Cable Impedance
The characteristic impedance of a transmission line is given by Zo = square root of ( L / C ), where L is inductance per unit length, and C is capacitance per unit length. This is the input impedance of either an infinite length of the cable, or more practically of a finite length terminated by an impedance equal to the characteristic impedance.
In the following I mostly ignore resistive losses in the cable, and I assume light speed travel for voltage steps, but in practice we need to multiply this by the 'velocity factor' which could be somewhere between 0.6 and 0.98 depending on choice of dielectric and construction.
For a parallel pair of circular cross-section wires each with diameter d, with separation D between their centres, Zo = 120 cosh-1 ( D / d ) ohms. This is only accurate for a range of frequencies. At very high frequency where the wavelength is comparable to D there may be significant loss of energy by radiation perpendicular to the cable. At low frequencies, e.g. 100 kHz or less, the current in the wires is no longer confined to a thin surface layer (the skin depth), and consequently there is a magnetic field inside the conductor, and the inductance must include the effect of this.
Radio frequency coaxial transmission lines usually have impedance 50 ohms or 75 ohms. There are optimum values for minimum attenuation, maximum power capacity or breakdown voltage. 75 ohms is preferred for low signal applications with poor signal to noise ratios, such as TV aerial cables. Twin wire rf lines are often 300 ohms. Cable impedance is generally some sort of logarithmic function of dimensions such as conductor spacing, and so even large changes in dimensions have little effect on impedance, and if some unusual value such as 4 ohms is required, then the dimensions may need to be inconveniently large or small to achieve this. The common 'zip cord' used as audio speaker cables may have impedance around 100R.
As an example consider a line with characteristic impedance 100R. If this is used to connect an 8 ohm load to an amplifier with 0.1 ohm output resistance, there will be reflection of energy back from the load, and energy will be reflected back and forward between source and load, reducing in amplitude at each reflection. This may seem a serious problem, in audio applications surely transients will be smeared and lose their initial shape. The reason why this is not a problem can be illustrated by looking at the worst possible effect, with a loss-free cable driven by an infinite impedance source, connected to an infinite impedance load, and with a current step input signal having a zero rise-time, starting at zero and stepping up to a small constant current. The result will be that the initial transient will reflect backwards and forwards forever with no loss, and the output voltage across the load must surely be regarded as an extreme distortion of the signal. The following diagram shows what happens:
The output is a continuous series of steps, which appears to have little similarity to the initial single step applied. However, this signal can be regarded as the sum of two individual signals, one a linear ramp, the other a sawtooth wave. For a 3 metre length of cable this sawtooth will have a fundamental frequency of 50MHz.
The ramp is just the signal which would be produced across a single capacitor with the current step applied:
The output from this incorrectly terminated cable therefore differs from the voltage across a single capacitor only by a 50MHz sawtooth. In practice of course there is no such thing as a zero rise-time step, but the result can be used to reach a similar conclusion for any real signal.
It is easy to generalise the result by observing that any arbitrary wave shape can be expressed as an infinite sum of step functions, in the same way that an infinite series of sine waves can be combined via a Fourier integral to generate any wave. For any wave it therefore follows that with high source and load impedances the cable effect is identical to a single ideal capacitor possibly with frequency components at and above 50MHz. Assuming linearity, which is reasonable for copper cables, then if there are no components above 50MHz in the original signal, which is almost certainly true for audio applications, then no high frequency components will appear at the output, and the entire effect is identical to a single capacitor. This is not an entirely complete description because we have left out resistive losses, and also we have only considered high source and load impedances. As it happens AIM-Spice has a lossless transmission line model, so it is easy enough to look at this and other sources and loads.
Here is the open ended line driven by a current step already explained above. The step is 10mA, and for some unknown reason Spice adds small overshoots, but there is otherwise good agreement with the version I worked out prior to trying simulations. The output would continue to increase indefinitely, but for that to happen the current source would need to be able to provide an unlimited output voltage, which real current sources are unable to do, so in reality the 'continuous' ramp would level off eventually at some finite voltage.
