What Is the Role of the Second Term in Magnetic Energy Functional Analysis?

[Hassan2]

Dear all,

I'm reading a paper on finite element magnetic field analysis. Basically there are two approaches to this. One is to use Maxwell equation and the other is to define an energy functional, discretize the problem and minimize the functional with respect to the unknowns.

The paper takes the second approach. The functional is given as:

$E=\int_{V}(\int_{0}^{B}H.dB)dv-\int_{V}J.Adv$

J is the source current density and A is the magnetic vector potential ( $B=\nabla \times A$)

The first term is the stored energy. My question is about the second term. It seems to be the magnetic energy given to the system. I need your help to interpret and understand the second term because in electromagnetic textbooks, the second term (divided by 2, and without the negative sign) is proved to be equal to the first term in the linear case. I'm totally confused with this energy functional.

Your help is appreciated.

Hassan

 

[eljose79]

Dear Hassan,

Thank you for sharing your thoughts and questions about the paper on finite element magnetic field analysis. The two approaches mentioned, using Maxwell's equations or defining an energy functional, are both commonly used in this field and can yield accurate results when applied correctly.

In the energy functional approach, the functional represents the total energy of the system. The first term, as you mentioned, is the stored energy which represents the energy stored in the magnetic field. This is equivalent to the first term in the linear case that is typically discussed in electromagnetic textbooks.

The second term, however, is the energy dissipated due to the presence of the source current density. This can also be seen as the work done by the source current in creating the magnetic field. This term is often neglected in linear cases because it is assumed that the source current is small enough to have a negligible effect on the overall energy of the system. However, in non-linear cases, this term must be taken into account as it can significantly affect the final results.

I hope this helps clarify the role of the second term in the energy functional. If you have any further questions or need more clarification, please do not hesitate to ask.