Reference

The Essex Echo 1995: Electrical Signal Propagation & Cable Theory Page 5

The detailed characteristics of the dielectric material are also important, as the model shows that the dielectric supports the majority of the signal during its transportation along the cable (which can take many passes if the cable is not optimally terminated). Dielectric loss has been cited as a contributory factor, which can be modeled as an equivalent frequency-dependent but low-conductivity σd, where σd = ωε (power factor). Power factors vary from typically 0.0005 to 0.05 (see Skilling [2]). The attenuation and phase constants then follow, as αd = 0.05ωε√µ (Power factor) and βd = ε√µ. However, it is difficult to see how these results affect audio cables from this simplistic appraisal. A more detailed study of the permittivity of dielectrics, some of which are nonlinear, is required.

Conclusions
The basic elements of our model are now complete, wherein we propose that the internal loss fields that propagate within the conductors are at least partially responsible for some claimed anomalies. The points to emphasize are as follows:

1: The loss field propagates at right angles to the axis of the cable; ie, radially into the conductors.

2: The loss field gives rise to the corresponding internal current distribution along the axis of the conductor (Jσ = σEσ). Note for the loss component that, although the direction of propagation is radial, the E-Barσ field is at right angles to the direction of propagation of the radial loss wave, and is along the conductor axis. This induces an axial conduction current, and is the component of current normally experienced.

3: The velocity of propagation within the conductor (copper) is both very slow and frequency-dependent. As a consequence, different frequencies propagate at different velocities; ie, the material is highly dispersive.

4: The velocity of the loss field is directly dependent upon σ and µ, which should be noted for magnetic materials. Usual analysis assumes σ to be a smooth and continuous function. However, crystal boundaries suggest discontinuities in σ, such that the conductors appear more like stranded, though disjointed wire, where such discontinuity represents a point of at least partial reflection and field redistribution. It is anticipated that if the microcurrents within the conductor could be observed, they might well show aspects of chaotic activity.

5: There is a problem even if σ is a linear but discontinuous function. However, nonlinearity due to partial semiconductor diode boundaries could lead to a complicated, frequency-interleaved intermodulation.

6: Stranded conductors without individual strand insulation appear to be a poor construction when viewed by this model, as the loss field propagates against the strands and experiences discontinuities in air/copper boundaries that are inevitably random. This is comparable to a large-scale granularity where crystal boundaries possibly represent a similar structure at the microlevel, but within the copper. A single strand of large-crystal copper or multiple strands of insulated wire—the quality of this dielectric will be important—will behave more as a simple impedance.

Conventional theory and actual conductor performance merge: At a diameter of around 0.8mm, the conductor becomes closer to a low-valued ideal resistor at audio frequencies.

7: Irregularities in cable construction and directional wave properties in the dielectric could lead to differences in the E-Barσ field patterns, hence current distribution within the conductors depending upon which end is the source. (Perhaps current vortices form, like whirlpools in a stream of water? The Chaos model?)

The exact nature of the loss field would in principle exhibit differences, thus allowing the cable to have a directional characteristic in that the error is not mirror-symmetric. Slight variations in diameter, for example—or, indeed, internal crystal structure—might well occur in manufacture due to a mechanical stress field. Such effects, however, would appear to lie in the domain of errors of errors. They would necessarily be of an extremely subtle nature.

8: Since boundary conditions, hence the loss field, are indirectly affected by all materials within the cable construction including surface oxidation of the conductor, we would expect each element to contribute to performance.

9: The time taken for the field to propagate to the skin depth δ is longer at low frequencies. Thus, thick conductors would appear more problematic at low frequencies, showing a greater tendency to time dispersion; although, to counter this, the overall error is lower.

10: δ = √[2/(ωµσ)). Magnetic conductors therefore have µ-dependent skin depths, and µ will exhibit nonlinear hysteresis. This needs investigation, as it suggests magnetic conductors should be avoided.

11: It appears cable defects have their greatest effects under transient excitation rather than within the pseudo steady-state of sustained tones. Transient edges are effectively time-smeared or broadened (albeit by a small amount), where this dispersion is a function of both the signal and the properties and dimensions of the conductors.

Amplitude frequency-response errors in the steady-state are at a level that is insignificant when listening to steady-state tones. However, their significance when mapped via the error function onto transient signals may well be of greater concern, with possibly a greater significance in stereo listening. In this sense, we support comments made by John Atkinson (footnote 5) on the importance of maintaining transient integrity at the beginning and end of sequences of sound, rather than worrying about slight relative level errors in the pseudo–steady-state of a sustained tone, or a slight change in harmonic balance.

It's the old story of measuring a system's frequency and phase responses with insufficient accuracy to extract the true system error, and then misinterpreting the significance of that error: check out the error function (footnote 6).

12: Axial propagation within the dielectric is usually not considered important at audio frequencies, as interconnects are generally much shorter than a wavelength, even at 20kHz. However, we have directed our attention to the loss field within the conductors, where, due to the slow velocity, cable dimensions comparable to the wavelength are significant. It is suggested that this viewpoint is usually not considered, where "skin depth" is rarely presented as a propagation or diffusion phenomenon.

Measurements of some loudspeaker cables have confirmed that skin depth must be considered if an accurate estimate of the series impedance is to be made. This is more noticeable for high-capacitance, low- (external) inductance cables where, proportionally, the impedance-related effects of skin depth are a greater fraction of the total series impedance.

Summary
We conclude from these observations that conductors should be sufficiently thin that only a fraction of a wavelength at the highest audio frequency is trapped within the conductors, although multiple, separately insulated (Litz) strands can usefully lower the series impedance due to the internal field in the conductors. Whether the overall impedance is lowered depends on the conductors' geometry and spacing, both of which define the series inductance due to the external field components.

Interleaving multiple conductors, both in Litz fashion and for send-and-return conductors, can lower susceptibility to external fields (eg, hum pickup) and lower the external series inductive component. This is an interesting balancing trick, especially when combined with EMC requirements. The external propagating fields should be distributed as uniformly as possible over the whole surface of the conductor, and the composite cable should be tightly wrapped to prevent external mechanical vibration from modulating the characteristic impedance (shaking wires, coils, and interconnects in loudspeaker systems, for example).

This article has tried to describe a more rigorous model (finely etched with a little speculation) for cable systems by reviewing some fundamental electromagnetic principles. It is important not to make engineering simplifications too prematurely when evolving a model. We have made some approximations, however, as field patterns can be highly complicated and depend upon cable geometry and internal material behaviour at a molecular level. (I keep thinking of current vortices.) Nevertheless, there is sufficient evidence to suggest that a cable's performance is not as simple as it first appears, often because the operation is viewed too simplistically, and because our notions of lumped circuit elements (discrete R, C, and L) warp our thinking, especially with respect to skin depth.

For me, the most striking observation is the slow, frequency-dependent velocity of a wave traveling in a conductor. Also, high conductivity and permeability make the conductor appear much larger on the inside, and crystal boundaries act as partitions within that space. TARDIS, Transient And Resistance DIStortion—watch out for the Stephen Spielberg version!

Footnote 5: John Atkinson, "Comment," HFN/RR, February & March 1985, Vol.30 Nos.2 & 3.

Footnote 6: Hawksford, ibid.

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