Analytical Field Calculation
In the coil cross-sections and 3-D coil ends the current flow is modeled
by line currents in the positions of SC strands in the cables. This
chapter treats the calculation of electromagnetic fields from line
currents via the Biot-Savart law and the resulting electromagnetic
forces on the conductors.
2-D analytical field calculation
Main Options
| Option |
Description |
| Axi-Symmetry |
Regard (x)- and (y)-coordinates as (z)- and (r)-coordinates and solve the Maxwell Equations in cylindrical coordinates. To model axi-symmetric 3-D cases in 2-D. |
-
With the "Axi-Symmetry"-option all conductors in the cross-section
must be positioned in the upper half-plane. They are then interpreted as
current-loops with the $x$-axis as axis of rotation.
-
The "Axi-Symmetry"-option produces a plot of the axial and radial
field-components over the $z$-position. The fields are plotted at
different radii between the center of the solenoid and the coil. These
plots however do not consider fields due to iron magnetization!
| Option |
Description |
| Grading of Current Density |
Take into account the inhomogeneous current density in keystoned cables. |
| Self Field in Strands |
For the calculation of the fields at strand-level: take into account also the self field. We define as the self-field the maximum value of the field generated by a strand on its surface, i.e. Istrand/2πrstrand. |
| Self and Mutual Inductance |
Calculate self- and mutual inductances between layers. The output is written into a table in the .output-file. |
| Quench and Temp. Margin |
Calculate the distance to the critical surface in the position of every strand. |
| Peak Field in Coil |
Calculate the field at strand-level. |
-
To switch the "Grading of Current Density"-option 'off' in
keystoned cables allows for a better comparison of ROXIE results with
other, FEM-based field calculation programs which would model the
current by a homogeneous current density.
-
The "Quench and Temp. Margin"-option uses the fit of the critical
surface given in the roxie.cadata-file's REMFIT block. The "Peak
Field in Coil"-option also prints a margin to load line for every
block. This option uses the linear approximation of the critical surface
around a working point. Note that, in order to obtain good results,
both, the fit and the linear approximation must be entered in
roxie.madata- or roxie.cadata-file according to measurements!
Be aware that the method used with the "Self and Mutual
Inductance"-option is only applicable in the absence of nonlinear
magnetic material. Otherwise the "Mutual Inductances in nl.
Circuits"-option must be chosen in the "Optimization Algorithm"-field
of the "General Information". The nonlinear self inductance is
evaluated using the SINDU- and SINDUD-options in the "Global
Values"-menu of the "Objectives". The latter options are used with
the "Transfer Function"-option.
Switching to 'on' the "Peak Field in Coil"-option has many
implications.
-
The forces upon each conductor are calculated.
-
The position of each conductor on the loadline is determined and the
maximum per block is written to the .output-file. This
functionality uses the linear approximation of the critical surface.
-
Plots of the blocks' position on the loadline and of the forces on
the conductors are automatically plotted when the "Postscript
Plots"-option is switched 'on'.
-
"Peak Field in Coil" is required to be switched 'on' for a number
of other options, e.g., quench margins, inductances, plots of fields
and currents in the coils, time-transient calculations, ...
| Variable |
Description |
| Radius of Harmonic Analysis |
Radius for Fourier Decomposition of the radial component of the field. |
| Inner Radius of the Iron Yoke |
For mirroring method. |
| * Relative Permeability of Yoke* |
For mirroring method. |
| * Highest Order of Multipole Coeff.* |
Calculate the coefficients of the Fourier Series up to this order. |
| * Type of Coil/Ref. Field* |
For relative multipole coefficients in units $10^{-4}$. They are related to the specified field component in |
Design variables
Layers:
| Variable |
Description |
| CURNLH |
Current in specified block. |
Coil Blocks (Cross-Section):
| Variable |
Description |
| TEMPBL |
Operation temperature in block. |
- The TEMPBL-option sets the operation temperature in specified
blocks, e.g., to test the impact of an inhomogeneous cooling on the
margins to quench or on persistent currents. The TEMPBL-option is used
with features that use the critical-current fit function and not with
those that use the linear approximation thereof.
