum. The code also calculates the electric field on element facets that lie on the region boundaries. It then goes to some effort to arrange them in a logical order. Here’s an example of the first portion of the table that appears in FieldList.DAT:

Identified  102 facets
Maximum field value of    2.5310E+06
occurs at z =    1.6848E+01  r =    1.0876E+00
on the boundary with Region No   1

Z            R            D           |E|           Ez           Er      NBorder
=====================================================================================
1.6997E+01   1.0134E-01   0.0000E+00   2.5182E+06  -2.5174E+06  -6.3852E+04     1
1.6988E+01   2.9815E-01   1.9705E-01   2.5280E+06  -2.5210E+06  -1.8840E+05     1
1.6968E+01   4.9752E-01   3.9744E-01   2.4856E+06  -2.4663E+06  -3.0928E+05     1
...

In this example, EStat does a very job of ordering the facet data. On the electrode, the list starts on the axis at the tip and travels along the surface to the left-hand boundary. For the insulator, the list covers the vacuum face, moving outward in radius. Each data line contains 1) the average coordinates of the facet and the distance from the start point, 2) the magnitude and components of the electric field and 3) the number of the region that shares the boundary facet.

An outstanding feature is that you can generate electric field listings along surfaces of arbitrary shape. For example, if you want to know the field along a curve in vacuum, divide the vacuum region into two vacuum regions separated by the desired path.

Here are links to input files for the example:

fieldlist.min
fieldlist.ein
fieldlist.scr

Please use this link for more information on EStat: http://www.fieldp.com/estat.html.