# Transmission line A **transmission line** is a cable or other guiding structure built to carry a radio-frequency or fast-changing electrical signal from one point to another as a travelling electromagnetic wave, rather than as a single voltage present everywhere along it at once. The distinction matters once the line's length is a non-negligible fraction of the signal's wavelength: a circuit drawn as a handful of lumped [[Resistor|resistors]], [[Capacitor|capacitors]] and [[Inductor|inductors]] no longer describes it, because the [[Voltage]] and [[Electric_current|current]] now depend on position as well as time. Coaxial cable, twisted pair, microstrip on a circuit board, and the open-wire lines that once carried telegraph traffic across continents, are all transmission lines in this sense. The theory began with very long telegraph and submarine cables, where a pulse sent down thousands of kilometres of wire arrived smeared out in time long before anyone cared about radio. It matters most today in [[Radio-frequency_engineering]], where wavelengths run from metres to millimetres and a line only centimetres long already behaves as a distributed circuit: the feed line between a transmitter and its [[Dipole_antenna|antenna]], a [[Telecommunications|telecommunications]] trunk between switches, and the printed strip connecting chips on a board are all designed by transmission-line rules. The primary microsim on this page puts a sinusoidal source behind a length of lossless line terminated in an adjustable load. Setting the load's resistance and reactance, and dragging the line's electrical length in wavelengths, builds a standing wave on the line and spins the impedance seen at the input around the [[Smith_chart|Smith chart]] as the length changes. ## Overview A transmission line is characterised by a characteristic impedance, set by its geometry and by the material between its conductors, and by a propagation velocity close to the speed of light in that material. Below some fraction of a wavelength (commonly a tenth), a short cable behaves as a single node and ordinary circuit theory is accurate enough; beyond that, the same cable must be treated as a chain of infinitesimal sections, each with its own series resistance, inductance, shunt conductance and capacitance, and solved as a [[Wave|wave]] problem rather than a node problem. RF designers exploit the same fact deliberately: a short length of printed line, cut to a calculated fraction of a wavelength, can act as an inductor, a capacitor, a filter section or an impedance transformer, without a single discrete component. ## History The mathematical treatment of a long conductor as a distributed circuit predates radio. Mid-nineteenth-century engineers laying submarine telegraph cables found that a signal sent down thousands of kilometres of insulated copper arrived badly smeared in time, a distortion traced to the cable's own distributed resistance and capacitance acting as a diffusing, low-pass system.[^cite-cable] Extending the model to include series inductance and shunt conductance, and showing that a particular ratio between the four parameters lets a line pass a pulse without distorting its shape, is credited to the same broad period of telegraph engineering.[^cite-heaviside] That distortionless condition is still called the Heaviside condition (below), and the general four-parameter equations are still called the telegrapher's equations. ## The four terminal model For the underlying mathematics, a transmission line can be treated as a black box with two input terminals and two output terminals (a four-terminal, or two-port, network). An ordinary circuit analysis of the box, based on lumped resistance and reactance alone, is exact only in the zero-length limit; a real line of finite length adds a phase shift and an impedance transformation that depend on its electrical length, not merely on the components at its ends. The four-terminal view is what lets a transmission line be folded into the same matrix formalism used for any other two-port network (impedance, admittance or scattering parameters, covered below), once its own parameters have been derived from the telegrapher's equations. ## Telegrapher's equations A short element of line of length dx carries series resistance R dx and series inductance L dx along its length, and shunt conductance G dx and shunt capacitance C dx across it, all specified per unit length. Applying Kirchhoff's voltage and current laws across that element and letting dx shrink to zero gives a pair of coupled partial [[Differential_equation|differential equations]] linking the voltage and current along the line to their derivatives in space and time. These telegrapher's equations are a circuit-theory shortcut for [[Maxwell's_equations|Maxwell's equations]], valid for a guided wave confined near two conductors whose separation is small compared with a wavelength. ### Special case of a lossless line Setting the resistance and conductance terms to zero collapses the telegrapher's equations to a pair of coupled wave equations in voltage and current, each satisfying the ordinary [[Wave_equation]] with propagation speed `v = 1/sqrt(L*C)`, the same equation solved by a uniform [[Plane_wave|plane wave]] in an unbounded medium. The ratio of voltage to current in a wave travelling one way alone is a real number, the characteristic impedance `Z0 = sqrt(L/C)`, independent of frequency and length. This idealisation is the one the sim's standing-wave panel uses. ### General case of a line with losses Restoring R and G gives a propagation constant that is genuinely complex at any [[Angular_frequency|angular frequency]] ω, `gamma = sqrt((R + j*w*L)*(G + j*w*C)) = alpha + j*beta`, whose real part α attenuates the wave with distance and whose imaginary part β sets the phase, and hence the wavelength, on the line. Because α and β generally both depend on frequency, a lossy line disperses a wideband signal, and a modulated pulse's envelope then moves at the [[Group_velocity]] of its components rather than at any one component's phase velocity. ### Special, low loss case When R and G are small compared with ωL and ωC, the normal case for a well-built RF line below its lossy upper limit, the attenuation and characteristic impedance can be expanded to first order, giving a nearly frequency-independent Z0 and an attenuation constant that rises roughly linearly with the resistive and dielectric loss terms. This is what makes it reasonable to call the characteristic impedance of a coaxial cable a single number, such as 50 Ω or 75 Ω, even though a real cable is never perfectly lossless. ### Heaviside condition A line satisfying `R/L = G/C` propagates every frequency at the same speed and attenuates every frequency by the same factor, so a complicated waveform keeps its shape as it travels even though it loses amplitude uniformly. Long telegraph and telephone lines were once deliberately loaded with extra series inductance to approach this condition, trading some attenuation for freedom from the shape-distorting dispersion a line far from it otherwise imposes on a wideband pulse. ## Input impedance of transmission line Looking into a length of transmission line toward a load, the impedance measured at the input is not simply the load impedance; it is transformed by the line's electrical length. This transformation is the single fact that most distinguishes line behaviour from ordinary circuit behaviour, and it is the one the sim is built to show directly: dragging the line-length control sweeps the input impedance around a circle on the [[Smith_chart|Smith chart]] centred on the real axis, closing the loop every half wavelength.[^cite-sc-rotate] ### Input impedance of lossless transmission line For a lossless line of length ℓ and characteristic impedance Z0 terminated in a load ZL, the input impedance is `Z_in = Z0*(Z_L + j*Z0*tan(beta*l))/(Z0 + j*Z_L*tan(beta*l))`. Because tan is periodic, Z_in repeats with period λ/2, so the load impedance seen at the input is exactly the same every half wavelength; a line an exact multiple of a half wavelength long is therefore invisible to matching. ### Special cases of lossless transmission lines Four terminations turn the general formula into results simple enough to use without a calculator, and a fifth length turns any load into its own reciprocal. #### Half wave length At `l = lambda/2`, `tan(beta*l) = 0` and Z_in = Z_L exactly: a half-wavelength line reproduces its load impedance at the input regardless of the line's own characteristic impedance, which is why a half-wave line is sometimes used simply to relocate a known impedance without disturbing it. #### Quarter wave length At `l = lambda/4`, the tangent term diverges and the formula reduces to `Z_in = Z0^2/Z_L`: the quarter-wave line is an impedance inverter, and choosing `Z0 = sqrt(Z_source*Z_load)` turns it into a matching transformer between two different real impedances. The sim's readout computes this figure live as the load resistance is adjusted. #### Matched load When `Z_L = Z0`, the input impedance equals Z0 at every length, because no power reflects from the load in the first place; the line is then invisible to the source in the sense that its length no longer matters. #### Short A short-circuit load, `Z_L = 0`, makes the input impedance purely reactive at every length, `Z_in = j*Z0*tan(beta*l)`, cycling between an inductive short line and a capacitive one every quarter wavelength. #### Open An open-circuit load likewise gives a purely reactive input impedance, `Z_in = -j*Z0*cot(beta*l)`, the mirror image of the shorted case shifted by a quarter wavelength; the sim shows both as total reflection with |Γ| = 1, one in phase and one inverted. ## Matrix parameters Because a length of transmission line is a linear two-port network, it can be described by any of the standard matrix parameter sets used throughout [[Electrical_engineering|circuit theory]], and converting between them is a matter of algebra once one set is known from the telegrapher's equations. ### Transmission parameters The ABCD, or transmission, parameters relate the input voltage and current directly to the output voltage and current, which makes them the natural choice for cascading several two-port sections (several lengths of line, or a line and a lumped matching network) by simple matrix multiplication. ### Admittance parameters The Y parameters express each port current as a linear combination of both port voltages, convenient when elements are more naturally combined in parallel, as shunt stubs on a line typically are. ### Impedance parameters The Z parameters do the reverse, expressing each port voltage as a combination of both port currents, and are the natural set for elements combined in series along the line. ### Scattering parameters The S parameters describe a line not in terms of total voltage and current but in terms of forward- and backward-travelling wave amplitudes, which is why they are the parameters actually measured on RF hardware: a network analyser launches a wave and measures what returns, and the reflection coefficient at a line's own input, `Gamma_L = (Z_L - Z0)/(Z_L + Z0)`, is nothing but its S11. Shifting the reference plane along a lossless line rotates the phase of every S parameter without changing its magnitude.