Current:
| Variable |
Description |
| CURNTB |
Current, all blocks effected. |
| CURNTH |
Current, only specified block. |
| CURNTC |
Current, only specified conductor. |
| CURNTS |
Current, only specified strand. |
| CURNTD |
Short circuit current (IB1 positive, IB2 negative)???. |
| CURNTF |
Current factor for all blocks. |
| CURRFH |
Current factor, only specified block. |
| CURRBH |
Current factor plus/minus one, only specified block. Binary operator to switch current in block between '+ ' and '- '. For use with genetic algorithm. Input 0,1 => factor 1, -1. |
Quench, Inductance:
| Variable |
Description |
| TURNS |
Number of turns per conductor (for inductance calculation). The strands in a cable are usually connected in parallel (Rutherford type cable). If they are connected in series, then the inductance increases by the number of turns per conductor. |
Additional Field:
| Variable |
Description |
| ADDX |
Constant induction (in tesla) in $x$-direction. |
| ADDY |
Constant induction (in tesla) in $y$-direction. |
Other:
| Variable |
Description |
| XCOIL |
$x$-displacement of the measurement coil (harmonic analysis). |
| YCOIL |
$y$-displacement of the measurement coil (harmonic analysis). |
| FCOIL |
Turning of the measurement coil (harmonic analysis). |
| DELLI |
Ellipticity of coil. Simulate a deformation of the coil and the mandrel. |
| R |
Radius of iron yoke for mirroring method. |
| CONPHI |
Constant current shell up to angle phi. |
| COSPHI |
Cosine current shell up to angle phi. |
| ELLPHI |
Intersecting ellipses up to angle phi. No longer supported. |
| IRISB3 |
$b_3$ correction for iris plot (units). |
| IRISB5 |
$b_5$ correction for iris plot (units). |
| IRISB7 |
$b_7$ correction for iris plot (units). |
| IRISB9 |
$b_9$ correction for iris plot (units). |
| IRIERR |
Maximum error in plot (units). Maximum on the legend of an iris plot. |
| GDFZQMA |
Good-field zone quality criterion (units) to calculate the radii of the good-field zone as used with the GDFZRMA, GDFZRMI, and GFZWID options in the "Objectives"-block below. |
| MAXW |
Radius for calculation of forces with Maxwell stress tensor. Calculate the forces on objects inside a circle, centered at $x=0$, $y=0$ with this radius. |
- For a comment on the "GDFZQMA"-option see the remark on the
"GDFZRMA", "GDFZRMI", and "GFZWID" options in the
"Objectives"-section below.
Objectives
Normal Multipoles:
| Variable |
Description |
| B |
Field. |
| BR |
Field related to the main component. |
| BQUEN |
Field at short-sample current (quenchfield). |
Skew Multipoles:
| Variable |
Description |
| A |
Skew field components. |
| AR |
Skew field related to the main component. |
Global Values:
| Variable |
Description |
| NIB |
$N\,I/B_\mathrm{ref.}$ |
| GFZRMA |
Good-Field Zone outer radius. |
| GFZRMI |
Good-Field Zone inner radius. |
| GFZWID |
Good-Field Zone width (outer radius minus inner radius). |
| MARGMI |
Minimum margin to quench. Calculated from linear approximation of critical surface. |
| XCOIL |
$x$-Displacement of the measurement coil. |
| YCOIL |
$y$-Displacement of the measurement coil. |
| DELLI |
Elliptical deformation of the coil on the mandrel, compare the DELLI-option in the "Design Variables". |
| SINDU |
Self inductance. |
| SINDUD |
Differential self inductance. |
| TORQUE |
Torque from Maxwell stress tensor, compare the MAXW-option in the "Others"-menu of "Design Variables". |
- The "GDFZRMA", "GDFZRMI", and "GFZWID" options
calculate the inner and outer radii of a zone, in which the field is of
good quality. The quality is defined in terms of the sum of unwanted
field (other than main component field). If this unwanted field, related
to the main component (in units $10^{-4})$, is below the value specified
in the "Design Variables" as GDFZQMA, then this radius has 'good
field'. For a dipole, only an outer radius exists, delimiting a circular
zone of 'good field'. For higher-order multipolar fields the good-field
zone is generally delimited by an inner and an outer radius. The inner
radius is due to lower-order field errors. The good-field calculations
are closely related to the iris-plots, in that the radii belong to
circles that can be inscribed into the iris plot, delimiting a specified
color region. The radii can be plot in the "Plotting Information 2D".
