Frequency-Dependent Inductance and Winding Loss Model for Gapped Foil Inductors

Abstract

For comprehensively optimizing high-frequency foil inductors, calculating the increased inductance and the additional eddy current losses due to a two-dimensional magnetic fringing field caused by air gaps in the centre leg is mandatory. This paper proposes an analytical model that accurately calculates the inductance and the additional eddy current losses in gapped inductors with foil windings. The analytical field expressions are combined into closed-form winding loss and inductance formulas and verified by measurements. Furthermore, the frequency dependency of the inductance due to the shielding effect of the foil conductors is being discussed.

Analytical Winding Loss and Inductance Models for Gapped Inductors With Litz or Solid Wires

“…which means, that the imaginary part of the inductance resembles a core loss resistor R c = ωL ′′ depending on the frequency and the imaginary part of the relative permeability. Z 1 refers to (18) computed with ( 13) and ( 15), Z 2 is computed with the compact models (3) and ( 7), in addition the partial resistances R f (38) and R 1D (37), and Rc (19), which is the core resistor. b) R /R DC of IND 2: Z 1 is computed with ( 13) and ( 17), the compact models (3) and ( 11) are used to compute Z 2 , and the partial resistances are calculated using (41), (40), and (19).…”

“…In contrast to mentioned literatur, this paper proposes formulas for the actual losses by considering the complete inductor geometry, instead of the losses per unit length. The presented formulas are not applicable to foil conductors, because foil conductors act as eddy current shields and must be modelled accordingly [19].…”

Frequency-Dependent Equivalent Circuit Parameter Calculation of Gapped Multiwinding Inductors

“…In addition, ( 16) is given to satisfy (13) in the Elayers, where the magnetic field is assumed to be independent of the frequency. The analytical magnetic field expressions considering the air gap fringing field are obtained from (18).…”

“…*Eddy current losses in open circuited windings: Existing models in literature [18,19] are not capable of correctly taking the additional air gap field induced eddy current losses in open-circuited windings into account. As shown in Fig.…”

“…I. * Self resistance of the individual winding calculated with methods from literature[18,19] considering an air gap but not the impact of any additional open-circuited winding layers in the core window. ** Self inductance of each winding computed with the manufacturer's A L value.…”