Transmission Media: Pair, Coax, Waveguide & Fibre

Every guided medium is the same object — a structure that confines a wave and takes a cut of it per unit length — and yet each one is stopped by a different thing. Twisted pair runs out of signal-to-crosstalk while it still has plenty of signal. Coaxial cable runs out of copper: skin effect makes its loss climb as $\sqrt{f}$ forever, and its impedance is a fixed compromise between the geometry that minimises that loss and the geometry that survives high power. Rectangular waveguide has no loss problem at microwave frequencies but simply will not propagate below a cutoff set by its width. Optical fibre has neither a cutoff nor a crosstalk floor, and after coherent detection removed dispersion from the critical path it is limited by one number: about 0.2 dB per kilometre.

Knowing which of those four failure modes binds is the whole of medium selection. Adding shielding to a cable that is attenuation-limited buys nothing; adding an amplifier to a link that is crosstalk-limited buys nothing; putting a wider waveguide on a link that is already single-mode buys a lower cutoff you did not want.

flowchart LR
  F["Rising frequency"] --> S["Skin effect<br/>R grows as sqrt f"]
  F --> D["Dielectric loss<br/>G grows as f"]
  F --> X["Capacitive and inductive<br/>coupling between pairs"]
  S --> A["Attenuation"]
  D --> A
  A --> M["Shrinking SNR"]
  X --> M
  M --> R["Reach collapses"]

The primary line parameters #

Any two-conductor line — pair, coax, PCB trace — is described by four primary parameters per unit length: series resistance $R$, series inductance $L$, shunt conductance $G$, shunt capacitance $C$. Everything measurable about the line follows from them through the telegrapher's equations. This classification of line types and their primary parameters is the part of the classical transmission-line theory that has not aged at all.

$$\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}, \qquad Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$$

Two regimes matter, and both are visible in real cable specifications.

Below a few hundred kHz the line is RC-dominated: $R \gg \omega L$, so $Z_0 \approx \sqrt{R/(j\omega C)}$ — complex, and rising as frequency falls. This is why long telephone loops need loading coils and why $Z_0$ specifications for data cable only start at 1 MHz.

Above roughly 1 MHz the line is LC-dominated: $\omega L \gg R$ and $\omega C \gg G$, so

$$Z_0 \to \sqrt{\frac{L}{C}} \quad \text{(real, frequency-flat)}, \qquad \alpha \to \underbrace{\frac{R}{2}\sqrt{\frac{C}{L}}}_{\text{conductor}} + \underbrace{\frac{G}{2}\sqrt{\frac{L}{C}}}_{\text{dielectric}}$$

The two loss terms have different frequency laws, and that difference is the single most useful fact about metallic media. Skin effect confines current to a surface layer of depth $\delta = \sqrt{2/\omega\mu\sigma}$, so $R \propto \sqrt{f}$ and the conductor term rises as $\sqrt{f}$. Dielectric loss has $G \propto f$ (constant loss tangent), so the dielectric term rises linearly in $f$. Below a few GHz copper loss dominates and attenuation looks like $\sqrt{f}$; past that the dielectric term takes over and the curve steepens.

That is not a theoretical shape — it is literally how cabling standards write the limit. The ISO/IEC Class E (Category 6) channel insertion loss over 100 m with four connectors is specified as

$$\mathrm{IL}(f) = 1.05\left(1.82\sqrt{f} + 0.0169\,f + \frac{0.25}{\sqrt{f}}\right) + 4 \times 0.02\sqrt{f} \quad \text{dB}, \; f \text{ in MHz}$$

$1.82\sqrt{f}$ is the skin-effect term, $0.0169f$ the dielectric term, $0.25/\sqrt{f}$ a low-frequency correction, the $1.05$ a cable-length allowance and the last term four connectors.

Twisted pair #

It is not the loss that kills it #

Insertion loss and crosstalk move in opposite directions with frequency, and they cross. Insertion loss climbs roughly as $\sqrt{f}$; NEXT — near-end crosstalk, the disturbance coupled onto a victim pair measured at the same end as the offending transmitter — degrades at about 15 dB per decade. Where the two curves meet, the near-end echo of your own transmitter is as strong as the signal arriving from the far end.

