Skin effect

"Skin depth" redirects here. For the depth (layers) of biological/organic skin, see skin.

Skin effect is the tendency of an alternating electric current (AC) to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an average depth called the skin depth. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor. The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth may be much smaller. Increased AC resistance due to the skin effect can be mitigated by using specially woven litz wire. Because the interior of a large conductor carries so little of the current, tubular conductors such as pipe can be used to save weight and cost.

Cause

Skin depth is due to the circulating eddy currents (arising from a changing H field) cancelling the current flow in the center of a conductor and reinforcing it in the skin.

Conductors, typically in the form of wires, may be used to transmit electrical energy or signals using an alternating current flowing through that conductor. The charge carriers constituting that current, usually electrons, are driven by an electric field due to the source of electrical energy. An alternating current in a conductor produces an alternating magnetic field in and around the conductor. When the intensity of current in a conductor changes, the magnetic field also changes. The change in the magnetic field, in turn, creates an electric field which opposes the change in current intensity. This opposing electric field is called “counter-electromotive force” (counter EMF). The counter EMF is strongest at the center of the conductor, and forces the conducting electrons to the outside of the conductor, as shown in the diagram on the right.

An alternating current may also be induced in a conductor due to an alternating magnetic field according to the law of induction. An electromagnetic wave impinging on a conductor will therefore generally produce such a current; this explains the reflection of electromagnetic waves from metals.

Regardless of the driving force, the current density is found to be greatest at the conductor's surface, with a reduced magnitude deeper in the conductor. That decline in current density is known as the skin effect and the skin depth is a measure of the depth at which the current density falls to 1/e of its value near the surface. Over 98% of the current will flow within a layer 4 times the skin depth from the surface. This behavior is distinct from that of direct current which usually will be distributed evenly across the cross-section of the wire.

The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalised to conductors of any shape by Oliver Heaviside in 1885. The 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 even at mains frequencies (50 – 60 Hz) in AC electrical power transmission and distribution systems. Although the term "skin effect" is most often associated with applications involving transmission of electrical currents, the 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.

Formula

The AC current density J in a conductor decreases exponentially from its value at the surface J_S according to the depth d from the surface, as follows:

$J=J_\mathrm{S} \,e^{-{d/\delta }}$

where δ is called the skin depth. The skin depth is thus defined as the depth below the surface of the conductor at which the current density has fallen to 1/e (about 0.37) of J_S. In normal cases it is well approximated as:

$\delta=\sqrt{{2\rho }\over{\omega\mu}}$.

where

ρ = resistivity of the conductor

ω = angular frequency of current = 2π × frequency

μ = absolute magnetic permeability of the conductor^[1]

A more general expression for skin depth which is more exact in the case of poor conductors (non-metals) at high frequencies is:^[2][3]

$\delta= \left({1 \over \omega}\right) \left\lbrace \left( {{\mu\epsilon} \over 2}\right) \left[ \left(1+\left({1 \over {\rho\omega\epsilon}}\right)^2\right)^{1/2} -1\right]\right\rbrace^{-1/2}$

where $\epsilon$ is the electric permittivity of the material. Note that in the usual form for the skin effect, above, the effect of $\epsilon$ cancels out. This formula is valid away from strong atomic or molecular resonances (where $\epsilon$ would have a large imaginary part) and at frequencies which are much below both the material's plasma frequency (dependent on the density of free electrons in the material) and the reciprocal of the mean time between collisions involving the conduction electrons. In good conductors such as metals all of those conditions are insured at least up to microwave frequencies, justifying this formula's validity.

This formula can be rearranged as follows to reveal departures from the normal approximation:

$\delta= \sqrt{{2\rho }\over{\omega\mu}} \; \; \sqrt{ \sqrt{1 + \left({\rho\omega\epsilon}\right)^2 } + \rho\omega\epsilon}$

At frequencies much below $1/\rho \epsilon$ the quantity inside the radical is close to unity and the standard formula applies. For instance, in the case of copper this would be true for frequencies much below 10¹⁸ Hz.

However in very poor conductors at sufficiently high frequencies, the factor on the right increases. At frequencies much higher than $1/\rho \epsilon$ it can be shown that the skin depth, rather than continuing to decrease, approaches an asymptotic value:

$\delta \approx {2 \rho} \sqrt{\epsilon \over \mu} \qquad (\omega \gg 1/\rho \epsilon)$

This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much 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=3000m). However as the frequency is increased well into the megahertz range, its skin depth never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors such as undoped silicon, the skin effect doesn't need to be taken into account in most practical situations: any current is equally distributed throughout the material's cross-section regardless of its frequency.

