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In this paper, an analytical optimization of solid-round-wire windings conducting sinusoidal current is performed. New closed-form analytical equations are derived for the normalized solid-round-wire diameter to achieve minimum ac winding losses for sinusoidal current. Approximations of Dowell's equation are used to derive expressions for normalized diameter at the local minimum of winding ac resistance and normalized critical diameter. An equation for the winding ac resistance at the local minimum is derived. Additionally, equations for the normalized diameter at the local maximum of ac resistance and the normalized boundary diameter between low- and medium-frequency ranges are given. The formulas enable inductor and transformer designers to minimize winding loss without utilizing finite-element method analysis. Experimental verification of the presented theory is given.