The IEEE 802.3an link-segment limits make the crossover explicit. Pair-to-pair NEXT loss is specified as

$$\mathrm{NEXT}(f) = -20\log_{10}\!\left(10^{\frac{65.3 - 15\log_{10} f}{-20}} + 10^{\frac{83 - 20\log_{10} f}{-20}}\right)$$

Evaluating that alongside the Class E insertion loss above:

Insertion-loss figures at 1, 16, 100, 250 and 625 MHz are the published Class E 100 m channel values (the standard applies a low-frequency floor, which is why 1 MHz reads 4.0 dB rather than the 2.3 dB the formula alone gives); 500 MHz is evaluated from the same formula, and the crosstalk columns from the 802.3an link-segment formulas. A Category 6 channel is only specified to 250 MHz — the extension to 625 MHz is the 802.3an link-segment proposal. ACR-N goes negative somewhere between 100 and 250 MHz — the raw near-end echo is larger than the received signal for the whole top half of a Category 6 channel's band.

This is why Category 6A is a physically bigger cable than Category 6 rather than a better-terminated one. You cannot equalise your way out of a neighbour's transmitter; you have to increase the separation between cables, tighten the twist schedules so adjacent cables decorrelate, or screen them.

The categories, and what the numbers actually mean #

The specified-bandwidth column is industry nomenclature: TIA-568.2-D and ISO/IEC 11801 are both paywalled, so those figures are not quoted from the standards themselves. Every reach and application figure in the table, by contrast, comes from IEEE 802.3 working documents read directly.

The two interesting rows are the ones where reach moved.

Category 6 and 10GBASE-T. The 802.3an draft supported Class E to 100 m conditionally — subject to alien-crosstalk-to-insertion-loss requirements — while unscreened Category 6 with no alien-crosstalk mitigation was good for only 55 m. Screened Class E made 100 m with no mitigation at all. The reach of a Category 6 channel at 10 Gb/s is therefore not a property of the cable; it is a property of how the cable was bundled.

Category 8 and 30 m. 25GBASE-T and 40GBASE-T are specified over up to 2 connectors and at least 30 m. Doubling the bandwidth from 1 GHz to 2 GHz costs roughly $\sqrt{2}$ in insertion loss per metre while crosstalk keeps degrading, and the budget simply runs out at a third of the traditional length. Category 8 is a top-of-rack jumper standard, not a horizontal-cabling standard.

Coaxial cable #

Coax removes the crosstalk problem by construction: the return current flows on the inside surface of the outer conductor, and at any frequency where the shield is more than a few skin depths thick, the field outside is decoupled from the field inside. What is left is attenuation, and a geometry choice.

$$Z_0 = \frac{1}{2\pi}\sqrt{\frac{\mu}{\varepsilon}} \, \ln\frac{b}{a} = \frac{60}{\sqrt{\varepsilon_r}} \ln\frac{b}{a} \; \Omega$$

with $a$ the inner conductor radius and $b$ the inner radius of the shield. That single degree of freedom, $b/a$, has to satisfy several conflicting objectives at once.

Why 50, and why 75 #

Minimising attenuation is a genuine optimisation. Widening the gap ($b/a$ up) reduces the current density on the centre conductor but lengthens the field path; narrowing it does the reverse. Differentiating the conductor-loss expression gives a stationary point, and its location depends on whether the outer conductor is much more conductive than the inner or comparably so. Ellingson's derivation gives $b/a = e \approx 2.72$ in one limit and $b/a \approx 3.59$ in the other, i.e.

$$Z_{0,\text{min-loss}} \approx \frac{59.9\;\text{to}\;76.6}{\sqrt{\varepsilon_r}} \; \Omega$$

The famous "77 Ω is the lowest-loss coax" is the equal-conductivity end of that range, for an air dielectric. The competing objective is power handling: peak power before dielectric breakdown scales as $V^2/Z_0$ with $V$ capped by the maximum field at the inner conductor surface, and that optimum sits near 30 Ω for air-filled line. There is no impedance that is good at both.

The arithmetic mean of 30 and 77 is 53.5 Ω and the geometric mean is 48 Ω — 50 Ω is the round number in that gap, and for the solid-polyethylene dielectric that actually got used ($\varepsilon_r = 2.25$) the minimum-loss impedance falls to about 51 Ω anyway, so 50 Ω is close to both criteria for real cable.

75 Ω is usually explained as "the minimum-loss impedance", which is only true for air. For the foam dielectrics used in CATV cable ($\varepsilon_r \approx 1.43$) the minimum-loss point is nearer 64 Ω; 75 Ω is better read as a compromise between low loss and cable flexibility, since lower impedance forces a fatter centre conductor. It is also convenient that receive-only distribution never needs power handling.

Attenuation in practice #

1694A values are the datasheet dB/100 ft converted at 3.281; LMR-400 values come from the datasheet's own attenuation model $(K_1\sqrt{f} + K_2 f)$ dB/100 ft with $K_1 = 0.1235129$, $K_2 = 0.0002626$. Bars are log-scaled to the 1694A column.

