Skin (Effect) in the Game

We love to pretend like our components are perfect. Resistors don’t have capacitance or inductance. Wires conduct electricity perfectly. The reality, though, is far from this. It is easy to realize that wire will have some small resistance. For the kind of wire lengths you usually encounter, ignoring it is acceptable. If you start running lots of wire or you are carrying a lot of current, you might need to worry about it. Really long wires also take some time to get a signal from one end to the other, but you have to have a very long wire to really worry about that. However, all wires behave strangely as frequency goes up.

Of course there’s the issue of the wire becoming a significant part of the signal’s wavelength and there’s always parasitic capacitance and inductance. But the odd effect I’m thinking of is the so-called skin effect, first described by [Horace Lamb] in 1883. [Lamb] was working with spherical conductors, but [Oliver Heaviside] generalized it in 1885.

Put simply, when a wire is carrying AC, the current will tend to avoid traveling in the center of the wire. At low frequencies, the effect is minimal, but as the frequency rises, the area in the center that isn’t carrying current gets larger. At 60 Hz, for example, the skin depth for copper wire — the depth where the current falls below 1/e of the value near the surface — is about 0.33 inches. Wire you are likely to use at that frequency has a diameter less than that, so the effect is minimal.

However, consider a 20 kHz signal — a little high for audio unless you are a kid with good ears. The depth becomes about 0.018 inches. So wire bigger than 0.036 inches in diameter will start losing effective wire size. For a 12-gauge wire with a diameter of 0.093 inches, that means about 25% of the current-handling capacity is lost. When you get to RF and microwave frequencies, only the thinnest skin is carrying significant current. At 6 MHz, for example, copper wire has a skin depth of about 0.001 inches. At 1 GHz, you are down to about 0.000081 inches. You can see this (not to scale) in the accompanying image. At DC, all three zones of the wire carry current. At a higher frequency, only the outer two zones carry significant current. At higher frequencies, only the outer zone is really carrying electrons.

So What?

There are a few practical issues to think about. Manufacturers that create cables for high-frequency use spend a lot of time perfecting the surface of the wire since a small imperfection on the surface that isn’t significant to the entire diameter of the wire might be a very important part of the current-handling capacity at a high frequency.

Cables that are copper clad — that is, that have a steel core and a copper surface — will conduct DC differently than AC at the design frequency. Some very high power applications save weight and cost by using hollow tubes instead of solid conductors. This is very common in transmitting coils that use what appears to be copper plumbing tubes. Electric power stations often use tube conductors, also, and benefit from the ability of a tube to stretch across a long distance without as much support as a heavier wire. Sometimes these tubes are silver plated since the plating will carry most of the current and silver is a good conductor but relatively expensive.

To mitigate skin effect, you can use litz wire for frequencies up to about 1 MHz. This is wire woven from many separate, insulated conductors. In addition to providing multiple “skins” for the current to flow, the weaving follows certain patterns to minimize the proximity effect between the wires. If you’ve ever taken apart an old AM radio with a ferrite rod antenna inside, you’ve probably seen litz wire as most of those antennas employ litz windings. Carbon nanotube threads can also work and are not as limited in frequency.

Another place this shows up is in welding rods. An iron rod will work fine for welding at DC, but not for high-frequency welding. The reduced current-handling capacity will cause the welding power to dissipate in heat throughout the rod instead of causing an arc.

The Math

The theory behind skin effect is that the change in current causes a change in magnetic field and that generates a reverse voltage (“back EMF”) in the wire. This reverse is strongest in the center of the wire and it forces the electrons away from the center. This is the same effect, by the way, that causes metals to reflect electromagnetic waves.

There is a complex formula for the skin depth that depends on the material’s resistivity, magnetic permeability, and other exotic values. However, as a rule of thumb, you can ignore the skin effect when the frequency is at or below 124 divided by the square of the wire diameter in thousandths of an inch. Beyond that point, assume resistance will increase about 3.2 times for every 10X increase in frequency.

For example, consider a two-inch piece of #18 wire. At DC, the wire would have a 1.06 mΩ resistance. The diameter in mils is 40.3. Squaring 40.3 and dividing it into 124 gives 76 kHz. Suppose you were going to use the wire at 100 MHz. Since this is just an estimate, consider that 100 kHz is three decades below 100 MHz and you can figure the resistance will be 1.06 x 3.2 x 3.2 x 3.2 = about 35 millohms. If you do the long math, the real answer comes out to just under 41 milliohms, so that’s not bad.

If you really want to look at the hairy math, you might enjoy the video below.

In Practice

Most of the time, you won’t care much about skin effect. That’s what makes it so insidious. We all know batteries, for example, don’t behave like ideal voltage sources and so we work around it. But wires are pretty good, until they aren’t. Long wires, high frequencies, and high currents can all conspire to make the pretty schematic an ugly circuit. Skin effect is just one of the reasons wires don’t behave like we wish they would.