[^cite-sc-refplane] ### Variable definitions Across all four sets the same handful of symbols recur: Z0 for characteristic impedance, γ for the propagation constant, ℓ for physical length, and β for the phase constant that turns a physical length into an electrical one measured in radians or wavelengths. ## Coupled transmission lines Two or more lines run close enough together interact through the electric and magnetic fields that leak between them, so a signal on one line induces a smaller, delayed replica on its neighbour; this is the mechanism behind unwanted crosstalk on a crowded circuit board and, exploited deliberately, the mechanism behind a directional coupler. ### Coupled matrix parameters A pair of coupled lines is described by an even-mode and an odd-mode characteristic impedance rather than a single Z0, corresponding to the two field patterns possible when both lines are driven together or with opposite polarity; the full scattering matrix of a coupled section follows from combining the two single-line solutions in those two modes. ## Practical types The telegrapher's equations do not care what the conductors look like, so long as R, L, G and C per unit length can be assigned to the geometry; the variety of transmission line types in use is a variety of ways to get a convenient, low-loss, well-shielded or easily manufactured characteristic impedance out of some particular cross-section. ### Coaxial cable A centre conductor surrounded by a cylindrical shield, with dielectric between them, confines its fields almost entirely inside the shield, which is why coaxial cable radiates and picks up interference far less than an open pair of wires; its characteristic impedance follows from the shield-to-centre diameter ratio and the dielectric constant between them. ### Planar lines Printed conductors on a dielectric substrate are transmission lines whose geometry is set by a photolithographic mask rather than a cable extrusion, the same process that patterns a [[Semiconductor_device|semiconductor device]], which is what makes them practical inside an [[Integrated_circuit|integrated circuit]] or on a printed circuit board. #### Microstrip A single conducting strip over a continuous ground plane, with substrate between them, is the most common planar line because only one side of the board needs patterning; part of its field passes through the substrate and part through the air above, so its effective dielectric constant averages the two. #### Stripline Sandwiching the strip between two ground planes, with dielectric filling the gap, keeps its field inside a uniform medium and gives a cleaner electromagnetic mode than microstrip, at the cost of a more complex, multilayer board. #### Coplanar waveguide Placing the ground conductors in the same plane as the signal strip keeps everything accessible from one face of the board, making it easy to mount a component across the gap without a via to the reverse side. ### Balanced lines A balanced line carries its signal as the difference between two conductors, neither grounded, rather than as a single conductor referenced to a shield. #### Twisted pair Twisting the two conductors together makes interference picked up from outside couple almost equally into both wires, so a receiver sensing only the difference between them cancels most of it out; the same twisting also sets the pair's characteristic impedance and its coupling to neighbouring pairs in a cable bundle. #### Star quad Four conductors twisted together and connected as two diagonally opposite pairs cancel external interference even more thoroughly than a simple twisted pair, at the cost of a more complex cable, and were long favoured for the most interference-sensitive telephone and broadcast-audio circuits. #### Twin-lead A pair of parallel conductors held apart by a continuous plastic ribbon, historically used to bring a broadcast antenna signal into a receiver, trades the twisted pair's interference rejection for a simple, low-loss line at ordinary television and FM broadcast frequencies. #### Lecher lines A pair of parallel wires or rods used as a resonant measuring line is a Lecher line: a standing-wave pattern set up on it, and probed with a sliding short or detector, gives the wavelength of a radio-frequency source directly from the spacing between voltage nodes, one of the oldest deliberate uses of transmission-line standing waves. ### Single-wire line A single conductor with the return path through the earth or through a distant ground plane, historically used for early telegraph and some [[Electric_power_transmission|power-distribution]] schemes, is the degenerate case of a transmission line in which one of the two conductors is the general environment rather than a dedicated wire. ## General applications Beyond simply carrying a signal from one place to another, an