2c:null 2d:["$","title",null,{"children":"Smith Chart | Mike Bostock | Observable"}] 32:T1435,viewof gamma = { const mouseR = svg``; const mouseX = svg``; const mouseG = svg``; const chart = html``; function label([x, y]) { return transform([x * 1.025, y * 1.025]); } chart.onmousemove = event => { let point = chart.createSVGPoint(); point.x = event.clientX; point.y = event.clientY; point = point.matrixTransform(chart.getScreenCTM().inverse()); mouseG.setAttribute("x2", point.x); mouseG.setAttribute("y2", point.y); point = chart.value = transformInvert([point.x, point.y]); point = smithInvert(point); if (point === undefined) { mouseR.removeAttribute("r"); mouseX.removeAttribute("d"); } else { const [R, X] = point; const cr = resistanceCircle(R); mouseR.setAttribute("cx", transform(cr)[0]); mouseR.setAttribute("cy", transform(cr)[1]); mouseR.setAttribute("r", scale(cr[2])); mouseX.setAttribute("d", reactanceArc(X, 0, Infinity)); } chart.dispatchEvent(new CustomEvent("input")); event.preventDefault(); }; chart.value = [0, 0]; return chart; }33:[] 30:{"id":"71b1a9479d99f6c1","slug":"smith-chart","title":"Smith Chart","thumbnail":"5ca8ec9e956a98c816c176c6a3d69080a75d2b4118d66f2f9058480ae154e4e3","access_level":"public","create_time":"$D2019-07-04T17:02:46.809Z","update_time":"$D2022-01-08T19:13:57.686Z","view_time":null,"first_public_time":"$D2019-07-06T21:19:32.439Z","has_unpublished_edits":null,"fork":false,"stars":54,"starred":false,"owner":{"type":"team","id":"8c2e47252fd4995c","login":"mbostock","name":"Mike Bostock","avatar_url":"https://avatars-next.observableusercontent.com/7dbba99ced40bd5ab8013c99b1a11da61231bce321ba1148a9d16ad0f121550a","tier":"pro_2026"},"version":1143,"publish_version":1143,"role":null,"authored":false,"authors":[{"type":"individual","id":"074c414ad1d825f5","login":"mbostock","name":"Mike Bostock","avatar_url":"https://avatars-next.observableusercontent.com/7dbba99ced40bd5ab8013c99b1a11da61231bce321ba1148a9d16ad0f121550a","tier":null}],"members":[{"type":"individual","id":"074c414ad1d825f5","login":"mbostock","name":"Mike Bostock","avatar_url":"https://avatars-next.observableusercontent.com/7dbba99ced40bd5ab8013c99b1a11da61231bce321ba1148a9d16ad0f121550a","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":32,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Smith Chart","cells":[{"id":0,"value":"md`# Smith Chart\n\nThis [chart](https://en.wikipedia.org/wiki/Smith_chart) plots the [reflection coefficient](https://en.wikipedia.org/wiki/Reflection_coefficient) ${tex`\\Gamma`} and the impedance ${tex`Z = R + jX`}, with ${tex`R`} being the resistance (the real part) and ${tex`X`} the reactance (the imaginary part). The grid shows lines of constant ${tex`R`} and constant ${tex`X`} . Smith charts are convenient for impedance matching by minimizing ${tex`|\\Gamma|`} , the distance to the origin.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":369,"value":"tex`\\Gamma = ${gamma[0].toFixed(3)} ${gamma[1] >= 0 ? \"+\" : \"-\"} j${Math.abs(gamma[1]).toFixed(3)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":918,"value":"tex`|\\Gamma| = ${Math.hypot(...gamma).toFixed(3)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":881,"value":"{\n const Z = smithInvert(gamma);\n if (!Z) return tex`Z = \\text{undefined}`;\n return tex`Z = ${Z[0].toFixed(3)} ${Z[1] >= 0 ? \"+\" : \"-\"} j${Math.abs(Z[1]).toFixed(3)}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"md`Circles of constant resistance ${tex`R`} have radius ${tex`\\frac{1}{R+1}`} and are centered at ${tex`\\langle \\frac{R}{R+1},0 \\rangle`}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":932,"value":"resistanceCircle = R => [R / (R + 1), 0, 1 / (R + 1)]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":935,"value":"md`Circles of constant reactance ${tex`X`} have radius ${tex`\\frac{1}{|X|}`} and are centered at ${tex`\\langle 1,\\frac{1}{X} \\rangle`}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"reactanceCircle = X => [1, 1 / X, 1 / Math.abs(X)]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":938,"value":"md`To compute ${tex`\\Gamma`} for a given ${tex`Z`}, we need to intersect the circle of constant resistance and the circle of constant reactance. Note that the two circles typically intersect in two places; this implementation is carefully designed to pick the point we want.