Figure 1 illustrates the problem with internal voids. The steady state solution addresses a square volume of uniform conductivity with a void spherical bubble at the center. The upper and lower boundaries in z have different fixed temperatures. We expect to see a solution like the lower one where the void makes a local perturbation such that lines of thermal flux go around it. Instead, in the top solution the void acts like a fixed-temperature surface with T = 0 °C.

Figure 1. Thermal solutions, uniform conductor between fixed-temperature plates with an insulating bubble. Top: Incorrect node assignment. Bottom: Valid solution.

The resolution lies in how region numbers are assigned to nodes and elements in MetaMesh. Here is an excerpt from the MetaMesh script for the example:

RegName(1): Conductor
RegName(2): Void
RegName(3): BoundDn
RegName(4): BoundUp
 ...

PART
  Region: Conductor
  Name: SolutionVolume
  Type: Box
  Fab:   2.00000E+00  2.00000E+00  2.00000E+00
END

PART
  Region: Void
  Name: Void
  Type: Sphere
  Fab:   2.50000E-01
  Surface Region Conductor
END

The first Part section fills the entire solution volume with the conductive material (region 1). The program sets the region numbers of all nodes and elements to RegNo = 1. The next Part section adds the spherical void (region 2). When an internal region is defined in MetaMesh, the convention is to overwrite existing element and node region numbers in the shared volume. In this case, nodes on the spherical boundary assume the region number of the void, which has undefined material properties in the HeatWave solution. In other words, the boundary nodes will remain at the default HeatWave condition of fixed-temperature of T = 0 deg-C.

The solution is to add the following command to the void part section:

Coat Conductor Conductor

The first region number following the Coat keyword is a region adjacent to the current part. MetaMesh collects nodes that lie on this intersection and changes their region to the second region number. In this case, all nodes on the surface of the sphere will have the material properties of the conductor. The coat command was included in the MetaMesh script to obtain a valid solution (lower part of the figure).

Use this link to download the input files for the demo calculation: InternalVoidDemo.zip. If you have questions or comments, please contact us at techinfo@fieldp.com

To understand transparent electrodes, we need to review some features of electric-field solutions and particle tracking in HiPhi and OmniTrak. The mesh created by MetaMesh consists of hexahedron elements, small divisions of the volume. Nodes are the vertices of the hexahedrons. In MetaMesh, region numbers are assigned to both the elements and nodes. The solution of Poisson’s equation involves determining electrostatic potential values at the nodes, influenced by the assigned of dielectric constant to the elements. Nodes inside and on the surface of an electrode are assigned a value that remains unchanged during the HiPhi solution process. Therefore, the region identity of elements inside the electrode does not affect the field solution. During particle tracking, OmniTrak stops a particle if it enters a Material region. Because the nodes have no volume, the code checks the region identity of elements. Therefore, if we constructed an electrode where the region number of nodes had the Material property but the region number of elements had the Vacuum property, then 1) the electric-field solution would be correct, and 2) particles would pass through without stopping. We can create such a region using the Coat command in MetaMesh.

Figure 1 shows an example solution, SOLIDDEMO. Protons accelerate across the Dee gap of a cyclotron with a cylindrical plasma column on the downstream side. Excerpts from the MetaMesh input file are listed at the end of the article. The solution includes the following regions:

RegName(1): Vacuum
RegName(2): Upstream
RegName(3): Downstream
RegName(4): Plasma

Region 1 is the standard vacuum propagation region, treated as a dielectric in HiPhi with εr = 1.0. Regions 2 and 3 are the fixed-potential electrodes. The important step is to assign a unique region number to the elements of the plasma column, even though they have the same physical property as the vacuum elements. MetaMesh uses this region number to identify nodes on the surface of the plasma.

