A-V Formulation in Two Dimensions

This section is based on the General Case of the A-V Formulation, where the Maxwell Equations are approximated in Maxwell Quasi Static.

Now, we will see an approximation on A-V Formulation in two dimensions for Superconductors.

1. Transient Case

1.1. A-V Formulation

This section recalls the A-V Formulation :

Thus \(\Omega\) the domain contains the superconductor domain \(\Omega_c\) and non conducting materials \(\Omega_c^C\) (\(\mathbf{J} = 0\)) like the air for example. Also \(\Gamma = \partial \Omega\) is the bound of \(\Omega\), \(\Gamma_c = \partial \Omega_c\) the bound of \(\Omega_c\), \(\Gamma_D\) the bound with Dirichlet boundary condition and \(\Gamma_N\) the bound with Neumann boundary condition, such that \(\Gamma = \Gamma_D \cup \Gamma_N\)..

We introduce :

  • Magnetic potential field \(\mathbf{A}\) : \(\textbf{B} = \nabla \times \textbf{A}\)

  • Electric potential scalar : \(\nabla \mathbf{V} = - \textbf{E} - \frac{\partial \textbf{A}}{\partial t}\)

In this example we only consider a bulk cylinder without transport current so the electrical potential scalar can be ignored and we have :

  • Electrical field : \(\textbf{E}= - \frac{\partial \textbf{A}}{\partial t}\)

A Formulation
\[\left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \sigma \frac{\partial \textbf{A}}{\partial t} &=& 0 \text{ on } \Omega & \text{(A form)} \\ \mathbf{A} \times \mathbf{n} &=& 0 \text{ on } \Gamma_D & \text{(D)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 \text{ on } \Gamma_N & \text{(N)} \end{matrix} \right.\]

with \(\sigma\) expressed with the E-J power law for the HTS :

\[\sigma=\frac{J_c}{E_c} \left(\frac{||\textbf{E}||}{E_c}\right)^{(1-n)/n}=\frac{J_c}{E_c} \left(\frac{||-\partial_t \textbf{A}||}{E_c}\right)^{(1-n)/n}\]

with :

  • \(J_c\): the critical current density \((A/m^2)\)

  • \(E_c\): the threshold electric field \((V/m)\)

  • \(n\): material dependent exponent

1.2. Differential Formulation

The model uses the A Formulation on geometry in two dimensions.

The magnetic flux density \(\textbf{B}\) as only two component, \(B_x\) and \(B_y\), and the magnetic potential field as only one component \(A_z\).

So, \(\mathbf{B} = \begin{pmatrix} B_x \\ B_y \\ 0 \end{pmatrix}\) and \(\mathbf{B} = \nabla \times \mathbf{A}(x,y)\), so \(\mathbf{A} = \begin{pmatrix} 0 \\ 0 \\ A_z \end{pmatrix}\)

We have \(\nabla \times \mathbf{A} = \begin{pmatrix} \frac{\partial A_z}{\partial y} \\ -\frac{\partial A_z}{\partial x} \\ 0 \end{pmatrix}\) and \(\nabla \times \left( \nabla \times \mathbf{A} \right) = \begin{pmatrix} 0 \\ 0 \\ -\Delta A_z \end{pmatrix}\)

The A Formulation becomes :

\[ - \frac{1}{\mu} \Delta A_z = -\sigma \frac{\partial A_z}{\partial t} \hspace{2cm} \text{(A form 2D)}\]

Boundary conditions :

With Dirichlet conditions (D), we have on \(\Gamma_D\) :

\[\begin{eqnarray*} \mathbf{A} \times \mathbf{n} = 0 \Longleftrightarrow \begin{pmatrix} - A_z \, n_y \\ A_z \, n_x \\ 0 \end{pmatrix} = 0 \end{eqnarray*}\]

However, as \(\mathbf{n} = \begin{pmatrix} n_x \\ n_y \end{pmatrix}\) isn’t zero, we have :

\[ A_z = 0 \text{ on } \Gamma_D \hspace{2cm} \text{(D 2D)}\]

