# Smith chart The **Smith chart** is a circular graphical calculator, plotted on a disk of unit radius, that [[Radio-frequency_engineering|RF]] engineers use to solve [[Transmission_line|transmission-line]] and impedance-matching problems without working through the underlying complex arithmetic term by term. It plots the reflection coefficient a load or a point along a line presents, and overlays that plot with a grid of curves representing the corresponding normalized resistance and reactance, so that a point's position on the disk can be read directly as an impedance. Moving along a length of line rotates a point on the chart at a fixed radius, and adding a series or shunt circuit element moves it along one of the grid's own curves, which is what makes the chart useful for matching-network design: a network is built up move by move, watching where each step lands, rather than solved as one large equation. The primary microsim on this page lets the reader drag a load point anywhere on the disk and then walk it down a lossy or lossless transmission line with two sliders, watching the point trace the chart's characteristic rotating, and optionally spiralling, path while a standing-wave-ratio gauge updates. ## History The chart is named for Phillip H. Smith, an engineer at Bell Telephone Laboratories, who described it in a 1939 article in *Electronics* magazine as a graphical calculator for transmission-line problems, and followed it with an expanded version in 1944[^smith1939]. Smith developed the chart to replace the slide-rule and tabular methods then used to design matching networks and antenna feed lines for growing [[Telecommunications|telecommunications]] systems, at a time when the [[Transistor|transistor]] did not yet exist and every complex-ratio calculation had to be done by hand or by a mechanical calculating aid. Regional variants are also attributed to other engineers — a Volpert–Smith chart in Russian-language literature and a Mizuhashi chart in Japanese literature — though the primary attribution and dates for those variants are not established in the sources cited here[^regional-names-cn]. The chart predates the digital computer, and although software has long since taken over the numerical work it was invented to avoid, RF engineers still use it, now generated and annotated by that same software, because plotting a design's progress on the chart makes trade-offs in a matching network visible in a way a table of complex numbers does not. ## Overview A load's impedance `Z_L` is normalized to the line's characteristic impedance `Z_0` as `z_L = Z_L / Z_0`, and the chart plots the corresponding [[Complex_analysis|complex]] reflection coefficient `Gamma = (z_L - 1)/(z_L + 1)`[^b068-gamma]. Because every passive load has non-negative resistance, every possible `Gamma` for a passive circuit lies within the unit circle, `|Gamma| <= 1`; the rim itself, `|Gamma| = 1`, represents a purely reactive load with no resistance at all, an open circuit sits at the chart's far right, a short circuit at its far left, and the centre point, `Gamma = 0`, represents a load perfectly matched to `Z_0`[^b068-rim]. Over that disk the chart draws two families of curves — circles of constant normalized resistance and arcs of constant normalized reactance — so a point's position can be read as an impedance directly, without separately working through the arithmetic that produced it, arithmetic that ultimately traces back to the transmission-line form of [[Maxwell's_equations|Maxwell's equations]]. Inductive loads, whose reactance is positive, plot in the top half of the chart, and capacitive loads, whose reactance is negative, plot in the bottom half[^b068-halves]. Beyond a single load, the same disk displays several related RF quantities at once — admittance, [[Voltage|voltage]] standing-wave ratio, and reflection or transmission coefficients — which is why the chart survived the arrival of the pocket calculator and the computer: one diagram still shows more at a glance than a column of separately computed numbers does. ## Mathematical basis ### Normalization Working in normalized impedance, `z = Z/Z_0`, rather than in a load's actual ohms, is what lets one chart serve every line regardless of its characteristic impedance: a 50-ohm load on a 50-ohm line and a 75-ohm load on a 75-ohm line normalize to the same point, `z = 1`, the chart's centre, even though their actual impedances differ. The chart's radial and angular grid is drawn once, in this normalized coordinate, and a design in any characteristic impedance is plotted on it by dividing every impedance in the problem by that line's own `Z_0` before touching the chart. ### Reflection coefficient and line propagation The reflection coefficient at the load, `Gamma_L = (z_L - 1)/(z_L + 1)`, is not the reflection coefficient a matching network further back along the line sees. Moving a distance toward the generator changes only the coefficient's phase on a lossless line, rotating it clockwise at fixed radius: `Gamma_in = Gamma_L * exp(-j*2*theta)`, where `theta` is the line's electrical length in radians, itself proportional to the line's physical length and to the signal's [[Angular_frequency|angular frequency]][^b068-gin]. Because a full rotation corresponds to `theta = pi`, or half a wavelength, the impedance any lossless line presents repeats every half wavelength along its length; a quarter-wavelength section instead rotates the point exactly halfway around the disk, which inverts the normalized impedance, `z_in = 1/z_L`, the basis of the quarter-wave