Figure 1. Geometry of the SOLIDDEMO example

The Part sections to define solution volume and electrodes are unexceptional. The Part section for the plasma is the interesting one:

PART
  Region: Plasma
  Name: Plasma
  Type: Cylinder
  Fab:   5.00000E-01  2.00000E+00
  Shift:   2.00000E+00  0.00000E+00  0.00000E+00
  Rotate:    90.00    0.00    0.00  XYZ
  Surface Region Vacuum
  Coat Vacuum Downstream
END

The Surface command shapes facets on the boundary of the Plasma and Vacuum regions to conform to the cylindrical shape. The Coat command specifies that all nodes on the boundary between Plasma and Vacuum elements should be reassigned to the fixed-potential Downstream region. The plot of Figure 2 in a plane normal to the plasma column shows the assignment of nodes. This is exactly the condition that we want.

Figure 2. Assignment of nodes in the SOLIDDEMO example. Zoomed plot in a plane normal to the plasma column.

Figure 3 (a plot in the mid-plane normal to the plasma column) shows the distortion of fields resulting from the presence of the plasma along with sample orbits of 5 keV protons crossing the gap with an applied accelerating voltage of 10 kV. Similar techniques may be applied to 2D solutions with EStat and Trak. In this case, a fixed potential line region on the surface of the cylinder is created after the definition of the dielectric cylinder volume. For OmniTrak users, here is a link to download the input files for the SOLIDDEMO example: http://www.fieldp.com/myblog/examples/soliddemo.zip. If you have questions or comments, please contact us at techinfo@fieldp.com

Figure 3. Electrostatic potential and particle orbits near the plasma column in the mid-plane normal to the column.

* File: SOLIDDEMO.MIN
GLOBAL
  XMesh
   -5.00000E+00  5.00000E+00  1.00000E-01
  End
  YMesh
   -5.00000E+00  5.00000E+00  1.00000E-01
  End
  ZMesh
   -5.00000E+00  5.00000E+00  1.00000E-01
  End
  RegName(  1): Vacuum
  RegName(  2): Upstream
  RegName(  3): Downstream
  RegName(  4): Plasma
END
PART
  Region: Vacuum
  Name: SolutionVolume
  Type: Box
  Fab:   1.00000E+01  1.00000E+01  1.00000E+01
END
PART
  Region: Upstream
  Name: ElectUpTop
  Type: Box
  Fab:   3.50000E+00  1.00000E+00  1.00000E+01
  Shift:  -3.25000E+00  1.50000E+00  0.00000E+00
END
 ...
PART
  Region: Plasma
  Name: Plasma
  Type: Cylinder
  Fab:   5.00000E-01  2.00000E+00
  Shift:   2.00000E+00  0.00000E+00  0.00000E+00
  Rotate:    90.00    0.00    0.00  XYZ
  Surface Region Vacuum
  Coat Vacuum Downstream
END
ENDFILE

Corona discharges in gases occur only in when electrodes have strong two- or three-dimensional variations (as in Figure 1). Corona (crown in Latin) is a pattern of bright sparks near a point or edge of an electrode. In such a region, the electric field is enhanced above the breakdown limit so that spark discharges occur. Taking the dimension of the corona region as dc, the approximate condition for breakdown is given by Eq. 9.49. In the equation, <E> is the
average electric field in the corona region. A self-sustained breakdown cannot be maintained by the low electric field in the bulk of the gap. Current in the low-field region is conducted by a Townsend (or dark) discharge, with the corona providing the source current. The system is stabilized by the high resistivity of the dark discharge. A relatively constant
current flows, even though the sparks of the corona fluctuate, extinguish, and reform rapidly. Inspection of Eq. (9.49) shows that for constant geometry, the size of the corona region grows with increasing electrode voltage. The voltage drop in the highly ionized corona is low; therefore, the voltage drop is concentrated in the dark discharge region. At some elevated voltage, bulk breakdown occurs and the electrodes are shorted.

Figure 1. Physical basis for a corona discharge at a pointed electrode.

Substituting air parameters in Eq. 9.49 at one atmosphere (p = 760 torr) and solving for the average electric field leads to the following relationship:

<E> ≈ 438/ln(1340 dc)  [kV/cm]

The electric field depends weakly on the average size of the breakdown region. Very small electrode protrusions may not cause corona because the length of the resulting enhanced field is not be long enough to sustain a breakdown. For dc = 1.0 cm, the critical field is <E> ≈ 60 kV/cm. The number applies to dry air at sea level. The danger level for non-ideal conditions is generally taken as <E> ≈ 35 kV/cm.

