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.configfile 2d.cfg
The command line to run this case in 3D is
mpirun np 4 feelpp_toolbox_thermoelectric case "github:{path:toolboxes/electric/busbar}" case.configfile 3d.cfg
2. 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

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.

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:

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 \(\mathbf{j}\)

k1 for the \(H_1\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)