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BusBar
In this example, we will model a busbar simply modeled as rectangular parallelepiped.A differential electrical potential is applied to the entry/exit of the busbar. We note respectively \(V_0\) the electrical potential on the entry and and \(V_1\) on the 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/busbar}" --case.config-file 2d.cfgThe command line to run this case in 3D is
mpirun -np 4 feelpp_toolbox_thermoelectric --case "github:{path:toolboxes/electric/busbar}" --case.config-file 3d.cfg2. Geometry
The busbar conductor consists in a rectangular cross section extruded along the x axis.+
In 2D, the geometry is as follow:
In 3D, this is the same geometry, but extruded along the z axis.
3. Input parameters
| Name | Description | Value | Unit | |
|---|---|---|---|---|
\(Lx\) |
internal radius |
1 |
\(m\) |
|
\(Ly\) |
external radius |
2 |
\(m\) |
|
\(Lz\) |
angle |
\(\pi/2\) |
\(rad\) |
|
\(V_D\) |
electrical potential |
9 |
\(V\) |
3.1. Model & Toolbox
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This problem is fully described by the Electric model, namely a poisson equation for the electrical potential \(V\) with Dirichlet boundary conditions on entry /exit.
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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 for the electrical probleme are introduced as simple Dirichlet boundary conditions for the electric potential on the entry/exit of the conductor. For the remaining faces, as no current is flowing througth these faces, we add Homogeneous Neumann conditions.
| 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 the electric potential \(V\), the current density \(\mathbf{j}\) and the electric field \(\mathbf{E}\). // presented in the following figure.
5. Verification Benchmark
The analytical solution is given by:
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:
From Gauss law we have: \(\mathbf{j} = \sigma\,\mathbf{E} = -\sigma \nabla V\). It follows:
with \(S=Ly*Lz\).
We will check if the approximations converge at the appropriate rate:
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k+1 for the \(L_2\) norm for \(V\)
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k for the \(H_1\) norm for \(V\)
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k for the \(L_2\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)
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k-1 for the \(H_1\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)
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