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SC-Related Time-Transient Effects

This chapter introduces ROXIE features to simulate superconductor magnetization, the persistent currents (PCs), as well as Eddy-Current effects on the strand level, the interfilament coupling currents (IFCCs) and on the conductor level, the interstrand coupling currents (ISCCs) in Rutherford-type cables. The PC models have been developed at CERN whereas the analytical models for IFCCs and ISCCs are based on models by M. Wilson. A network model for more accurate modeling of ISCCs is equally available.

All models in this chapter have in common that they should be used with the "Layer Definition"-option in the "Main Options". For the use of semi-analytical models, the "Symmetry: 0: Gen., 1: in1, 2: 2in1"-option in the "Time Transient Effects"-widges allows to make use of coil symmetries nevertheless.

The term 'SC magnetization' is used for all quantities in ROXIE's semi-analytical models, also for Eddy-Current effects such as IFCCs and ISCCs. These effects are modeled as magnetizations on the strand/cable level.

Semi-analytical models for SC magnetization

The main material parameters for the SC magnetization models are found in the roxie.cadata-file which is editable via the "Open cable data window (.cadata)"-entry in the "Run"-menu. Main input parameters are found in the "Time Transient Effects"-widget.

Math Equations

The analytical formula to compute the strand-magnetization from PCs is given by:

\[ \begin{aligned} M = \sum_{i=1}^n M_i = \sum_{i=1}^n \int_{q_i}^{q_{i+1}} m_i(q) \, \mathrm{d}q. \end{aligned} \]
\[ \begin{aligned} M_i &= \frac{4r\xi}{\pi} \int_{q_i}^{q_{i+1}}J_{\mathrm c}(B(q))(1-q)^2 \, \mathrm{d}q \\ &= \frac{4r\xi\mathcal{F}}{\mu_0 \pi} \int_{q_i}^{q_{i+1}} \frac{(1-q)^2}{\sqrt{B(q)}} \, \mathrm{d}q, \end{aligned} \]
\[ \begin{aligned} M_i &= \frac{4rB(q)}{5\pi\mathcal{F}^2 \mu_0^2 \mathcal{H}^3} \left. \left[B_{\rm ext}^3+\xi\mathcal{H}\mathcal{F}\mu_0 \left(\left(5-4q+\frac{5}{4}q^2\right)\xi\mathcal{H}\mathcal{F}\mu_0-\left(q-4\right)B_{\rm ext}^{3/2}\right)\right]\right|_{q=q_i}^{q=q_{i+1}} \, \, \, \end{aligned} \]

The IFCCs are evaluated from:

\[ \begin{aligned} M_{\mathrm{f}} = \lambda_{\mathrm{w}}\, \partial_{t}B\, \left(\frac{l_{\mathrm {w}}}{2\pi}\right)^{2} \underbrace{\frac{1}{\rho_0 + \rho_1 B}}_{\rho_{\mathrm {eff.}}}. \end{aligned} \]

The user is required to provide the wire filling-factor λw, the wire twist-pitch lw and the effective resistivity ρeff which consists of a constant part ρ0 and a coefficient due to magneto resistance ρ1.

ISCCs are calculated as the sum of the following components:

Math Equations
\[ \begin{aligned} M^{\perp}_\mathrm{c} &= \frac{1}{120}\,\frac{\partial_{t}B^{\perp}}{R_\mathrm{c}}\,l_\mathrm{c}\, N (N-1) \,\frac{c}{b}, \\ M^{\perp}_\mathrm{a} &= \frac{1}{3}\,\frac{\partial_{t}B^{\perp}}{R_\mathrm{a}}\,l_\mathrm{c}\,\frac{c}{b}, \\ M^{\parallel}_\mathrm{a} &= \frac{1}{8}\,\frac{\partial_{t}B^{\parallel}}{R_\mathrm{a}}\,l_\mathrm{c}\,\frac{b}{c}. \end{aligned} \]

The user provides the cable twist-pitch lc, the contact- and adjacent resistances, Rc, Ra and the cable dimensions b (narrow side - the mean value is taken for keystoned cables) and c (broad side).

Main options

Option Description
Time Transient Perform a time-stepping, evaluate superconductor magnetization.

Design variables

Layers:

Variable Description
FILDIL Filament Diameter in layer (in \mum).

Coil Blocks (Cross-Section):

Variable Description
FILDIA Filament Diameter in block (in \mum).
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.

Magnetization:

Variable Description
STRPRI Print info of specified strand(s).
ABSCIS Field harmonic versus excitation- (B_n versus I) plot (see objectives): take I of specified block number (default is block 1).

