Skin effect

Skin effect has practical consequences in the analysis and design of radio-frequency and microwave circuits, transmission lines (or waveguides), and antennas.

It is also important at mains frequencies (50–60 Hz) in AC electric power transmission and distribution systems.

Although the term skin effect is most often associated with applications involving transmission of electric currents, skin depth also describes the exponential decay of the electric and magnetic fields, as well as the density of induced currents, inside a bulk material when a plane wave impinges on it at normal incidence.

The imaginary part of the exponent indicates that the phase of the current density is delayed 1 radian for each skin depth of penetration.

One full wavelength in the conductor requires 2π skin depths, at which point the current density is attenuated to e−2π (1.87×10−3, or −54.6 dB) of its surface value.

In good conductors such as metals all of those conditions are ensured at least up to microwave frequencies, justifying this formula's validity.

However, in very poor conductors, at sufficiently high frequencies, the quantity inside the large parentheses increases.

This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much larger than the skin depth itself.

For instance, bulk silicon (undoped) is a poor conductor and has a skin depth of about 40 meters at 100 kHz (λ = 3 km).

The conclusion is that in poor solid conductors, such as undoped silicon, skin effect does not need to be taken into account in most practical situations.

When skin depth is not small with respect to the radius of the wire, current density may be described in terms of Bessel functions.

The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by:[6]: 38

This impedance is a complex quantity corresponding to a resistance (real) in series with the reactance (imaginary) due to the wire's internal self-inductance, per unit length.

It can be shown that this will have a minor effect on the self-inductance of the wire itself; see Skilling[9] or Hayt[10] for a mathematical treatment of this phenomenon.

However, the presence of a second conductor in the case of a transmission line reduces the extent of the external magnetic field (and of the total self-inductance) regardless of the wire's length, so that the inductance decrease due to skin effect can still be important.

For instance, in the case of a telephone twisted pair, below, the inductance of the conductors substantially decreases at higher frequencies where skin effect becomes important.

is not changed by the skin effect and is given by the frequently cited formula for inductance L per length D of a coaxial cable:

Although the geometry is different, a twisted pair used in telephone lines is similarly affected: at higher frequencies, the inductance decreases by more than 20% as can be seen in the following table.

Iron wire is impractical for AC power lines (except to add mechanical strength by serving as a core to a non-ferromagnetic conductor like aluminum).

At a few kilohertz, an iron welding rod would glow red hot as current flows through the greatly increased AC resistance resulting from skin effect, with relatively little power remaining for the arc itself.

[17] Conductive threads composed of carbon nanotubes[18] have been demonstrated as conductors for antennas from medium wave to microwave frequencies.

Tubular conductors are typical in electric power switchyards where the distance between supporting insulators may be several meters.

This technique is particularly used at VHF to microwave frequencies where the small skin depth requires only a very thin layer of silver, making the improvement in conductivity very cost effective.

Skin effect itself is not actually combatted in these cases, but the distribution of currents near the conductor's surface makes the use of precious metals (having a lower resistivity) practical.

Recently, a method of layering non-magnetic and ferromagnetic materials with nanometer scale thicknesses has been shown to mitigate the increased resistance from skin effect for very high-frequency applications.

[citation needed] As experiments have shown, this has potential to greatly improve the efficiency of conductors operating in tens of GHz or higher.

A practical consequence is seen by users of induction cookers, where some types of stainless steel cookware are unusable because they are not ferromagnetic.

A layer of silver 3 μm thick evaporated on a piece of glass is thus an excellent conductor at such frequencies.

In copper, skin depth can be seen to fall according to the square root of frequency: In Engineering Electromagnetics, Hayt points out that in a power station a busbar for alternating current at 60 Hz with a radius larger than one-third of an inch (8 mm) is a waste of copper,[20] and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons.

This effect was first noticed by Heinz London in 1940, who correctly suggested that it is due to the mean free path length of the electrons reaching the range of the classical skin depth.

Distribution of current flow in a cylindrical conductor, shown in cross section. For alternating current , current density decreases exponentially from the surface towards the inside. Skin depth, δ, is defined as the depth where the current density is just 1/e (about 37%) of the value at the surface; it depends on the frequency of the current and the electrical and magnetic properties of the conductor.
Induction cookers use stranded coils ( Litz wire ) to reduce heating of the coil itself due to skin effect. The AC frequencies used in induction cookers are much higher than standard mains frequency ‒ typically around 25–50 kHz.
Cause of skin effect. A main current I flowing through a conductor induces a magnetic field H . If the current increases, as in this figure, the resulting increase in H induces separate, circulating eddy currents I W which partially cancel the current flow in the center and reinforce it near the skin.
Current density in round wire for various skin depths. Numbers shown on each curve are the ratio of skin depth to wire radius. The curve shown with the infinity sign is the zero frequency (DC) case. All curves are normalized so that the current density at the surface is the same. The horizontal axis is the position within the wire with the left and right extremes being the surface of the wire. The vertical axis is relative current density.
The internal component of a round wire's inductance vs. the ratio of skin depth to radius. That component of the self inductance is reduced below μ /8 π as skin depth becomes small (as frequency increases).
The ratio AC resistance to DC resistance of a round wire versus the ratio of the wire's radius to the skin depth. As skin depth becomes small relative to the radius, the ratio of AC to DC resistance approaches one half of the ratio of the radius to the skin depth.
Four stages of skin effect in a coax showing the effect on inductance. Diagrams show a cross-section of the coaxial cable. Color code:
black - overall insulating sheath
tan - conductor
white - dielectric
green - current into the diagram
teal - current coming out of the diagram
Dashed black lines with arrowheads are magnetic flux ( B ). The width of the dashed black lines is intended to show relative strength of the magnetic field integrated over the circumference at that radius. The four stages are:
  • A non-energized
  • B low frequency
  • C middle frequency
  • D high frequency
There are three regions that may contain induced magnetic fields: the center conductor, the dielectric and the outer conductor. In stage B , current covers the conductors uniformly and there is a significant magnetic field in all three regions. As the frequency is increased and the skin effect takes hold ( C and D ) the magnetic field in the dielectric region is unchanged as it is proportional to the total current flowing in the center conductor. In C , however, there is a reduced magnetic field in the deeper sections of the inner conductor and the outer sections of the shield (outer conductor). Thus there is less energy stored in the magnetic field given the same total current, corresponding to a reduced inductance. At an even higher frequency, D , the skin depth is tiny: All current is confined to the surface of the conductors. The only magnetic field is in the regions between the conductors; only the external inductance remains.
Skin depth vs. frequency for some materials at room temperature, red vertical line denotes 50 Hz frequency: