The Science of Cables

Electrical measurements are totally irrelevant.

The LCR wire model is applicable only with air as a dielectric. Makes perfect sense in that application. Once that wire is encapsulated in varnish, enamel, lacquer or dielectrics of varying consists then all bets are off. “Wire” then becomes an electrical system that is very different from the raw metal.

Let's see how. Let's start by simply accepting the engineers' favorite factors for cables: R (resistance), C (capacitance), and L (inductance). They say that those things make a difference, and it's true. Among other things, they can affect both the power level passed by a cable and its frequency response.

In thinking about this, remember a couple of things. First, that "resistance" is a term best applied to DC (direct current), and that with AC (alternating current) signals like music, wherever there is capacitive or inductive reactance (as there always is in cables), the more correct term to use is probably "impedance" (Z).

Second, remember that inductance, one of those favored factors, "…is the property of an electrical conductor by which a change in electric current through it induces an electromotive force (voltage) in the conductor." (read more HERE) What that means is that any current flowing through a conductor causes an electromagnetic field to form around that conductor (and out to infinity, in accordance with the "inverse-square law"), and that, when the direction of that current changes (as it does every time the AC current changes polarity) the field collapses and the collapsing field creates a voltage ("back EMF") that opposes the flow of incoming new signal. The higher the frequency, the more of an effect this has on the sound.

Remember, also, that capacitance, another of the "favored factors", is the ability to store energy (read more HERE), and that a capacitor is formed any time two conductors ("plates") are brought together, separated by a non-conductor "dielectric" (read more HERE). What that means is that every cable is, by definition, a capacitor, with its two conductors (positive and negative or "going" and "coming") being the plates, and the insulation between them being the dielectric.

Now, have you ever noticed that, when electronics designers or engineers call for a capacitor to be included in a circuit, they not only specify its capacitance, but also its type? (Ceramic, Film and paper, Polymer, air gap, mica, tantalum, and many, many others (read more HERE and HERE). If factors other than just the measured capacitance of a capacitor are important (and can make a performance difference) in other types of capacitors, how can those exact same things not make a difference in cables?

The amount of capacitance—and of inductance—in any cable or other capacitor is largely determined by how far apart the "plates" are spaced, with those two factors in a sort of "seesaw" balance:  The more capacitance there is, the less inductance, and vice versa.

Another very major factor is the material that the dielectric is made of:  For cables, virtually all of the various non-conductive elements are part of the dielectric. This is important in two ways: The first is that different dielectric materials have different dielectric constants (the ratio of the capacitance of a capacitor in which a particular insulating material is the dielectric, to its capacitance in which a vacuum is the dielectric, read more HERE). Or, to put it most simply,  the dielectric constant of a material is a number that shows how much energy any given volume of it  can store as compared to that same volume of vacuum. By way of illustration, the dielectric constant of a hard vacuum is 1.0, while balsa wood (a little stiff for most cables) is 1.4. Teflon® (there are several varieties) is around 2.0.  Polyethylene is around 2.2, and PVC and TPR (thermoplastic rubber), the two most popular cable insulators, by far, can have dielectric constants of as high as 6.8 (read more HERE).

Whatever the amount of stored energy, when the signal carried by the cable changes polarity, all of that energy is dumped into the signal path, canceling some of the incoming signal, or actually creating out-of-phase artifacts. This results in either the loss of low-level information or the actual creation of new information, either of which would be surprising if it didn't affect the sound!

The other important thing about dielectrics (which, remember, include the non-conducting elements of a cable) is their "dump rate"—how quickly they can release stored energy after the signal changes polarity so that they can start storing the new signal energy coming in. Dump rates vary wildly, with some materials, PVC, for example, being quite (relatively) slow and others (Teflon®, for example) being very, very fast. This can make a definitely audible difference (with faster dump rates that affect the incoming signal for less time being much preferred), and, surprisingly, the dump rate of a material and its dielectric constant are not directly proportional. Polyethylene is only around ten percent higher in dielectric constant than the best form of Teflon®, but has—while still vastly faster than PVC —an audibly slower dump rate, to lose low-level detail and "muddy" its sound.

There are still other factors that affect the sound of a cable—its "geometry"; the type, purity and crystal structure of the metal in its conductors:  the existence of mysterious (and perhaps mythical) micro-diodes at the crystal junctures of copper; the self- and mutual-inductance of both the cable and its connectors; how much and what kind of metal those connectors are made of, and many more, but I'm out of space for now.

I would argue that on a slightly handwaving way, you can study "electricity" using classical physics. After all, much of it can be understood through Maxwell's equations. However, if you do this, you simply have to consider the charge densities, dielectric constant, magnetic permeabilities, et cetera as black boxes.

Quantum mechanics kicks in if you want to understand why a material is a conductor or an insulator. In solids, you can treat these questions through the study of electronic band structures, which relies heavily on Bloch's theorem and also on the Fermi-Dirac statistics. These are all elements from quantum mechanics. Therefore, if you want to understand how these charge densities behave on a microscopic level, you need QM. In typical systems, we can recover more empirical notions of of electricity such as Ohm's law from quantum treatments.

Now, if you really want to study how electromagnetic fields and electrons interact with each other (in detail), you will have to go further and you need to consider quantum electrodynamics (QED). This will give you the most detailed description of how photons and electrons interact. I would argue, however, that QED is often a bit of an overkill for condensed matter or atomic physics problems (not always, but often). Therefore you will find many "effective models", which can significantly simplify things.

Among these effective models, you have for example the Hubbard model, which includes interactions between electrons, without explicitly including the fact that these interactions are mediated via the electromagnetic field. You have a whole zoo of similar models, so I will not go into all of them. The main point is that they usually focus on the behaviour of the electrons and do not explicitly consider couplings to the electromagnetic field. There are, however, models which study the response of the material to the electromagnetic field. Usually this leads you to models which treat quasi-particles, such as plasmons, polarons, and polaritons. I am a bit out of my field of expertise here, but I believe that these can be used to derive the parameters that go into Maxwell's equations. Note, however, that these models are still not explicitly considering full-scale quantum electrodynamics.

As I get the feeling that you are also interested in the radiation side of the story, let me shift gears a little. Radiation actually is a very old problem in quantum physics. It lies at the basis of the probabilistic interpretation of the theory and motivated Heisenberg to develop his matrix mechanics. Light-matter interactions in that time were narrowly connected to atomic physics and spectroscopy, later molecular and nuclear physics joined in, covering a range from microwaves to gamma-rays in the electromagnetic spectrum. Now, if we really want to understand in depth how the electrons (or nucleons for nuclear physics) in these systems interact with electromagnetic fields, we must again divert to QED.

Nevertheless, also on the side of the electromagnetic field, there are effective models. These can be found, for example, in quantum optics. In these models, you typically make serious simplifications on the level of the "matter" and focus on the electromagnetic field. Typically, the interaction between light and matter generates some type on nonlinear effects in the electromagnetic field, so I would argue that the vast majority of models in nonlinear optics are models where you had some type of interaction with matter, which you coarse-grain out. Note, however, that these effective descriptions do not even require quantum mechanics to make sense. You can usually do nonlinear optics using Maxwell's equations. If you want to see effective models of the quantum side of the electromagnetic field in action, you have to turn to quantum optics, where you usually include matter (like a "two-level atom") in a more explicit way, see for example the Jaynes-Cummings model.

With this little excursion into the realm of optics, you may notice that there was not a lot of "electricity". The reason why we did not really get into that, is because it is horribly difficult. The treatment of models in condensed-matter theory, which only deal with the interacting electrons are complicated on their own and so is the theory of the quantum and nonlinear effects in the electromagnetic field. There is, however, one additional playground which we can explore. You may wonder what happens when we consider quantum properties of the electromagnetic field and combine them with macroscopic conductors and insulators. This is done in what is called Macroscopic quantum electrodynamics, which can be used to study for example the Casimir effect.

To conclude, let me stress that genuine quantum effects in the electromagnetic field itself (so everything related to light et cetera) are quite rare in day to day life. The electromagnetic radiation effects that are related to electronics and electricity is described really well by Maxwell's equations. However, if you really want to understand what happens in materials through which your electricity flows, on a microscopic level, you will have to consider the quantum models of condensed-matter physics.

Disclaimer: None of these fields is really my speciality, so I would be happy if a condensed-matter physicist or a quantum optician could provide more details or corrections if necessary.