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 .
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 .
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
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 .
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
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):