Photo credit: Title graphic [Mariusz.stepien] CC BY-SA 3.0

75 thoughts on “Skin (Effect) in the Game

  1. Hey I wonder if anyone has thought of that, we’re not really in an expanding universe, we’re the atoms near the center of a giant wire, maybe it’s a toroid, looped back, randomly vortexing across nothing, because the big bang was a minion fart gun…

      1. “…but for some reason the North Americans cling on to their Inches.”
        … must resist .. urge … to … dive down the NSFW double entendre rabbit hole of clinging to ones inches…
        Phew.. it was hard.. but I think I managed..

      2. To be fair there are three countries on the planet that currently use the imperial measurement system:

        Liberia – Liberian government has begun transitioning away from use of imperial units to the metric system.
        Myanmar – Since October 2013 has been preparing to adopt the International System of Units.
        United States of America – too costly and would be confusing.

          1. @ǝıuuɐ
            I have drank a pint of Molson. The problem is Canada’s proximity to the US, close to the border is mostly the imperial system, north east it is metric, it takes a few generations for people to ultimately think in metric and the US is a source of hindrance.

    1. [raises hand] I’m in the US, and I’d like metric too. Come on, 0.000081 inches? Even people who are accustomed to inches can’t visualize that. It’s 2.05 micrometers. At least some people can visualize that.

    2. After reading the dimensions of wires in decimal inches about four times, I simply skipped the rest of the article. Not even the conversion in brackets like many reputable articles have – very poor editorial skills.

    1. Even if copper was the same price as aluminum, they’d still use aluminum. Per unit weight, aluminum has half the resistance of copper. The only metals that do better don’t fare well in oxygen environments (eg, sodium).

      Silver has the lowest resistance per unit volume, which is why it’s used to mitigate skin effect.

        1. I think that was because of thermal expansion. I understand that aluminum wiring would eventually walk itself out of connections – it also doesn’t play well when twisted with or screwed onto copper. The corrosion, etc would cause arcing, fires, death. So yeah, it’s not too popular with home insurance companies.

          1. I understood a big reason was it is reasonably soft. Using a common screw terminal if you overtighten you crush the conductor, reducing the cross section of the conductor, raising resistance, and… poof.

        2. That’s largely because both homeowners and sometimes electricians would use the wrong outlets with aluminum wire. Aluminum has a very different coefficient of expansion than copper. So outlets rated for use with aluminum wire MUST be used. One wrong outlet, and eventually it WILL make a poor connection that oxidizes and gets hot, possibly causing a fire.

    1. There are solutions that don’t require anyone to learn new systems or move from their comfort zone. I posted one above another that works well is the universal metric translator script….Now we can focus on the meat of the articles rather than pedantic bickering over system uses.

        1. J’ecrit dans le Francais quand je voudrais démontrer mon Francais est tres mal.

          Though I too have a zone of comfort east of france and enjoy finding italian money in the couch. ;-)

    2. You know, I used to have the same argument with the Russians. Most of us in the US that are engineers can flip back and forth pretty effortlessly. But here’s the sad fact: over here, we have wire in AWG and that specification is in inches. You can convert, but there is a penalty for that in a lot of cases if the conversion isn’t exact. Keeping significant digits is sometimes challenging, too, although there are good rules of thumb. So local custom or more poetically, when in Rome… I use metric when I lay out my own PCBs, for example. But when I’m using things where the core measurement is in Imperial units than I stick with that too.

      What amuses me is how — and it seems worse in the last few years — that everyone feels the need to impose their idea of “right” on everyone else in virtually every venue. By your logic, since China accounts for 20% of the world population, I’m an idiot for using dollars instead of yuan. Oh sure, you can convert, but the majority of the world uses yuan, so….

      On the other hand, I don’t have a burning need to convert you. I personally like metric. But I also like using the units that make the most sense for the situation at hand.

      1. I think people tend to “impose” their unit because it makes sense to them. I would love to be able to use imperial unit, sure conversion methods exist but inches, feet, oz, etc.. Doesn’t make any sense to me.

        With practice it would be easy, but everything I read/see are with the SI metric system

        1. Well, that’s not exactly my point. My point is, that conversions are not always exact. We used to run into this all the time. Where a dimension was designed (rightly or wrongly) in Imperial and the metric conversion was just a little inexact for one reason or another. Sometimes that adds up or has other surprising effects. So when I’m talking AWG wire I used Imperial and, honestly, you should too in that case. Not because Imperial is better, but because — in this case — that’s the unit that the standard calls out and the manufacturer uses. If I’m dealing with mm size wire, then I switch gears to that. If I for some reason had to deal with wire specified in wavelengths of neon atom emissions, then I’d cope with that. But conversions are not the way to go.

