Magnetic Inductance Report

The MagneticInductanceReport computes the magnetic inductance matrix of a single or multi-coil system. This matrix characterizes how magnetic fields generated by currents in coils induce voltages in the same or in other coils.

Inductance describes the ability of a coil to store magnetic energy when an electric current flows through it. It also quantifies how a time-varying current in one coil induces a voltage in another via electromagnetic coupling.

When an electric current flows through a coil, it generates a magnetic field. If this current changes over time, the resulting variation in the magnetic field induces a voltage — an effect described by Faraday’s law of induction.

In a system with multiple coils and no eddy currents, the voltage \(V_i\) induced in coil \(i\) due to time-varying currents \(I_j\) in all coils is given by:

\[V_i = \sum_j L_{ij} \frac{\mathrm{d}I_j}{\mathrm{d}t} \quad.\]

Here: \(V_i\) is the total induced voltage in coil \(i\), \(I_j\) is the current in coil \(j\), \(L_{ij}\) is the inductance between coil \(i\) and coil \(j\), and \(\frac{\mathrm{d}I_j}{\mathrm{d}t}\) is the time derivative of the current in coil \(j\).

The inductance matrix \(L_{ij}\) captures both

Self-inductance \(L_{ii}\): The voltage induced in a coil due to a change in its own current. It quantifies the coil’s ability to store magnetic energy and resist changes in current.
Mutual inductance \(L_{ij}, \quad i \ne j\): The voltage induced in one coil due to a changing current in a different coil. It measures the magnetic coupling between coils and is central to electromagnetic interactions like transformer action.

Magnetic Energy

The magnetic energy stored in the system is expressed by the inductance matrix:

\[W = \frac{1}{2} \sum_{i,j} L_{ij} I_i I_j\]

Usage Example

from mufem.electromagnetics.coil import MagneticInductanceReport

help(MagneticInductanceReport)

Help on class MagneticInductanceReport in module mufem.electromagnetics.coil:

class MagneticInductanceReport(mufem.ReportTrait)
 |  Method resolution order:
 |      MagneticInductanceReport
 |      mufem.ReportTrait
 |      pybind11_builtins.pybind11_object
 |      builtins.object
 |
 |  Methods defined here:
 |
 |  __init__(...)
 |      __init__(self: mufem.electromagnetics.coil.MagneticInductanceReport, name: str, marker: mufem.Marker) -> None
 |
 |      Constructor with name and marker.
 |
 |  evaluate(...)
 |      evaluate(self: mufem.electromagnetics.coil.MagneticInductanceReport) -> Value
 |
 |      Calculates and returns the magnetic inductance of a coil.
 |
 |  get_name(...)
 |      get_name(self: mufem.electromagnetics.coil.MagneticInductanceReport) -> str
 |
 |      Returns the name of the report.
 |
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |
 |  __dict__
 |
 |  ----------------------------------------------------------------------
 |  Static methods inherited from pybind11_builtins.pybind11_object:
 |
 |  __new__(*args, **kwargs) class method of pybind11_builtins.pybind11_object
 |      Create and return a new object.  See help(type) for accurate signature.