• Document useful general results on the effective resistance and ohmic losses in wires at high frequency.
• Show how to set up calculations of magnetic field distributions in wires with the two-dimensional code Nelson.

To start, let’s review some basic equations for a long wire with cross-section area Aw (m2) and conductivity σ (S/m). At low frequency, the current density is uniformly distributed across the wire and the resistance per length given by Eq. 1. If the wire carries an alternating current I0 cos(2πf t) at frequency f, Eq. 2 gives the time-averaged power dissipation.At high values of f, the magnetic field associated with the current flow is inhibited from penetrating the metal wire, concentrating the current density near the wire surface. The effective effective cross-section area of the wire is therefore smaller than Aw, increasing the resistance per length and the ohmic power loss for a given current. The penetration distance for magnetic fields, called the skin depth, is given Eq. 3. The quantity μ is the magnetic permeability, given by μ = (1.257E-6) μr, where μr is the relative magnetic permeability. The relative permeability equals 1.0 for most wire metals.

The skin depth allows us to define the high and low-frequency regimes. For a circular wire of radius rw, Eq. 4 gives the condition for enhanced ohmic losses. In the following calculation, we’ll use the example of a circular copper wire 1.0 mm in radius. The conductivity of copper is σ = 5.814E7 S/m. Using Eq. 3, the frequency corresponding to δ = 0.5 mm is 17.4 kHz. For a wire area Aw = 3.142E-6 m2, Eq. 1 gives the steady-state resistance R = 5.474E-3 Ω/m. For a drive current amplitude I0 = 1.0 A, the low-frequency power loss is P = 2.738E-3 W/m.

The program Nelson finds field magnetic distributions in the frequency domain — all quantities (including drive currents) vary harmonically at the same frequency. The input files to define the mesh and to control the calculation are circular.min and circular.nin. Figure 1 shows the mesh with dimensions in millimeters. The wire extends an infinite distance out of the page. The large air region was included so that the calculation could easily be modified for non-circular wires — the distant boundary has little effect on the magnetic fields inside a wire.

Figure 1. Mesh for the Nelson circular wire calculation.

In the Nelson control script circular.nin, the boundary is a surface of fixed vector potential (flux conserving boundary) and the air region has μr = 1.0 and σ = 0.0 S/m. The copper wire has μr = 1.0, σ = 5.814E7 S/m and a drive current I0 = 1.0 A at a phase of 0.0°. The calculation of eddy currents in the wire uses the multistage, self-consistent method described in Chap. 4 of the Nelson manual. Figure 2 shows lines of B. At low frequency, |B| inside the wire increases linearly with distance from the axis, consistent with uniform current density. At high frequency, the magnetic flux density is concentrated on the surface.

Figure 2. Lines of magnetic flux density in and around a circular copper wire at low and high frequency.

To make a quantitative comparison, we can use the automatic volume integral in the Nelson Analysis menu to find the total power dissipation per length along z. Results of calculations at several frequencies are plotted in Fig. 3. The theoretical steady-state value is shown as a dashed line. As expected, power losses increase significantly in the range 10-20 kHz. At high frequency, the current is confined to a thin layer on the wire surface. In this case. the power loss increases approximately as √f, proportional to 1/δ.

Figure 3. Power dissipation per meter in a copper wire of radius 1.0 mm as a function of frequency with a drive current amplitude of 1.0 A.

It is important to recognize that the numerical results for the 1.0 mm copper wire may be applied to circular wires of any diameter or composition. It is not necessary to repeat the calculation for each special case. The strategy is to recognize scaling laws and to incorporate them into a universal curve. As an example, the green line of Fig. 4 plots the power level relative to the steady-state value as a function of the ratio of the wire dimension to the skin depth. There are two advantages to applying scaling laws:

• Generality – the graphed data may used to determine the effective resistance of any circular wire.
• Identification of trends – it’s evident that the power loss is proportional to 1/δ at high frequency.

Figure 4. Universal curve for high-frequency wire resistance, power relative to the steady-state value plotted as a function of the wire dimension normalized by the skin depth. Circular and square wire cross sections.