Resistance

The effective resistance due to a current confined near the surface of a large conductor (much thicker than δ) can be solved as if the current flowed uniformly through a layer of thickness δ based on the DC resistivity of that material. We can therefore assume a cross-sectional area approximately equal to δ times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter D large compared to δ, has a resistance approximately that of a hollow tube with wall thickness δ carrying direct current. Using a material of resistivity ρ we then find the AC resistance of a wire of length L to be:

$R\approx {{L \rho} \over {\pi (D-\delta) \delta}} \approx {{L \rho} \over {\pi D \delta}}$

The final approximation above assumes $D \gg \delta$.

A convenient formula (attributed to F.E. Terman) for the diameter D_W of a wire of circular cross-section whose resistance will increase by 10% at frequency f is:

$D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}$

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.

Material effect on skin depth

In a good conductor, 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 remains lower even with the reduced skin depth. However this means that there is less reduction in A.C. resistance when substituting a metal of higher conductivity, compared to the reduction of D.C. resistance, when its diameter is larger than the skin depth for that frequency.

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. However being ferromagnetic its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220 micrometres at 60 Hz. Iron wire is thus useless for A.C. power lines. The skin effect also reduces the effective thickness of laminations in power transformers, increasing their losses.

Iron rods work well for direct-current (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 as current flows through the greatly increased A.C. resistance resulting from the skin effect, with relatively little power remaining for the arc itself. Only non-magnetic rods can be used for high-frequency welding.

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. With the skin effect having little effect on each of the thin strands, the bundle does not suffer the same increase in AC resistance that a solid conductor of the same cross-sectional area would due to the skin effect.^[4]

Litz wire is often used in the windings of high-frequency transformers to increase their efficiency by mitigating both skin effect and proximity effect. Large power transformers are wound with stranded conductors of similar construction to litz wire, but employing a larger cross-section corresponding to the larger skin depth at mains frequencies.^[5] Conductive threads composed of carbon nanotubes^[6] have been demonstrated as conductors for antennas from medium wave to microwave frequencies. Unlike standard antenna conductors, the nanotubes are much smaller than the skin depth, allowing full utilization of the thread's cross-section resulting in an extremely light antenna.

High-voltage, high-current overhead power transmission lines often use aluminum cable with a steel reinforcing core; the higher resistance of the steel core is of no consequence since it is located far below the skin depth where essentially no AC current flows. In other applications, solid conductors are replaced by tubes, completely dispensing with the inner portion of the conductor where little current flows. This hardly affects the AC resistance but considerably reduces the weight of the conductor.

Solid or tubular conductors may also be silver-plated to take advantage of silver's higher conductivity. 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. Silver or gold plating is similarly used on the surface of waveguides used for transmission of microwaves. This reduces attenuation of the propagating wave due to resistive losses affecting the accompanying eddy currents; the skin effect confines such eddy currents to a very thin surface layer of the waveguide structure. The skin effect itself isn't actually combated in these cases, but the distribution of currents near the conductor's surface makes the use of precious metals (having a lower resistivity) practical.

Examples

Skin depth vs. frequency for some materials, red vertical line denotes 50 Hz frequency:
Mn-Zn - magnetically soft ferrite
Al - metallic aluminium
Cu - metallic copper
steel 410 - magnetic stainless steel
Fe-Si - grain-oriented electrical steel
Fe-Ni - high-permeability permalloy (80%Ni-20%Fe)

We can derive a practical formula for skin depth as follows:

$\delta=\sqrt{{2\rho }\over{(2 \pi f) (\mu_0 \mu_r)}} \approx 503\,\sqrt{\frac{\rho}{\mu_r f}}$

where

δ = the skin depth in metres

μ_r = the relative permeability of the medium

ρ = the resistivity of the medium in Ω·m, also equal to the reciprocal of its conductivity: ρ = 1 / σ (for Copper, ρ = 1.68×10^-8 Ω·m)

f = the frequency of the current in Hz

Gold is a good conductor with a resistivity of 2.44×10^-8 Ω·m and is essentially nonmagnetic: μ_r = 1, so its skin depth at a frequency of 50 Hz is given by

$\delta = 503 \,\sqrt{\frac{2.44 \cdot 10^{-8}}{1 \cdot 50}}= 11.1\,\mathrm{mm}$

Lead, on the other hand, is a relatively poor conductor (among metals) with a resistivity of 2.2×10^-7 Ω·m, about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or $\sqrt{9} = 3$ times that of gold.

