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.

In reviewing the application, it occurred to me that there was a much easier approach to model such a structure, based on technology that we had already incorporated in the two-dimensional Mesh program. The idea is to assign elements in a plane to regions according to values in a photographic image. The advantages are easy setup and high reliability. In this article, I will describe new MetaMesh capabilities for analyzing photographs and illustrate them with an example of a layer in a printed circuit.

Figure 1. Sample printed circuit, PNG format.

Figure 1 is a photograph of the test geometry. The image has only two colors — white represents thin conductors. In the benchmark example, conductors (Region 2) are mapped to a 1.0 mm layer in a uniform solution volume (air, Region 1). The first step is to set the interpretation of photographs. The following entries appear in the GLOBAL section of the MetaMesh script:

INTERVALS  Lightness
  50.0  100.0  2
END

The construct states that photographs will be interpreted in terms of lightness values (0.0 to 100.0 percent). Lighter areas will be assigned to Region 2. An INTERVAL section may contain up to 250 data lines — in this case, only one interval is needed. Alternatively, assignment may be via hue values (0.0 to 360.0 degrees) in the image.

Parts are basic building blocks in MetaMesh. The concept is flexible — parts range from simple geometric solids to complex STL shapes. We have added a new PHOTO part type. The following section defines a photo part in the example:

PART
  Type Photo PC01.PNG 0.00  0.00  50.00  50.00
  Name PC
  Fab 1.00
END

In the workbench frame, the photograph image is mapped into an x-y plane centered at z = 0.0 with thickness Δz set by the fabrication parameter. The resulting solid can be moved to any position or orientation with optional SHIFT and ROTATE commands. In the example, the photographic image is contain in the file PC01.PNG. (MetaMesh also handles BMP and PCX formats.) The information is mapped to an area with spatial dimensions xmin = 0.0, ymin = 0.0, xmax = 50.0 and ymax = 50.0, independent of the image pixel dimensions. Figure 2 shows the resulting mesh in the slice plane z = 0.0, while Figure 3 shows a three view with perspective. The entire structure was created with the two simple statements listed above.

Figure 2. MetaMesh screen shot, slice in the plane z = 0.0.

Figure 3. Three dimensional view, printed-circuit example.

Photographic analysis is a powerful new feature of MetaMesh. It greatly simplifies the representation of multi-layer printed circuits and striplines for electromagnetic analysis. Complex printed-circuit assemblies can be built in two steps:

1. Set up a foundation mesh with element layers of the appropriate thicknesses.
2. Use photographic layouts to set conductors or dielectrics in each layer. If there are different components in a single layer, they can be represented with different colors in the photograph.

The photographic method has unlimited applications. Information from up to 250 photographs may be included in a MetaMesh script. The technique applies to any three-dimensional structure with photographic slice data, such as MRI scans.

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.

1) If the installed programs give an activation error message, the first step is to check whether you are on a proxy server. You can check this with your browser (the browser must have a record of proxy settings to communicate with the Internet).

Firefox: Click on Tools/Options and then the Advanced tab. Activate the Network tab, In the Connection box, click on the Settings button. If you are on a proxy server, the settings will be listed in the dialog. Note the IP and port values for the HTTP proxy.

Internet Explorer: Click on Tools/Internet options. Choose the Connections tab and then click on the LAN settings button. The proxy settings are listed in the dialog.

2) Field Precision programs will recognize the proxy server if you set the environmental variable HTTP_PROXY. To set a temporary value for testing, use the Set command from the command prompt (cmd.exe):

set http_proxy=value

If the proxy server does not require a user name and password, the value of HTTP_PROXY has the following form:

http://proxyname:port
Example: set http_proxy=http://proxy.myprovider.net:8080

Use the following form if the proxy server requires a user name and password:

http://user:password@host:port
Example: set http_proxy=http://jsmith:ghRt334@proxy.company.com:8001/

Check whether you can launch mesh.exe or metatmesh.exe.

3) If the programs run, you should set a permanent environmental variable. Procedures vary slightly in different versions of Windows — here is the one for NT. Run the Control Panel and click on System. Activate the Advanced tab to bring up the dialog shown in the figure. Click the button Environmental variables to show the second dialog. Click the appropriate New button to set a system-wide variable or one that will apply only to your user name. Fill in the variable name HTTP_PROXY and add a value as above.

Setting an environmental variable in Windows

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.

Program: TriComp
PlotStyle: Spatial
Outline: ON
Grid: ON
Scientific: OFF
FixedPoint: OFF
Vectors: OFF
XGMin:  3.500000E+00
XGMax:  9.000001E+00
...

The file contains all necessary parameters to reconstruct spatial plots (in the z-r or x-y planes). The file includes the following information for spatial plots:

• Style information, like whether to display a reference grid, element outlines or field vectors.
• Boundaries of zoomed views
• The plot type (e.g., mesh, filled contour, contour lines,…)
• The plotted quantity.

You can also save information to reconstruct scan plots:

• Style information (number of scan points, symbol display,…)
• Coordinates of the scan start and end Limits of the scan.
• Plotted quantity.