The $\sqrt{f}$ law is visible and so is its breakdown. From 100 MHz to 1 GHz — a factor of 10 — 1694A's loss rises 3.23×, against $\sqrt{10} = 3.16$: pure skin effect. From 1 GHz to 6 GHz, a factor of 6, it rises 2.81× against $\sqrt{6} = 2.45$: the dielectric term is now contributing. This is exactly the $K_1\sqrt{f} + K_2 f$ split that Times Microwave publishes as a formula.

Waveguide #

Around 10 GHz the "bigger cable" argument terminates. Extrapolating LMR-400's own attenuation model past its rated range to 10 GHz gives roughly 50 dB per 100 m, and a rigid air-dielectric line of the same class only claws back a fraction of that. Waveguide sidesteps the problem by deleting the centre conductor — no thin wire means no concentrated current density — at the price of a hard low-frequency limit.

A hollow metal pipe cannot support a TEM wave; it supports TE and TM modes, each with a cutoff frequency below which the propagation constant is imaginary and the field is evanescent. For the dominant $\mathrm{TE}_{10}$ mode of an air-filled rectangular guide with broad wall $a$:

$$f_c = \frac{c}{2a}$$

Standard guides have an aspect ratio near $a{:}b = 2{:}1$ (WR-90 is 0.900 × 0.400 in, so 2.25:1). $\mathrm{TE}_{20}$ cuts on at exactly $2f_c$ regardless of $b$, and $\mathrm{TE}_{01}$ at $c/2b \ge 2f_c$, so single-mode operation is available over at most one octave. The published band is narrower still — it starts around $1.25 f_c$, where dispersion and loss have settled, and stops around $1.9 f_c$ to keep margin against higher-order modes.

Cutoffs computed from $f_c = c/2a$ with $a$ = the WR number in hundredths of an inch; bands are the standard published ranges.

Attenuation in industry-standard copper guide runs from 0.11 dB/m for WR-90 at 10 GHz to 2.7 dB/m for WR-10 at 90 GHz — that is 11 dB/100 m at X-band, against roughly 50 dB/100 m for good flexible coax extrapolated to the same frequency. The scaling is unkind: because both $1/a$ and $1/\delta$ grow with the design frequency, wall-loss attenuation goes as

$$\alpha \propto f_{\text{design}}^{3/2}$$

which is why a nine-fold jump from WR-90 to WR-10 costs a factor of about 27 in dB/m. Waveguide is not a low-loss medium in absolute terms; it is a low-loss medium at frequencies where nothing metallic else works, and it is used in the last few metres of radar and satellite front ends, not for distance.

Optical fibre #

Fibre changes the question. There is no crosstalk floor, no skin effect, and the guiding structure is dielectric, so the loss mechanisms are scattering and molecular absorption rather than resistance.

Single-mode versus multimode #

The number of guided modes depends on the normalised frequency $V = (2\pi a/\lambda)\,\mathrm{NA}$, with $a$ the core radius; below $V = 2.405$ only the fundamental mode propagates. For standard single-mode fibre that means a core around 8–10 µm — ITU-T G.652.D specifies a mode field diameter of 8.6–9.2 µm at 1310 nm in a 125.0 µm cladding, with a cable cutoff wavelength at or below 1260 nm so that everything in the O-band and above is single-moded.

Multimode fibre uses a 50 or 62.5 µm core, which admits hundreds of modes with different path lengths. Modal dispersion is the resulting spread of arrival times, and for a step-index profile it is brutal:

$$\frac{\Delta\tau}{L} \approx \frac{n_1 \Delta}{c} \approx 49\;\mathrm{ns/km} \quad (n_1 = 1.48,\; \Delta = 1\%)$$

A near-parabolic graded index makes the high-angle modes spend their long path in lower-index glass where they travel faster, which cancels the first-order term and leaves

$$\frac{\Delta\tau}{L} \approx \frac{n_1 \Delta^2}{8c} \approx 0.06\;\mathrm{ns/km}$$

nearly three orders of magnitude better — a factor of about 800. Every multimode fibre in service is graded-index for this reason, and the residual profile error is exactly what the OM grades measure.

The three windows #

Silica's attenuation is the sum of two curves going opposite ways. Rayleigh scattering from frozen-in density fluctuations falls as $\lambda^{-4}$; infrared absorption from Si–O vibrational modes rises steeply past about 1600 nm. Their sum has a minimum near 1550 nm. Sitting on top is the OH⁻ overtone absorption peak at 1383 nm, which historically split the usable spectrum in two.