If you have questions or comments, please contact us at techinfo@fieldp.com

dim FArray() as FolderItem

These statements appear in the calling routine:

redim FArray(-1)
CollectFiles(FTarget)

The recursive subroutine has the form:

sub CollectFiles(FItem as FolderItem)
Dim n as Integer
Dim f as FolderItem

For n = 1 to FItem.Count
  f = FItem.Item(n)
  if f.directory then
    CollectFiles(f)  // Recursive call
  else
    FArray.append f
  end if
next
end sub

The subroutine CollectFiles calls itself to whatever depth is necessary. Although the approach is succinct and guaranteed to astound the hoi polloi, there is a significant drawback. What if FTarget is extremely large and the user becomes bored? How can the collection process be stopped? Killing the thread that performs the collection in the middle of the recursion leaves an indefinite operation stack, generally followed by strange program behavior.

The point is that the same function can be accomplished with a safe non-recursive routine. Such a routine can be aborted at any time with no consequences. Recursion gives no advantages in speed or resource efficiency, so why climb a cliff if you can walk around? Here is my solution. I have divided the folder items for files and directories in separate global arrays:

Dim DirectoryCollection() as FolderItem
Dim FileCollection() as FolderItem

The non-recursive subroutine has the compact form:

sub CollectFiles(FTarget as FolderItem)
Dim NStart,NEnd as integer
Dim n,m as integer
Dim FItem,f as FolderItem

// Reset the arrays
redim DirectoryCollection(-1)
redim FileCollection(-1)

// Set the initial state
DirectoryCollection.Append FTarget
NStart = 0
NEnd = UBound(DirectoryCollection)

// Main loop, repeat until there are no additional directories
do
  // Loop through directories at the current level
  for n = NStart to NEnd
    FItem = DirectoryCollection(n)
    // Add items for the current directory
    for m = 1 to FItem.Count
      f = FItem.Item(m)
      if f.directory then
        DirectoryCollection.Append f
      else
        FileCollection.Append f
      end if
    next
  next
  NStart = NEnd + 1
  NEnd = UBound(DirectoryCollection)
loop until (NStart > NEnd)
end sub

If you have questions or comments, please contact us at techinfo@fieldp.com

Gas insulation is used is used extensively in high-voltage power transmission for two reasons:

1. Gas can recover its insulating properties after a breakdown.
2. It’s free.

In this application, it is critical to ensure that voltages and electric fields are well below levels that can lead to a self-sustained breakdown. To derive the breakdown conditions, first consider the one-dimensional gas-filled gap of Figure 1. The electrodes have separation d and the applied voltage is Vo. There is no charge flow between the electrodes at low voltage. We want to determine how a large interelectrode current can be initiated at high voltage.

Figure 1. One-dimensional gas-filled gap.

Suppose that a few electrons are produced on the negative electrode (possibly caused by photo-emission from cosmic rays or ultraviolet light). The small source current density leaving the electrode is represented by jo. The electrons are accelerated by the applied field and move between the widely spaced gas molecules, as shown in Figure 2a. In a collision with a molecule, an electron is strongly deflected and much of its kinetic energy is absorbed. As a simplification, assume that electrons are accelerated between molecules and lose all their directed energy in a collision. The parameter λ is the mean free path, the average distance between collisions. The mean free path is inversely proportional to the density of gas molecules. In turn, the density is proportional to the pressure, p, so that λ ~ 1/p .

Figure 2. Processes in a gas-filled, high-voltage gap. a) Electron migration in the presence of an electric field. b) Liberation of electrons on the negative electrode by ion recombination.

The average energy gained by an electron between collisions is given by Eq. 9.33. An election may ionize a molecule in a collision if ΔT is high enough (ΔT > 30 eV). In the ionization process, the excess kinetic energy of the electron drives an electron out of the molecule, leaving a positive ion. The two electrons move forward under the influence of the field, producing further electrons. The current density increases geometrically along the electron drift direction. The ions drift in the opposite direction and do not contribute to ionization in the gas. The motion of drifting ions is dominated by collisions; generally, they cannot reach high enough velocity to eject electrons in a collision with a molecule.