          In fact, since I grew up when Carter tried to get the US metric, I would say that was the reason it failed. We were taught to memorize conversion factors. But not only is that not handy but you don’t get a “feel” for what the units are. It is like using a foreign language phrase book. You might get by, but you aren’t going to get fluent like that. You need total immersion.

          So once again, I’ll assert: Adapt your units to the task at hand. Don’t adapt the task at hand to your units.

          1. By definition, No. 36 AWG is 0.005 inches in diameter, and No. 0000 is 0.46 inches in diameter. But since inches are legally defined as 25.4 mm, it perfectly valid to say that No. 36 AWG is 127 µm in diameter, and No. 0000 is 11.684 mm in diameter.

          2. Some places legally defined pi as 3, too. There’s an old story about a farmer who sees a man poking around his barn. He goes out to ask the stranger what he wants. The stranger whips out a badge and says, “Listen! I’m from the state tax office and this badge means I don’t have to tell you when I’m going in your barn. Now I’m going to go over to that shed across the field.” The farmer says, “You should do that.” The agent gets irate and once again waves his badge in the farmer’s face. “I don’t NEED your permission as long as I’m carrying this badge.” The farmer relents.

            The agent opens the gate, enters the field, and attracts the attention of the farmer’s old bull. The bull, enraged, starts charging at the agent who begins running and screaming for help. The farmer cups his hand to his mouth and says, “Show him the badge! Show him the badge!”

            So, yes, no one is arguing you can’t do conversions. By the way, there is a serious byproduct of the “legal” decision to make the inch 2.54cm in 1959. The American inch changed by 2 millionths of an inch and the UK inch by 1.7 millionths of an inch. So in practical situations, you probably don’t care. But in very precise machining, that’s an issue and you have to know if you are dealing with plans drawn with the old standards.

            A recent manned spacecraft design I was involved with had mixed units in the cockpit. I had tried to push for SI throughout, but the pilots really like the Imperial units they use and they win every argument.

            Anyway, I’ll stop talking about it, but I will also continue to match the units to the design units of the system I’m talking about.

      2. I’m an aerospace engineer, so we need to deal with imperial units a lot.

        It’s not the inch itself that is the problem, one could argue that the size of the meter is just as arbitrary as the inch.

        What we really hate is that is doesn’t scale well to very large or very small sizes (fractions, mills, etc.), and that many “standards” just don’t make any sense. A 16mm^2 wire has twice the cross section area of a 8mm^2 wire, and for most practical purposes, twice the current capability. A general rule of thumb is about 7A per square millimeter.

        With AWG wires, there is no such thing. I think 3 sizes usually is approximately twice the area, but not always, and I wouldn’t know for any other factor. Also, solid wire and stranded wire of the same AWG does not have the same cross section area; I’m not sure how the number of strands factors into this.

        Take a 1″ PVC pipe. Q: Which dimension would be 1 inch? A: NONE OF THEM! Even the inner diameter is larger than 1″.

        We tend to call these “retard units” because frankly, the US is behind the rest of the world, which is exactly what retarded means.

        1. 20% is more than anyone else, so actually yes more people are in China than any other country. India is getting closer though.

          I often say that if there is reincarnation I was probably Chinese in another life. Then again… if there is reincarnation, statistically, everyone has been Chinese at some point.

      3. Skin effect is a scientific observation, science uses the international standards, there is not conversion from inches.

        I can only conclude at this point that this is yet again a deliberate trolling article trying to get lots of comments which translates in interest and links and which eventually translates in money.

        Incidentally, throwing some mention of Russians in to get people even more enervated also only works in the US (and some eastern european states if done right). But sure, audience is a majority US so go for it.

  2. For everyone, who complain about imperial units, last rule of thumb can be translated to
    > However, as a rule of thumb, you can ignore the skin effect when the frequency is at or below 80_000 divided by the square of the wire diameter in mm.

      1. Say Dax (or other readers for that matter), do you happen to know if the material matters when doing an antenna groundplane? I mean if you for instance use aluminium for the antenna does it matter if the groundplane is/contains iron?

  3. Local != International, your comparison don’t make sense.

    While english is actually still recognized worldwide as the standard language for international exchanges, SI decimal unit is incontestably the worldwide standard unit system for international exchanges. It’s really that simple.

    A lot of peoples make effort to speak engligh to be able to exchange worldwide, USA peoples only have to make the effort to use SI decimal unit to get the same level.

      1. Also known as ‘ISO containers’ and they come with official ISO specifications both in imperial and metric, albeit imperial being rounded more.

        And ironically the first standard set for containers was done by the ‘Bureau International des Containers’ in Paris for international transport, and they still set the standards. So the fathers of metric set standards in feet, sigh.

