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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. Real cables have resistive losses which must also be taken into account.

Radio frequency coaxial transmission lines invariably have impedance 50 ohms or 75 ohms. These values are compromises, there being certain 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 usually 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 characteristic impedance can be worked out theoretically, the method I once had to learn is called 'conformal transformation', but fortunately there are only a few common arrangements of conductors and the relevant equations are well known. 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, as explained in the transmission line article. 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 an effect from this.

As an example, a pair of 1mm diameter wires with 2mm between their centres will have characteristic impedance 158 ohms. 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 cables, then no high frequency components will be included, 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. At other impedances the cable effect could be inductive or resistive, but in any case reflections are not a problem at audio frequencies.

The conclusion is that only the equivalent capacitance, inductance or resistance of the line are important at audio frequencies, and unless one of these is unusually high there will be no significant effect. Matching the 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.