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Brooks' Bits: The Skinny on Skin Effect, Part 2

In Part 1 of this series (located here), I described how skin effect is all about current density. The units of current density are current/area2, for example amps/cm2. If we multiply current density by cross-sectional area, we get current.

So here are two propositions. They are well accepted in electronics, and are useful models, even if they are not exactly correct, as we shall see. They are also mutually exclusive!

At high enough frequencies, so that the skin effect is in play:

Proposition 1There is a depth below the surface called the skin depth (sd). It is given by the formula:

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.

The skin depth defines a cylindrical cross-section between the surface of the (circular) conductor and the skin depth. This cylindrical cross-section has a higher resistance (per unit length) than does the full conductor, because the cross-sectional area of the cylinder is smaller. The current flows uniformly through this cylindrical cross-section. By calculating the area of this cylindrical cross-section, we can calculate the (increased) resistance of the conductor caused by the skin effect.

Proposition 2The current density is highest at the surface and declines exponentially according to the (normalized) formula (I’ll explain what I mean by “normalized” in a moment):

where 

J = current density            sd = skin depthd = distance in from the surfacee = base of the natural logarithm (2.718)

These relationships are graphed in Figure 1. The graphs in Figure 1 are normalized. That means we have unit current density on the vertical axis and unit area on the horizontal axis. If we want to apply these (normalized) graphs to a real-world situation with, say, 20 ma of current through a conductor with an 8 mm2 area, we would multiply the vertical axis by 2.5 ma/mm2 and the horizontal axis by 8 mm2. Equation 2 is normalized in the same manner. It applies to unit current and unit area.

Three graphical areas are shown in Figure 1. The horizontal green rectangle shows the current density curve for very low frequency, when there is no skin effect to worry about. It shows uniform current density from the circumference (x=0) down to the center of the conductor (x=1). If we have a high-frequency signal with a skin depth, sd, equal to 0.125 (units), then the blue rectangle shows the assumed current density uniformly distributed down to the skin depth, and no current further into the conductor (Proposition 1). The black curve shows a current density curve, Equation 2 (Proposition 2). Under the assumptions we have shown in Figure 1, the current density curve equation would be:

[Eq.3]

Figure 1. Current density relationships.

If each curve represents current density, then the area under the curve represents the total current. For example, a current of one unit per square area times one unit of area (the area of the green rectangle) gives us one unit of current. Similarly, the product of 8 units/square times 0.125 square (the blue rectangle) also gives us one unit of current. Similarly, the area under the current density curve should give us one unit of current. We would normally find this area by calculus integration.

These three areas are shown in Figure 2.

Figure 2. The area under each curve represents the total current.

The area under each curve should give us the total current, equal to one unit in these instances. The green curve (Figure 2a) does so almost by definition. The area under the blue curve does so, too. But it must, because we factored the horizontal axis by 1/sd and we multiplied the vertical axis by sd. Therefore, the area is the same by design. So the real question is whether the area under the current density curve (Figure 2c) equals one. And the answer to that is, well, almost! If skin depth is a small percentage of the radius (say below 30% or 40%) the approximation is very good. But if the skin depth is deeper than that, then the entire accuracy of these assumptions starts falling off.

Now, I am NOT saying all this to disparage the skin depth calculations we all use. I am saying this to point out that these are all models, not reality, and the models are pretty good. For small skin depths they are very good. For deeper skin depths, they are good, but with a potential error of as much as a few percent. Nevertheless, for the kinds of accuracy we need in skin effect calculations, they are usually plenty good enough, and well within the margins of error of other measurements and calculations we have to make at the same time.

So, to summarize what we have learned about skin effect to this point:

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. This is described by Equation 2.
3. There is an imaginary depth under the surface of the conductor called the skin depth (sd) given by Equation 1.
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.

In the next installment of this series, we will explore concepts such as the crossover frequency and the proximity effect.

Technical Note: The exact solution for the skin effect in round conductors relies on Bessel functions. To learn more about that, do a search on “skin effect Bessel” and you’ll get hundreds of hits. In a private e-mail exchange with Prof. Todd Hubing, Michelin Professor of Vehicle Electronics, Clemson University, in June, 2012, Professor Hubing said this:

Bessel functions only apply to round wires. Even for round wires, the exponential approximation is very good. For planes, the decay is exactly exponential. If the conductor is thicker than a few skin depths, the skin depth approximation is very accurate. If the conductor thickness is less than a skin depth*, the current distribution is approximately uniform and the actual thickness should be used to calculate the resistance. For thicknesses greater than one but less than three skin depths, we don't have a good closed form expression, but the result falls somewhere between the other extremes and can still be estimated with reasonable accuracy.

*  Ed. Note: Refer to the crossover frequency in Part 3 of this series.

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.