Highly magnetic materials have a reduced skin depth due to their large permeability μ_r as was pointed out above for the case of iron, despite its poorer conductivity. A practical consequence is seen by users of induction cookers, where some types of stainless steel cookware are unusable because they are not ferromagnetic.^[7]

At very high frequencies the skin depth for good conductors becomes tiny. For instance, the skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a micron:

Conductor	Skin depth (μm)
Aluminum	0.80
Copper	0.65
Gold	0.79
Silver	0.64

Thus at microwave frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver 3 μm thick evaporated on a piece of glass is thus an excellent conductor at such frequencies.

In copper, the skin depth can be seen to fall according to the square root of frequency:

Frequency	Skin depth (μm)
60 Hz	8470
10 kHz	660
100 kHz	210
1 MHz	66
10 MHz	21
100 MHz	6.6

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 one-third of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons.

Skin effect reduction of the self inductance of a conductor

Refer to the diagram below showing the inner and outer conductors of a coaxial cable. Since the skin effect causes a current at high frequencies to flow mainly at the surface of a conductor, it can be seen that this will reduce the magnetic field inside the wire, that is, beneath the depth at which the bulk of the current flows. It can be shown that this will have a minor effect on the self inductance of the wire itself; see Skilling^[8] or Hayt^[9] for a mathematical treatment of this phenomenon.

Note that the inductance considered in this context refers to a bare conductor, not the inductance of a coil used as a circuit element. The inductance of a coil is dominated by the mutual inductance between the turns of the coil which increases its inductance according to the square of the number of turns. However when only a single wire is involved, then in addition to the "external inductance" involving magnetic fields outside of the wire (due to the total current in the wire) as seen in the white region of the figure below, there is also a much smaller component of "internal inductance" due to the magnetic field inside the wire itself, the green region in figure B. In a single wire the internal inductance becomes of little significance when the wire is much much longer than its diameter. The presence of a second conductor in the case of a transmission line requires a different treatment as is discussed below.

Due to the skin effect, at high frequencies the internal inductance of a wire vanishes, as can be seen in the case of a telephone twisted pair, below. In normal cases the effect of internal inductance is ignored in the design of coils or calculating the properties of microstrips.

Inductance per length in a coaxial cable

Let the dimensions a, b, c be the inner conductor radius, the shield (outer conductor) inside radius and the shield outer radius respectively, as seen in the crossection of figure A below.

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, blue=current coming out of the diagram, dashed blue lines with arrowheads=magnetic flux (B). The width of the dashed blue lines is intended to show relative strength of the magnetic field integrated over the circumference at that radius. The four stages A, B, C and D are non-energized, low frequency, middle frequency and high frequency respectively. 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.

For a given current, the total energy stored in the magnetic fields must be the same as the calculated electrical energy attributed to that current flowing through the inductance of the coax; that energy is proportional to the cable's measured inductance.

The magnetic field inside a coaxial cable can be divided into three regions, each of which will therefore contribute to the electrical inductance seen by a length of cable.^[10]

The inductance $L_{cen} \,$ is associated with the magnetic field in the region with radius $r < a \,$, the region inside the center conductor.

The inductance $L_{ext} \,$ is associated with the magnetic field in the region $a < r < b \,$, the region between the two conductors (containing a dielectric, possibly air).

The inductance $L_{shd} \,$ is associated with the magnetic field in the region $b < r < c \,$, the region inside the shield conductor.

The net electrical inductance is due to all three contributions:

$L_{total} = L_{cen} + L_{shd} + L_{ext}\,$

$L_{ext} \,$ 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:

$L/D = \frac{\mu_0}{2 \pi} \; ln \! \left( \frac {b}{a} \right) \,$

At low frequencies, all three inductances are fully present so that $L_{DC} = L_{cen} + L_{shd} + L_{ext}\,$

At high frequencies, only the dielectric region has magnetic flux, so that $L_{\infty} = L_{ext}\,$

Most discussions of coaxial transmission lines assume they will be used for radio frequencies, so equations are supplied corresponding only to the latter case.