Objectives

Magnetization data:

Variable Description
MSTR Magnetization in filament.
BSTR Magnetic induction in filament.
MSTRT Magnetization modulus.
BSTRT Magnetic modulus.
MSTRF Angle of magnetization.
BSTRF Angle of magnetic induction.
AB Skew and normal harmonics in one plot. Currently not supported.
DTRF Current factor of block as function of time.
  • The MSTR*- and BSTR*- and DTRF-options plot excitation, field and resulting strand magnetization. The x-axis data varies between the chosen analytical model: IFCCs and PCs-option 1 and 3 yield plots over the excitation current (of the first block); ISCCs and PCs-option 4 plot over the time. For the latter two options, the MSTRF- and BSTRF-options yield the angular information of the respective strand's magnetization and excitation.

Plotting Information 2-D

Coil fields:

Variable Description
MX SC magnetization (x-component).
MY SC magnetization (y-component).
|M| SC magnetization (modulus*sign).
MMOD SC magnetization (modulus).
M SC magnetization vectors.
P Losses per strand in W/m. The losses due to filament/strand/cable-magnetization is computed as \frac{\mathrm{d}\mathbf{M}\cdot \mathbf{B}}{\mathrm{d}t}. For the ISCC magnetization model this yields the Ohm's losses.
PINT The integral \int \mathbf{M}\cdot\mathrm{d}\mathbf{B} is calculated and updated at every time step. The result is the integrated magnetization losses for PCs, IFCCs and the ISCC magnetization model in J/m.
BPERP B perpendicular to broad face of the conductor.
BPARA B parallel to broad face of the conductor.

Bn strand contr. of M:

Variable Description
M1 B_1 contribution of SC magnetization.
M2 B_2 contribution of SC magnetization.
M3 B_3 contribution of SC magnetization.
M4 B_4 contribution of SC magnetization.
M5 B_5 contribution of SC magnetization.
M6 B_6 contribution of SC magnetization.
M7 B_7 contribution of SC magnetization.
M8 B_8 contribution of SC magnetization.
M9 B_9 contribution of SC magnetization.
M10 B_{10} contribution of SC magnetization.
M11 B_{11} contribution of SC magnetization.

Time Transient Effects

Options

Option Description
IFCC (Wilson) Interfilament Coupling Currents - analytical model by M. Wilson.
ISCC (Wilson Analytic) Interstrand Coupling Currents - analytical model by M. Wilson.
Nonlinear Inner Iterations Only with BEMFEM calculations of a nonlinear iron yoke are 'on'. This option makes ROXIE recalculate the contribution of the iron yoke at every step of the inner magnetization iteration. If 'off' the nonlinear iron yoke is only calculated before the first and after the last step of the inner iteration.
Plotting Magn. Fields Only The harmonic analysis is done only from the fields due to SC magnetization. The perturbation of the field quality due to SC magnetization is calculated.

  • The "IFCC (Wilson)"-option uses a formula to evaluate the total strand magnetization which does not take into account those Eddy-Current loops that close in the outer copper coating of the strand, i.e., it assumes a highly resistive barrier between the filaments and the outer coating.

  • The "ISCC (Wilson Analytic)"-option a homogeneous magnetization in each conductor. The nature of ISCCs is better represented in a network model, see Section 9.2{reference-type="ref" reference="sec:transientnetwork"}.

Parameters

Variable Description
PC: 0:None; 1,3:1D; 4:Vector Persistent Current (PC) calculations. '0': no PC calculation; '1','3': two implementations of the same 1D persistent current model; '4': vector hysteresis model for field-changes in modulus and direction.
Symmetry: 0: Gen., 1: 1in1, 2: 2in1 To use time-transient effects, you should use the layer-option which generates the full coils. If there is a symmetry, then you may specify only the blocks first quadrant (in the right aperture) in the "Block spec. (Peak fields, Forces, FEM plots)"-widget. SC-magnetization is then only evaluated in these blocks. The contribution of the other blocks in the layer is considered automatically.
Start Time for Loss Calculation Start time for loss calculation (in seconds).
End Time for Loss Calculation End time for loss calculation (in seconds).
Start Time for Multipole Variation The BIRD- and BIRRD-options in the "Objectives"-table calculate the multipole variation during a transfer function or during a transient calculation. For the latter, the time-frame for the variation can be given.
Start Time for Multipole Variation The BIRD- and BIRRD-options in the "Objectives"-table calculate the multipole variation during a transfer function or during a transient calculation. For the latter, the time-frame for the variation can be given.
Maximum Number of Iterations Maxim number of iterations in the determination of the SC magnetization.

Time-Grid definition

The table for the definition of a stepping time-grid has the following columns.

Variable Description
No Number of interval.
Ts Start time of interval (seconds).
Te End time of interval (seconds).
Steps Number of time steps in interval.