transformer. Converting the rotated coefficient back to an impedance runs the same relation in reverse, `z_in = (1 + Gamma_in)/(1 - Gamma_in)`, the input-impedance equation a matching design ultimately wants answered at whatever reference plane it is built from. ### Geometry of the Z chart The chart's grid follows from the same relationship between `Gamma` and `z`, which is a bilinear (Mobius) transform and therefore maps every straight line in one plane to a circle in the other[^b068-bilinear]. Each circle of constant normalized resistance `r` maps to a circle in the `Gamma` plane centred at `(r/(1+r), 0)` with radius `1/(1+r)`, and each line of constant normalized reactance `x` maps to an arc of a circle centred at `(1, 1/x)` with radius `1/|x|`. Every resistance circle and every reactance arc passes through the point `Gamma = 1`, the open-circuit point, since an infinite-resistance line touches it regardless of `r`. The interior of the chart, where `|Gamma| < 1`, corresponds to positive resistance, and the horizontal diameter of the disk, where reactance is zero, is purely resistive from the short circuit at one end to the open circuit at the other[^b068-halves]. ### Impedance–admittance conversion Because admittance is the reciprocal of impedance, `y = 1/z`, and a half-turn rotation of the chart is exactly the transformation `z -> 1/z` described above, the same grid read upside down (rotated by `pi`) serves as an admittance chart: constant-conductance circles and constant-susceptance arcs replace constant-resistance circles and constant-reactance arcs, with the sign of the susceptance term flipped relative to the reactance convention[^b068-admittance]. This lets a single plotted point be reinterpreted as either an impedance or an admittance without leaving the page, which matters because some circuit elements are more naturally described in one than the other. ### Circuit elements and topology Building a matching network is a sequence of moves on the chart, one per element. Adding a series [[Inductor|inductor]] or [[Capacitor|capacitor]] to a load moves its point along the constant-resistance circle already passing through it, since a series element changes reactance without changing resistance; adding a shunt element instead moves the point along the constant-conductance circle passing through it, read on the admittance chart[^b068-series-shunt]. Neither kind of move can cross the open-circuit or short-circuit point, since no finite series or shunt element can create an infinite discontinuity where none existed[^b068-cross]. A matching network is therefore designed by alternating series and shunt moves, each chosen to walk the starting point toward the chart's centre, `Gamma = 0`, where the line finally sees a matched load. ### Plotting examples A lossless line's standing-wave ratio is fixed once its load is fixed, `SWR = (1 + |Gamma|)/(1 - |Gamma|)`, and does not change as a line-length control moves the point around its circle, since rotation at constant radius leaves `|Gamma|` untouched; a load presenting `|Gamma| = 0.5`, for instance, gives an SWR of 3 no matter how far along the line it is read[^b068-gin]. Adding loss to the line turns that constant-radius circle into an inward spiral toward the centre, since each pass down a lossy section shrinks `|Gamma|` a little further, and the standing-wave ratio the same load produces falls correspondingly as the observation point moves back from it[^b068-loss]. ## Applications ### Distributed-component matching Matching a load to a line using only lengths of [[Transmission_line|transmission line]] and stubs — short sections of line left open or shorted at their far end to present a pure reactance — is the classical, purely graphical use of the chart: a single series line length combined with a single shunt open- or short-circuited stub can move almost any load to the chart's centre, and the two required lengths can be read directly off the intersections the load's constant-conductance circle makes with the unit-conductance circle after rotation. This distributed approach suits microwave frequencies well, where a convenient length of line is physically short and a stub is cheaper and more reliable than a wound or discrete component. ### Lumped-element circuits At lower radio frequencies, or wherever a physically short line length is impractical, the same chart-walking method matches with discrete inductors and capacitors instead of line lengths and stubs: a series element moves a point along a constant-resistance circle, and a shunt element moves it along a constant-conductance circle, exactly as in the distributed case, so the same graphical procedure designs a two-element lumped network as readily as it designs a stub match. Because a lumped design needs no particular physical line length, it remains the natural choice below the frequency at which a quarter-wavelength stub becomes inconveniently long to build. ## Variations and extensions ### Admittance chart A chart printed and gridded directly in admittance, rather than reused by mentally rotating the impedance chart, is common enough to be sold and taught as its own variant, particularly where a design works mostly in shunt elements and would otherwise need that rotation performed by eye at every step. Combined impedance-admittance charts, printing both grids faintly on the same disk, are also common in RF design and [[Frequency_response|frequency-response]] simulation software and textbooks, so that neither conversion needs to be done by hand at all[^b068-admittance]. ### Negative resistance Standard