Skin depth

Updated November 18, 2010

Click here to go to our page on conductivity

Click here to go to our page on transmission line losses

Why are there two conductors on this high power line? Skin depth!!!

Additional explanations of skin depth and its effects on microwave circuits can be found here:

RF sheet resistance (first read about "DC" sheet resistance here)

RF sheet resistance examples

High permeability materials

Surface roughness

Skin depth is one of the most fundamental issues in microwave engineering, but it also affects lower frequencies, even 60 cycle AC power. The skin depth equation is as relevant to microwave engineering as Newton's law (F=mA) is to physics.

This is one of the best discussions of skin depth effects anywhere, including the entire searchable microwave paper collection of the IEEE. We know, because we did the search! Along with this discussion we offer a download that calculates equivalent RF sheet resistance for metals that have up to three different layers. If you are a thin film vendor and would like to sponsor this outstanding page and beat your competitors to the punch, contact us!

By the way, we use the metric system for this discussion, and in the download. Why? because there are no popular units of bulk resistivity in the English system. Regarding skin depths and thin film thicknesses, if you want to convert from microns to micro-inches exactly, divide microns by 0.0254. There are approximately 40 micro-inches in one micron, that's close enough when you are dealing with plating.

Here is a clickable index for this web page:

The mystery of skin depth - a timeless story

Skin depth - what is it?

About that power line...

The mystery of skin depth - a timeless story

The topic of skin depth is a misunderstood one, but we are going to end that right here and now. Every time microwave engineers encounter a new frequency band or a new metalization scheme, there is some question over RF losses in transmission lines such as microstrip. Often "experts" start in on the discussion, and it goes like this...

"Hey Bert, the gain on that new design stinks, what's the problem? Did you change anything from the last design?"

"Nothing major, Ernie. Except we did go to a new thin film vendor... outsourced to Pakistan. There was a language barrier, don't know if they were able to understand the plating specification. The assemblers love it, they say the gold plating is 'hard as a rock'!"

"Oh jeez, sounds like a skin depth problem. We're just process people, we don't know nothing about RF. Go get the Wally, the old fart who saved that low frequency X-band job for us 20 years ago!"

Later when Wally shows up for work around 10:30 AM, and is interrupted from his daily manicure with a pocket knife...

"Oh Great One, can you tell us if we have a problem with the deep mystery of skin depth?"

"While you are filling my coffee cup, let me scratch up one of your parts with my disgusting but still very cool Microwaves101 pocketknife... yes, here's the problem, it's nickel plating!"

"Yes, we knew about that when it was in process, but we fixed it by tripling the gold plating requirement. There's 300 microinches of gold there, it's gotta be low loss!"

"You don' unnerstand... it don't matter if there's more gold there than in Priscilla Presley's wedding ring! You're in a pickle and that there gold ain't gonna fix it no how! If you whippersnappers would get off that newfangled E-M software baloney and learn you some fundamentals, maybe our stock price would rise enough so I could afford to retire, dagnabbit! I gotta go, it's time for my nap now!"

The same conversation is heard many times around the country each year, and thousands of man-hours are wasted, often with expensive design-of-experiments where egghead test engineers try to measure the effects of different plating schemes, or worse, EM freaks try to model the effects using electromagnetic analysis software. By reading this web page the heartbreak of skin depth need not affect anyone's pay raise ever again, even if you are "just a process guy".

Skin depth - what is it?

Skin depth is a measure of how far electrical conduction takes place in a conductor, and is a function of frequency. At DC (0 Hz) the entire conductor is used, no matter how thick it is. As you double the cross-sectional area of a wire, the DC resistance per unit length decreases by half, as you'd expect according to Ohm's law. At RF frequencies, the effect that conductor thickness has on its conductance is nonlinear (actually, a negative exponential.) There is a limitation on the conductance that you can achieve, and increasing the thickness of precious metals to reduce losses RF can be a waste of money if you don't know what you are doing.

One common misconception about skin depth is which surface of a conductor is carrying the RF current. It is always (mostly*) the surface nearest the media in which the EM wave propagates. Illustrated below in a cross-section of microstrip, the RF currents are highest in the lower surface of the microstrip line. That is why the first metal is most important to conductivity in a microstrip multi-metal stack-up. Please note that for waveguide, the opposite is true; it doesn't matter what you underplate with, the surface plating is what counts.

* There are some fields on the top of microstrip line, but at much lower intensity. We propose to analyze this in April 2009 and create a page on the topic. Here's a page on microstrip loss which helps explain the concept.

Skin depth equation

The well-known equation is for skin depth given below. Note that skin depth
() is a function of only three variables, frequency (f ), resistivity (), and relative permeability (_R).

Bulk resistivity is a measure of how resistive a material is. It is the reciprocal of bulk conductivity. Click here to look up your favorite metals' conductivities. And click here to try out our skin-depth calculator.

Believe it or not, conductors can become thinner at higher frequencies with little impact on circuit loss, because the skin depth shrinks with frequency. Although the sheet resistance of metalization always increases with frequency, the RF loss per wavelength can actually decrease. Thus 150 microinches of gold on alumina thin-films at X-band can be thinned to half that for millimeterwave applications to save money without compromising performance. A branchline coupler at Q-band will have less loss in dB than one designed for L-band! Millimeterwaves are so cool...

It's time for some Microwaves101 Rules of Thumb!

Rule of thumb: always plan on providing at least five skin depths of low-loss conductor. This will keep more than 99% of your electrons happy and provide good performance without wasting precious metals.

Rule of thumb: when using copper-clad boards, you usually won't have to give skin depth another thought, because you will have at least 700 microinches of copper (1/2 ounce) on both sides, which is five skin depths down to 330 MHz.

About that power line...

Most engineering problems only consider skin depth at microwave frequencies. Let's look at a wider spectrum:

Notice that when you plot different conductors this way, they don't look much different. At 60 Hertz, no matter what the conductor, the skin depth is about 1 centimeter. That means that a power line 10 centimeters in diameter will satisfy the "five skin depths" rule (radius is 5 centimeters). Bigger than that and you are wasting copper. So to decrease losses on long high-tension runs, power companies sometimes hang double, and even triple wires for each power phase.

But wait! There's more to that power line than meets the eye. This information came from Jeff, in November 2010. Thank you sir!

There may be some small savings on resistance due to skin depth but considering there is no electric field between the adjacent lines (in the same phase), this would not justify the added expense. The reason for using multiple cables on a single phase is to increase the equivalent radius (aka GMR, geometric mean radius) of the cable. This in turn both raises the distributed capacitance and increases the breakdown voltage (the sharper the edge, the lower the breakdown voltage). Thus, two small, cheap cables are electrically the same as a thick, expensive cable.

Where skin depth does help in 60 Hz power transmission is in cable strength. Rather than make the cables out of pure aluminum, a steel core is added for strength. Since steel has a non-unity relative permeability, this has a significant effect on skin depth. The 60 Hz power guide is kept mainly in the aluminum outer cladding while the core provides the physical backing for the cable.

Long ago, there was great debate about whether power should be distributed as direct or alternating current. Read about it here!