AC losses in HTS tape demo

Model definition

Components	Geometric info
air	radius = 8 cm
tape	width = 4 mm, heigth = 95 μm, length = 1 cm
copper layer	thickness = 20 μm
silver layer	thickness = 2 μm
YBCO layer	thickness = 1 μm
hastelloy layer	thickness = 50 μm
domain	length = 1 cm
YBCO cross-section	$A_{ybco} = 3.96\cdot10^{-9}$ m$²$

Material Data

Magnetic permeability ($\mu$)

• all domains: $\mu_0$

Electric resistivity ($\rho$)

• Hastelloy: $10^{-6}~\Omega$m
• Silver: $10^{-8}~\Omega$m
• Copper: $10^{-8}~\Omega$m
• YBCO:
  • $\rho=\frac{E_c}{J_c}\left(\frac{||J||}{J_c} \right)^{n-1}$
    • $E_c=100~\mu$V/m
    • $n=30.5$
    • $J_c=2.85\times 10^{10}$ [A/m$^2$]
    • $I_c=J_c\cdot A_{ybco}$
  • $E\approx E^0+\frac{\partial E}{\partial J}(J-J^0)$

Inputs case 1

• Frequency $f=50$ Hz

• Operation current $I_{op}(t)=0.8I_c\cdot$sin$(2\pi f t)$

• External magnetic flux density $B_{ext}(t)=0$ [T]

Inputs case 2

• Frequency $f=50$ Hz

• Operation current $I_{op}(t)=0$ A

• External magnetic flux density $B_{ext}(t)=20\cdot$sin$(2\pi f t)$ [mT]

Output results

• Joule losses in the ybco region, and in the normalconducting region

$$\begin{equation} P_i(t)=\int_{\Omega_{i}}\boldsymbol{E}(t)\cdot \boldsymbol{J}(t)~\rm{d}\Omega \end{equation}$$

• Field visualizations

Step-by-step guide

Here you’ll find a step-by-step tutorial on how to simulate this in Quanscient Allsolve.

Step 1 - Create geometry

1. Start with a new project and name it:  

2. Upload the mesh file “htstape.msh”:  

Step 2 - Define shared regions

1. Proceed to the Properties tab to define regions and materials.  

2. Click on the + icon next to Shared regions to define physical regions for later convenience.

   • Select geometric entities by clicking the geometry visualization. Finalize the selection by clicking `Apply
     • Note: After defining the air region, hide the air volume by clicking the button indicated by the arrow, and then clicking the air volume `
   • Define the following shared region volumes: air, copper, silver, hastelloy, ybco and tape (including all the tape layers)
   • In addition define the tape region excluding ybco layer and name it normalconducting
   • Define the following shared region surfaces from the HTS tape cross-section: s_copper, s_silver, s_hastelloy, s_ybco, and s_tape including of all the previously defined surfaces  

Step 3 - Define material properties

1. Click on the + icon next to Materials and select Air, Silver, Copper and YBCO from the material list with Confirm and select the corresponding already defined shared volume to it. Finalize defining a material with Apply.

   • Change Electric conducticity to 1e8 forSilver and Copper.  

2. For the Hastelloy, we need to create a new material.  

3. Name it to hastelloy and add the following properties with +: Magnetic permeability: $\mu_0$ and electric conductivity: $10^6$ S/m.  

Step 3 - Define shared expressions

1. Assing correct values to YBCO_Jc=2.85e10, and YBCO_n=30.5
2. Create a new shared expression by clicking + icon next to Shared expressions, and define $A_{ybco}$=3.96e-9
3. Similarly, define shared expressions: freq and $I_{op}$. Define also $B_{ext} = 0$ T  

Step 3 - Define the physics, boundary conditions and sources

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

2. Click on the + icon to add a new physics. Select Magnetism $\varphi$ and Magnetism $\boldsymbol{H}$.

• Apply air region for Magnetism $\varphi$
• Apply tape region for Magnetism $\boldsymbol{H}$  

3. Add Constraint for Magnetism $\varphi$.  

4. Choose a point region from the boundary of the air domain and set the value to 0.0  

5. Add External field for Magnetism $\varphi$. Assign $B_{ext}$ to $y$-direction at the infinity boundary of the air domain.  

6. Add Lump I/V cut for Magnetism $\varphi$, by selecting the curves going around the tape cross-section. Assign $I_{op}$ as the applied total current.  

7. Add $\boldsymbol{H}-\varphi$ coupling for Magnetism $\boldsymbol{H}$.  

Step 5 - Apply simulation settings

1. Click on + icon next to the Simulations to add a simulation.  

2. Under the SIMULATIONS SETTINGS:

   • Name: Simulation: case 1
   • Analysis Type: select Transient
   • Time stepping algorithm: select Implicit Euler
   • Start time: 0 s, End time: 0.02 s, Time step: 0.00016 s
   • Solver mode: Direct solver
   • Node Count = 1 and Node Type = 1 CPU, 16GB
   • Click on Apply button to confirm the settings.  

3. Click on Mesh under Simulation: case 1 and select Mesh 1.  

Step 6 - Simulation outputs

1. Click on + icon next to Outputs under Simulation: case 1. Now select Custom under Value outputs.

   • Define Joule loss integrals Pnc and Pybco:
     • Pybco: integrate(reg.ybco,transpose(E)*j,2)
     • Pnc: integrate(reg.normalconducting,transpose(E)*j,2)
   • Net currents:
     • Itot: lump.I
     • Iybco: integrate(reg.s_ybco,on(reg.ybco,compz(j)),2)  

2. To visualize $\boldsymbol{J}$ in the YBCO region, create Field output similarly as in the image.  

Step 7 - Custom modifications to the script

1. Go to Script under Simulation: case 1 and enable scripting mode.  

2. Change interpolation order of $\varphi$ to 1.  

3. Replace line 101 with the following linearization to enable Newton Raphson method:

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.ybco, dofe * (qs.curl(qs.tf(fld.H)) - var.curl_tf_Hs))
form += qs.integral(reg.normalconducting, 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))

 

4. Finally:

• change tolerance to 1e-4
• replace the field output region to reg.s_ybco  

4. Now Save script and click on Simulation: case 1 and then on Run Simulation button. The simulation status changes from Not run to Running and after completion to Success.  

Step 5 - Visualizing the simulation results

1. To visualize $\boldsymbol{J}$, add visualization, select j and use similar settings as shown in the image.