Electromagnetism. Collected Comments.

This is a collection of pieces from the 'Latest News' pages, edited a little to make them flow together better:

My own speaker cables are an unknown brand, they look like fairly cheap 'zip-cord', which came with the speakers when I bought them secondhand, and I never found any reason to change them. I accept that for a given speaker almost any cable will add some frequency dependent amplitude and phase variations which in extreme cases could be audible, but anything beyond that I suspect is 'marketing'. Some of the more expensive cables probably are 'extreme cases' and really do sound different, or in some cases cause instability in badly designed amplifiers.

There are two types of distortion, linear and non-linear. The non-linear variety adds new frequencies such as intermodulation products, and can be detected with the usual distortion measuring equipment, and if it can be detected at all for a copper cable it is invariably found to be down near the limits of measurement (For example see Cable distortion and dielectric biasing debunked by Bruno Putzeys.)

Various mechanisms have been proposed for cable nonlinearity, ranging from unlikely 'micro-diodes' to known effects such as magnetoresistance. My own view is that there is no point looking for or inventing different explanations unless some reliable measurements can provide data sufficient to distinguish between them. Even for the known effects it is difficult to find any convincing calculations of the distortion levels to be expected in typical copper speaker cables.

Linear distortion adds no new frequencies, so each frequency component can only be changed in level or phase, so frequency dependent amplitude and phase variations are the only effect possible.Some years ago I wrote a piece in my physics section, Cable Impedance, showing how even the worst possible example, with a current step driving a lossless open ended line, which produces a continuously repeated upward step output having no resemblance to the single input step, is just the same effect as a single capacitor for band-limited audio signals. I recently noticed that AIM-Spice includes a lossless transmission line model, so I have extended that page to include a few more examples, together with amplitude and phase plots to show that nothing bad happens within the audio range.

Maybe sometime I will extend the article further to include resistive losses and properties of dielectrics, but I am fairly sure nothing important would be revealed. I will continue to use my cheap zip-cable, but of course anyone using speakers with extreme low impedance dips or amplifiers with stability problems may need to select cables more carefully.I wrote a piece some time ago about conduction in metals, but realise some of that is at least simplified if not actually wrong. The part about skin depth certainly needs improving. It is based on the explanation in Feynman's Lectures Vol.2 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.

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 at times 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. The reason why it is not a problem I am sure has been explained somewhere. I think it should be easy to explain by reference to the animated plot.

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. So, why are the delayed and reversed direction currents at different levels in the wire not a problem?

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.There are other ways of looking at this. The 60Hz example has a time period of 16.7msec, enough time for the internal field to travel 50mm into the wire, if it was thick enough. Using a more typical 1mm diameter wire the 'slow' internal field will reach the centre after 167usec, at which time the surface 60Hz wave has only changed phase by 3.6 degrees. At higher frequencies the refractive index is lower and the internal field travels faster, so it gets to the centre of the wire quicker, but the phase of the surface wave has changed more. Anyway the internal field is not some highly delayed version of the signal, it is just phase shifted a little, with higher phase shift at higher frequency. The internal energy storage with phase shift increasing with frequency is equivalent to just an internal inductance, in effect in series with the external inductance, though not a simple fixed inductance, its value changes with frequency. The reducing skin depth at high frequencies means the internal energy starts to drop when the skin depth becomes comparable to the radius of the wire, so the internal inductance falls. A good account of this internal inductance can be found at Electromagnetic Waves In Matter.

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.

I maybe need to explain why the internal inductance is in effect in series with the external inductance, otherwise we could imagine the internal and external fields are two separate signals travelling along the cable. We could say the internal field is caused by the external field, but there are perhaps equally good reasons to say the external field is caused by the electrons in the wire, so maybe causality arguments are not helpful. Either way the two signals are linked and must travel along the cable at the same speed. The electrons in the wire need not travel along at the speed of the external field, it is only changes in electron density which match the speed of the field.

Also to be explained is how the field travelling into the metal is related to the field travelling along the wire, I wrote something about that in the transmission line page.

I maybe need to mention that exact solutions of this sort of problem are often extremely difficult, and explanations of the sort I sometimes attempt are usually over-simplistic and only close to the truth under a limited range of conditions, so it may in practice be easier to just measure what happens under the conditions of interest.I updated the page about cable skin depth to include the points made earlier on this page. One problem is that I said the surface field is determined by the instantaneous total current through the cable, which implies faster than light communication, how else can the current at the centre of the wire have any effect at the surface without a time delay. Given the low speed of electromagnetic energy inside the conductor the delay could be expected to be significant. I added a few words to try to explain that also, but I need to think more about that. More generally one of Maxwell's equations gives this 'instantaneous' effect, this relates the line integral of the magnetic field round a loop to the total current through the loop. The integral round a loop is equal to the sum of the integrals round two smaller loops or the sum for any number of smaller loops we choose to divide the original loop into. For any two adjoining loops the common boundary is counted twice in opposite directions in adding the integrals, so these all cancel and only the single outer loop counts, and includes the effects of all the inner areas. I also need to add something about the other term in the Maxwell equation which is the rate of change of the electric field, which is sometimes confusingly referred to as 'displacement current'. If the current at some depth in the wire changes then there must be a changing electric field, so this term needs to be included if we want to understand more completely how the surface field is related to a changing internal current.

