Skin Depth

This page about skin depth is based on the explanation in 'The Feynman Lectures on Physics, vol.2' by R.P.Feynman, (Addison-Wesley 1967) Chapter 32, which involves the refractive index of the metal. There is an explanation in Wikepedia in terms of circulating eddy currents cancelling the current at the centre of a wire, but I think that is not the best way to understand it.

The skin depth effect in metals such as copper is usually thought of as a high frequency effect, but of course it depends what you mean by 'high'. What matters is the refractive index of the metal. Well above a certain frequency known as the 'plasma frequency' the refractive index is real, and the metal is transparent. Below that frequency the index has an imaginary component, and because of this an e-m wave will be attenuated as it travels through the metal. For copper the plasma frequency is about 1012Hz, so for transmission line problems we are only concerned with the 'low frequency' behaviour.

In the case of a twin wire transmission line carrying a signal there will be resistive losses in the wires, and to replace this loss energy must be supplied from the surrounding field. To accelerate the conduction electrons there must be a component of the electric field, E, parallel to the wire, and the magnetic field, B, is in closed loops round the cable. The direction of flow of energy is given by the 'Poynting Vector' which is proportional to the vector cross-product ExB and this has a component into the wire at its surface. To maintain a uniform current distribution throughout the wire the energy must reach the centre of the wire with little attenuation, and it is the refractive index of the metal which determines the rate of attenuation.

To summarise the effect, the refractive index at low frequencies is:

Refractive index, n = ( 1-i ).sqrt( k/f )
'sqrt' means the square root, k is a constant, f is frequency in Hz, and i is the square root of -1.
For copper the value of k is about 5 x 1017 sec.-1.

The electric field E in the conductor is given by E = Eo exp (iw(t - nx/c))
where w is angular frequency, Eo is the peak electric field component along the wire at its surface, t is time, and x is the distance into the metal.

The imaginary component of n multiplied by i gives a real exponential term, and so the field level falls exponentially inside the metal, and the skin depth is conventionally defined as the depth at which the field has fallen to e-1, about 0.37, of its original level. The value of n is a function of the frequency, and putting this into the equation for the electric field gives a value for the skin depth inversely proportional to the square root of the frequency. In other words, increasing the frequency by a factor of 4 reduces the skin depth by half.

To be more precise we should take account of the shape of the conductor, for example a cylindrical conductor will have a different skin depth effect to a wide flat conductor. The solution is given around page 20 of the following link:
Skin depth of Cylindrical Conductor. This involves the solution of Bessel's equation of order zero. The following ignores such details.

The skin depth of copper at 20kHz is about 0.47mm, and for the resistance of the wire at 20kHz to be double the d.c. value the radius must be 1.4mm. (This applies to solid core wires, and for lower radius wires the skin depth gives a lower relative increase in resistance, and so is less significant.) The d.c. resistance per metre is about 0.0028 ohms, so the 20kHz resistance is 0.0056 ohms, half of this contributed by the skin effect.

To take an extreme example, suppose 10 metres of this wire is used in a speaker cable to drive a speaker with impedance as low as 2 ohms at 20kHz ( fortunately very rare ). The cable will have two conductors giving total added skin effect resistance of 0.056 ohms. In series with the 2 ohm speaker resistance this gives attenuation by a factor 0.97, i.e. -0.24 dB.

I studied the account in the Feynman Lectures over 40 years ago, but I only recently learnt that the skin depth is just the wavelength of the signal divided by 2 pi. The relevant wavelength is inside the conductor, and for example in copper at 60Hz the skin depth is 8mm and the wavelength is just 5cm. The wavelength in a vacuum is 5000km, and the ratio of wavelengths tells us the refractive index of copper at 60Hz is around 100 million. The velocity of the electromagnetic field into the interior of the conductor is therefore a surprisingly slow 3m/sec. One result of this is that the surface current can change polarity while the current further below the surface is still in the original direction. In other words the current can for a significant time be in different directions at different depths. A good animated plot can be found at Some Skin Effect Notes (But there is at least one error. Equations 5,6 don't combine to give eq.7, there is a factor of c missing somewhere, but I hate 'Gaussian units' so I'm not checking further, the plots still look ok.) Note that the magnetic field H lags the current density J by 45 degrees. Note also that the animation looks the same for a wide range of frequencies because of the fixed ratio between skin depth and wavelength, and because the horizontal axis is in multiples of skin depth rather than actual distance.
It may be tempting to think this must cause problems in audio cables, if part of the current is determined by what happened maybe as much as a msec earlier would that not 'smear transients'? I think that was actually suggested in a published article some years ago (Hawksford?) The reason why it is not a problem I am sure has been explained somewhere, but I will try to give a simple explanation here.

We can think of the slow 'diffusion' of the field into the conductor as just the effect of the internal inductance of the wire, and a good source of information about this is Electromagnetic Waves In Matter which covers both sinewaves and transients, but I don't see any explanation there for why the delayed and possibly reversed current at increasing depths has little or no effect on the signal.
Looking again at the animated plot at Some Skin Effect Notes the points on the plots at the left hand side oscillating between +1 and -1 represent the field levels at the surface and outside the conductor, and that field may travel close to light-speed carrying energy to the load at the end of the cable, while the internal variations travel into the wire at much lower speed, e.g. 3m/sec for a 60Hz signal.
The total current can be found by integrating the current density through a cross-section, and the time delays and current reversals may seem to complicate this, but one of Maxwell's equations, for the curl of the magnetic field H, tells us that the line integral of H round the circumference of the wire is proportional to the instantaneous total current along the wire, so the value of H at the surface tells us that the total current including delays and reversals is proportional to the surface current with just the 45 degree phase difference between J and H at the surface mentioned previously. The animation assumes a wire of large diameter compared to the skin depth, which is rarely the case in audio frequency applications, so the less delayed current near the left side of the plot is all we are concerned about and the resulting phase shift may be far less than 45 deg.
But maybe this is looking the wrong way round, the current at some depth is determined by what the field was doing some time earlier, so in terms of cause and effect it may be more correct to say the field changes the current to make it obey the equation. The field however can only change the current at the surface, it can no longer change the internal field and current which are slowly travelling into the interior. The field therefore changes the surface current in such a way that the total current including delayed lower levels matches the external field, and so the total current and the external field are correct representations of the instantaneous signal level at that location, and there are no delayed effects distorting the field, only the small phase shifts mentioned earlier.

A possibly important question is whether the added phase shift is a linear function of frequency so that there is a constant time delay. An interesting link covering this question is Effects of wire diameter and spacing where Fig 14 shows that group delay is virtually flat up to 25kHz for 1mm dia wires, but is not so flat for 2mm or greater, but even then the change in delay at 25kHz is mostly under 50nsec, so fairly harmless. The results shown there are for a 3m length of cable driving a 8R load, but the derivation is not shown, so it is not entirely clear which effects have been included.

A frequently referenced treatment of twin wire effects by Butterworth from the 1930s is allegedly wrong, a possibly more accurate treatment from 1971 is The Proximity Effect In Systems Of parallel Conductors..., but it's heavy reading and includes a program listing in Fortran IV; 'for use on the IBM 360 computer', which I used myself back in 1968 in a computing course. Happy memories.

I maybe need to mention that exact solutions of this sort of problem are often extremely difficult, and explanations of the sort I attempt here are usually over-simplistic and only close to the truth under a limited range of conditions, so it may in practice often be easier to just measure what happens under the conditions of interest.

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