Electron multiplication in a gas with an applied electric field is characterized by α, the first Townsend coefficient [J. S. Townsend, Electricity in Gases, Phil. Trans. A193, 129 (1900)]. This parameter is defined by Eq. 9.34. Consider an element of length Δx at the position x. The quantity n(x) is the density of electrons at x. According to the definition of α, the total number of additional electrons produced in a volume with cross-sectional area dA and length Δx is given by Eq. 9.34a. Dividing both sides of the equation by dAΔx, and taking the limit of small Δx leads to Eq. 9.35. The solution of the equation implies an exponential electron density variation. Expressed in terms of the current density of electrons (assuming that the average electron drift velocity is independent of x), Eq. 9.35 implies the result of Eq. 9.36 if the negative electrode is located at x = 0. The current density arriving at the positive electrode is given by Eq. 9.37.

Although the amplification factor may be high, Eq. 9.37 does not imply that there is an insulation breakdown. The current terminates if the source term is removed. A breakdown occurs when current flow is self-sustaining, or independent of the assumed properties of the source. In the breakdown mode, the current density rapidly multiplies until the voltage supply is shorted (discharged). Ion interactions at the negative electrode introduce a mechanism to maintain a self-sustained discharge. Although the positive ions do not gain enough energy to ionize gas molecules during their transit, they may generate electrons at the negative electrode through secondary emission (Fig. 2b). This process is parametrized by the secondary emission coeficient γ. The coefficient is high at high ion energy (> 100 keV). The large value of reflects the fact that ions have a short stopping range in solid matter and deposit their energy near the surface. Ions in a gas-filled gap have low average energy. The secondary-emission coefficient has a nonzero value for slow ions because there is available energy from recombination of the ion with an electron at the surface. The secondary emission coefficient for a zero-velocity ion is in the range γ ~ 0.02.

With ion interactions, the total electron current density leaving the negative electrode consists of the source term plus a contribution from secondary emission. We define the following current densities: jo is the source, jc is the net current density leaving the negative electrode, ji is the ion current density arriving at the negative electrode, and ja is the total electron flux arriving at the positive electrode. Each ionizing collision creates one electron and one ion. By conservation of charge, the ion current is given by Eq. 9.38. The equation states that ions and electrons leave, the gap at the same rate in the steady state. The portion of the electron current density from the negative electrode associated with secondary emission is given by Eq. 9.39. Equation 9.39 implies the result of Eq. 9.40. By the definition of the Townsend coefficient α, we can write the total electron current arriving at the anode as Eq. 9.41. Combining Eqs. 9.40 and 9.41 leads to the result of Eq. 9.42.

Comparing Eq. (9.42) with (9.37), the effects of ion feedback appear in the denominator. The current density amplification becomes infinite when the denominator equals zero.  When the conditions are such that the denominator of Eq. (9.42) is zero, a small charge at the negative electrode can initiate a self-sustained discharge.  The discharge current grows rapidly over a  time scale on the order of the ion drift time, heating and ionizing the gas. The end result is a spark (a high-current-density plasma channel). Equation 9.43 gives the condition for spark formation.

Calculation of the absolute value of α as a function of the gap parameters involves complex atomic physics. Instead, we will develop a simple scaling relationship for α as a function of gas pressure and electric field. Such a relationship helps to organize experimental data and predict breakdown properties in parameter regimes where data is unavailable. The ionization coefficient is proportional to the number of collisions between an electron and gas atoms per centimeter traveled, or α ~ 1\λ.  The ionization coefficient also depends on the average electron drift energy, Eλ. The ionization rate is proportional to the fraction of collisions in which an electron enters with kinetic energy greater than the ionization energy of the atom, I. The properties of particles traveling through a random distribution of collision centers is treated in texts on nuclear physics. The familiar result is that the distance between collisions is a random variable that follows the Poisson distribution of Eq. 9.44. The energy of a colliding electron is, by definition, T = xeVo/d. The fraction of collisions in which an electron has T > I is given by the integral of Eq. 9.44 from x = Id/eVo to infinity. The result of these considerations is that α is well described by the scaling law of Eq. 9.45. The quantities A and B are determined from experiments or detailed collision theory. Equation 9.45a, the gap voltage for sparking, follows from substitution of Eq. 9.45 into 9.43. Solving for the sparking voltage Vs give the main result, Eq. 9.46.

