# Skin depth calculation for non-sinusoidal currents

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). 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?

• May I suggest using the frequency domain to calculate this? Apr 6 '20 at 21:45

I would have thought that the complete waveform that makes up the PWM synthesis of a triangle wave would have needed to be broken down to fourier constituents. That seems to be what is proposed in the answer by Curd in the previous question.

The paper by C. R. Sullivan seems to present an approximation for a specific purpose. You may need to analyze that paper in more detail and perhaps look at the supporting literature to determine if it is suitable for your purpose. I believe that a Fourier analysis of the actual waveform is the only way to get an exact skin depth. That says that the PWM frequency does influence the skin depth. However a method based on Sullivan may provide sufficient accuracy.

• Thanks for the answer. It is indeed a different approach and simplification, but based on the Fourier series (as far as I understand it). According to the author, this simplification would not lead to accurate results for a "square current", for example. In a DC-DC Wanlder such a current does not exist. The reason why I avoid Fourier is, as the author also says: Finding Fourier coefficients and then summing the infinite series can be tedious. I was hoping to find an easier way. xP
– Noah
Apr 6 '20 at 16:00

The edge rate (the rise time) is crucial.

The E&M classic by Jackson has predictions on this.

In the lab, I measured front--back coupling through standard PCB foil (35 microns, 1.4 mils) as an erfc(time) response with 160 nanoseconds propagation

"erfc" is the response through a long lossy delay line, or a large cascade of RC lowpass filters

and that number --- 160 nanoseconds --- does agree with Jackson.

Notice this measurement is THROUGH a sheet of copper, not along a wire.

I used drive coil and sense coil

The test waveform was 100KHz, the rise/fall times about 10 nanoseconds.

The effective frequency for DC-DC converters is:

$$f_{eff} = \frac{f_{s}}{2\pi(D-D^2)}$$

the above approach is not wrong, this equation is a simplification of it. See also: https://docs.rs-online.com/376e/0900766b815cb9f5.pdf

Therefore, the skin depth is:

$$\delta = \sqrt\frac{2(D-D^2)\rho_{w}}{\mu_{0}\mu_{R}f_{s}}$$