## Abstract

Purpose - The purpose of this paper is twofold. First, it aims to study the proximity-effect power loss in the foil, strip (rectangular), square, and solid-round wire inductor windings. Second, it aims to optimize the thickness of the foil, strip, square wire windings, and the diameter of the solid-round-wire, the minimum of winding AC resistance and the minimum of winding AC power loss for sinusoidal inductor current. Design/methodology/ approach - The methodology of the analysis is as follows. First, the winding resistance of the single-layer foil winding with a single turn per layer and uniform magnetic flux density B is derived. Second, the single-layer foil winding with uniform magnetic flux density B is converted for the case, where the magnetic flux density B is a function of x. Third, the single-layer winding is replaced by the winding with multiple layers isolated from each other. Fourth, transformation of the multi-layer foil winding into different conductor shapes is performed. For the solid-round-wire windings, the results of the derivation are compared to Dowell's equation and verified by measurements. Findings - Closed-form analytical equations for the optimum normalized winding size (thickness or diameter) at the global or local minimum of winding AC resistance are derived. It has been shown that the AC-to-DC winding resistance ratio is equal to 4/3 (F_{Rv}=4/3) at the optimum normalized thickness of foil and strip wire winding h_{opt}/δ_{w}. The AC-to-DC winding resistance ratio is equal to 2 (F_{Rv}=2) at the local minimum of the square wire and solid-round-wire winding AC resistances. Moreover, it has been shown that for the solid-round wire winding, the proximity-effect AC-to-DC winding resistance ratio is equal to Dowell's AC-to-DC winding resistance ratio at low and medium frequencies. The accuracy of equation for the winding AC resistance of the solid-round wire winding inductors has been experimentally verified. The predicted results were in good agreement with the measured results. Research limitations/implications - It is assumed that the applied current density in the winding conductor is approximately constant and the magnetic flux density B is parallel to the winding conductor (b>>h). This implies that a low- and medium-frequency 1-D solution is considered and allows the winding size optimization. This is because the optimum normalized winding conductor size occurs in the low- and medium-frequency range. The skin-effect winding power loss is much lower than the proximity-effect winding power loss and therefore, it is neglected. Originality/value - This paper presents derivations of closed-form analytical equations for the optimum size (thickness or diameter) that yields the global minimum or the local minimum of proximity-effect loss. A significant advantage of these derivations is their simplicity. Moreover, the paper derives equations for the AC-to-DC winding resistance ratio for the different shape wire windings, i.e. foil, strip, square and solid-round, respectively.

Original language | English |
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Pages (from-to) | 1793-1814 |

Number of pages | 22 |

Journal | COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering |

Volume | 31 |

Issue number | 6 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

## Keywords

- Eddy currents
- Inductors
- Optimization
- Optimization techniques
- Proximity effect
- Winding loss