Equation 9.46 is known as Paschen’s law [F.Paschen, Wied.  Ann. 37, 69 (1889)] for gas breakdown. The sparking voltage is a function of the product pd and constants that depend on the gas properties. The values of A and B are relatively constant over a wide voltage range when the average electron energy is less than I. Furthermore, Vs is insensitive to γ; therefore, the results are almost independent of the electrode material. Figure 3 shows a normalized lot of Vs versus pd. The sparking voltage reaches a minimum value Vsmin at a value pdmin. Voltage hold-off increases at both low and high values of pd. Voltage hold-off is high at low pd because there is a small probability that an electron will strike a molecule while traveling between electrodes. At high values of pd, Vs ~ pd . In this regime, electrons undergo many collisions, but the mean free path is short. Few electrons gain enough energy to produce an ionization.

Figure 3. Normalized Paschen curve: sparking voltage versus the product of pressure and gap width.

Equation 9.46 can be rewritten as Eq. 9.47. The table below shows the spark parameters for some common gases. The minimum sparking voltage is high for electronegative gases like oxygen and low for gases with little probability of electron capture like helium and argon. In electronegative gases, there is a high probability that electrons are captured to form negative ions. The heavy negative ions cannot produce further ionization, so that the electron is removed from the current multiplication process. If you have questions or comments, please contact us at techinfo@fieldp.com.

• You can tailor shapes to coordinate with mesh divisions.
• You can change parameters in MetaMesh scripts with your own programs to create control loops and extended run sequences.
• If you’re an infrequent 3D CAD user, you can avoid the anguish of relearning the program twice a year.
• Getting modified CAD shapes from your mechanical engineer can be time consuming.
• You only have to build a model once. After that, you can generate an infinite set of related shapes just by typing in a few numbers.

If that last point sounds intriguing, read on. In this article, I’ll show how to build a spreadsheet that can generate a entire class of shapes simply by changing parameters.

As an example, consider the rounded box of Fig. 1. The box has full widths of Wx, Wy and Wz along the axes of the fabrication space. The edges and corners have fillets of radius R. The top part shows a Geometer view of the box hanging on a rod with color-coding by parts. The following parts are required to construct the shape:

• 8 spheres of radius R shifted to the corners.
• 12 cylinders of radius R to define the edges with rotations and translations to position them.
• 6 boxes (slabs) to define the faces.

Looks like a lot of work. But remember, you only have to do it once!

Figure 1. Rounded box model. Top: Geometer display of the component parts. Bottom: Finished mesh, output from MetaMesh.

The key is to use a spreadsheet to create MetaMesh script commands directly. Figure 2 shows my spreadsheet. Notice that comment lines start with an asterisk, following the MetaMesh script convention. The defining parameters are gathered at the top. To create the complete assembly, all you need do is set the four values. I also included the option of setting the region name of the assembly as well as that of the surrounding medium as strings in the parameter section. This information is copied to the REGION and SURFACE commands of all the parts. The content cells for the NAME command have entries like

=CONCATENATE($B$9;"DDD")

The highlighted cell in Fig. 2 shows typical information to set the required shifts and rotations to build the box. I won’t attempt to describe the content of every cell. Rather, I’ll give a link below so you can download the spreadsheet in Open Office Calc and Microsoft Excel format.

Figure 2. Initial section of the RoundedBox spreadsheet.