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":69,"value":"function circleCircleIntersect([ax, ay, ar], [bx, by, br]) {\n const dx = bx - ax, dy = by - ay, d = Math.hypot(dx, dy);\n const x = (dx * dx + dy * dy - br * br + ar * ar) / (2 * d);\n const y = Math.sign(ay) * Math.sqrt(ar * ar - x * x);\n return [ax + (dx * x + dy * y) / d, ay + (dy * x - dx * y) / d];\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":973,"value":"md`And we must handle a few degenerate cases, such as ${tex`R = \\infty`} or ${tex`X = 0`}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":144,"value":"function smith([R, X]) {\n if (isNaN(X) || isNaN(R)) return;\n if (R === Infinity || Math.abs(X) === Infinity) return [1, 0];\n if (X === 0) return [(R - 1) / (R + 1), 0];\n return circleCircleIntersect(reactanceCircle(X), resistanceCircle(R));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":293,"value":"md`We can likewise implement the reverse projection, computing ${tex`Z`} for a given ${tex`\\Gamma`}. Given some point ${tex`\\Gamma = \\langle x,y \\rangle`}, it lies on the constant resistance circle for some ${tex`R`}:\n\n${tex.block`(x - \\frac{R}{R + 1})^2 + y^2 = (\\frac{1}{R + 1})^2`}\n\nSolving for ${tex`R`}:\n\n${tex.block`R = \\frac{-x^2 - y^2 + 1}{x^2 - 2x + y^2 + 1}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":352,"value":"md`Likewise ${tex`\\Gamma`} lies on the constant reactance circle for some ${tex`X`}:\n\n${tex.block`(x - 1)^2 + (y - \\frac{1}{X})^2 = (\\frac{1}{X})^2`}\n\nSolving for ${tex`X`}:\n\n${tex.block`X = \\frac{2y}{(x - 1)^2 + y^2}`}\n\n(Thanks to Wolfram Alpha for solving these equations algebraically.)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":212,"value":"function smithInvert([x, y]) {\n if (Math.hypot(x, y) > 1) return;\n return [\n (1 - x ** 2 - y ** 2) / (x ** 2 - 2 * x + y ** 2 + 1),\n 2 * y / ((x - 1) ** 2 + y ** 2)\n ];\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":991,"value":"md`These two helper functions compute the SVG path data for lines of constant resistance and reactance, respectively.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":456,"value":"function resistanceArc(R, X1, X2) {\n const r = scale(1 / (R + 1));\n const [x1, y1] = transform(smith([R, X1]));\n const [x2, y2] = transform(smith([R, X2]));\n if (Math.sign(X1) !== Math.sign(X2)) {\n const [x0, y0] = transform(smith([R, 0]));\n return `\n M${x1},${y1}\n A${r},${r} 0 0,${0 < X1 ? 0 : 1} ${x0},${y0}\n A${r},${r} 0 0,${X2 < 0 ? 0 : 1} ${x2},${y2}\n `;\n }\n return `\n M${x1},${y1}\n A${r},${r} 0 0,${X2 < X1 ? 0 : 1} ${x2},${y2}\n `;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":453,"value":"function reactanceArc(X, R1, R2) {\n const [x1, y1] = transform(smith([R1, X]));\n const [x2, y2] = transform(smith([R2, X]));\n if (X === 0) {\n return `\n M${x1},${y1}\n L${x2},${y2}\n `;\n }\n const r = scale(1 / Math.abs(X));\n return `\n M${x1},${y1}\n A${r},${r} 0 0,${X < 0 ? 1 : 0} ${x2},${y2}\n `;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":844,"value":"md`We also need an invertible transform between from normalized coordinate space in ${tex`[-1, +1]`} to screen coordinates in pixels.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":715,"value":"function transform([x, y]) {\n return [(x + 1) * (width / 2 - margin) + margin, (1 - y) * (width / 2 - margin) + margin];\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":717,"value":"function transformInvert([x, y]) {\n return [(x - margin) / (width / 2 - margin) - 1, 1 - (y - margin) / (width / 2 - margin)];\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1047,"value":"function scale(r) {\n return (width / 2 - margin) * r;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1036,"value":"width = 975","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1038,"value":"margin = 20","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1137,"value":"md`## Further Reading\n\nIf you want to learn more about Smith charts, here are some additional resources.\n\n* Wikipedia’s [entry on Smith charts](https://en.wikipedia.org/wiki/Smith_chart)\n* Antenna Theory’s [introduction to Smith charts](http://www.antenna-theory.com/tutorial/smith/chart.php)\n* Aliaksi Chapyzhenka’s [React component](https://github.com/AI6YP/smith-chart)\n* Germinal Camps’ [Vue.js component](https://github.com/germi/vue-smith-chart)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1"}} 31:"$30:body"