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In this study, an analytical optimisation of the foil, strip, square and solid-round-wire winding inductors conducting sinusoidal current is performed. The Ampère law is used to derive analytical equations for the AC-to-DC winding resistance ratio of different shape inductor windings valid at low and medium frequencies. These equations are used to perform optimisation of windings to obtain the global minimum of the winding AC resistance of the foil and strip wire windings and the local minimum of the winding AC resistance for the square and solid-round-wire windings. Derivations of AC-to-DC winding resistance ratio and winding AC resistance based on Ampère's law for the solid-round-wire windings are compared to Dowell's equation. Results of the predicted winding AC resistance based on Ampère's law for the solid-round-wire windings are validated by experimental results.

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Foil Winding Resistance and Power Loss in Individual Layers of InductorsThis paper presents an estimation of high-frequency winding resistance and power loss in individual inductor layers made of foil, taking into account the skin and proximity effects. Approximated equations for power loss in each layer are given and the optimal values of foil thickness for each layer are derived. It is shown that the winding resistance of individual layers significantly increases with the operating frequency and the layer number, counting from the center of an inductor. The winding resistance of each foil layer exhibits a minimum value at an optimal layer thickness. The total winding resistance increases with the total number of layers.

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In this study, an analytical optimisation of solid-round-wire windings conducting both the dc and ac non-sinusoidal periodic currents is performed. A closed-form analytical equation is derived for the normalised solid-round-wire diameter to achieve the minimum power loss for inductors conducting ac non-sinusoidal periodic currents superimposed on the dc component. The low-and medium-frequency normalised winding ac resistance for the nth harmonic frequency is derived and used to obtain the normalised total-power-valley diameter at the local minimum of the winding dc and ac power losses. Additionally, an equation for the local minimum of the winding dc and ac power losses is derived. A high-frequency approximation of Dowell's equation at the nth harmonic frequency is used to derive the normalised total-power-critical wire diameter at which the total winding power loss (dc and ac) is equal to the total winding power loss at the local minimum. A design procedure of the inductor with an optimised winding diameter operating in pulsewidth-modulated dc-dc buck converter in discontinuous conduction mode is presented. Experimental verification of the presented theory and comparison of the total winding power loss of the inductors with different solid-round-wire gauge are performed.

This study presents analytical winding power loss minimisation of foil inductors conducting ac harmonic currents with and without dc offset. The approximated equation for the ac winding power loss is derived and used to determine the optimum foil thickness of inductors operating with multi-harmonic ac currents. An approximated equation for the total winding power loss is also derived and used to determine the optimum foil thickness of inductors operating with multi-harmonic ac currents superimposed on the dc current. The design procedure for inductors with minimum total winding power loss is presented for a pulsewidth-modulated dc-dc boost converter operating in discontinuous conduction mode. The theoretical predictions have been verified by measurements of inductors with four different foil thicknesses.