As an example, suppose we have a 1/8″‘ diameter stainless steel rod and we want to find the frequency limit such that the power loss is no more than 50% of the low-frequency value. An inspection of Fig. 4 shows that δ must be greater than rw/2.5. With σ = 1.240E6 S/m and rw = 1.59 mm, the frequency corresponding to δ = 0.636 mm is f = 503.7 kHz. To check the result, I set up another Nelson calculation with modified wire radius and conductivity. For I0 = 1.0 A, the predicted steady-state power dissipation is 50.77 mW/m. The Nelson calculation gives 50.77 mW/m at f = 1.0 Hz and 76.28 mW/m at 503.7 kHz. The ratio of the two power levels is 1.50.

The results for circular wires could have been derived analytically with some effort. The true advantage of a numerical approach is that complex shapes are no more difficult to handle than simple shapes. To illustrate, I set up a run for square copper wires by changing the shape of the wire region in the Mesh input file. I used a wire with side lengths D = 2.0 mm. Figure 5 shows lines of B at high frequency. The universal power dissipation curve for square wires is plotted as the blue line in Fig. 4.

Figure 5. Lines of magnetic flux density in and around a square copper wire at 40 kHz.

Figure 1. Definition of coil box parameters

Figure 1 defines the geometric parameters of the coil box:

• The coil cross section has width W1 in the x-y plane and W2 in z.
• The coil is approximated by N1 current filaments in the x-y plane and N2 filaments in z.
• The BAR models have length D1 in x and D2 in y.
• The ELBOWR models have inner radius R1 and outer radius R2 and extend 90°.
• I is the total current carried by the filaments.

In the standard position, the current flows in the x-y plane in the direction of positive θ (counterclockwise) when view from the +z direction. You can download the Magwinder sample input file coilbox_example.cdf which creates the structure for a specific set of parameters. The equations for the model parameters are listed as comment lines.

Figure 2. Spreadsheet for automatic calculation of coil box parameters

I have also set up the spreadsheet shown in Fig. 2. To use it, fill in the parameter values and then copy and paste the cells starting at GLOBAL as text into a MagWinder script (CDF). Here is a link for a file for the OpenOffice spreadhsheet (coil_box.ods). If you enjoy purchasing office software, here is a link to an Excel file (coil_box.xls).

Figure 3. Resulting current elements with polarity indicators projected to the x-y plane (N1 = 3)

Figure 3 shows the resulting assembly projected in the x-y plane. The plot illustrates the new Magwinder capability to display the polarity of current elements. They are represented as sperm cells swimming in the direction of the head. There are two ways to reverse the current direction.

• Use a negative coil current, I → -I.
• Rotate the COIL 90° about the x or y axis.

Figure 4. Quadrupole assembly with four coil boxes

You can use the basic assembly multiple times in a Magwinder script. For example, Fig. 4 shows a quadrupole magnet. The standard text from the spreadsheet was pasted into four COIL sections, and then each coil was assigned a specific rotation and translation.

Example of a popup menu in RealBasic

This article shows how to set up a popup menu using the current routines of RealBasic. An important feature is setting the enable status of menu entries. Except in the simplest programs, entries must be disabled depending on the program context to avoid illegal operations (i.e., save a entity that hasn’t yet been created).

The example produces the popup menu shown in Fig. 1. Notice that some entries are grayed because rulers do not exist when the program opens. The code listed below is included in the MouseDown event of the associated window. Note the comment numbers added to the Append statement documenting the order of entries (separators don’t count). These indices are used in the code lines that set the enable status. It’s important to include the test

if (HitItem <> nil) then
end if

to avoid an error if the user clicks on a disabled menu entry.

