Maxwell's equations for electrical systems
Introduction
Maxwell's equations are the fundamental laws of electromagnetism.
They relate the density of the electric charges q and the density of the electric
current in a domain to the fields which result from them:
- the electric field strength
and the electric flux density
- the magnetic flux density
and the magnetic field strength
General form of the equations
The general form of Maxwell's equations in domains without moving bodies is the following:
Maxwell-Gauss : (1)
Maxwell-Faraday : (2)
Conservation of magnetic flux density : (3)
Maxwell-Ampere: (4)
Other equations
With these equations, the following constitutive laws of materials are added:
Characteristics of the conducting media : (5)
Characteristics of the magnetic media : (6)
Characteristics of the dielectric media : (7)
where :
- σ is the conductivity of the material (in S)
- μ is the permeability (in H/m)
- ε is the permittivity (in F/m)
Separation
In the case of low frequency AC fields , the equations in the electric fields and
and the equations of the magnetic fields
and
can be decoupled.
Thus, there are Maxwell's equations for the electrical systems and Maxwell's equations for the magnetic systems, respectively:
- a set of equations for the electric fields, and
- another set of equations for the magnetic fields.
This separation of the electrical and magnetic systems depends on: materials, work frequencies, size of the study domain. It is usually possible for technical devices working in the range of frequency f < 1 to 10 GHz.
Form of equations for an electrical system
For an electrical system, we suppose that the magnetic field does not modify the current
distribution in the conductors. This assumption remains true as long as
the frequency does not exceed a certain limit. This results in the null value of the
term
in equation (2).
Thus, the equation can be written in the following way:
Maxwell-Gauss : (1)
Maxwell-Faraday : (2)
Maxwell-Ampere : (4)
Characteristic of the conducting media : (5)
Characteristic of the dielectric media : (7)