BusBar

In this example, we model a busbar as a rectangular conductor. A differential electrical potential is applied to the entry and exit.

1. Running the case

The command line to run this case in 2D is

mpirun -np 4 feelpp_toolbox_electric --case "github:{path:toolboxes/electric/cases/busbar}" --case.config-file 2d.cfg
Case option
--case "github:{path:toolboxes/electric/cases/busbar}"
Case config file option
--case.config-file 2d.cfg

The command line to run this case in 3D is

mpirun -np 4 feelpp_toolbox_electric --case "github:{path:toolboxes/electric/cases/busbar}" --case.config-file 3d.cfg
Case option
--case "github:{path:toolboxes/electric/cases/busbar}"
Case config file option
--case.config-file 3d.cfg

2. Geometry

The busbar conductor consists in a rectangular cross section extruded along the x axis in 2D. In 3D, this is the same geometry, but extruded along the z axis.

3. Input parameters

Name Description Value Unit

\(Lx\)

length

1

\(m\)

\(Ly\)

width

2

\(m\)

\(Lz\)

thickness (3D)

0.5

\(m\)

\(V_D\)

electrical potential

9

\(V\)

3.1. Model & Toolbox

  • This problem is described by a steady electric conduction model for \(V\) with Dirichlet boundary conditions on entry/exit.

  • toolbox: electric

3.2. Materials

Name Description Marker Value Unit

\(\sigma\)

electric conductivity

omega

\(4.8e7\)

\(S.m^{-1}\)

3.3. Boundary conditions

The boundary conditions are imposed electric potential on entry/exit (V0, V1). On the other boundaries, zero normal current is applied through the natural homogeneous Neumann condition.

Marker Type Value

V0

Dirichlet

0

V1

Dirichlet

\(V_D\)

Lside, Rside, top*, bottom*

Neumann

0

*: only in 3D

4. Outputs

The main fields of concern are:

  • electric potential \(V\)

  • current density \(\mathbf{j}\)

  • electric field \(\mathbf{E}\)

  • Joule losses density

5. Verification Benchmark

The analytical solution is given by:

\[V(x) = V_0 + \left(\frac{x}{Lx}-1\right) (V_1-V_0)\]

Note that we may get an expression for the resistance \(R\) of the busbar from this equation. We recall that \(R\) is defined as \(R = V_D/I\) where \(I\) stands for the total current flowing in the busbar (\(V_D\) corresponds to the differential applied voltage).

By definition:

\[I = \int_{V0} \mathbf{j} \cdot \mathbf{n} \,d\Gamma\]

From Gauss law we have: \(\mathbf{j} = \sigma\,\mathbf{E} = -\sigma \nabla V\). It follows:

\[R = \frac{1}{\sigma} \frac{Lx}{S}\]

with \(S=Ly*Lz\).

We will check if the approximations converge at the appropriate rate:

  • k+1 for the \(L_2\) norm for \(V\)

  • k for the \(H_1\) norm for \(V\)

  • k for the \(L_2\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)

  • k-1 for the \(H_1\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)