Second and third window figures are ITU-T G.652 representative concatenated-link values for the 1260–1360 and 1530–1565 nm regions; the 850 nm figure is the attenuation used in IEEE's 10GBASE-S link model.

The specification limits are looser than what fibre actually achieves. G.652.D caps the cable attenuation coefficient at 0.40 dB/km anywhere from 1310 to 1625 nm, and at 0.30 dB/km across 1530–1565 nm. Crucially it also caps 1383 ± 3 nm at 0.40 dB/km after hydrogen ageing — that is the "low water peak" requirement, and it is why the E-band is usable on modern fibre and was not on 1990s fibre.

The floor is much lower than the spec. The lowest reported silica fibre loss is 0.1419 dB/km at 1560 nm (0.1424 at 1550), achieved with a germanium-free, fluorine-co-doped silica core to suppress Rayleigh scattering and a 147 µm² effective area, an improvement of 4 mdB/km over the prior record. Submarine cable systems are built from fibre near that floor because every 0.01 dB/km compounds over 8000 km.

Dispersion, and the tradeoff it forces #

The two windows disagree. G.652 fibre has its zero-dispersion wavelength between 1300 and 1324 nm — but its attenuation minimum is at 1550 nm, where chromatic dispersion is a nominal 17 ps/(nm·km) (the specification allows 13.3 to 18.6). You get low dispersion or low loss, not both.

Chromatic dispersion spreads a pulse in proportion to the source's spectral width. For an externally modulated laser the spectral width is set by the modulation itself, $\Delta\lambda \approx \lambda^2 B/c$, so requiring the spreading to stay below a fraction $\epsilon$ of a bit period gives

$$L \le \frac{\epsilon\, c}{D \lambda^2 B^2}$$

The $B^{-2}$ is the whole problem. With $D = 17$ ps/(nm·km), $\lambda = 1550$ nm and $\epsilon = 0.5$, this gives about 590 km at 2.5 Gb/s, 37 km at 10 Gb/s, and 2.3 km at 40 Gb/s. Every quadrupling of rate cuts uncompensated reach by sixteen. The 1990s answers were all forms of paying it back: transmit at 1310 nm and eat the loss, insert dispersion-compensating fibre with the opposite sign, or use dispersion-shifted fibre that moves $\lambda_0$ into the C-band (which then created a four-wave-mixing problem in WDM systems, and is why G.652 outlived it).

Coherent detection removed the dispersion wall #

A coherent receiver mixes the incoming field with a local oscillator and digitises amplitude and phase on both polarisations. Chromatic dispersion is a linear, static, exactly-known all-pass filter; once you have the field rather than just its intensity, you can invert it in the DSP.

The numbers say how completely this changed things. The OIF 400ZR interface carries 400 Gb/s using single-carrier dual-polarisation 16-QAM at about 59.84 GBd with a concatenated FEC whose post-FEC error floor is below $10^{-15}$. Its specified chromatic dispersion tolerance is roughly 2400 ps/nm — at 17 ps/(nm·km), that is 141 km of ordinary G.652 fibre, handled in silicon, with no compensating fibre anywhere in the link. And the specified reach for the amplified DWDM application is 120 km or less, which is a loss budget, not a dispersion budget. The binding constraint moved back to attenuation.

Multiplexing then does the rest. ITU-T G.694.1 anchors the DWDM grid at 193.1 THz with spacings of 12.5, 25, 50 and 100 GHz, plus a flexible grid. The conventional C-band, 1530–1565 nm, is 195.94 to 191.56 THz — 4.38 THz of usable spectrum. On the 75 GHz grid that a 59.84 GBd signal needs, that is about 58 channels; at 400 Gb/s each, roughly 23 Tb/s on one fibre pair over 120 km, from pluggable modules in a router faceplate. See multiplexing for how the grid itself is organised.

The research frontier is further out again: 800 Gb/s on a single wavelength over 7865 km, and 41 Tb/s over 291 km of unrepeatered C-band transmission.

The comparison #

Bars are log-scaled: width $= 8\log_{10}(\text{value}/10) + 8$ px, with value in MHz·km. Twisted-pair rows use the standardised category bandwidth and reach. Coax rows assume a 20 dB budget at 6 GHz, from the attenuation table above. The waveguide row is the WR-90 published band over the length that costs 20 dB at 0.11 dB/m. Multimode rows are the standardised effective modal bandwidth. The single-mode row is C-band optical spectrum over the OIF 400ZR amplified reach.

Choosing #