The Analytical Formula for Calculating the Self-Inductance for the Circular Coil of the Rectangular Cross-Section With a Non-Uniform Current Density

Abstract

In this article we give an analytical formula for calculating the self-inductance for circular coils of rectangular cross-section which has a nonuniform current density. Recently, the formula for calculating this important electromagnetic quantity was published in the form of the single integral whose kernel function was a sum of elementary functions. However, a new formula is obtained in the form of elementary functions, single integrals, and the complete elliptic integral of the first, second, and third kind. Although its development looks tedious, we obtain a rather user-friendly expression for the calculation. From the general case, the self-inductance of the thin disk coil (pancake coil) with the nonuniform current is obtained in a remarkably simple form. The results of this work are compared with different known methods, and all results are in the excellent agreement. Our approach has not been found in the literature.

Fast Analytic–Numerical Algorithms for Calculating Mutual and Self-Inductances of Air Coils

“…The articles by Babic and his various co-authors [14][15][16][17][18][19][20][21] also give solutions to many basic problems of calculating the mutual and self-inductances of different coils in any mutual position.…”

“…The method presented in the previous section was used to calculate mutual and selfinductances for rectangular coils in any of their mutual positions. Formulas (21) and (22) were used to calculate the magnetic flux linking the secondary coil.…”

“…The described method also gives very accurate results when calculating mutual and self-inductances, as well as field distributions for spiral coils, helical coils, very thin coils [31,34], and coils with nonuniform current density distributions [21]. This requires the proper selection of discretization parameters of the coil areas and the magnetic flux integration areas, as well as the selection of equivalent radii when modeling spiral or Archimedean coils [35,36].…”

“…Using Formulas (17), (18), (21), and (22), it is also possible to determine the distribution of a magnetic field generated by systems of any number of cylindrical or rectangular coils arranged arbitrarily in space. Figure 13 shows, for example, the distribution of the magnetic field (the normal component of the magnetic induction vector) over circular and rectangular coils depending on the thickness of the electrically conductive area.…”

Fast Analytic‐Numerical Algorithms for Calculating Self and Mutual Inductances of Air Coils

“…Methods for calculating the self and mutual inductances of different air coils have been presented in many articles [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”