if IsContextualClick then
// ===============================
// Define menu extries
// ===============================
dim base as new MenuItem
base.Append( new MenuItem( "Create ruler" ) )  '1
base.Append( new MenuItem( "Analyze graph" ) )  '2
base.Append( new MenuItem( "Analyze drawing" ) )  '3
base.Append( new MenuItem( MenuItem.TextSeparator ) )
base.Append( new MenuItem( "Flip ruler" ) )  '4
base.Append( new MenuItem( "Hide/show rulers" ) )  '5
base.Append( new MenuItem( "Close ruler" ) )  '6
base.Append( new MenuItem( "Save ruler" ) )  '7
base.Append( new MenuItem( "Load ruler" ) )  '8
base.Append( new MenuItem( MenuItem.TextSeparator ) )
base.Append( new MenuItem( "Instruction manual" ) )  '9
base.Append( new MenuItem( "About FP UniScale" ) )  '10
base.Append( new MenuItem( MenuItem.TextSeparator ) )
base.Append( new MenuItem( "Exit program" ) )  '11

// ===============================
// Set the enabled status
// ===============================
if (RulersHidden) then
base.item(4).enabled = False   'Flip ruler
base.item(5).enabled = True    'Hide/show rulers
base.item(6).enabled = False   'Close ruler
base.item(7).enabled = False   'Save ruler
base.item(8).enabled = False   'Load ruler
else
// Ruler1 closed, Ruler2 closed
if ((not RulerOpen1) and (not RulerOpen2)) then
base.item(4).enabled = False  'Flip ruler
base.item(5).enabled = False  'Hide/show rulers
base.item(6).enabled = False  'Close ruler
base.item(7).enabled = False  'Save ruler
base.item(8).enabled = True   'Load ruler
// Ruler1 open, Ruler2 closed
elseif (RulerOpen1 and (not RulerOpen2)) then
base.item(4).enabled = True   'Flip ruler
base.item(5).enabled = True   'Hide/show rulers
base.item(6).enabled = True   'Close ruler
base.item(7).enabled = True   'Save ruler
base.item(8).enabled = True   'Load ruler

// Ruler1 closed, Ruler2 open
elseif ((not RulerOpen1) and RulerOpen2) then
base.item(4).enabled = True   'Flip ruler
base.item(5).enabled = True   'Hide/show rulers
base.item(6).enabled = True   'Close ruler
base.item(7).enabled = True   'Save ruler
base.item(8).enabled = True   'Load ruler
// Ruler1 open, Ruler2 open
else
base.item(4).enabled = True   'Flip ruler
base.item(5).enabled = True   'Hide/show rulers
base.item(6).enabled = True   'Close ruler
base.item(7).enabled = True   'Save ruler
base.item(8).enabled = False  'Load ruler
end if
end if

// ===============================
// Carry out actions
// ===============================
dim hitItem as MenuItem
hitItem = base.PopUp
if (HitItem <> nil) then
select case (HitItem.Text)
case("Analyze graph")
// Action code
case("Analyze drawing")
// Action code
case("Create ruler")
// Action code
case ("Flip ruler")
// Action code
case ("Load ruler")
// Action code
case ("Save ruler")
// Action code
case ("Close ruler")
// Action code
case("Instruction manual")
// Action code
case("About FP UniScale")
// Action code
case("Exit program")
// Action code
end select
end if
return true
end if

Figure 1. The Gerber Variable Scale (circa 1970)

Nowadays, practically everything I measure is on a computer screen. For this application, it’s almost impossible to use my old Gerber scale, although I have tried many times. For a couple decades, my dream has been to make something I could use on the computer that would be as useful and (hopefully) as simple. I tried several freeware screen rulers, but the results were disappointing. They were designed to measure literal distances on the computer screen (in pixels, mm, …) rather than the lengths of objects displayed on the screen. The situation was analogous to having a ruler that could measure the size of a sheet of paper, but not the dimensions of data printed on the paper.

I am happy to announce that I have just completed my vision of the ideal screen measurement system, the Field Precision Universal Scale. With it, you can create custom rulers calibrated to graphical objects displayed on your computer screen by any program. The Universal Scale has other analysis features that extend well beyond simple rulers. The program is available for free download as a service of Field Precision to the science and engineering communities. The official release date is January 1, 2012. You can get a beta copy now at http://www.fieldp.com/fpuniscale.html.

Figure 2. Universal Scale main window and sample ruler.

