QuarterTurn
In this example, we will compute the magnetic field generated by a stranded conductor. The geometry of the conductor is chosen such that we can derive the analytical expression of the magnetic field.
1. Problem
The conductor \(\Omega\) consists in a rectangular cross-section torus, as in section thermoelectric, considered only in 3D. The geometry also contains an external domain which is an approximation of \(\mathbb{R}^3\setminus\Omega\).
| Name | Description | Value | Unit | |
|---|---|---|---|---|
\(r_i\) |
internal radius |
\(1.10^{-3}\) |
m |
|
\(r_e\) |
external radius |
\(2.10^{-3}\) |
m |
|
\(2*H\) |
height |
\(2.10^{-3}\) |
m |
The current density \(\mathbf{j}\), in \(A/m^2\)can be viewed as an input parameter, or given by the thermoelectric model.
The boundary conditions are given below:
- \(\mathbf{A} = \mathbf{0}\) at the infinity
- \(\mathbf{A} \times \mathbf{n} = 0 \) on symetry plane
2. Results
For simple conductor geometry, analytical expressions of the magnetic field along the Z-Axis may be found in physics textbooks.
The expression of the magnetic field in \(\mathbb{R}^3\) is more difficult to derive but may be found in several papers, like \cite{Jackson99}.
As a classical result, we consider only the magnetic field along the Z-Axis, which analytical expression is given bellow: