IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 6, NOVEMBER zyxwvut 1985 zyxwvutsr NUMERICAL MODELING OF RESIDUAL MAGNETIC PHENOMENA* zyxwv S. S. Udpa', Y. Lord' and Sun Yu-shi- 2165 Abstract zyxwvutsrqpo - This paper describes a two step method for numerically modeling residual magnetostatic field phenomena which closely mimics the first and second quadrant behavior of the host ferromagnetic material. The method is illustrated by predicting residual leakage fields around a rectangular slot in a steel bar. INTRODUCTION Residual magnetic phenomena play an important role in the operation of many magnetoelectric devices, including magnetic tape equipment and electrical machinery excited with permanent magnets. In nondestructive testing applications, steel parts to be tested are initially magnetized with a dc current of appropriate magnitude which, when removed, leaves residual leakage fields around defects present at the surface of the specimen. Such leakage fields can be detected using any flux sensitive device such as a Hall plate, moving coil or magnetic tape, although in industrial applications of the technique, a fine magnetic powder often mixed withfluorescent material is used as the defect detection mechanism. A major topic for research relates to the subject of, defect characterization involving the deduction of defect characteristics on the basis of leakage field measurements. A theoretical framework to predict the leakage field for a given defect becomes necessary in as much as it provides a reference for comparison with signals obtained from actual measurements i n the field in lieu of a defect prototype which is oftewdifficult and expensivetoproducein a laboratory. Existing models can be characterized as belonging to one of two categories. Analytical models are largely based on the assumption that residual leakage fields around surface cracks can be represented by equivalent magnetic dipoles. Zatsepin and Shcherbinin Ill postulate uniform surface magnetic charges over the sides of rectangular slots to obtain expressions for the horizontal and vertical components of the leakage field arising as a result of crack, hairline or lap type defects. Despite the simplicity .of the model and its ability to yield leakage field profiles similar to experimentally observed signals, the approach suffers from several disadvantages. In addition to the fact that the method is qualitative, as the value of the distributed "magnetic chargeft is not known, the inherent assumptions of the model are phenomenologically incorrect. The model not only ignores saturation effects and fringing but also treats the ferromagnetic material as nonferromagnetic with unit relative permeability. In addition the model treats active and residual leakage fields identically despite the fact that the underlying physical processes are different [2,31. - *This work has been supported in part by the Army Research Office +The authors are with Colorado State University, Fort Collins, CO 80523 . -The author is with the Nanjing Aeronautical In- stitute, Nanjing. P.R.C. Among the earliest tousenumericaltechniques to model permanent magnets using finite difference techniques were Reichert 141 and Harrold 151. Binns et a1 and others [6-101 developed a finite element formulationinvolving the use of the equation B = F'H f Mo where = Po(l+X) represents the apparent permeability zy x is the incremental susceptibility, and Mo = B,. the remanent flux density Using this constitutive relationship, Binns et zy dl and Kamminga 16.91 solve the governing Poissons' equations using finite element techniques. The weakness of the method lies in the assumption of the BfH curve being single valued thereby ignoring hysteresis and the need for specifying an equivalent current density to account for the "residual field'' . Campbell et a1 1111 have developed finite element models using a magnetic scalar potentialfunction. This paper describes a finite element formulation for a model which follows the underlying physical process closely [31. The model relies on a two step procedure splitting the specimenmagnetizationprocess into two phases based on the material behavior i n the first two quadrants of the corresponding B/H characteristics. The application described in this paper involves the prediction of a residual leakage field profile around a slot in a steel bar. FINITE EL3MENT MODEL Consider a steel bar containing a rectangular slot, as shown in Figure 1; the bar is initially magnetized by passing a dc current. The governing equation relating to the magnetostatic case can be derived from Maxwell's equations as: V X zyxwvutsr (-0 X A) = -J 1 u (2) where the vector magnetic potential A is given by B = Curl A, Figure 1: Cross-sectional view of a ferromagnetic bar with slot 0018-9464/85/1100-2165$01.~@1985 IEEE