SC 001 - Superconductor AC Loss

Model definition

In this tutorial, a superconducting wire is simulated using the $H-\varphi$ formulation. The wire consists of twisted superconducting filaments embedded into a copper matrix. The whole modelling domain with an air cylinder around the wire is shown in the figure below.

 

Component	Dimension
Air diameter	10 mm
Copper core 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 filaments. The losses over a volume of interest can be computed as

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

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Material Data

Magnetic permeability ($\mu$):

• all domains: $\mu_0$

 

Electric resistivity ($\rho$):

• Copper: $10^{-10}~\Omega$m
• Superconducting filaments:
  • $\rho=\frac{E_c}{J_c}\left(\frac{||J||}{J_c} \right)^{n-1}$
    • Crictical electric field strength, $E_c=100~\mu$V/m
    • Exponent, $n=30$
    • Critical electric current, $I_c=100$ A
    • Total cross-section of superconducting filaments, $A_{sc}=3.4541\times 10^{-7}$ m$^2$
    • Critical electric current density, $J_c=I_c/A_{sc}$

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Source

The problem is sourced by applying the total current of

$$I(t)=0.8I_c\sin(2\pi ft),$$

where the frequency $f$ is 50 Hz

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Step-by-step guide

Step 1 - Create geometry

1. Start with a new project.  

2. Upload the geometry as a .step file and finish by confirming the model changes.  

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Step 2 - Define regions, materials and shared expressions

1. Proceed to the Properties section to define Shared region volumes for superconducting filaments, air and copper with the + button.

2. Choose Air, Copper and YBCO from the Materials with the + button.

Power law for YBCO material

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

3. Assign the created Shared regions to the corresponding Materials with Select volume.  

4. Define new Shared expressions: YBCO_Ic, f, YBCO_Asc, and Iop as specified in the Material data section.

Shared expression	Value
YBCO_Ic	$100$
f	$50$
YBCO_Asc	$3.4541\times 10^{-7}$
Iop	$0.8$*YBCO_Ic*$sin(2$*$pi$*$f$*$t)$

 

5. Modify the values of the predefined shared expressions YBCO_Jc and YBCO_n.

Shared expression	Value
YBCO_Jc	YBCO_Ic / YBCO_Asc
YBCO_n	30

 

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Step 3 - Define the physics and apply the source

1. Proceed to the Physics section to define physics and interactions.  

2. Click on the + icon to add a new physics. Select Magnetism 𝑯 and Magnetism 𝜑.

3. Apply sc and copper volumes for Magnetism 𝑯 by Add volume and apply air region for Magnetism 𝜑.  

4. Apply 𝑯-𝜑 coupling for Magnetism 𝑯 and Apply a point Constraint with value 0 for Magnetism 𝜑 at the external boundary of the air domain.  

5. To Apply the electric current source:

• a) add Lump I/V cut for Magnetism 𝜑.
• b) Add curve by selecting the curve regions forming a closed loop around the cross-section of the copper region.
• c) Use Iop for the value of the current.  

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Step 4 - Create mesh and set up the simulation

1. Proceed to the Simulations section and add mesh with +

2. Apply & mesh with the settings shown in the image  

3. Add a simulation with + and Apply the settings shown in the image. Set the previously generated mesh for the simulation by clicking on Mesh under Simulaiton 1 and by choosing Mesh 1.  

4. Define Custom value output named SC loss and Cu loss for computing the Joule losses in the filaments and copper respectively. The output expression for SC loss is integrate(reg.sc, transpose(E) * j, 4). The output expression for Cu loss is integrate(reg.copper, transpose(E) * j, 4).  

5. Navigate to the Script and enable the scripting mode. Replace the code of line under # Mangetism H formulation 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 5 - Running the simulation and checking the results

1. Click on Not run next to Simulation 1 to start the simulation.  

2. To follow the simulation in progress, Click on Logs under the Results.  

3. The losses in the superconductor and copper are visualized in Plotting 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

[2] Allsolve demo project of Superconductor AC losses

[3] Youtube tutorial of Superconductor AC losses

[4] Newton Linearization

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