With Neumann conditions (N), we have on \(\Gamma_N\) :

\[\begin{align} \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} = 0 \Longleftrightarrow & \begin{pmatrix} \partial_y A_z \\ \partial_x A_z \\ 0 \end{pmatrix} \times \begin{pmatrix} n_x \\ n_y \\ 0 \end{pmatrix} = 0 \\ \Longleftrightarrow & \begin{pmatrix} 0 \\ 0 \\ \partial_y A_z \, n_y + \partial_x A_z \, n_x \end{pmatrix} = 0 \\ \Longleftrightarrow & \hspace{1.5cm} \frac{\partial A_z}{\partial \mathbf{n}} = 0 & \text{(N 2D)} \end{align}\]

The A Formulation becomes :

A Formulation in two dimensions approximation
\[\text{(A 2D)} \left\{ \begin{matrix} \sigma \frac{\partial A_z}{\partial t} - \frac{1}{\mu} \Delta A_z &=& 0 \text{ on } \Omega \hspace{2cm} \text{(A 2D)} \\ A_z &=& 0 \text{ on } \Gamma_D \hspace{2cm} \text{(D 2D)} \\ \frac{\partial A_z}{\partial \mathbf{n}} &=& 0 \text{ on } \Gamma_N \hspace{2cm} \text{(N 2D)} \end{matrix} \right.\]

1.3. Weak Formulation

In this subsection, we write the weak formulation of equation (A 2D) the A-V Formulation in two dimensions.

By making product \(\phi \in H^1(\Omega)\) and integration of (A 2D) on \(\Omega\) :

\[ \int_{\Omega}{ \left( \sigma \frac{\partial A_z}{\partial t} - \frac{1}{\mu} \Delta A_z \right) \, \phi \ dxdy } = 0\]

By Formula of Green :

\[ \int_{\Omega}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdy } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdydz } = \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}} d\Gamma }\]
\[ \int_{\Omega}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdydz } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdydz } = \int_{\Gamma_D}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}} } + \int_{\Gamma_N}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}} }\]

We impose the boundary conditions :

  • Dirichlet : \(A_z = 0\) on \(\Gamma_D\) (D 2D)

  • Neumann : \(\frac{\partial A_z}{\partial \mathbf{n}} = 0\) on \(\Gamma_N\) (N 2D)

Finally, we have :

Weak formulation of A Formulation in two dimensions approximation
\[ \int_{\Omega}{ \sigma \frac{\partial A_z}{\partial t} \, \phi \ dxdy } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdy } = 0 \hspace{1cm} \text{(Weak AV 2D)}\]

1.4. Time Discretization

In this subsection, we use the time discretization by backward Euler method on the (Weak A 2D).

We discretize in time the problem with the time step \(\Delta t\).

We note \(f^n(\mathbf{x}) = f(n\Delta t, \mathbf{x})\), for \(n \in \mathbb{N}\).

We have the approximation with backward Euler method : \(\frac{\partial A_z}{\partial t} \approx \frac{A_z^{n+1}-A_z^n}{\Delta t}\).

The equations (Weak A 2D) becomes :

Time Discretization of A Formulation in two dimensions approximation
\[\small{ \int_{\Omega}{ \sigma \frac{A_z^{n+1}}{\Delta t} \, \phi \ dxdy } + \int_{\Omega}{ \frac{1}{\mu} \nabla A_z^{n+1} \cdot \nabla \phi \ dxdy }= \int_{\Omega}{ \sigma \frac{A_z^n}{\Delta t} \, \phi \ dxdy } \hspace{14cm} \text{(Disc A 2D)} }\]

2. Magnetostatic Case

This section presents the A-Formulation in two dimensions in stationary case.