Excitation function definition

The table for the definition of excitation functions for each block of conductors has the following columns. Each block can be assigned a number of successive excitation functions over time intervals that need not be the same as the stepping intervals in the table above. At every step of the stepping table each block must be assigned one (and only one) excitation function.

Variable Description
No Number of excitation function definition.
Ts Start of excitation interval.
Te End of excitation interval.
Function Predefined excitation functions.
1: Linear ramp, from A(T_\mathrm{s}) to B(T_\mathrm{e}),
2: A+B\cos(C t + D),
3: Parabolic from A(T_\mathrm{s}) to B(T_\mathrm{e}) with acceleration C, \frac{C}{2}t^2 + \beta t + \gamma, where \beta and \gamma are calculated to match A and B at T_\mathrm{s} and T_\mathrm{e}.
4: Exponential from A(T_\mathrm{s}) with derivative B,
5: Quadratic from A(T_\mathrm{s}) with maximum slope C and linear slope increase during B\,\Delta T, compare Fig. 9.1. Parameter D is '1': ramp up, '2': ram up and down, '3': ramp down.
A Function parameter A.
B Function parameter B.
C Function parameter C.
D Function parameter D.
Blocks,Layers With the "Symmetric Coil"-option blocks are assigned the excitation function in the specified time interval. With the "Layer Definition"-option, individual layers are assigned excitation functions.
  • Function number 5 with D=2: In this case the \Delta T in Fig. 9.1{reference-type="ref" reference="fig:gsiexcitationfunction"} is set to (T_\mathrm{e}-T_\mathrm{s})/2.
Quadratic excitation function as used in fast-ramping magnets.

Network model of Interstrand Coupling Currents

For the simulation of interstrand coupling currents an electrical-network model is implemented in ROXIE. In this model, interstrand coupling currents are not treated as an additional magnetization of the SC conductors (as in the above analytical model) but as additional (positive and negative) line currents in the position of each strand.

The network model of ISCCs cannot be used with nonlinear iron.

Plotting Information 2-D

Current distribution:

Variable Description
ICC Interstrand coupling currents.

Time transient effects

Options:

All analytical SC magnetization models must be 'off'.

Option Description
ISCC (Network Model) Network model (without inductance matrix for interstrand coupling current determination.
LICC + Mut. Inductances Compute and use the inductance matrix in the network model.
Plotting Magn. Fields Only The harmonic analysis is done only from the fields due to ISCCs. The perturbation of the field quality due to ISCCs is calculated.

Parameters:

The "Symmetry"-parameter should be set to zero and all blocks be specified in the "Block spec. (Peak fields, Forces, FEM plots)"-widget.

Time-grid definition:

The same time-grid definition is used as for the analytical models.

Excitation Function Definition:

The excitation function is defined in the same way as for the analytical models.

Quench Calculation

ROXIE features a new quench simulation tool. The tool is currently under development. A first version is available in ROXIE 9.3. This version corresponds to the IEEE paper [@Schwerg:2007lr] by Nikolai Schwerg. It is highly recommended to read this paper before using the algorithm.

To do quench simulation, the following options need to be set

  • "Quench Calculation" in "Global Information".

  • "Self and Mutual Inductance" in "Global Information".

  • "Quench and Temperature Margin" in "Global Information".

  • "Peak Field in Coil" in "Global Information".

  • "Time Transients" in "Main Options". For a description of the "Time Transients"-widget in the context of quench, see Section 4.3{reference-type="ref" reference="sec:quench_ex"}.

  • "Layer Definition" in "Main Options".

  • All conductors must be specified in the "Block spec." widget.

  • A winding scheme needs to be specified, see Section 4.3{reference-type="ref" reference="sec:quench_ex"}.

Quench calculation with ROXIE automatically produces a number of files for post-processing with gnuplot:

  • The roxie.dat file contains about 30 quantities over time, one row per time-step. The description is found in the last line of the file.

roxie.QUENCH is an event-history of the quench simulator.

  • .xdat-files contain relevant data per conductor and time-step.

  • .xsp-files are overwritten in each time step. By calling

gnuplot load "filename.xsp"

you can plot cross-section data.

  • .gnu-files are gnuplot macros to display .xdat and files and the roxie.dat file. They are loaded by

gnuplot load "filename.gnu"

The files can also be viewed during a simulation run. Update a view by typing the gnuplot command "e".

  • Overview.gnu shows the characteristic quantities (voltages, current, resistances, and temperature).

  • Powers.gnu shows the dissipated power over the coil cross-section.

  • Xsections.gnu shows cross-section data (temperature margin, field, resistances, and temperature).

  • Voltages.gnu shows voltages versus time (terminal voltage, induced voltage, ohmic voltage, dump resistor voltage, diode voltage).

