TRANSMISSION LINES

Every circuit in this module so far has treated a wire as a perfect connector. Voltage at one end equals voltage at the other, instantly. Current is the same everywhere along its length. That is the lumped model, and it is a lie — but a very good one, almost always.

The lie breaks when the wire is long enough that the signal's travel time across it becomes comparable to the signal's own period. A rule of thumb: when the line length exceeds roughly λ/10, the voltage at the near end and the voltage at the far end can differ by tens of percent at the same instant. Pulse edges that should arrive instantly instead propagate, at a speed close to — but always below — the speed of light. Reflections return from the far end. Standing waves set themselves up in the middle. An engineer who ignores this ships a product that looks fine on a benchtop meter and rings uncontrollably on an oscilloscope.

At audio frequencies a household wire is fine; at 2.4 GHz a ten-centimetre trace is already three-quarters of a wavelength in polyethylene and needs design. Between those two extremes is the grey zone where most digital electronics now lives: data rates high enough that every PCB trace longer than a few centimetres has to be treated as a proper line, impedance-controlled, terminated, routed. The wire has become a transmission line, and the rules change.

The man who wrote the rules was not an academic. Oliver Heaviside left school at sixteen, taught himself mathematics from a corner of his brother's house in Camden Town, and worked for years as a telegraph operator on the Anglo-Danish cable. He never held a university post, never earned a degree, published in electrical journals rather than physics ones, lived most of his adult life in poverty, and was written off by half the establishment as a crank. He was also the first person on Earth to understand what Maxwell had actually said.

Maxwell's 1873 Treatise on Electricity and Magnetism presented his theory as twenty coupled equations in twenty variables — quaternions, potentials, component-wise. Heaviside, working alone at a kitchen table, rewrote it in the vector notation every physics student learns today and reduced it to the four compact equations now painted on the side of university buildings. In the process he invented operator calculus for circuits (Laplace transforms, basically), coined the word "impedance," introduced the step and impulse functions that bear his name, and — the subject of this page — derived the equations of a long telegraph line. He did all of this because his day job was keeping telegrams legible after they had travelled a thousand miles of cable, and nobody else was going to do it for him.

His model is deceptively simple. Take any small slice dx of a wire pair and give it four properties per unit length:

A real line is an infinite chain of these cells. In the lossless limit (R = G = 0) the math comes out clean and every essential phenomenon still shows up — so we start there.

Write Kirchhoff's voltage and current laws across one cell, take the limit as dx → 0, and two coupled partial differential equations fall out:

In plain words: the voltage along the line changes because the current through the series inductance is changing (the ∂I/∂t drives a back-EMF per metre). The current changes along the line because some of it is being diverted into the shunt capacitance (a rising ∂V/∂t demands a charging current). Two ordinary circuit laws made infinitely local.

Differentiate the first with respect to x, the second with respect to t, and substitute. Both V and I obey the one-dimensional wave equation, with propagation velocity

Signals on the line propagate as waves. A pulse launched at the left end arrives at the right end a time τ = length/v later, undistorted in the lossless limit. For vacuum-dielectric coax, v approaches c; for a typical polyethylene-filled cable, v ≈ 0.66·c, because the dielectric slows the wave down. This is the first fact every high-speed-link designer learns: the signal moves at roughly two-thirds of light-speed, and that budget must fit inside the rise-time of the signal. A 30 cm cable has a one-way delay of about 1.5 ns — comparable to a modern logic edge, already long enough to matter.

The second essential number is the line's characteristic impedance:

Read Z₀ as: if you launch a sinusoidal wave at one end of an infinitely long line, the ratio V/I at every point along it equals Z₀. The wave needs voltage to drive it through the series L and current to charge the shunt C; the ratio of the two, at the one frequency-independent value that makes both local equations happy, is √(L/C).

For 50 Ω coax (L ≈ 250 nH/m, C ≈ 100 pF/m) the geometry is tuned by cable manufacturers so that √(L/C) = 50. For 75 Ω CATV cable, L/C is chosen higher. The impedance is entirely a geometric property of the cross-section: inner-conductor diameter, outer-shield diameter, dielectric permittivity. You can change the length of the cable without changing Z₀, because both L and C scale with length in the same way and the ratio is preserved. A hundred metres of 50 Ω cable is still a 50 Ω cable.

Now terminate the far end with a real load. If the load impedance Z_L happens to equal Z₀, the wave arrives, deposits its energy in the load, and disappears. If Z_L ≠ Z₀, the wave cannot satisfy both boundary conditions at once (V/I = Z_L at the load, V/I = Z₀ on the line), so part of it reflects.

Γ (uppercase gamma) is the voltage reflection coefficient: the fraction of the incident amplitude that travels back toward the source, signed.

Matching is the engineering discipline of high-speed electronics. Every Ethernet jack, every antenna feed, every PCB trace carrying a gigahertz signal has been designed so that the source sees Z₀ and the load looks like Z₀ — otherwise, power is wasted and echoes ring in the line long after the edge has passed.

A mismatched line driven by a continuous sinusoid carries both a forward wave and its reflection. The two interfere. At some positions the crests add and the envelope peaks at V₊(1 + |Γ|); at positions a quarter wavelength away they cancel and the envelope bottoms out at V₊(1 − |Γ|). The pattern stands still — hence the name.

The single number that captures the severity of the mismatch is the standing wave ratio:

SWR = 1 is a flat line — matched. SWR = ∞ is a pure standing wave — full reflection. Amateur-radio operators watch an SWR meter in the shack to decide whether their antenna is healthy; most rigs will fold back their power above SWR ≈ 3 to protect the finals. SWR is handy because it is easy to measure (slide a probe along a slotted line, find max and min) and it captures mismatch with a single dimensionless number that does not depend on the length of the cable.

Transmission-line physics is the substrate of the modern electromagnetic world. The 50 Ω coax on every lab bench, the 75 Ω cable in every wall, the striplines on every gigahertz PCB, the feedline up to every cell tower, the twinax pairs inside every server rack, the undersea telegraph cables Heaviside himself worked on, the trans-Atlantic fibres that replaced them a century later (optical, but the math rhymes) — all of them are his model made physical. Impedance mismatches in a USB-C cable show up as eye-diagram closure; mismatches in an MRI receive coil show up as lost signal-to-noise. Wherever a wire has become long, his self-taught mathematics runs it — delivered, as he delivered all of it, without the blessing of any institution that might have claimed credit.