For more information on the theoretical aspects of the good-field zone,
contact Nikolai Schwerg (nikolai.schwerg@cern.ch).
Peak Fields:
| Variable |
Description |
| PEAK |
Peakfield in the block. |
| LOADLI |
Percentage on the load line. |
Magnetization Data:
| Variable |
Description |
| AB |
Skew and normal in one plot. No longer supported. |
Solenoid Data:
The solenoid options are available with the "Axi-Symmetry"-option 'on'
in the "Main Options".
| Variable |
Description |
| SOLBXM |
$\max |
| SOLBYM |
$\max |
| SOLBXD |
$\Delta |
| SOLVOL |
Total coil volume. |
| SOLVOB |
Coil volume of specified block. |
Block spec. (Peak fields, Forces, FEM plots)
In this data line the blocks are specified in which to do peak-field
calculations, force-calculations and more. The following format is
allowed:\
1 4 7-9 10\
Geometry:
| Variable |
Description |
| YOKE |
Imaging iron yoke. |
| DISPLV |
Displacement vectors in Blocks. |
Aperture:
| Variable |
Description |
| QUAL |
Field quality in aperture. Deviation from pure field is calculated in every point from Biot-Savart law. |
| IRIS |
Like QUAL but deviation calculated from the field harmonics (faster). |
| GFZ |
Good-field zone. For more information, see remarks on the "GDFZRMA", "GDFZRMI," and "GFZWID" options in the "Objectives" section above. |
| MATR |
Field vectors in cross-section. Modulus represented by arrow size. |
| MATRC |
Field vectors in cross-section. Modulus represented by color code. |
| MATRP |
Like MATR but only field from SC-magnetization (PCs, ISCCs analytic model, IFCCs). |
-
Note that the MATR, MATRC- and MATRP-options can be operated
as such or with the "Field-Vector Matrix"-option from the "Interface
Options". The option lets the user define the matrix spacing and
produces an output file. Furthmore, the reduced field from numerical
field calculations is only taken into account if the "Field-Vector
Matrix"-option is used.
-
The QUAL-option evaluates the formula
$f_{ij}=1-\frac{B(\mathbf{x}{ij})}{B$
in a matrix of 100x100 points (200x200 for $360^\circ$ plots) over the
plotting range. The color-scheme has one color for every $0.1$ units of
$10^{-4}$. With the 20-color legend, the maximum field-deviation
displayed is $2$ units of $10^{-4}$.}}(\mathbf{x}_{ij})
-
The IRIS-option works similarly to the QUAL-option. The difference
is that the field is not calculated in every matrix point from
Biot-Savart law but it is generated from the Fourier-Series expansion.
This method is faster. By default the legend encompasses $8$ units of
$10^{-4}$. Each color therefore represents $0.4$ units of $10^{-4}$. For
the IRIS-option, the upper bound of the legend can be set using the
IRIERR-option in the "Other:"-menu of the "Design Variables".
