I have some questions regarding the skin depth of wires.
In this forum it was already asked how the skin depth can be determined for a square voltage (e.g. inductor voltage of buck-converters) (link: [square wave skin depth](https://electronics.stackexchange.com/questions/359979/square-wave-skin-depth)). Unfortunately, the question could not be answered. I have an approach and would like to know if it is correct in your opinion. 

In general, the skin depth is defined with

$$\delta = \sqrt\frac{\rho\_{w}}{\pi\mu\_{0}\mu\_{R}f}$$

with \$\rho\_{w}\$ as the resistivity of the conductor (\$1.72 \cdot 10^{-8}\$ \$\Omega m\$ for copper at room temperature). \$f\$ is the frequency of a sinusoidal current in the winding in \$Hz\$, \$\mu\_{0}\$
is the permeability of free space (\$4\pi\cdot10^{-7}\$ \$\frac{H}{m}\$) and \$\mu\_{R}\$ is the relative magnetic permeability of the conductor ( for copper it is approx. 1). Now, the only problem in this equation for an inductor of a buck converter is the frequency \$f\$. For non-sinusoidal currents and combinations of dc and ac current, this equation can be used if the frequency is replaced by the effective frequency (see <http://power.thayer.dartmouth.edu/papers/pesc1997a.pdf> Appendix C). The effective frequency of a triangular current waveform can be calculated with
$$f\_{eff} = \frac{rms(\frac{di\_{L}(t)}{dt})}{2\pi\cdot I\_{L,RMS}}$$
with \$i\_{L}(t)\$ as the inductor current in \$A\$ and \$I\_{L,RMS}\$ as the RMS inductor current in \$A\$.
Fortunately, the inductor current is
$$di\_{L} = \frac{V\_{L}(t)}{L} \cdot dt$$ 
with \$U\_{L}\$ as the inductor voltage in \$V\$ and \$L\$ as conductivity in \$H\$. With this we get
$$f\_{eff} = \frac{rms(V\_{L}(t))}{2\pi\cdot I\_{L,RMS}\cdot L}$$
If we take the buck converter as an example, we get
$$f\_{eff} = \frac{(V\_{i}-V\_{o})\sqrt{D}+V\_o\sqrt{(1-D)}}{2\pi\cdot I\_{L,RMS}\cdot L}$$
It can be seen that the effective frequency does not depend on the switching frequency and that the maximum results at a duty cycle of 0.5 and maximum output voltage. The minimum effective frequency results at a Example: \$V\_{o} = 250\$ \$V\$, \$V\_{i} = 500\$ \$V\$, \$L=200\$ \$\mu H\$, \$I\_{L,RMS}=113\$ \$A\$, \$D = 0.5\$. For the effective frequency we get
$$f\_{eff} = \frac{(500 - 250)\sqrt{0.5}+ 250 \sqrt{0.5}}{2\pi\cdot 113\cdot 0.0002} = 2490Hz$$
Finnaly, the minimum skin depth is
$$\delta = \sqrt\frac{1.72 \cdot 10^{-8}}{\pi\cdot4\pi\cdot10^{-7}\cdot 2490}=1.33mm$$
My questions: Is my approach to calculate skin depth for the indcutor wire in DC-DC converters correct? If it is: Why is the skin depth not depending on my switching frequency?