The following gnuplot macros produce postscript files

  • Report.gnu reports the most characteristic quanities over time and in the cross-section.

  • roxie.dat.gnu Illustrates the roxie.dat file (see above).

Note that all the output files can take several hundreds of megabytes of disc space!

Global information

Option Description
Quench Calculation Perform a quench simulation.
Winding Scheme Input Change the conductor numbering in the preview window according to winding scheme.

  • For the quench algorithm the user is required to supply the winding scheme of the magnet's coils. The "Winding Scheme Input"-option serves to debug the winding-scheme input.

  • The winding-scheme input is documented in an example in Section 4.3{reference-type="ref" reference="sec:quench_ex"}.

Design variables

Quench, Inductance:

Variable Description
QUENCH Conductor number in which quench originates. Use the conductor number from the preview window with the "Winding Scheme Input"-option 'off'.
QLOSST Time constant of interstrand coupling currents.
HEATTT Heat transfer coefficient from turn to turn per unit length.
HEATER Heater delay (in seconds) for specified conductor (covered by the heater). Use the conductor number from the preview window with the "Winding Scheme Input"-option 'off'.
HEATP0 Power per conductor per length (either exponentially decaying or constant).
HEATTA Time constant (in seconds) for exponential heater power (needs HEATP0 to be set).
DUMPR Dump resistor (ohm).
DUMPT Delay (in seconds) for switching-in of the dump resistor.
QDIOD Diode threshold voltage (V).
QDIODR Diode forward resistance (ohm).
QUTH Quench detection threshold voltage (V) (on the terminal of the magnet).
QMI Maximum number of time-steps.
QCD Number of discretization steps for field re-calculation (based on magnetic energy change) (default = 50).
QIMF Lower threshold (in terms of a fraction of the initial current) to stop quench calculation (default = 0.001).
RUNGES Minimum time step size (seconds) for adaptive Runge-Kutta (default = 0.0001).
RUNGEM Maximum time step size (seconds) for adaptive Runge-Kutta (default = 0.1).
RRR RRR for conductors in layer.
  • "HEATTT" is the heat conductance per unit length between adjacent conductors (in \mathrm{W}\mathrm{m}^{-1}\mathrm{K}^{-1}).

If the "HEATER" variable is not specified for a conductor (or set to zero), then the conductor is not covered by a heater.

A conductor covered by a quench heater will quench after the heater delay if "HEATP0" and "HEATTA" are not set. If "HEATP0" and "HEATTA" are set, then the conductor quenches when the heaters have supplied enough energy to increase the conductor temperature above the critical value.

If "HEATP0" is set, then a constant heating power is transferred to a conductor covered by a quench heater.

If, in addition to "HEATP0", "HEATTA" is specified, then the heating power decays exponentially with "HEATTA" as a time constant.

If "QUTH" is not set, then the quench is assumed to be detected immediately. Otherwise, heater- and dump-resistor delays ("HEATER" and "DUMPT") are counted from the time, when the quench detection voltage is reached at the magnet terminals.

If the "QDIOD" value is not specified or set to zero, then the magnet is immediately short-circuited. The "QUTH" value is then obsolete and the quench is immediately detected.

The magnetic field is not calculated in every time-step of the quench calculation, but only at "QCD" excitation levels. The value must be sufficiently high to resolve the nonlinear iron-yoke response, but sufficiently small to allow for a reasonably fast simulation.

The "RUNGES" and "RUNGEM" values determine the bounds for the adaptive time-step algorithm. If both are set to the same value, the time-stepping is no longer adaptive.

Note that for plotting graphs from the "Objectives" widget, no more than 200 points can be considered. If you want to plot, e.g., the terminal voltage over time from the "Objectives"-table, "QMI" must not have a value above 199. Note however that alternative post-processing based on gnuplot exists (see above) that does not suffer this limitation.

Objectives

Quench:

Variable Description
T Temperature in specified conductor (0 = peak temperature).
I Overall current.
R Total effective resistivity of the magnet (including dump resistor).
MIITS MIITS \int I^2\,\mathrm{d}t in units 10^6\,\mathrm{A}^2\mathrm{s}.
V Potential difference between the specified conductor and input terminal, which is connected to ground (NOR = 0 yields the terminal voltage).
VMAX Peak voltage between any conductor and ground.
VDUMP Voltage on the dump resistor.
  • Note that for plotting graphs from the "Objectives" widget, no more than 200 points can be considered. If you want to plot, e.g., the terminal voltage over time from the "Objectives"-table, "QMI" must not have a value above 199. Note however that alternative post-processing based on gnuplot exists (see above) that does not suffer this limitation.

Plotting Information 2-D

Coil Fields:

Variable Description
T Conductor temperature.
V Conductor potential to ground.