Estimated reading time: 3 minutes

The Skinny on Skin Effect, Part 3: Crossover Frequency

In Part 1 and Part 2 of this series (located here and here) we learned that:

1. At higher frequencies, current tends to flow at the outer surfaces of the conductor rather than uniformly throughout the conductor. The principle behind this effect is inductance.
2. When skin effect comes into play, the current density is highest at the surface of the conductor and declines exponentially towards the center of the conductor.
3. There is an imaginary depth under the surface of the conductor called the skin depth (sd).
4. We model currents, when the skin effect comes into play, as being uniform down to the skin depth, then not flowing at all below the skin depth. We know this is not exactly true, because of point 2, but this model works with a high degree of accuracy.

The formula for the skin depth was presented in Part 2 of the series and is repeated here:

where:

ρ = resistivity (0.6787 uOhm-in)ω = angular frequency = 2*π*fμ = Absolute magnetic permeability of the conductor (3.192*10-8 weber/amp-in)f = frequency in Hz.

It is instructive to look more closely at this formula. First, note that it does not depend on the dimensions of the conductor, or even the conductor’s shape! Skin depth is purely a function of frequency. That leads to some conclusions that are not particularly intuitive. Here are two of them:

1. Suppose the skin depth is calculated to be deeper than the radius of the wire or half the thickness of the trace. In that case, the calculation is meaningless. It simply means that the current is flowing uniformly throughout the entire cross-sectional area. Skin depth only has meaning if it is less than the radius of a wire of half the thickness of a trace.
2. Skin effect comes into play at lower frequencies for thicker traces than it does for thinner traces. Half the thickness of a thick trace is a larger number (therefore lower frequency) than it is for a thinner trace. For example, the skin depth at 35 MHz is about 0.44 mils. This means that the skin depth is affecting current for a 1.0 oz. trace (whose half-thickness is 0.65 mils), but not for a 0.5 oz. trace (whose half-thickness is 0.325 mils.)

A graph of skin depth as a function of frequency is shown in Figure 1.

Figure 1. Skin depth as a function of frequency.

It is instructive to look at a concept called the crossover frequency. That is the frequency where the skin effect just comes into play. Consider the trace shown in Figure 2 (See Note 1). Let d be the skin depth at some given frequency. The crossover frequency is where d just equals Th/2.

Figure 2. Rectangular trace.

[Eq. 2]    d = 2.6/f 0.5 = Th/2   

[Eq 3]     f = 2.62 *4 /Th 2  =  27/Th2

Where:

Th is in inches and f is in Hz, ORTh is in mils and f is in Mhz

We can construct a graph that shows the skin effect region as a function of frequency and trace thickness. I have done that in Figure 3, with trace thickness shown in mils and in ounces. The skin effect region, the area where the skin effect is coming into play, is the region to the upper right of each curve.

Figure 3. The skin effect region is the area above the curve.

UltraCAD’s PCB Trace Calculator (Figure 4, see Note 2) is one of the available tools that do skin effect calculations.

Figure 4. UltraCAD’s PCB Trace Calculator.

Notes

1.  The crossover frequency analysis for a circular conductor is analogous. The result is:

f = 27/D2  where f is the crossover frequency and D is the diameter of the conductor.

2.   The calculator is available from www.ultracad.com.

Douglas Brooks has an MS/EE from Stanford University and a Ph.D. from the University of Washington. He has spent most of his career in the electronics industry in positions of engineering, marketing, general management, and as CEO of several companies. He has owned UltraCAD Design Inc. since 1992. He is the author of numerous articles in several disciplines, and has written articles and given seminars all over the world on signal integrity issues since founding UltraCAD. His book, Printed Circuit Board Design and Signal Integrity Issues was published by Prentice Hall in 2003. Visit his website at www.ultracad.com.