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Brooks' Bits: The Skinny on Skin Effect, Part 1

OK. I know. I just couldn’t resist that title!

In a previous series of articles I pointed out that there are only three primary passive components in electronics; resistors, capacitors, and inductors (see Note 1). These components occupy a spectrum, with capacitors and inductors at the opposite ends, and resistance being a special case right in the middle. Capacitors (pure ones) have an impedance that decreases with frequency, and a voltage phase shift that lags current by – always – 90 degrees.

Inductors (pure ones) have an impedance that increases with frequency, and a voltage phase shift that leads current by – always – 90 degrees. Resistors (pure ones) have an impedance that is independent of frequency and a voltage phase shift of – always – zero degrees compared with current.

Now we know that the skin effect causes the resistance of a conductor to increase with frequency. But this statement is not exactly true. Resistance is not changing. The resistance of a conductor is given by Equation 1:

R = ρ*L/A                    [Eq. 1]

where: R= the resistance of the conductor            ρ = the resistivity of the conductor material            L = the length of the conductor, and            A = cross-sectional area of the conductor.

Looking at that formula, what is changing with frequency? The resistivity is not changing. The physical cross-sectional area of the conductor is not changing. The length is not changing. So why does the resistance, in fact, change?

The answer is that where the current is flowing is what is changing. At very high frequencies the current tends to flow along the outer circumference of the conductor, but not down the centerline. Therefore, the effective cross-sectional area of the conductor is changing. Therefore, we have a higher current density through a smaller portion of the conductor, resulting in a higher voltage drop. This behaves as if the resistance of the conductor itself is changing. Skin effect is all about current density – where the current is flowing.

Where Current Flows

We usually think of a conductor (wire or trace) as being purely resistive. But, in fact, conductors have some (small) inductance. When a voltage is first applied to a conductor, no current flows. This is because the changing driving current creates a changing magnetic field around the conductor that induces a current in the opposite direction which resists the driving current that created it in the first place. In the very next instant of time, the changing magnetic field weakens slightly and a very small net current flows in the forward direction. This is the inherent nature of inductance. An interesting question is: Where does this net current flow?

Intuitively we can picture three possibilities: (a) the net current flows uniformly across the entire cross-sectional area of the conductor; (b) the net current flows down the centerline of the conductor; or (c) the net current flows on the outside circumference of the conductor. The determining factor is the strength of the changing magnetic field. It turns out that the induced magnetic field is strongest along the centerline of the conductor and weakens inversely with the square of the distance from the centerline. Therefore, the changing field is weakest around the circumference.

Therefore, when the first increment of current starts to flow, it flows in a thin band around the circumference. As more net current flows, the band around the circumference begins to thicken. The current density is highest around the circumference, but lower-current densities exist as we look further into the conductor. Only when the magnetic field stops changing (DC) does the current flow uniformly across the entire cross section of the conductor. Figure 1 illustrates this relationship.

Figure 1. Very-high-frequency currents flow primarily at the circumference of a copper conductor, an effect called the skin effect. The shading represents the current density.

In high-frequency situations, the current starts flowing along the circumference of the conductor, and then with successive increments of time starts flowing further into the cross-sectional area. But before the current density stabilizes (as a uniform current density), the current changes direction. Then the effect starts all over again. At very low frequencies, much of the current flows through most of the conductor. But at very high frequencies, the current never flows much below the circumference. Thus, at higher frequencies:

1. The effective cross-sectional area is lower,
2. The current density is higher,
3. The voltage drop is higher, and
4. Therefore, the resistance appears to be higher.

Why Do We Care?

The first reason we might care about skin effect is that it impacts any calculation that involves Ohm’s Law (E = iR). Thus, the voltage drop across a conductor and the power dissipated within the conductor (and therefore loss to the circuit) will increase with frequency.

This leads directly to the second reason: The skin effect can impact trace current/temperature effects. Although it is rare for large power supply currents to operate at very high frequencies, it is not impossible. If that happens, then skin effect needs to be considered.

Finally, our model for transmission lines (Note 2) is a lossless one. Under that condition, the characteristic impedance, Zo, is resistive and terminating the trace is straightforward. But if there are skin effect losses, the transmission line no longer meets the conditions for being “ideal” and the termination issue becomes much more difficult. We will discuss this in a subsequent part of this series.

You may be aware of the rule of thumb that says to find the effective resistance of a circular conductor with skin effect, we calculate the cross-sectional area of the cylinder formed by the outer circumference of the conductor and the skin depth. This assumes that all of the current is flowing uniformly throughout this cross-sectional area. But you may have already intuitively figured out that the current density really follows an exponential path, being highest at the surface of the conductor and lowest along the centerline. It is not uniform anywhere!

This will be the topic of Part 2 of this series.

Notes

1. See my three-part series “Resistance, Reactance, and Impedance,” available here.

2. See “Current Flow on Traces, Part 1: Transmission Lines,” available here.

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.