#### APPENDIX

###### A. LOSS CALCULATION

The expression for Fr used here may be derived by first calculating loss in a conducting cylinder in a uniform field, with the assumption that the field remains constant inside the conductor, equivalent to the assumption that the diameter is small compared to a skin depth. This results in power dissipation P in a wire of length

where B is the peak flux density. This is equal to the first term of an expansion of the exact Bessel-function solution [19].

Combining this with the assumption of a trapezoidal field distribution results in (2). For configurations in which the field is not zero at one edge of the winding, a factor is used to account for the resulting change in losses, where = Bmin/Bmax [9].

###### B. COMPARISON WITH EXPANSION OF DOWELL SOLUTIONS

Equation (2) is similar to the expression for the first terms of a series expansion of the exact one-dimension solution,

where p is the number of layers and is the ratio of effective conductor thickness to skin depth. For a large number of layers (equivalent to the assumption, above, of a trapezoidal field distribution), this reduces to . The usual expression for is

where beq/bb, heq and beq are the height and breadth of an "equivalent" rectangular conductor, and N is the number of turns per layer. Based on equal cross sectional area, beq=heq= dc. This results in

where hb is the height of the bobbin area allocated to this winding. The number of layers is p=. Substituting these expressions for p and into the simplified version of (17), and using =, we obtain

the same as (2), except for the substitution of bb for bc, and the addition of a factor of . This discrepancy can be explained by comparing (16) to the equivalent expression for a rectangular conductor

where s is the side of a square conductor. Equating these two, we obtain . Thus, it appears that using an equivalent square conductor with sides equal to would be a more accurate approximation than the equal area approximation that is usually used [11].

###### C. NON-SINUSOIDAL CURRENT WAVEFORMS

Non-sinusoidal current waveforms can be treated by Fourier analysis. The current waveform is decomposed into Fourier components, the loss for each component is calculated, and the loss components are summed to get the total loss:

where Ij is the rms amplitude of the Fourier component at frequency j. From (2) it can be seen that

Defining Fr-tot by P=I2tot-rms Fr-tot Rdc leads to

This can also be written as

where

One may calculate this effective frequency for a non-sinusoidal current waveform and use it for analysis of litz-wire losses, or for other eddy-current loss calculations. Note that this applies to waveforms with dc plus sinusoidal or non-sinusoidal ac components. The results will be accurate as long as the skin depth for the highest important frequency is not small compared to the strand diameter.

A triangular current waveform with zero dc component results in an effective frequency of 1.1031, where 1 is the fundamental frequency. Once the effective frequency of a pure ac waveform has been calculated, the effective frequency with a dc component can be calculated by a re-application of (26):

Finding Fourier coefficients and then summing the infinite series in (26) can be tedious. A shortcut, suggested but not fleshed out in [4], can be derived by noting that

so that

The primary limitation of effective-frequency analysis is that it does not work for waveforms with more substantial harmonic content. For instance, the series in (26) does not converge for a square wave. Similarly, the derivative of a square wave in (29) results in an infinite rms value. A Bessel-function-based description of loss may be necessary. However, in practice leakage inductance prevents an inductive component from having perfectly square current waveforms. A square wave with finite-slope edges leads to a finite value of eff. If the skin depth for this effective frequency is not small compared to the strand diameter, the simple analysis of loss in (2) will still give accurate results, and the analysis of litz-wire stranding given here is still accurate.

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