Skin effect
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The skin effect is the tendency of an alternating electric current (AC) to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. That is, the electric current tends to flow at the "skin" of the conductor. The skin effect causes the effective resistance of the conductor to increase with the frequency of the current. Skin effect is due to eddy currents set up by the AC current.
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[edit] Introduction
The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalized to conductors of any shape by Oliver Heaviside in 1885. The skin effect has practical consequences in the design of radio-frequency and microwave circuits and to some extent in AC electrical power transmission and distribution systems. Also, it is of considerable importance when designing discharge tube circuits.
The current density J in an infinitely thick plane conductor decreases exponentially with depth d from the surface, as follows:
- Failed to parse (Cannot write to or create math output directory): J=J_\mathrm{S} \,e^{-{d/\delta }}
where δ is a constant called the skin depth. This is defined as the depth below the surface of the conductor at which the current density decays to 1/e (about 0.37) of the current density at the surface (JS). It can be calculated as follows:
- Failed to parse (Cannot write to or create math output directory): \delta=\sqrt{{2\rho }\over{\omega\mu}}
where
- ρ = resistivity of conductor
- ω = angular frequency of current = 2π × frequency
- μ = absolute magnetic permeability of conductor Failed to parse (Cannot write to or create math output directory): = \mu_0 \cdot \mu_r
, where Failed to parse (Cannot write to or create math output directory): \mu_0
is the permeability of free space (4π×10−7 N/A2) and Failed to parse (Cannot write to or create math output directory): \mu_r is the relative permeability of the conductor.
The resistance of a flat slab (much thicker than δ) to alternating current is exactly equal to the resistance of a plate of thickness δ to direct current. For long, cylindrical conductors such as wires, with diameter D large compared to δ, the resistance is approximately that of a hollow tube with wall thickness δ carrying direct current. That is, the AC resistance is approximately:
- Failed to parse (Cannot write to or create math output directory): R={{\rho \over \delta}\left({L\over{\pi (D-2\delta)}}\right)}\approx{{\rho \over \delta}\left({L\over{\pi D}}\right)}
where
- L = length of conductor
- D = diameter of conductor
The final approximation above is accurate if D >> δ.
A convenient formula (attributed to F.E. Terman) for the diameter DW of a wire of circular cross-section whose resistance will increase by 10% at frequency f is:
- Failed to parse (Cannot write to or create math output directory): D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}
The increase in AC resistance described above is accurate only for an isolated wire. For a wire close to other wires, e.g. in a cable or a coil, the ac resistance is also affected by proximity effect, which often causes a much more severe increase in ac resistance.
[edit] Material Effect on Skin Depth
Skin depth varies as the inverse square root of the conductivity. This means that better conductors have a reduced skin depth. The overall resistance of the better conductor is lower even though the skin depth is less. This tends to reduce the difference in high frequency resistance between metals of different conductivity.
Skin depth also varies as the inverse square root of the permeability of the conductor. In the case of iron, its conductivity is about 1/7 that of copper. Its permeability is about 10,000 times greater however. The skin depth for iron is about 1/38 that of copper or about 220 microns at 60 Hz. Iron wire is worthless as a conductor at power line frequencies. Skin effect reduces both the effective thickness of laminations in power transformers and their losses.
Iron rods work well for dc welding but it is impossible to use them at frequencies much higher than 60 Hz. At a few kilohertz, the welding rod will glow red hot from skin effect losses but will barely have enough power available to sustain an arc. Only Non-magnetic rods can be used for high frequency welding.
[edit] Effect on impedance of round wires
For isolated round wires with radius Failed to parse (Cannot write to or create math output directory): R
on the order of or smaller than Failed to parse (Cannot write to or create math output directory): d
, the assumption of exponential decrease of Failed to parse (Cannot write to or create math output directory): J
with depth Failed to parse (Cannot write to or create math output directory): \delta is no longer valid. In this case, Failed to parse (Cannot write to or create math output directory): J must be found by solving
- Failed to parse (Cannot write to or create math output directory): \frac{d^2J}{dr^2} + \frac{1}{r} \frac{dJ}{dr} = j \omega \mu \sigma J
If we transform variables from Failed to parse (Cannot write to or create math output directory): r
to Failed to parse (Cannot write to or create math output directory): j^{-1/2}r
, this equation has the form of a zeroth-order Bessel equation. Using the boundary condition Failed to parse (Cannot write to or create math output directory): J(R) = J_S
and considering that Failed to parse (Cannot write to or create math output directory): J must be finite at Failed to parse (Cannot write to or create math output directory): r = 0 for a solid wire, the solution to this equation is
- Failed to parse (Cannot write to or create math output directory): J(r) = J_S \frac{J_0(\sqrt{-2j}r/\delta)}{J_0(\sqrt{-2j}R/\delta)} = J_S \frac{\mathrm{Ber}(\sqrt{2}r/\delta) + j \mathrm{Bei}(\sqrt{2}r/\delta)}{\mathrm{Ber}(\sqrt{2}R/\delta) + j \mathrm{Bei}(\sqrt{2}R/\delta)},
where Failed to parse (Cannot write to or create math output directory): J_0(x)
is the zeroth order Bessel function of the first kind, and Failed to parse (Cannot write to or create math output directory): \mathrm{Ber}(x)
and Failed to parse (Cannot write to or create math output directory): \mathrm{Bei}(x)
are Kelvin functions.