Figure 2 shows the main window with a ruler displayed. Note that the program comes in a nice leather case. The push buttons on the left call up the three main functions:

• Create custom rulers that may be moved around the screen over other applications.
• Measure points and digitize curves of graphs in publications and reports.
• Measure dimensions on drawings and diagrams in documents.

Figure 3 shows a screenshot of a drawing analysis. In the remainder of this article, I’ll describe some interesting features of the program.

Figure 3. Set up to analyze dimensions of a diagram in a publication.

• My Gerber Variable Scale may lie in a drawer for months between uses. When I take it out, I never have doubts about how it works. I tried to maintain the same user-friendliness in the Universal Scale. The challenge is that the program does a lot more than slide back-and-forth. I limited the program to few well-defined functions with minimal bells-and-whistles. Contextual instructions are displayed at each step, so it’s not necessary for users to memorize procedures.
• Measurements with physical devices used to be fun in the old days, so I gave the program a retro look.
• Calibration sheets are the key to program operation. They are translucent windows that act like tracing paper that can be moved over any displayed graphical material on the computer screen. Mouse clicks in the measurement sheet are used to calibrate rulers, graphs and drawings and to measure points.
• I applied my experience with normal coordinates in finite-element analysis to measurements on graphs. The program uses a generalized trapezoid coordinate system. This means that you can obtain good measurements from badly-scanned graphs (rotations, distortions,…).

Footnotes

[1] Unfortunately, it appears that the Gerber Variable Scale is no longer available for purchase. As a matter of historical interest, here is a link to the original manual: GerberVariableScale.pdf.

[2] Oscillographs were Polaroid photographs of oscilloscope traces, the only way to record fast transient data before digitizers.

The concept of a magnetic circuit is often useful to estimate achievable magnetic flux density in a laboratory magnet (i.e., a system with an iron flux conductor, a small air gap and a drive coil). The theory is described in Sect. 5.7 of my book Principles of Charged Particle Acceleration (available for free download at http://www.fieldp.com/cpa.html. The relationships follow from Ampere’s law and the conservation of magnetic flux. The issue is clouded for circuits driven by permanent magnets. In fact, I have to admit that Sect. 5.8 in my book is misleading. In reality, the gap field in a permanent magnet circuit depends on how the assembly is constructed.

Figure 1. Permag calculations of permanent magnetic circuits.

The issue can best be understood with an example. Figure 1 shows a cylindrical magnetic circuit for a PerMag calculation. The air gap and permanent magnet are cylindrical with 2.0 cm radius. The gap length is Lg = 1.0 cm and the magnetic length is Lm = 2.0 cm. The ideal permanent magnet has Br = 1.0 tesla and the iron has μr = 500. The intuitive view of a magnetic circuit is that the lines of B are forced to flow in the high-reluctance iron (i.e., the iron is a magnetic conductor). In this case, the behavior of the circuit should not depend on the relative position of the permanent magnet and we expect the same gap field for the two geometries illustrated in Fig. 1. Under the assumption of contained flux, the theory in Sect. 5.8 predicts an average gap field:

Bg = Br/(1 + Lg/Lm)

For parameters of calculation, formula implies Bg = 0.67 tesla.

The PerMag calculation gives a gap field of about Bg = 0.57 tesla for case a. The result is consistent with theory if we consider that there are flux losses from leakage and fringing flux. On the other hand, the gap field for the bottom configuration is only 0.15 tesla, a significant difference. Clearly, the idea that B lines are “contained” by the iron is wrong. We need a better understanding of the psychology of flux lines.

A line of magnetic flux density emerging from the left-hand side of the permanent magnet wants to find its way to the right hand side the easiest way possible (i.e., the path of least reluctance). For case a, the B lines emerge directly into the air gap. Most of them follow the easiest path — cross the short gap and return through the iron. A fraction of the lines take a shortcut to the outer iron piece or return directly through air to the upstream side of the magnet.

In contrast, the B lines in case b must travel a relatively long distance through the iron stalk before reaching the air gap. The reluctance of the radial gap between the inner and outer iron pieces is less than that of the working gap. The result is that B lines cut out early and most of the flux never reaches the working gap. The example has an important practical implication. If the goal is to produce the maximum flux in an air gap of a permanent magnet circuit, the magnets should be adjacent to the gap.