To add the part to a MetaMesh script, just copy and paste all the active cells of the spread sheet. This is where script directives come in handy. You can shift and rotate all 26 parts as a rigid body. Here’s an excerpt from a test script:

#SHIFT 0.00 23.75 0.00
* Corners
PART
 Region:    RoundedBox
 Name:    RoundedBoxDDD
 Type:    Sphere
 Fab:    0.5
 Shift:    -2.5    -1.75    -2
 Surface    Region    Air
END
PART
 Region:    RoundedBox
 Name:    RoundedBoxUDD
 Type:    Sphere
 Fab:    0.5
 ...

 Fab:    5    3.5    1
 Shift:    0    0    -2
 Surface    Region    Air
END
PART
 Region:    RoundedBox
 Name:    RoundedBoxZD
 Type:    Box
 Fab:    5    3.5    1
 Shift:    0    0    2
 Surface    Region    Air
END
#ENDSHIFT

Here a link to zip file that contains the spreadsheets and the full MetaMesh script: RoundedBox.piz. Change the piz suffix to zip after you download it. Consider a standard parametric model whenever you find yourself building the same shapes multiple times. I gave similar pointers for working with Magwinder in an article last year. If you have questions or comments, please contact us at techinfo@fieldp.com.

After 30 years, I remain convinced that the fixed two-window file manager is the best choice for organizing a hard disk. The advantages are especially apparent when you run technical programs like ours and have to keep track of extensive sets of data files. Although several excellent commercial and freeware programs are available[1], we decided to create our own program (FP File Organizer) for the following reasons:

• The file-management functions that we are developing will be added to major upgrades of our program launchers (tc.exe., amaze.exe, gambet.exe and xenos.exe). These programs will be distributed with the February 2013 update.
• We can easily add custom features to facilitate work with our technical codes.
• We designed the interface with care to make it easy to learn the program and to perform common operations.

Figure 1. Screenshot of FP File Organizer.

In this article, I’ll discuss the interface design and some of the program features. Figure 1 shows a screenshot. As in other commander-type programs, there are two windows that point to different directories (folders) or drives. The locations are determined by the navigation controls at the top. The principle of operation is simple. One of the windows is active (indicated by the red light). A window is made active by selecting a file or directory or by clicking an associated control. The Delete, Rename and Edit file commands work on selected files in the active window. The Copy and Move commands transfer selected files in the active window to the other window. Within this basic organization, our program has several unique features.

• Notice in Fig. 1 the absence of a menu and toolbar at the top, common features of Windows programs that I dislike. All the controls of FP File Organizer are clearly visible at fixed locations.
• 99.9% of the work you perform with a file manager involves the fundamental operations copy, move, delete, rename and create-directory. Other available file managers are involved in an arms race of advanced features that tend to accumulate on the toolbar. Working with other programs involves continually seeking important controls in a sea of small icons. We will add future advanced features for FP File Organizer to to a popup menu, with no changes to the basic control layout.
• All controls are easily-identified push buttons at logical positions. Folder navigation controls are grouped at the top. The fundamental file controls are placed between the windows at the focal point in the middle of the program window to emphasize their function.
• Other file managers attempt to put too much information in the left and right windows. FP File Organzer shows the file directory name plus one item of context-sensitive information. Directories are always collected at the top the listing, with shading and names in uppercase letters. The order of the lists depends on the current sort mode. For sort-by-name, directories are listed in alphabetic order with their creation date and files are organized alphabetically with their exact byte size. For sort-by-extension, the files are grouped alphabetically by their extension and then by name, again with the byte size. Finally, in sort-by-date both directories and files are ordered first by creation date and then alphabetically by name showing the creation date.
• With regard to navigation, I find I spend too much time stepping back to the same disk location I used earlier. To minimize effort, FP File Organizer supports a saved-location list. If you expect to return to a directory, click Add location to put it on the list. Locations are easily removed when you move on to another task. The program also stores a list of previous locations for each window so you can move quickly back-and-forth between directories. The program remembers locations from the previous session.

If you have questions or comments, please contact us at techinfo@fieldp.com.

Footnotes

[1] The two-window file manager lives on, albeit with many more advanced features. I use FreeCommander, a freeware program available at http://www.freecommander.com/. An alternative is Turbo Navigator. It is out-of-print, but you can download the last version here: http://www.321download.com/LastFreeware/page6.html.