Coil Fields:
For all coil fields the "Peak Field in Coil"-option must be switched
'on'.
| Variable |
Description |
| A |
Vector potential. |
| BX |
Magnetic field ($x$-component). |
| BY |
Magnetic field ($y$-component). |
| *$ |
$B$ |
| B |
Magnetic field vectors. |
| BPERP |
$B$ perpendicular to the broad face of conductor. |
| BPARA |
$B$ parallel to the broad face of conductor. |
| MARG |
Margin to quench (in %). |
| MARGT |
Temperature margin (in K). |
- The MARG- and MARGT-options are calculated from the critical
current fit. Compare comment on the "Quench and Temp. Margin"-option
in the "Global Information" above.
Lorentz Forces:
For all force calculations the "Peak Field in Coil"-option must be
switched 'on'.
| Variable |
Description |
| FX |
Electromagnetic force in $x$-direction. |
| FY |
Electromagnetic force in $y$-direction. |
| *$ |
$F$ |
| F |
Force vectors. |
| FPERP |
$F$ perpendicular to the broad face of the conductor. |
| FPARA |
$F$ parallel to the broad face of the conductor. |
| FORC |
Electromagnetic forces on blocks. |
| FPN |
$F$$\parallel$ over $F$$\perp$. |
Current Distribution:
For all current representations the "Peak Field in Coil"-option must
be switched 'on'.
| Variable |
Description |
| I |
Current in strand. |
| JELE |
Current density in strand. |
| JCU |
Copper current density in strand. |
| JSC |
Superconductor current density in strand. |
| *$ |
$I$ |
| *$ |
$JEL$ |
| *$ |
$JCU$ |
| *$ |
$JSC$ |
Bn Strand Contribution of I:
For all harmonic representations the "Peak Field in Coil"-option must
be switched 'on'.
| Variable |
Description |
| B1 |
$B_1$ contribution of strand current. |
| B2 |
$B_2$ contribution of strand current. |
| B3 |
$B_3$ contribution of strand current. |
| B4 |
$B_4$ contribution of strand current. |
| B5 |
$B_5$ contribution of strand current. |
| B6 |
$B_6$ contribution of strand current. |
| B7 |
$B_7$ contribution of strand current. |
| B8 |
$B_8$ contribution of strand current. |
| B9 |
$B_9$ contribution of strand current. |
| B10 |
$B_{10}$ contribution of strand current. |
| B11 |
$B_{11}$ contribution of strand current. |
Interface options
| Option |
Description |
| Field-Vector Matrix (MAP) |
Define a field-vector matrix and produce a file. A widget opens in the GUI. The reduced field from numerical field calculations is taken into account, compare Section 5.1.1. |
| Field Along a Line (2-D,3-D) |
Calculate the field in $x$- and $y$- component along a file. Output is written to a postscript file and .output-file. An extra widget opens. |
| 2-D Field Map in Coil |
Write the field at every strand to a file, compare Section 5.1.6. |
| 2-D Line Currents |
Produce a filename.fila2-D-file which contains two tables: (1) a table with the corner points of the current-carrying areas and (2) a table with the position of the individual line currents in the model, compare Section 5.1.7. |
| Write Multipoles for Pp. |
Write the multipole components to a file for post-processing, compare Section 5.1.8. |
Levitation in 2-D
To calculate the levitation force-field, the "Field-Vector Matrix"-
and "Levitation (grad $B^2$)"-options are used. The algorithm
calculates the levitation forces in every point defined with the
"Field-Vector Matrix"-option, henceforth called the reference matrix.
A plot is produced that shows the forces in $x$- and $y$-direction in
the points of the reference matrix.
Objectives
Magnetic levitation:
| Variable |
Description |
| LEVDX |
Variation of force ($x$-component) over the reference matrix. |
| LEVDY |
Variation of force ($y$-component) over the reference matrix. |
| LEVYM |
Maximum levitation force ($y$-component) over the reference matrix. |
Aperture:
| Variable |
Description |
| QUAL2 |
Levitation force error on $F_y$. |
| QUAL3 |
Levitation force error on $F_y$ and $F_x$. |
- The QUAL2- and QUAL3-options evaluate the forces in $x$- and
$y$-direction in a matrix of 100x100 points (200x200 for $360^\circ$
plots) over the plotting range. The deviation of the force field from
the mean-value in the reference-matrix is evaluated. The color-scheme
has one color for every percent of deviation. With the 20-color legend,
the maximum displayed force-deviation 20 percent.