charts are drawn only for passive circuits, where resistance cannot be negative and every point lies within the unit circle. Active devices used throughout radio and [[Telecommunications|telecommunications]] hardware, such as oscillators and amplifiers built from a [[Transistor|transistor]] or similar device and operated close to instability, can present a negative resistance at a port, corresponding to points outside the unit circle, `|Gamma| > 1`, where the device delivers more power back down the line than was incident on it. Extending the chart's grid beyond the rim lets a designer plot such a device's reflection coefficient directly, and the extended chart is used both in oscillator design, where a negative-resistance port is deliberately terminated to sustain oscillation, and in stability analysis, where the boundary between stable and potentially unstable terminations for an active two-port is tested with tools such as the [[Nyquist_stability_criterion|Nyquist stability criterion]] and drawn as a circle on the same extended chart. ### Mapping onto a sphere Because the reflection-coefficient plane, extended to include the point at infinity, is topologically a sphere rather than a flat disk, the whole chart — including the region outside the unit circle that an ordinary printed chart leaves off the page — can be mapped onto the surface of a sphere by the stereographic projection standard elsewhere in [[Complex_analysis|complex analysis]]. On that sphere the open-circuit point and the point at infinity are simply two ordinary points rather than a special case at the edge of a page, and the passive region `|Gamma| <= 1` and the active region `|Gamma| > 1` become the sphere's two hemispheres, meeting at the equator formed by the original chart's rim. The construction is mostly of theoretical and teaching interest, since a flat printed or rendered chart is easier to plot on and read than a physical or rendered globe. ## Microsims The **Smith chart** sketch lets the reader drag a red point anywhere inside the disk to set a load `z_L`, then walk it down a transmission line with two sliders: line length, from 0 to 0.5 wavelengths, rotates the point clockwise around the disk exactly as the input-impedance formula above predicts, and line loss, from 0 to 6 [[Decibel|decibels]], shrinks that same rotation into an inward spiral. A reset button returns the load to a fixed default mismatch. Underneath the moving point the sketch draws the resistance circles and reactance arcs from their exact formulas rather than from a printed template, and reports a standing-wave-ratio gauge and a small diagnostics panel that update as either slider moves, alongside the normalized input impedance `z_in` the current line length and loss produce. *Try:* drag the load point out toward the rim of the disk, into the upper half where reactance is positive, then move the line-length slider through its full range and watch the point complete one clockwise circle at fixed radius while the SWR gauge stays fixed; only after that, bring the loss slider up from zero and watch the same circle pull inward toward the matched centre instead. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Smith_chart) : [Wikitube](https://en.wikitube.io/wiki/Smith_chart) Skeleton mirrored at revision 1374740590. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Transmission_line]] - [[Bode_plot]] - [[Nyquist_stability_criterion]] - [[Root_locus_analysis]] - [[Transfer_function]] - [[Frequency_response]] - [[Complex_analysis]] ## Footnotes The bilinear relationship between reflection coefficient and normalized impedance, and the circle geometry it implies, are worked from the standard forms cited below; the surviving text of the cited chapter's own worked numerical examples was lost in the source's own extraction, so the sketch's own defaults stand in for a worked example here. Page numbers below are PDF pages of the open edition cited. ## References [^smith1939]: Smith, P. H. "Transmission Line Calculator." *Electronics*, vol. 12, pp. 29-31, January 1939. [^regional-names-cn]: Citation needed: primary attribution and dates for the Volpert–Smith and Mizuhashi chart variants named in some regional literature. [^b068-rim]: 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. [^b068-gamma]: Steer, M. *Microwave and RF Design: Networks*. 2019, p. 70 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-gin]: Steer, M. *Microwave and RF Design: Networks*. 2019, p. 70 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-halves]: Steer, M. *Microwave and RF Design: Networks*. 2019, pp. 67, 70, 78 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-bilinear]: Steer, M. *Microwave and RF Design: Networks*. 2019, pp. 82-84 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-admittance]: Steer, M. *Microwave and RF Design: Networks*. 2019, p. 74 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-series-shunt]: Steer, M. *Microwave and RF Design: Networks*. 2019, pp. 75-76, 79 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-cross]: Steer, M. *Microwave and RF Design: Networks*. 2019, p. 76 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. [^b068-loss]: Steer, M. *Microwave and RF Design: Networks*. 2019, pp. 80-81 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/microwave-and-rf-design-networks . CC BY-NC. ## External links - Live p5.js microsim: https://editor.p5js.org/sciencenibber/full/GS26xlDMW - p5.js Editor source: https://editor.p5js.org/sciencenibber/sketches/GS26xlDMW <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->