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Superconducting Wire
Introduction
Current can flow in a superconducting wire with practically zero resistance, although factors including temperature, current density, and magnetic field can limit this phenomenon. This application solves a time-dependent problem of a current building up in a superconducting wire close to the critical current density. This application is based on a suggestion by Dr. Roberto Brambilla, CESI, Superconductivity Dept., Milano, Italy.
The Dutch physicist Heike Kamerlingh Onnes discovered superconductivity in 1911. He cooled mercury to the temperature of liquid helium (4 K) and observed that its resistivity suddenly disappeared. Research in superconductivity reached a peak during the 1980s in terms of activity and discoveries, especially when scientists uncovered the superconductivity of ceramics. In particular, it was during this decade that researchers discovered YBCO — a ceramic superconductor composed of yttrium, barium, copper, and oxygen with a critical temperature above the temperature of liquid nitrogen. However, researchers have not yet created a room-temperature superconductor, so much work remains for the broad commercialization of this area.
This application illustrates how current builds up in a cross section of a superconducting wire; it also shows where critical currents produce a swelling in the nonsuperconducting region.
Model Definition
The dependence of resistivity on the amount of current makes it difficult to solve the problem using the Magnetic Fields interface. The reason is that a circular dependency arises because the current-density calculation contains the resistivity, leading to a resistivity that is dependent on itself.
An alternative approach uses the magnetic field as the dependent variable, and you can then calculate the current as
The electric field is a function of the current, and Faraday’s law determines the complete system as in
where E(J) is the current-dependent electric field. The model calculates this field with the empirical formula
where E0 and α are constants determining the nonlinear behavior of the transition to zero resistivity, and JC is the critical current density, which decreases as temperature increases.
For the superconductor YBCO, this model uses the following parameter values (Ref. 1):
Parameter
Value
 E0
0.0836168 V/m
 α
1.449621256
 JC
17 MA
 TC
92 K
Systems with two curl operators are best dealt with using vector elements (edge elements). This is the default element for the physics interfaces in the AC/DC Module that solve similar equations. This particular formulation for the superconducting system is available in the AC/DC Module as the Magnetic Field Formulation interface.
For symmetry reasons, the current density has only a z-component.
The model controls current through the wire with its outer boundary condition. Because Ampère’s law must hold around the wire, a line integral around it must add up to the current through the wire. Cylindrical symmetry results in a known magnetic field at the outer boundary
This is applied as a constraint on the tangential component of the vector field.
Results and Discussion
The model applies a simple transient exponential function as the current through the wire, reaching a final value of 1 MA. This extremely large current is necessary if the superconducting wire is to reach its critical current density. Plotting the current density at different time instants shows the swelling of the region in which the current flows. This swelling comes from the transition out of the superconducting state at current densities exceeding JC. Figure 1 presents a plot of the current density at t = 0.1 s.
Figure 1: The current density at 0.1 s.
Reference
1. R. Pecher, M.D. McCulloch, S.J. Chapman, L. Prigozhin, and C.M. Elliotth, “3D-modelling of bulk type-II superconductors using unconstrained H-formulation,” 6th European Conf. Applied Superconductivity, EUCAS, 2003.
Application Library path: ACDC_Module/Devices,_Superconducting/superconducting_wire
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select AC/DC > Electromagnetic Fields > Vector Formulations > Magnetic Field Formulation (mfh).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Geometry 1
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
Keep all the default values.
Circle 2 (c2)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 0.1.
4
Click  Build All Objects.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file superconducting_wire_parameters.txt.
Definitions
Define a step function that will be used in the expression of the superconductor characteristic.
Step 1 (step1)
In the Definitions toolbar, click  More Functions and choose Step.
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
I1
I0*(1-exp(-t/tau))
A
 
H0phi
I1/(2*pi*sqrt(x^2+y^2))
A/m
 
In order to simplify the application of the boundary condition, add a cylindrical coordinate system.
Cylindrical System 2 (sys2)
In the Definitions toolbar, click  Coordinate Systems and choose Cylindrical System.
Magnetic Field Formulation (mfh)
1
Click the  Show More Options button in the Model Builder toolbar.