Figure 1. Two-dimensional Trak model of a high-current electron-beam gun

It is relatively easy to set a narrow gap in a two-dimensional Trak mesh. The task is more challenging for three-dimensional OmniTrak meshes, where large numbers of elements lead to long run times. One purpose of this article is to demonstrate how to a create a cathode surface with good facets by optimizing the fitting logic for parts of the cathode assembly. A second activity is to confirm that Trak and OmniTrak generate almost identical results for a test geometry.

The benchmark electron gun of Fig. 1 has a flat cathode with 2.0 mm radius. The applied voltage on the cathode and focusing electrode is -54 kV. The focusing electrode shape gives some beam convergence in the acceleration gap and a modest divergence after passing through the extraction aperture. The combined effects of the focusing electrode profile and the 0.2 mm gap give a beam with good current-density uniformity and low emittance. A Trak calculation gives current density je = 16.4 ± 0.8 A/cm² at the cathode and total current 2.062 A. The top plot in Fig. 2 shows the phase-space distribution (r, r’) at z = 30.0 mm. Over-focusing at the edge is relatively small.

Figure 2. Phase-space distributions (r,r') for the extracted beam at z = 30.0 mm

To minimize run time in OmniTrak, the calculation is performed in the first quadrant of the x-y plane (x ≥ 0.0 mm, y ≥ 0.0 mm) with symmetry boundaries at x = 0.0 mm and y = 0.0 mm for both the fields and electron trajectories. The element sizes near the emission surface have the same dimensions as those in the Trak calculation (Δx = Δy = 0.050 mm, Δz = 0.125 mm). The mesh had 1.63 million elements. Figure 3 shows a slice view of the mesh geometry near the cathode surface.

Figure 3. Detail of the mesh for the OmniTrak calculation in the plane y = 0.0 mm

The cathode assembly is formed from three parts:

• A turning to represent the cathode, with outline specifications taken from the Trak input file. The part is associated with the CATHODE region (fixed potential equal to -54 kV) in the HiPhi and OmniTrak calculations.
• A turning to represent the focusing electrode, again with outline vectors from the Trak input file (FOCUS region with -54 kV fixed potential).
• A turning to represent the gap, with inner radius 2.0 mm and outer radius 2.2 mm. As shown, the shape extends a few elements past the cathode surface. The part is assigned to the GAP region, which has the same physical properties in the electrostatic calculation as the VACUUM region (εr = 1.0).

I employed two techniques to ensure that the facets on the cathode surface were flat and well-formed. The first is simple: pick a foundation mesh that minimizes the work that MetaMesh must perform. In this case, I made sure that a foundation element boundary occurred at z = 0.0 mm, the position of the cathode surface. Here is the complete mesh specification in z:

ZMesh
-2.00  -0.25   0.250
-0.25   0.50   0.125
0.50  30.00   0.250
End

If the element boundary were at an arbitrary position (z ≠ 0.0 mm), then it would be necessary for MetaMesh to perform thousands of needless operations to shift facets.

The second technique is to plan fitting operations carefully. Fitting of a part is controlled by the commands:

SURFACE REGION RegNo
SURFACE PART PartNo

In response to the first command, MetaMesh collects all facets of the part that are shared with elements that have region number RegNo. The program them shifts the facet nodes toward the theoretical outer surface of the part.  The second command form has similar function, except that MetaMesh picks facets adjacent to elements with part number PartNo.

Let’s consider how fitting works for the cathode assembly. Here is a summary of fitting commands for the contiguous parts of the cathode assembly:

PART
  Region: Gap
  Name: Gap
  Surface Region Cathode
  Surface Region Focus
END
PART
  Region: Cathode
  Name: Cathode
  Coat Vacuum Emission
END
PART
  Region: Focus
  Name: Focus
  Surface Region Vacuum
END

To begin, note the order of parts. The dielectric part appears before parts with fixed potential. With this order, any shared nodes are assigned the region number of the \textttCATHODE or \textttFOCUS parts, giving the correct fixed-potential condition on the electrode surfaces.

