# Skin Effect

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When an AC current flows through a conductor the current density is not uniform over the cross-section of the conductor as it is with DC. Instead, the current is concentrated towards the surface of the conductor as shown graphically in Figure 1. The current density is greatest at the surface and decreases exponentially as we move toward the centre of the conductor. This phenomenon is known as the “skin effect”, and it is caused by an outward force on the moving charges from the magnetic field that they produce.

We define a “skin depth” ẟ to be the depth at which the current density has fallen to 1/e (about 37%) of the density at the surface. This means that 63% of the current flows in the surface at or above the skin depth. The skin depth ẟ is given by Equation 1, where σ is the conductivity of the material, µ is its magnetic permeability and f is the AC frequency.

We can see from the equation that the skin depth reduces as the frequency rises. With a copper conductor, the skin depth at mains frequency (50Hz where I live) is just 9.3mm. It’s a little bit less than this at 60Hz. At 1MHZ the skin depth is just 0.06mm. Figure 2 shows a graph of skin depth vs frequency for a copper conductor.

The skin effect becomes very important in the design of switched mode power supply magnetics and explains why high-current high-frequency windings are often made of thin copper foil, multiple twisted insulated wires in parallel or Litz wire. Litz wire consists of many fine strands of insulated wire twisted or bundled together. The strands can be as small as 0.1mm in diameter and there can be many thousands of strands in a single wire as shown in Figure 3.

Even at mains frequencies we need to think about skin effect if we need very large currents. The skin depth of under 10mm suggests there is no point in a circular cross-section conductor of more than 15-20mm in diameter or else we are just wasting copper, since very little current will flow in the centre of the conductor. Instead, fixed installations tend to use rectangular cross-section bus bars.

Rectangular bus bars can be up to 200 mm in width but are rarely thicker than 6 mm – meaning no point in the conductor is more than 3 mm from the surface. For higher currents busbars are paralleled with air spaces between them. The rectangular cross-section has a couple of other advantages. First, they have higher surface area to cross-section ratio than the equivalent round or square conductor so lose heat to convection at a higher rate. Second, they are easy to connect by overlapping their large flat surfaces and securing with bolts. This, and the use of two bars in parallel, is illustrated nicely in Figure 4.

Where a flexible conductor is required, each conductor is made up of many thinner conductors are twisted together to make a single cable. There is quite some science in this since the aim is to make sure that each individual conductor weaves in and out over the length of the cable such that they are all close to the surface and in the interior about the same amount, ensuring they all see the same current density.

For this to work, the individual wires must also be slightly insulated from each other. This inter-strand resistance only has to be a few ohms so is usually accomplished via an oxidation or other chemical surface treatment.

Figure 5 shows the cross section of a typical high current cable. It illustrates a second technique used to ensure optimum use of the copper. In this case the cable is made up of five segments, each of which is one of the twisted bundles mentioned above squished into a wedge shape. Using multiple smaller bundles simplifies manufacture and makes the cable more flexible. It’s interesting to note that these techniques for overcoming skin effect in power cables were invented by Canadian Humphreys Milliken in 1933. He published in them in Patent #1,904,162 which is an easy read at only 3 pages long. It contains some nice diagrams showing just how the windings are arranged if that is of interest to you.

References
“Skin Effect.” In Wikipedia, June 7, 2022. https://en.wikipedia.org/w/index.php?title=Skin_effect&oldid=1091968787.

MWS Wire – Magnet Wire, Specialty Wire. “Litz Wire – Reduce AC Losses in High Frequency Windings.” Accessed June 21, 2022. https://mwswire.com/specialty-wire/litz-wire/. Humphreys, Milliken. Electrical cable. United States US1904162A, filed August 13, 1930, and issued April 18, 1933. https://patents.google.com/patent/US1904162A/en.

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