As the skin effect increases, the currents are concentrated near the outside of the inner conductor (r=a) and the inside of the shield (r=b). Since there is essentially no current deeper in the inner conductor, there is no magnetic field beneath the surface of the inner conductor. Since the current in the inner conductor is balanced by the opposite current flowing on the inside of the outer conductor, there is no remaining magnetic field in the outer conductor itself where $b < r < c \,$. Only L_ext contributes to the electrical inductance at these higher frequencies.

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.

Characteristics of telephone cable as a function of frequency

Representative parameter data for 24 gauge PIC telephone cable at 70°F

Frequency (Hz)	R (ohm/Kft)	L (mH/Kft)	G (uS/Kft)	C(nF/Kft)
1	52.50	0.1868	0.000	15.72
1k	52.51	0.1867	0.022	15.72
10k	52.64	0.1859	0.162	15.72
100k	58.41	0.1770	1.197	15.72
1M	141.30	0.1543	8.873	15.72
2M	196.03	0.1482	16.217	15.72
5M	304.62	0.1425	35.989	15.72

More extensive tables and tables for other gauges, temperatures and types are available in Reeve.^[11] Chen^[12] gives the same data in a parameterized form that he states is usable up to 50 MHz.

Chen^[12] gives an equation of this form for telephone twisted pair: $L(f) = \frac {l_0 + l_{\infty}(\frac{f}{f_m})^b }{1 + (\frac{f}{f_m})^b} \,$

See also

• Proximity effect (electromagnetism)
• Penetration depth
• Eddy currents
• Litz wire
• Transformer
• Induction Cooking
• Induction heating
• Magnetic Reynolds number

Notes

1. ^ The permeability μ can be found from μ_r , the relative permeability of the material by multiplying it by μ₀ , the permeability of free space: $\mu= \mu_0 \cdot \mu_r$.
2. ^ VanderVorst (2006)
3. ^ Jordan (1968, p. 130)
4. ^ Xi Nan; Sullivan, C. R. (2005), "An equivalent complex permeability model for litz-wire windings", Industry Applications Conference 3: 2229–2235, doi:10.1109/IAS.2005.1518758, ISBN 0-7803-9208-6, ISSN 0197-2618, http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F10182%2F32507%2F01518758.pdf%3Farnumber%3D1518758&authDecision=-203 
5. ^ Central Electricity Generating Board (1982). Modern Power Station Practice. Pergamon Press. 
6. ^ "Spinning Carbon Nanotubes Spawns New Wireless Applications". Sciencedaily.com. 2009-03-09. http://www.sciencedaily.com/releases/2009/03/090309121941.htm. Retrieved 2011-11-08. 
7. ^ If the permeability is low, the skin depth is so large that the resistance encountered by eddy currents is too low to provide enough heat
8. ^ Skilling (1951, pp. 157–159)
9. ^ Hayt (1981, pp. 434–439)
10. ^ Hayt (1981, p. 434)
11. ^ Reeve (1995, p. 558)
12. ^ ^{a b} Chen (2004, p. 26)

References

• Chen, Walter Y. (2004), Home Networking Basics, Prentice Hall, ISBN 0130165115 
• Hayt, William (1981), Engineering Electromagnetics (4th ed.), McGraw-Hill, ISBN 0070273952 
• 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.
• Ramo, Whinnery, Van Duzer (1994). Fields and Waves in Communications Electronics. John Wiley and Sons. 
• Reeve, Whitman D. (1995), Subscriber Loop Signaling and Transmission Handbook, IEEE Press, ISBN 0780304403 
• Skilling, Hugh H. (1951), Electric Transmission Lines, McGraw-Hill 
• Terman, F. E. (1943), Radio Engineers' Handbook, New York: McGraw-Hill . For the Terman formula mentioned above.
• Xi Nan; Sullivan, C. R. (2005), "An equivalent complex permeability model for litz-wire windings", Industry Applications Conference 3: 2229–2235, doi:10.1109/IAS.2005.1518758, ISBN 0-7803-9208-6, ISSN 0197-2618, http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F10182%2F32507%2F01518758.pdf%3Farnumber%3D1518758&authDecision=-203 
• Jordan, Edward (1968), Electromagnetic Waves and Radiating Systems, Prentice Hall, ISBN 978-0132499958 
• Vander Vorst, Andre; Rosen, Arye; Kotsuka, Youji (2006), RF/Microwave Interaction with Biological Tissues, John Wiley and Sons, Inc., ISBN 978-0471732778 

External links

• Skin Effect in HiFi Cables
• Skin Effect Relevance in Speaker Cables