2.1. Differential Equation

As we saw before \(\textbf{E}= - \frac{\partial \textbf{A}}{\partial t}\), and as \(\textbf{J}=\sigma(\textbf{E})\textbf{E}\). In order to keep the characteristics of the Superconductors, we need a new formula in the stationary case.

\[ - \frac{1}{\mu} \Delta A_z = J \hspace{2cm} \text{(A form 2D)}\]

Here instead of the E-J power which is traditionally used for superconductor we use the error function \(erf\) :

\[ J = J_c erf\left(\frac{-A_z}{A_r}\right)\]

where \(J_c\) is the critical current density and \(A_r\) is a parameter resulting from the combination of the parameter that control the stepness of the curve of erf and the time it takes to reach the AC excitation.

An external magnetic field \(B_0\) is applied by setting the boundary conditions for \(A_z\) on the air boundary. In order to have a magnetic field along \(y\) and as \(B_y=-\partial A_z/\partial x\), we have \(A_z=-B_0 x\) as Dirichlet boundary condition.

The A Formulation becomes :

A Formulation in two dimensions approximation
\[\text{(A 2D)} \left\{ \begin{matrix} - \frac{1}{\mu} \Delta A_z = J_c erf\left(\frac{-A_z}{A_r}\right) \hspace{2cm} &\text{(A form 2D)} \\ A_z =-B_0 x \text{ on } \Gamma_D \hspace{2cm} &\text{(D 2D)} \\ \frac{\partial A_z}{\partial \mathbf{n}} = 0 \text{ on } \Gamma_N \hspace{2cm} &\text{(N 2D)} \end{matrix} \right.\]

2.2. Weak Formulation

In this subsection, we write the weak formulation of equation (A form 2D) the A Formulation in two dimensions, by multiplying the equation by the test function \(\phi \in H^1(\Omega)\) and doing an integration of (A form 2D) on \(\Omega\) :

\[ \int_{\Omega}{ - \frac{1}{\mu} \Delta A_z \cdot \phi \ dxdy } = \int_{\Omega_c}{ J_c erf\left(\frac{-A_z}{A_r}\right) \cdot \phi \ dxdy }\]

By Formula of Green :

\[ \int_{\Omega}{ - \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdy } = \int_{\Omega_c}{ J_c erf\left(\frac{-A_z}{A_r}\right) \cdot \phi \ dxdy } + \int_{\Gamma_D}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}} } + \int_{\Gamma_N}{ \frac{1}{\mu} \frac{\partial A_z}{\partial \mathbf{n}} }\]

We impose the boundary conditions :

  • Dirichlet : \(A_z = -B_0 x\) on \(\Gamma_D\) (D 2D)

  • Neumann : \(\frac{\partial A_z}{\partial \mathbf{n}} = 0\) on \(\Gamma_N\) (N 2D)

Finally, we have :

Weak formulation of A Formulation in two dimensions approximation
\[ \int_{\Omega}{ - \frac{1}{\mu} \nabla A_z \cdot \nabla \phi \ dxdy } = \int_{\Omega_c}{ J_c erf\left(\frac{-A_z}{A_r}\right) \cdot \phi \ dxdy } + \int_{\Gamma_D}{ -B_0 x } \hspace{1cm} \text{(Weak A 2D)}\]

3. AC loss Calculation

Superconductor are materials that have no electrical resistance. But when they are subjected to time-varying magnetic field, they still show power loss dissipation. Because the superconductors operate at very low temperatures, this power dissipation is a real issue in lots of applications. This power dissipation is called AC loss and it really influences the efficiency and the cost of superconductors. That’s why creating models that can accurately predict the AC loss are essential for the design of superconductors, and therefore, for their future implementation in the technology market.

In the case of this example, the AC loss can be computed as :

\[ Q= -4\int_{\Omega}{ J A d\Omega}\]

With :

  • \(J\) the current density

  • \(A\) the magnetic potential field.

4. References

  • Finite-Element Formulation for Systems with High-Temperature Superconductors, Julien Dular, Christophe Gauzaine, Benoît Vanderheyden, IEEE Transactions on Applied Superconductivity VOL. 30 NO. 3, April 2020, PDF

  • A numerical model to introduce students to AC loss calculation in superconductors. Francesco Grilli and Enrico Rizzo 2020 Eur. J. Phys. 41 045203