2
In the Show More Options dialog, in the tree, select the checkbox for the node Physics > Advanced Physics Options.
3
Click OK.
When using the Magnetic Field Formulation physics with superconducting material, it is necessary to turn off the automatic divergence constraint, that can lead to instability.
4
In the Model Builder window, under Component 1 (comp1) click Magnetic Field Formulation (mfh).
5
In the Settings window for Magnetic Field Formulation, click to expand the Divergence Constraint section.
6
Clear the Activate divergence constraint checkbox.
Faraday’s Law 1
Set the constitutive relation for the default Faraday’s Law node to use Resistivity.
1
In the Model Builder window, under Component 1 (comp1) > Magnetic Field Formulation (mfh) click Faraday’s Law 1.
2
In the Settings window for Faraday’s Law, locate the Constitutive Relation Jc-E section.
3
From the Conduction model list, choose Electric resistivity.
Faraday’s Law 2
1
In the Physics toolbar, click  Domains and choose Faraday’s Law.
2
Select Domain 2 only.
3
In the Settings window for Faraday’s Law, locate the Constitutive Relation Jc-E section.
4
From the Conduction model list, choose E-J characteristic.
The Faraday’s Law feature will use the material data specified in the Superconductor material.
Set up the boundary condition for the magnetic field.
Tangential Magnetic Field 1
1
In the Physics toolbar, click  Boundaries and choose Tangential Magnetic Field.
2
Select Boundaries 1, 2, 5, and 8 only.
3
In the Settings window for Tangential Magnetic Field, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose Cylindrical System 2 (sys2).
5
Locate the Magnetic Field section. Specify the H0 vector as
0
r
H0phi
phi
0
a
Materials
Add the materials used in the model. For the domain surrounding the wire, create a material representing Air.
Air
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Air in the Label text field.
The Magnetic Field Formulation physics requires a finite resistivity in all domains.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1
1
Basic
Resistivity
res_iso ; resii = res_iso, resij = 0
rho_air
Ω·m
Basic
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1
1
Basic
For the superconductor, create a custom material that uses the ’E-J Characteristic’ model.
Superconductor
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Superconductor in the Label text field.
3
Select Domain 2 only.
Fill in the relative permittivity and permeability.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1
1
Basic
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1
1
Basic
Now add a subnode that provides the material model for the superconductor.
5
Click to expand the Material Properties section. In the Material properties tree, select Electromagnetic Models > E-J Characteristic.
6
Click  Add to Material.
7
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Property group
Electric field norm
normE
E0*(((normJ-Jc)/Jc)*step1((normJ-Jc)/1[A/m^2]))^alpha
E-J characteristic
Mesh 1
Proceed with creating the mesh. Use a finer mesh in the superconducting domain to resolve the current density.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.02.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Coarse.
4
Click  Build All.
Study 1
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
Click  Range.
4
In the Range dialog, type 0.005 in the Step text field.
5
In the Stop text field, type 0.1.
6
Click Replace.
To improve the accuracy of the time-dependent solution, specify a small initial time step and a maximum step size.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
Select the Initial step checkbox. In the associated text field, type 1e-9.
5
From the Maximum step constraint list, choose Constant.
6
In the Maximum step text field, type 1e-3.
7
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 node, then click Fully Coupled 1.
8
In the Settings window for Fully Coupled, click to expand the Method and Termination section.
9
From the Jacobian update list, choose Once per time step.
10
In the Study toolbar, click  Compute.
Results
Magnetic Flux Density (mfh)
The default plot, shown after the computation is completed, visualizes the norm of the magnetic flux density.
Create a plot group to visualize the current density.
Current Density
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Current Density in the Label text field.
Surface 1
1
Right-click Current Density and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Magnetic Field Formulation > Currents and charge > Current density - A/m² > mfh.Jz - Current density, z-component.
3
In the Current Density toolbar, click  Plot.
4
Click the Zoom In button on the Graphics toolbar two or three times to get a closer view of the wire.
Under the Export node, it is possible to create an animation of the evolution of the current density distribution.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Current Density.
4
Click the  Play button in the Graphics toolbar.
The animation can also be saved to file by selecting File from the Target list box then clicking Export.