In this article we give an analytical formula for calculating the self-inductance for circular coils of rectangular cross-section which has a non-uniform current density. Recently, the formula for calculating this important electromagnetic quantity was published in the form of the single integral whose kernel function was asum 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.
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