SC 004 - Superconductor AC Loss

In this tutorial, a high-temperature superconducting (HTS) wire is simulated using the H-φ formulation.

Model definition

The wire consists of twisted superconducting filaments embedded into a copper matrix. The whole modelling domain with an air cylinder around the wire is illustrated below.

Element	Dimension
Air cylinder diameter	10 mm
Copper cylinder diameter	535 μm
Filament diameter	350 μm
Domain length	10 mm

Output results

• Joule losses as a function of time in the copper and the superconducting filament regions. The losses over the volume of interest can be computed as

$$P(t)=\int\boldsymbol{E}(t)\cdot \boldsymbol{J}(t)~\rm{d} V.$$

Material Data

Magnetic permeability, $\mu$:

• all domains: $\mu_0$

Electric resistivity, $\rho$:

• Copper: $10^{-10} ~ \Omega \rm m$
• Superconducting filaments:
  • $\rho = \frac{E_{\rm c}}{J_{\rm c}} \left( \frac{||J||}{J_{\rm c}} \right) ^{n-1}$
    • Critical electric field strength, $E_{\rm c} = 100 ~ \rm μV/m$
    • Exponent, $n = 30$
    • Critical electric current, $I_{\rm c} = 100 ~ \rm A$
    • Total cross-section are of superconducting filaments, $A_{\rm sc} = 3.4541 \cdot 10^{-7} ~ \rm m^2$
    • Critical electric current density, $J_{\rm c} = I_{\rm c} / A_{\rm sc}$

Source

The problem is sourced by applying a total current of

$$I(t) = 0.8 ~ I_{\rm c} ~ \sin(2 \pi f t),$$

where the frequency $f$ is $50 ~ \rm Hz$.

Step-by-step guide

Here you’ll find a detailed step-by-step tutorial on how to simulate AC loss in a twisted filament HTS wire Quanscient Allsolve.

Step 1 - Import the geometry

1. Start with a new project and name it as

   SC twisted filament AC loss

2. Import the geometry as a .step file with default import options.

   File download link: twisted-superconductor.step

   

3. Confirm model changes.

Step 2 - Define shared regions

1. Go to the Properties section.

2. Define a shared region for air:

   Region name	Region type	Target
   air	Volume	Air cylinder

   

3. Define a shared region for copper:

   Region name	Region type	Target
   copper	Volume	Copper matrix

   

4. Define a shared region for the superconducting filaments:

   Region name	Region type	Target
   sc	Volume	SC filaments

   

Step 3 - Define materials

1. Assign the Air material to the air shared region:

   

2. Assign the Copper material to the copper shared region:

   

3. Assign the YBCO superconductor material to the sc shared region:

   

   Note: YBCO power law

   When the YBCO material is added, some predefined shared expressions corresponding to the YBCO power law are automatically created.

Step 4 - Define shared expressions

1. Define the following new shared expressions:

   Name	Description	Expression
   f	Working frequency [Hz]	50
   YBCO_Ic	Critical current [A]	100
   YBCO_Asc	Filament cross-section area [m^2]	3.4541e-7
   Iop	Operating current [A]	0.8 * YBCO_Ic * sin(2 * pi * f * t)

2. Modify the expressions of the following shared expressions:

   Name	Modified expression
   YBCO_Jc	YBCO_Ic / YBCO_Asc
   YBCO_n	30

Step 5 - Define physics and apply the current source

1. Go to the Physics section.

2. Add the Magnetism H physics:

   Physics	Target
   Magnetism H	Copper matrix and SC filaments

   

3. Add the Magnetism φ physics:

   Physics	Target
   Magnetism φ	Air cylinder (air shared region)

   

4. Add a Lump I/V cut interaction to Magnetism φ.

   Interaction name	Interaction type	Target	Value
   Current source	Lump I/V cut	a counter-clockwise loop at the top edge of the copper matrix	Iop

   

5. Add a Constraint interaction to Magnetism φ:

   Interaction name	Interaction type	Target	Value
   Gauge	Constraint	a point at the external boundary of the air domain	0

   

6. Add the H-φ coupling interaction to Magnetism H.

7. Before moving on, check that your physics tree matches the one below:

   

Step 6 - Generate a mesh

1. Go to the Simulations section.

2. Add a new mesh.

3. Set Mesh quality to Expert settings.

4. Set Used mesher to Basic.

5. Set Curvature enhancement to 25.

6. Generate the mesh and check the preview.

   

   

Step 7 - Select simulation settings

1. Add a new simulation.

2. Set Analysis type to Transient.

3. Select Transient settings:

   Timestep algorithm	Start time [s]	End time [s]	Timestep size [s]
   Implicit Euler	0	0.01	0.0001

4. Set Solver mode to Iterative solver.

5. Set Node count to 50.

6. Select Mesh 1 as the mesh for your simulation.

7. Define custom value outputs for computing Joule losses in the filaments and copper:

   Output name	Output type	Output expression
   SC loss	Custom value output	integrate(reg.sc, transpose(E) * j, 4)
   Cu loss	Custom value output	integrate(reg.copper, transpose(E) * j, 4)

8. Open the Script for your simulation.

9. Enable Scripting mode.

10. Replace the first autogenerated line under # Magnetism H with the following Newton-linearization [4]:

    rho = 1/par.sigma(df.j)
    dedj = rho*qs.eye(3) + (expr.YBCO_n-1.0)*rho/qs.max(df.j*df.j, 1e-40) * df.j * qs.transpose(df.j)
    dofe = rho*df.j + dedj * (qs.curl(qs.dof(fld.H))+var.curl_dof_Hs - qs.curl(fld.H)-var.curl_Hs)
    form += qs.integral(reg.sc, dofe * (qs.curl(qs.tf(fld.H)) - var.curl_tf_Hs))
    form += qs.integral(reg.copper, qs.inverse(par.sigma(df.j)) * (qs.curl(qs.dof(fld.H)) + var.curl_dof_Hs) * (qs.curl(qs.tf(fld.H)) - var.curl_tf_Hs))

    

Step 8 - Run the simulation and see results

1. Run the simulation.

2. To follow the simulation progress, open Logs.

3. The SC and Cu loss results can be seen in Plotting, even while the simulation is running:

   

References

[1] H-φ Formulation in Sparselizard Combined With Domain Decomposition Methods for Modeling Superconducting Tapes, Stacks, and Twisted Wires. https://doi.org/10.1109/TASC.2023.3240389

[2] Allsolve demo project of Superconductor AC losses. https://allsolve.quanscient.com/#/projects/demo/8fed82d1-5bf0-4c02-835b-94e65a60f847

[3] Youtube tutorial of Superconductor AC losses. https://youtu.be/B9QZZ5y7RpQ

[4] Newton Linearization. https://en.wikiversity.org/wiki/Nonlinear_finite_elements/Newton_method_for_finite_elements