This paper presents an accurate analytical explicit expression for the self-inductance of a flat pancake round coil made up of concentric turns. The expression is obtained by converting the semi-infinite integral representation for the mutual inductance between two arbitrary turns of the coil into a finite integral, and then by expanding the integrand into a series of Legendre polynomials. As a result, a sum of simpler integrals is obtained, whose analytical evaluation is straightforward. The self inductance is finally expressed as the sum of logarithmic functions, describing the contributions from the self-inductances of the single turns, plus the mutual-inductance terms originating from all the possible pairs of turns of the coil, each one given by a power series of the ratio between the radii of the turns. Numerical simulations are performed to illustrate the advantages of the proposed solution.
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