A General Relation Between Real and Imaginary Parts of the Magnetic Susceptibility

TL;DR

This paper derives a general theoretical relation between the real and imaginary parts of magnetic susceptibility in the Laplace domain, extending the Kramers-Kronig relations to broader conditions applicable to magnetic materials.

Contribution

It introduces a new general relation between real and imaginary parts of magnetic susceptibility in the Laplace domain, applicable to any causal, linear, time-invariant magnetic material.

Findings

Derived a general relation using complex analysis techniques.

Validated the relation as a generalization of Kramers-Kronig relations.

Applicable to various magnetic susceptibility functions in the Laplace domain.

Abstract

This paper is devoted to the study and the obtaining of the general relation between the real part and the imaginary part of the magnetic susceptibility function in the Laplace domain. This new theoretical technique is general, and can be applied to any magnetic material, that can be considered like causal and Linear time invariant (LTI). A discussion of the causality which is extensively used in Physics has been done. To obtain the relations, some important concepts like Titchmarsh's theorem and Cauchy's Theorem have been reviewed, which results in the integral of a analytic function, that is formed with the magnetic susceptibility used in the Laplace domain. The Cauchy Integral expression in the Laplace domain under certain conditions leads to a general relations between real and imaginary part of the magnetic susceptibility in the complex \textit{s}-plane. These new relationships allow the validation of the magnetic susceptibilility functions developed by different researchers, in the Laplace domain, not just the frequency response like the well known Kramers-Kronig relations. Under certain conditions in these new general relations, the well known K-K relations can be obtained as a particular case.