The following is a paper by H. Aspden published in the Journal of Applied Physics, vol. 23, pp. 523-528 (1951).
EDDY-CURRENTS IN SOLID CYLINDRICAL CORES HAVING NON-UNIFORM PERMEABILITY
Abstract: A method of estimating magnetization losses in solid cylindrical cores of ferromagnetic material is given. The conception of complex permeability is introduced to show the effects which hysteresis has upon the degree of flux penetration. A formula is given for the effective permeability which should be used in the classical treatment in which non-linearities have been ignored. An arbitrary case in which the permeability changes across the core section is considered and the solution is extended to a core which is homogeneous, but whose permeability changes with the degree of magnetization. In this way errors due to a non-linear B-H curve are reduced and the complex permeability notation is shown to lead to a more reasonable Steinmetz relationship. The eddy-current anomaly is mentioned and it is noted that the theory developed is suited to a consideration of the anomaly as explained in terms of time-lag effects.
Commentary: This paper was written during a period when the author was working for a Ph.D. degree which involved experimental research on the eddy-current anomaly. See the related paper: [1956a]. However, the detailed analysis based on complex magnetic permeability as covered by the mathematics in this paper in relation to a non-linear B-H magnetization curve was not included nor involved in the research thesis which was submitted in support of that Ph.D. degree.