Spatial versus scan plots in TriComp programs

The simplest application of saved views is to restore a complex plot when you reload the solution at a later time. We have included several features in our implementation to enable a higher level of data analysis:

• A saved view works even if a completely different solution than one use to create the view in loaded. In fact, views can be reloaded even if the user is running a different TriComp program. The current program analyzes the parameters and uses only ones that are relevant. For example, if a view-boundary or scan point is outside the current solution volume, the point is moved to the boundary. If a plot quantity is undefined, the first valid quantity  is used.
• A free-form parser is employed when a view is loaded. Therefore, a file need not contain a complete set of parameters. Changes are made only to the quantities listed in the file.
• Users can modify entries in the view file, either directly with an editor or via a Perl or Python script. You can include comments, change parameter values or remove parameter lines.

Another important feature is related to the fact that analysis functions of all TriComp postprocessors may be controlled by a text script. We have added the following  script command:

PLOT FSaveView FOutput Nx Ny
PLOT (DiodeRegion VIEW001 800 600)

The string FSaveView is the prefix of a saved view file. The string FOutput is the prefix of a plot file in BMP format. The integer parameters give the image resolution in pixels. In response to the command, the program loads the saved view file, applies the parameters to create a plot for the currently-loaded solution and saves it as a graphics file.

Here are some possible scenarios:

• You have a set of electric field solutions with different applied voltage and possibly different boundaries and you want to check electric field levels in a critical region (e.g., an electron source). Use the Zoom command to create the desired plot and save a view file. Remove all information lines except the view boundaries and load the view when you load a different solution.
• You have a set of magnetic field solutions and want precision plots of field lines passing through specific points. Add a field line interactively in PerMag and save a template view file. Then edit the view file to remove unnecessary information, change the number of field lines and add coordinates for the desired points.
• You want to create a large set of radial temperature scans moving along the axis of a probe assembly. In TDiff, create a template view file for one scan. Write a Python or Perl script with a loop of axial positions that modifies the axial scan coordinates in the view file and then runs TDiff in the analysis mode. The set of BMP files could then be stitched into an AVI movie with our Cecil_B utility.
• A sequence of annotated view files could be employed in a training session or educational demonstration.
• You want to show how the levels of magnetic flux induction change with changing coil currents, preserving magnitude scaling in the plots. After creating a set of solutions, load one of them adjust the view, plot style and plot quantity. Turn off autoscaling, specify values for BMin and BMax and then save a view file. Load the view for the other solutions.

Δr = B/B0.    [1]

The permeability characterizes the contribution of material currents to the total magnetic field.

An AC magnetic field at frequency f0 (Hz) can penetrate a block of iron or steel to a distance given by the skin depth:

δ= √[ρ/(π*μr*Δr*f0)].    [2]

where ρ is the volume resistivity in Ω-m. The steels used in transformers have high μr (several thousand) and low resistivity. For example, nickel steel has μr = 8000 and ? = 45 × 10[-7] Ω-m. At f0 = 1.0 KHz, the skin depth is only δ = 0.12 mm (4.7 mil). Therefore, steel structures used for AC magnets are laminated, fabricated from thin sheets separated by insulators. The orientation of the laminations follows the direction of the magnetic field in the magnetic circuit, as in the figure below. If the lamination thickness is smaller than δ, the field penetrates each lamination so that the magnetic properties are almost the same as for static fields.

Laminated core

In a practical finite-element calculation of a macroscopic device, we approximate cores with many thin lamina as a homogeneous object. In principle, a laminated core has an anisotropic permeability, with μr » 1 along the lamina and μr ~ 1 perpendicular to them. In practice, the field lines and lamina are generally in the same direction in the yokes and poles of magnetic circuits. In consequence, we can neglect the normal field component and use a single value of μr for the component parallel to the lamina.

The relative magnetic permeability of the insulating layers of a laminated core is μr ≈ 1.0. The fill fraction f is defined as the fraction of the core cross-section occupied by steel. In magnet calculations, we must account for the missing core material. To begin, suppose the system operates with magnetic flux density B in the lamina well below the saturation value. In this case, we can use a fixed value of μr throughout the core material. The effective magnetic permeability of the laminated core is defined as average magnetic flux density inside the core compared to the applied value.

<μr> = <B>/B0 = [μr*B0*f + B0*(1-f)]/B0 = μr*f + (1-f).     [3]

The second term on right-hand side is usually small compared to first. To include the effect of the fill fraction, we use a reduced value of relative permeability for the core. For example, if μr = 2000 and f = 0.80, then <μr> = 1600.

Next, consider a core represented by a magnetization curve. The curve has the form μr(B0) in Magnum and μr(<B>) in PerMag. For Magnum calculations, the code determines a local value of μr for a known value of B0. To represent a laminated core, we prepare a table where no changes are made to the independent values (B0) and μr values are adjusted according to Eq. [3]. For a PerMag calculation, must make two changes to the table. First, if the calculation were expressed as μr(B) (where B is the magnetic flux density inside the lamination), the dependent values of μr would be reduced according to Eq. [3]. On the other hand, the code uses the value of <B> for the interpolation, where

<B> = B*f + (B/μr)*(1-f)  μrB*f. [4]

Therefore, we should multiply values along the abscissa by f. Forexample, if a material saturates at a field of 2.0 tesla inside the lamination, saturation would occur at an average magnetic flux density <B> = 1.5 tesla for f = 0.75.

In summary, here is how to modify magnetization tables when there is a fill fraction f:

• For Magnum tables of μr(B0), multiply μr values by f.
• For PerMag tables of μr(B), multiply both μr and B values by f.