  4. I love how all the comments are complaining that it’s in inches so they don’t have a frame of reference, but how many would actually have a feel of how thick 25 micrometers is? To me both 25 um and 0.001 inches are equally unimaginable because I don’t have a frame of reference, how much thinner is that than a sheet of paper?
    No matter what units they use there will be someone who is unfamiliar with it, and you’re angry that that someone is you. Until Google adds a translate this page for units it’s up to each individual user to deal with the problem.

    1. “but how many would actually have a feel of how thick 25 micrometers is? ”

      Anyone who’s taken the entry level courses into engineering in metric. 25 µm is the surface roughness of a bar of steel cut with a cheap hacksaw.

      1. “Anyone who’s taken the entry level courses into engineering in metric.”

        Did you intend to really sound that condescending? Admittedly, my entry level engineering courses (and the following 4 years) were probably only 60%/40% metric/imperial, but you’d think after 20+ years experience I would have picked up that bit of trivia somewhere. I hadn’t. To me, 25 µm is still pretty unfathomable.

        I really don’t understand why people get so bent out of shape over imperial units. Conversion isn’t that hard (in fact it helps us look more brilliant in front of non-techie people), and as Al said earlier, sometimes imperial units just make more sense. Example: all of the European-designed metric packaging equipment I work on uses roller chain with a 12.7mm (aka 1/2 inch) pitch
        Sure… metric can be smoother to use generally, but it’s not hard to be flexible.

        1. Quick, without a calculator: how many gallons in 1000 cubic inches?!? Your life depends on it!

          A cubic meter contains 1000 liters. Still not convinced?

          You could argue I’m cherry-picking the units; name me a conversion that works trivially in the imperial system and doesn’t in metric.

          The point isn’t that the meter is better than the inch (both are arbitrary), the point is that we have a CONSISTENT system with a MINIMUM of arbitrary units.

          Example: 1 newton-meter times 2 pi radians per second equals 1 watt of mechanical power.
          Would you care to do the same in imperial?

          By the way, 25 µm is about a quarter of the thickness of a regular post-it note.With metric Vernier calipers, you can measure to an accuracy of 50µm (it’s possible, but 100µm is easier), so about half of that. I’d assume you learned to use Vernier calipers at some point in your 20-year career, so that should give you some idea.

          1. I don’t disagree with your points. The math usually is harder in the imperial system. But you seemed to have missed my point: Imperial unit usage doesn’t anger me. I do prefer metric and use it when needed, or when I have a choice, but I certainly don’t feel the need to argue about how one is better than the other. (see my last statement in my previous post)

            I don’t get upset when people speak a different language or use different currency either.

            As to your last point, of course I can use calipers and can measure accurately when needed… I had just never heard the relationship between 25 µm and the surface left by a cheap hacksaw before. How much is a cheap hacksaw? And is that measured in Euros?

  5. Now to make a broadband analogue frequency separator from a tapped rod that can cope with loads of current and high voltage.
    I’ve seen star shaped conductors used. Even pumping coolant through the hollow centre.
    Maybe if it was shaped like a pompom with the points close enough together, it would still allow the skin effect to take influence whilst still permitting access to the centre of the conductor.

  6. 28 thoughts on “Skin (Effect) in the Game”

    And ALL those comments (at my post time) here are about the units used in the article, NOT about the subject of the article: “Skin-Effect”.

    What a disservice to the nice skin-effect article here by Al Williams.

    We all deal with multiple units systems every day. I have NO problem with Mr. Williams using one units system to get his thoughts across. Units conversion is not a problem for me the way the article was written.

    And gee… I wanted to comment about issues with skin-effect and dissimilar materials such as PVD coatings on wires and PCB traces and how strange things happen with skin-effect when the coating/plating is over dissimilar metals. (Interstitial deposition coating is different from plating when it comes to skin-effect.)

    But never mind. The comments here are all about petty fights about the units used. I’m turned off & outa-here :-(

    1. I’m just continually amazed that so many people who’d probably describe themselves as smart, apparently have their world collapse and turn into gibbering wrecks when faced with a grade 5 math unit conversion.

        1. Perhaps he was too busy using apostrophes correctly. I am surprised at how many people must have such low self esteem that they feel the need to tear down practically every Hackaday post on trivialities.

  7. Reminds me of when I went for a tour of the local hydro power/pump station. The guy doing the tour pointed through a glass door at a bunch of buss bars and asked if I noticed anything strange. I wasn’t sure what to look at till he mentioned that they were Aluminium pipes. When I asked why they weren’t solid he mentioned the skin effect. Why waste money on metal that the current isn’t going to flow through.

  8. “0.000081 inches”

    Even without getting into an argument about the metric system, that’s a stupid number.
    Use scientific notation thanks.

    8.1 x 10^-5 inches.
    Or 81 microinches.
    Or 0.081 thou, if you must.

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