The next example shows the output of a 100R lossless line with a 1k load driven by a zero impedance generator producing a 1V voltage step with 1ns rise time. Again the line length is 3m, but in Spice the time delay is specified instead of length and here it is 10ns. Because the load is not correctly matched to the 100R line energy reflects back and forward. The incident and reflected waves add at the high impedance load, but the wave reflected back to the source gets inverted there by the low impedance signal source and so is subtracted, so the original 1V is alternately increased and decreased, producing a 'square wave' which in this example has frequency 25MHz. Because of the resistive load there is less than 100% reflection and the square wave is rapidly attenuated leaving just a constant 1V. Had there been no load the output would have alternated between 2V and zero indefinitely, but of course real lines are not lossless so that would be unrealistic.
The next example is more closely related to a typical speaker cable, with a 100R line feeding a 10R load. Here the relatively low load causes an inverted reflection but only sufficient to reduce the voltage to about 0.2V, and the reflected wave is again reflected and inverted at the signal source, so it once more has positive polarity when it reaches the load a second time, and the series of ever reducing reflections produce an increasing output which eventually levels off at 1V. If we filtered out the components at and above 50MHz we would get a smooth curve. Again the band-limited result can be achieved with a single reactive component, this time an inductor about 1uH in series with the load as shown in the second of the following diagrams. In both cases, either cable or inductor, it takes about 0.1usec for the output to get to 63% of the final 1V level, and in the case of the inductor this is equal to the time constant, L/R = 0.1 usec. The cable of course has a time delay of 10ns for the initial step to travel along the line before there is any output.
Using either a source or load impedance of 100R prevents these continuous reflections and we get a single step output rising to its maximum in a time equal to the rise time of the input step. The next example is just a 100R load. Here a 10mA current step is used, but the result is similar for a 1V voltage step input.
We could alternatively prevent reflections with a 100R source impedance, but we really don't want to do this to drive a 10R load because although there is a single step it is highly attenuated as shown next.
So, the reflections in speaker cables have typically the same effect as a series inductor, but what is the effect on frequency response and phase shift? These are shown next for a 8R load, and it can be seen that the frequency response is almost perfectly flat, and the phase shift reaches only -0.9 degrees at 20kHz.
The phase shift looks even less of a problem if we look at it on a linear frequency scale, then as seen next it is an almost perfectly straight line. A phase shift proportional to frequency is equivalent to a constant time delay, in this case 125 nsec, which is also equal to the time constant L/R for 1uH and 8R.
But what about driving a typical speaker, is the effect then greater? The slightly surprising answer appears to be 'no'. The next diagrams show a much simplified equivalent of a speaker load and the resulting amplitude and phase responses. Even the very small phase shifts seen here have more to do with the 0.01 ohm output impedance of the amplifier than the line, reducing the output impedance to zero gives almost perfectly flat responses. It should really be no surprise that the cable has no noticeable effect if we recall that the equivalent series inductance found when driving 8R was 1uH, but the speaker equivalent circuit used already has a series inductance of 2mH, i.e. 2000 times higher. More complex speakers with crossover networks may be less inductive at high frequencies, so closer to the 8R resistive load response.
The conclusion is that only the equivalent capacitance or inductance of the line, plus resistive losses, are important at audio frequencies, and unless one of these is unusually high there will be no significant effect. Matching the cable characteristic impedance to a nominally 8 ohm load may lead to inconvenient dimensions, and could have little if any point when a typical speaker may have impedance varying between 5R and 20R or more in the audio frequency range. There may be some advantage to adding a RC Zobel network at one or both ends of the line. This may have no significant effect at audio frequencies but it could help suppress interference pickup at radio-frequencies. There is also the question of amplifier stability, not all amplifiers are unconditionally stable with all combinations of speaker and cable. If there are any real audible differences when different cables are tried I believe these are most likely the results of either interference pickup or amplifier stability problems.
Footnote.
There are plenty of differing views to be found, not all emanating from those with cables to sell. One interesting example is by Cyril Bateman, Cables, Amplifiers and Speaker interactions. I have ignored resistive losses on the grounds that including them adds nothing useful for audio speaker cable applications, but Bateman says the resistive losses R and G are dominant at low frequencies, which I am not entirely convinced by. Then again, that 0.01 ohm amplifier output impedance did have more effect than the lossless cable for the simulated speaker load.
To see how unimportant the cable effects on phase are in practical use we need to look at the phase shift as a function of frequency for a typical speaker. We can get some idea how bad this is by looking at the square wave response. Here's one I did earlier:
This is a 'square wave' from a single full-range drive unit. There are certainly better responses, the Quad ESL was famous for producing something at least recognisable as a square wave, but anyway it appears a fraction of a degree phase error added by the speaker cable is the least of our worries.