The total current in the wire may be found by integrating Failed to parse (Cannot write to or create math output directory): J(r)
from 0 to Failed to parse (Cannot write to or create math output directory): R
. It may more easily be found by relating it to the derivative of the electric field at the surface of the wire via its magnetic field. Ampere's Law at the wire surface gives an azimuthal magnetic field
- Failed to parse (Cannot write to or create math output directory): H(R) = \frac{I}{2 \pi R}
Maxwell's Equations in cylindrical coordinates gives
- Failed to parse (Cannot write to or create math output directory): H(r) = \frac{1}{j \omega \mu} \frac{dE}{dr}
where the electric field Failed to parse (Cannot write to or create math output directory): E
points in the direction of the current. Equating these two functions at Failed to parse (Cannot write to or create math output directory): r = R gives
- Failed to parse (Cannot write to or create math output directory): I = - \frac{2 \pi R \delta J_S}{\sqrt{-2j}} \frac{J_0'(\sqrt{-2j}R/\delta)}{J_0(\sqrt{-2j}R/\delta)}
where the prime on the Failed to parse (Cannot write to or create math output directory): J_0
in the numerator indicates a first derivative, and we have used Failed to parse (Cannot write to or create math output directory): J(r) = \sigma E(r)
. The impedance in the wire is given by
- Failed to parse (Cannot write to or create math output directory): Z = \frac{E(R)}{I} = R' + j \omega L',
where Failed to parse (Cannot write to or create math output directory): R'
and Failed to parse (Cannot write to or create math output directory): L' are the resistance and inductance per unit length of the wire. Plugging in for Failed to parse (Cannot write to or create math output directory): E(R) and Failed to parse (Cannot write to or create math output directory): I gives
- Failed to parse (Cannot write to or create math output directory): Z = \frac{j R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R}) + j \mathrm{Bei}(\tilde{R})}{\mathrm{Ber}'(\tilde{R}) + j \mathrm{Bei}'(\tilde{R})}
- Failed to parse (Cannot write to or create math output directory): R' = \frac{R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R})\mathrm{Bei}'(\tilde{R}) - \mathrm{Bei}(\tilde{R})\mathrm{Ber}'(\tilde{R})}{\mathrm{Ber}'(\tilde{R})^2 + \mathrm{Bei}'(\tilde{R})^2}
- Failed to parse (Cannot write to or create math output directory): \omega L' = \frac{R_0}{\sqrt{2} \pi R} \frac{\mathrm{Ber}(\tilde{R})\mathrm{Ber}'(\tilde{R}) + \mathrm{Bei}(\tilde{R})\mathrm{Bei}'(\tilde{R})}{\mathrm{Ber}'(\tilde{R})^2 + \mathrm{Bei}'(\tilde{R})^2}
where the fundamental resistance Failed to parse (Cannot write to or create math output directory): R_0
and unitless scaled "radius" Failed to parse (Cannot write to or create math output directory): \tilde{R}
are given by
- Failed to parse (Cannot write to or create math output directory): R_0 = \frac{1}{\sigma \delta}
and
- Failed to parse (Cannot write to or create math output directory): \tilde{R} = \frac{\sqrt{2}R}{\delta}.
[edit] Mitigation
A type of cable called litz wire (from the German litzendraht, braided wire) is used to mitigate the skin effect for frequencies of a few kilohertz to about one megahertz. It consists of a number of insulated wire strands woven together in a carefully designed pattern, so that the overall magnetic field acts equally on all the wires and causes the total current to be distributed equally among them. This has the effect of reducing the effective permeability and increasing the skin depth.[1]
Litz wire is often used in the windings of high-frequency transformers, to increase their efficiency by mitigating both skin effect and, more importantly, proximity effect.
Large power transformers are wound with stranded conductors of similar construction to litz wire, but of larger cross-section. [2]
High-voltage, high-current overhead power transmission lines often use aluminum cable with a steel reinforcing core, where the higher resistivity of the steel core is largely immaterial.
In other applications, solid conductors are replaced by tubes, which have the same resistance at high frequencies but lighter weight. Very recently, researchers have been able to create extremely light cell-phone antennae using carbon-nanotubes[3], their performance attributed to Skin effect.
Solid or tubular conductors may also be silver-plated providing a better conductor (the best possible conductor except for superconductors) than copper on the 'skin' of the conductor. Silver-plating is most effective at VHF and microwave frequencies, because the very thin skin depth (conduction layer) at those frequencies means that the silver plating can economically be applied at thicknesses greater than the skin depth.
[edit] Examples
In copper, the skin depth at various frequencies is shown below.
| frequency | d |
|---|---|
| 60 Hz | 8.47 mm |
| 10 kHz | 0.66 mm |
| 100 kHz | 0.21 mm |
| 1 MHz | 66 µm |
| 10 MHz | 21 µm |
In Engineering Electromagnetics, Hayt points out that in a power station a bus bar for alternating current at 60 Hz with a radius larger than 1/3rd of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than 1/2 inch (12 mm) thick except for mechanical reasons. A thin film of silver deposited on glass is an excellent conductor at microwave frequencies.
[edit] See also
- Proximity effect (electromagnetism)
- skin depth
- "The Skin Effect Myth" for Tesla coils
- Surface wave
- Litz wire
[edit] References
- ^ [1]
- ^ Central Electricity Generating Board (1982). Modern Power Station Practice. Pergamon Press.
- ^ Spinning Carbon Nanotubes Spawns New Wireless Applications
- Hayt, William Hart. Engineering Electromagnetics Seventh Edition. New York: McGraw Hill, 2006. ISBN 0-07-310463-9.
- Nahin, Paul J. Oliver Heaviside: Sage in Solitude. New York: IEEE Press, 1988. ISBN 0-87942-238-6.
- Ramo, S., J. R. Whinnery, and T. Van Duzer. Fields and Waves in Communication Electronics. New York: John Wiley & Sons, Inc., 1965.
- Terman, F. E. Radio Engineers' Handbook. New York: McGraw-Hill, 1943. For the Terman formula mentioned above.