Interface Options
| Option |
Description |
| Levitation (grad B**2) |
This option must be 'on' in order to do levitation calculations. |
3-D Analytical Field Calculation
In this section we only document those options that are proper to 3-D
calculations and thus not available in 2-D.
| Option |
Description |
| 3-D Peak Field Calc. |
Calculate the field and forces on strand level. |
| 3-D Field Harmonics |
Calculate the integrated multipole components along a line. An extra widget opens in the GUI. |
-
For 3-D Peak field calculations is is also imperative to have the
"Peak Field in Coil"-switch 'on' in the "Global Information".
-
In "3-D Field Harmonics"-calculations a maximum number of 8
integration points is printed separately to the .output-file. Above 8
steps, only the integral harmonics are being printed. The orders of
multipoles to be plotted are specified in the "Objectives"-table. A
plot is then automatically generated when the "Postscript
Plots"-option is switched 'on'.
Design variables
Plotting:
| Variable |
Description |
| SCALIZ |
3-D field cones plotted in $-z$-direction regardless of the "Plot Imaged at z=0 plane". |
Additional Field:
| Variable |
Description |
| ADDZ |
Constant induction (in tesla) in $z$-direction. |
Other
| Variable |
Description |
| FSCAL |
Main field component (absolute value) to which relative 3-D integrated harmonics should be related when the "3-D Field Harmonics"-option is switched 'on' in the "Global Information 3-D"-widget. |
Objectives
Normal Multipoles:
| Variable |
Description |
| B3 |
Average field over end, calculated with the "3-D Field Harmonics"-option in the "Global Information 3-D"-widget. |
| B3R |
B3, related to main component, calculated with the "3-D Field Harmonics"-option in the "Global Information 3-D"-widget. |
Skew Multipoles
| Variable |
Description |
| A3 |
Average field over end, calculated with the "3-D Field Harmonics"-option in the "Global Information 3-D"-widget. |
| A3R |
A3, related to the main component, calculated with the "3-D Field Harmonics"-option in the "Global Information 3-D"-widget. |
Conductor Data
| Variable |
Description |
| PVAR |
Variation of pressure on the narrow face. |
Peak Fields
| Variable |
Description |
| PEAK3-D |
3-D Peak field in the block. |
| Option |
Description |
| 3-D Min. Field in Cond. |
Choose the filament with the lowest field in the peak-field conductor for the Roller-coaster plot. |
| 3-D Max. Field in Cond. |
Choose the filament with the highest field in the peak-field conductor for the Roller-coaster plot. |
- The "3-D Min. Field in Cond." and "3-D Max. Field in
Cond."-options produce so-called Roller-coaster plots. The option
"3-D Peak Field Calc." must be 'on'. For each block specified in the
"Block spec. (Peak fields, Forces, FEM plots)"-widget the conductor
with the largest peak-field is chosen. The field and its components are
plotted over the intersection number in $z$-direction and, in a second
plot, over the $z$-coordinate. The data is also written into tables in
the .output-file.
| Variable |
Description |
| P |
Pressure due to Lorentz forces on surfaces. |
| FXFZ |
$F_z$ on the broad side; $F_x$ on the narrow side. |
| FYFZ |
$F_z$ on the broad side; $F_y$ on the narrow side. |
| FRFF |
$F_r$ on the broad side; $F_\varphi$ on the narrow side. |
| B |
$B_\mathrm{min}$ on the broad side; $B_\mathrm{max}$ on the narrow side. |
| BMID |
Average $B$ on the broad and narrow sides. |
| JZ |
Current in $z$-direction. |
- The JZ-option works only if the "Peak Field in Coil"-option is
switched 'on' in the "Global Information".