I remember a quote, probably from Feynman, concerning the laws of electromagnetism, to the effect that 'there are no known non-trivial solutions to these equations'. The static field of a uniform charged sphere is an example of a trivial solution, and this sort of thing is quite common in physics, hence the well known jokes about 'spherical chickens in a vacuum'.

I remember a series of articles in Wireless World many years ago about 'displacement current' in capacitors. Having done a course on classical electromagnetic theory which never mentioned displacement current I was baffled then, and searching through my Physics text books I still found no mention. It appeared that this referred to the dE/dt term in one of Maxwell's equations, but why anyone would want to call this a current I never entirely understood. I thought maybe some quantum mechanical explanation involving vacuum polarisation and relative displacement of virtual electron-positron pairs could be invoked, but the term predates quantum mechanics. Maybe it's just one of those terms invented in the distant past which linger on long after the original motivation has faded. A Google search suggests that the term was invented by Maxwell, who believed it had something to do with stress in the 'aether'. If the capacitor uses a dielectric then there are actual currents because of the displacement of charges, which can be represented by a polarisation vector, but there is still that dE/dt term. The main text book used in my course was Feynman Lectures Vol-2, now available online. The equation first appears in section 1-4 in terms of 'circulation' and 'flux', but it looks wrong, 'the flux of electric current through S' I think should be either 'electric current through S' or 'flux of current density through S'. Curiously the correct equation does appear later in chapter 18, in Table 18.1.

In the piece about transmission lines I mentioned a problem which according to some sources remains unsolved: A uniformly accelerating electron is predicted to radiate electromagnetic energy, but then applying the 'equivalence principle' we would expect a charge held unaccelerated in a gravitational field to also emit radiation, and there is no evidence of that ever happening. I suggested that the solution is that to be equivalent the observer or detector in the gravity version needs to be in free-fall and so is accelerating relative to the charge, and will then observe radiated energy. I thought I was being clever to see that, but it turns out to be well known, e.g. Researchgate.

Anyway, it's still interesting, in the quantum mechanical version a virtual photon in an inertial reference frame can become a real photon for an accelerating observer. This is related to the Unruh Effect and 'Unrhu radiation', which in turn is related to Hawking Radiation from black holes.I was trying to choose a good 170mm speaker drive unit, I was looking at two options, one cheap the other expensive, and two of the differences are that the expensive one includes a cast chassis, and also a 'shorting ring' on the pole 'to reduce inductance'. I'm not sure how that works. If I ever understood this sort of thing it was long ago. I found an explanation involving 'Lenz's law' which I don't understand, it looks wrong:

'Lenz's law states that the direction of an induced e.m.f. will be such that if it were to cause a current to flow in a conductor in an external circuit, then that current would generate a field that would oppose the change that created it.'

Or:

'The minus sign in Faraday’s law of induction is very important. The minus means that the emf creates a current I and magnetic field B that oppose the change in flux — this is known as Lenz’s law'.

I'm sure that minus sign is the result of a sign convention for the positive direction round a loop having first chosen the positive direction through the loop. If we chose the opposite convention then the minus sign would instead appear in another of Maxwell's equations. Also, if the magnetic field changes at a constant rate the emf is constant and the induced current just causes a constant addition to the changing field. 'Opposing the change' doesn't then look like a good description, the rate of change is unaltered. A better wording would perhaps be that it 'adds a field in the opposite direction to the positive rate of change'.

In a speaker with shorting ring there are of course two coils; in effect a transformer with shorted secondary, which is the arrangement often suggested for measuring transformer primary leakage inductance by cancelling the rest of the inductance, so I guess it should work to some extent to reduce speaker coil inductance and associated nonlinearity, as advertised. However, one source claims that some of the lowest distortion drivers available don't have shorting rings.So why do I want to write pages about elementary electromagnetic theory? It was about 50 years ago when I studied the subject in any detail, so I forgot lots of it since then. What worries me is reading things on the internet, and thinking; 'that looks wrong' but not knowing whether it's me or someone else getting it wrong. Some parts I remember well enough and feel confident about, other parts need 'research' looking at multiple sources. It's surprising how much disagreement there is even on basic theory. Anything on audio sites about speaker cables needs approaching with caution, but even Feynman's Lectures had many errors, not all of which have been corrected in the online version, so just accepting anything on the internet is a bad idea, and writing things in short articles as I relearn them fits in with my original intent for the website mentioned at the top of the 'Latest News' page (3rd para).

There are already a few pages of em-theory related material, so maybe I will collect them into a separate section. Some need rewriting first. My page about skin depth still doesn't look entirely convincing, but other treatments I found also seem inadequate. The Wikipedia page is rather odd, e.g. it says the field 'forces the conducting electrons to the outside of the conductor' which must be wrong, or at least badly worded, at most it could accelerate conduction electrons more in some locations than others, the density of conduction electrons must be almost uniform throughout a conductor, a difference of even one part per million between two regions would cause huge fields and forces. I gave an example of the magnitude of charge variations in my transmission line page:

"In a 300 ohm transmission line carrying 1 amp the negative conductor has an excess of conduction electrons of the order of 1 part in 10^{12}...."

I need to check that sometime, I think I assumed 300V between the conductors, about 1cm apart, then one part per million would imply three hundred million volts across 1cm, which I imagine would have a rather dramatic effect.

The 'circulating eddy currents' in the diagram look a bit misleading too, adding them all together there are almost no actual circulating currents or radial currents, adjacent loops cancel along their common edge, apart from some higher order effects which increase at high frequencies. I really need to do some calculations.