I will point out some features of the fitting list and then explain how they work. The GAP part has its own region assignment and contains commands for fitting to both the CATHODE and FOCUS. The CATHODE part has no SURFACE command, while the FOCUS part has a single command for surfaces in contact with elements of the VACUUM region. The order is chosen to ensure that facets on the front and side faces of the cathode are shifted only in the x-y plane, not in z. Here is the logic:

• Fitting parts with sharp edges (like a cylinder or the annular shape of the GAP) involves ambiguity. Should the facet be moved toward one edge or the other? This is the reason why I extended the GAP past the CATHODE and FOCUS. The end is far enough away to ensure that MetaMesh moves facets to the inner and outer cylindrical surfaces when fitting the GAP. In other words, node displacements are purely in the x-y plane.
• There are two reasons why there is no SURFACE command for the CATHODE: 1) nodes on the front surface already lie at z = 0.0 and 2) the side surfaces have already been positioned through fitting of the GAP.
• The inner surface of the FOCUS has already been set by fitting of the GAP. The single SURFACE command specifies that front-surface facets (adjacent to VACUUM elements) should be adjusted. The GAP was defined as a separate region to ensure that only nodes on front surface facets are moved.
• The COAT command for the CATHODE resets the region number of the nodes of facets adjacent to VACUUM elements. The new region number has the emission property in OmniTrak. Because the GAP has its own region number, only facets on the front face of the CATHODE are set as emitters.

Figure 4 is a three-dimensional view of the completed cathode assembly. Facets on the front cathode face are flat and well-formed. The OmniTrak calculation gives a quadrant current of 0.512 A, corresponding to a total current 2.046 A. The bottom plot of Fig. 2 shows the phase-space distribution at z = 30.0 mm for 5024 model electrons. A comparison at this distance provides a sensitive test of correspondence. The distributions are almost identical. The beam is slightly larger in the OmniTrak calculation.

Trak and OmniTrak users can use this link to download input files for the calculations:

http://www.fieldp.com/myblog/examples/cathodegapdemo.zip

Figure 4. Three-dimensional view of the completed cathode and focus electrode.

One solution is to automate the process (i.e., prepare the update files and then let the computer upload autonomously for several hours). WinSCP is a good freeware utility to help implement automated FTP transfers. It is available at http://winscp.net/eng/index.php.

The documentation for WinSCP is a little convoluted, so I prepared an example based on my experience to help you set things up. To begin, download and install WinSCP. For convenience, put c:\program files\winscp on the path. Alternatively, copy the relatively small executable files to a location that is already on the path. I use a general directory c:\batch.

As an example, suppose you have the files

c:\distfiles\program01.exe
c:\distfiles\program02.exe
c:\distfiles\program03.exe
c:\distfiles\program04.exe
c:\distfiles\program05.exe

that should be copied to the directories

public_html/download/site01
public_html/download/site02
public_html/download/site03
public_html/download/site04
public_html/download/site05

on your server, automatically replacing any existing files with the same names. Suppose your site has the URL mycompany.com. The user name and password for FTP transfers is myuser and mypassword.

1) Use a text editor to create a file FTPUpload.bat in the directory c:\distfiles with the following single-line content:

winscp.exe /console /script=FTPUpload.txt

The batch file runs WinSCP in the console mode and calls up the script for instructions.

2) Create another file in the directory called FTPUpload.txt with the following content :

open ftp://myuser:mypassword@mycompany.com/public_html/download
option confirm off
cd site01
put program01.exe
cd ../site02
put program02.exe
cd ../site03
put program03.exe
cd ../site04
put program04.exe
cd ../site05
put program05.exe
exit

The first line opens a password-protected FTP session and transfers to a convenient directory. The second line ensures that the script will not pause for confirmation when replacing the old files. The subsequent lines migrate to directories and copy the appropriate files. Simply run the batch file and everything happens automatically. WinSCP maintains a console window to show the status of operations.

Note: If you are bothered by world-class hackers breaking into your office, be careful about storing the file FTPUpload.txt with your site password.