electrically long line can be used as a circuit element in its own right, because its input impedance and its transfer characteristics are both determined by controllable geometry rather than by a manufactured lumped part. ### Signal transfer The most direct use, connecting a source to a load with minimal loss and reflection, still requires the characteristic impedance to be matched at both ends; a badly mismatched line produces standing-wave voltage peaks that can exceed the source voltage and stress connectors and insulation a matched line would not. High-speed computer buses, governed by their own [[Communication_protocol|communication protocols]], are increasingly built as controlled-impedance lines for the same reason. ### Transmission line circuits Because a length of line looks, at its input, like an impedance set by its length and termination, sections of line serve directly as circuit elements at frequencies where a discrete inductor or capacitor would be inconveniently small or unpredictably parasitic. #### Stepped transmission line Joining sections of different characteristic impedance approximates a smooth, continuously tapered matching transformer with discrete, easily manufactured steps; more sections give a broader matching bandwidth at the cost of more physical length. #### Approximating lumped elements A short length of high-impedance line behaves approximately as a series inductor, and a short length of low-impedance line approximately as a shunt capacitor, which is why microwave filters are routinely built as an alternating sequence of narrow and wide printed strips rather than as an assembly of discrete inductors and capacitors. ### Stub filters An open- or short-circuited length of line, connected as a branch off the main line, presents a pure reactance at its junction that can be tuned by adjusting the stub's length; combining stubs at different points builds a [[Filter_(signal_processing)|filter]] out of transmission line alone. ### Pulse generation Charging a length of line and discharging it through a switch into a matched load produces a rectangular pulse whose duration is set purely by the line's round-trip travel time, a technique used in [[Radar|radar]] modulators that need a clean, precisely timed pulse without a fast electronic timer. ## Sound The same distributed-parameter mathematics applies to any medium that supports a travelling wave under a linear restoring force, including a column of air in a duct or pipe carrying [[Sound]]. An acoustic transmission line has its own characteristic impedance, set by the medium's density and the duct's cross-section rather than by inductance and capacitance, and the same reflection and standing-wave behaviour at an open or closed end that governs an electrical line's short and open cases governs a wind instrument's bore or a loudspeaker enclosure built as a folded acoustic line. ## Microsims The primary sketch models a lossless transmission line driven by a sinusoidal source and terminated in an adjustable complex load. Two sliders set the load's resistance and reactance and a third sets the line's electrical length in wavelengths; a checkbox starts and stops the standing-wave animation, and a preset button cycles through matched, open, short and partial-reflection loads. From these settings the sketch computes the load reflection coefficient Γ_L, the standing-wave ratio, the voltage standing-wave pattern along the line, and the input impedance, plotting the last of these as a point moving around a Smith chart as the length control is dragged. *Try:* Set the load to a short or an open and watch the reflection coefficient jump to the rim of the chart, |Γ| = 1, then drag the line length to a quarter wavelength and read off the inverted input impedance the readout computes. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Transmission_line) : [Wikitube](https://en.wikitube.io/wiki/Transmission_line) Skeleton mirrored at revision 1368798775. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Smith_chart]] - [[Dipole_antenna]] - [[Group_velocity]] - [[Wave_equation]] - [[Radio-frequency_engineering]] - [[Maxwell's_equations]] ## References [^cite-cable]: Citation needed: a primary source (author, paper or patent title, venue, year) for the diffusion-model analysis of early submarine telegraph cables has not been pinned to a Portal Book page in this pass. [^cite-heaviside]: Citation needed: a primary citation (paper or collected-works volume, with year) for the derivation of the distortionless-line condition has not been pinned to a Portal Book page in this pass. [^cite-sc-rotate]: Steer, M. *Microwave and RF Design: Networks*. 2019, pp. 72, 79-85 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^cite-sc-refplane]: Steer, M. *Microwave and RF Design: Networks*. 2019, p. 67 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. ## Further reading - Michael Steer. *Microwave and RF Design: Networks* (2019). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks - Steven Ellingson. *Radio Systems Engineering*, Revised First Edition (2023). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering ## External links - Transmission lines, live sketch: https://editor.p5js.org/sciencenibber/full/ISuJxIOBT - Transmission lines, editor source: https://editor.p5js.org/sciencenibber/sketches/ISuJxIOBT <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->