31:T5b7,{ const {index, select, x_axis} = graph1_options; const {i, u} = calculated_data[index - 1]; const f = select == "cc" ? (o => i * i * o.r) : (o => u * u / o.r); return Plot.plot({ insetBottom: 11, insetRight: 50, x: { label: "Estimated temperature (K) →", grid: true, }, y: { label: "↑ Power (P)", grid: true, domain: [0, Math.max(1.2 * calculated_data.at(-1).p, Math.min(3 * calculated_data.at(-1).p, f(calculated_data[3]) * 1.1))], }, marks: [ Plot.line(calculated_data, {x: "t", y: f, strokeDasharray: [11, 4]}), Plot.differenceY(calculated_data, {x: "t", y1: f, y2: "p", positiveFill: "#07f2", negativeFill: "#f082"}), Plot.text(calculated_data, Plot.selectLast({x: "t", y: f, text: _ => "Power\ndelivered", textAnchor: "start", dx: 3})), Plot.text(calculated_data, Plot.selectLast({x: "t", y: "p", text: _ => "Power required", textAnchor: "end", dx: -10, dy: 0})), Plot.dot([calculated_data[index - 1]], {x: "t", y: f}), Plot.text([calculated_data[index - 1]], {x: "t", y: f, text: o => `${o.i}A\n${o.u}V`, lineAnchor: "bottom", dy: -12}), Plot.line(bb_table.filter(o => 293 < o.temp && o.temp < 2700), {x: "temp", y: 0, z: null, dy: 5, stroke: "css", strokeWidth: 7}) ] }); }32:T454,----------------------------------------------------- ### Notes ^[a] This makes them an example of a PTC resistor — *Positive Temperature Coefficient*. Due to this property, light bulbs are sometimes used as crude current limiters. They aren’t very precise, but they are cheap, some can dissipate 100 W, and you used to be able to simply buy them in a supermarket. ^[b] This is only a rough estimate. Heat doesn’t enter and exit only via radiation. Other mechanisms include convection and conduction of heat through the support wires. On the other hand, the large amount of infrared radiation can be reflected back to the light bulb by reflectors around the bulb. ^[c] The X axis shows estimated temperature, which probably isn’t entirely accurate. However for every data point, the two power values on the Y axis are directly derived from measurements. And because our estimated temperature increases monotonically with voltage and current, our data points are definitely still in the right order.33:[{"id":"53948293d58e398704e645415ee60f1cc3b88f25573bfe71e6f2d6f84ca12490ddc15c0840465c658f19027eac61f39ed74ebc7765a8cedcbd7ab9dc6a5b72e0","href":"https://static.observableusercontent.com/files/53948293d58e398704e645415ee60f1cc3b88f25573bfe71e6f2d6f84ca12490ddc15c0840465c658f19027eac61f39ed74ebc7765a8cedcbd7ab9dc6a5b72e0","name":"light-u.webp","type":"image/webp","lastModified":1743914398848.327,"size":967266},{"id":"290e82c827f2476ea93affb76c8fae2e9042e659fdda86f2cc9a93182fc1286afdf0413103d229b830a71f346052d10870127926d9f7b8a862f040f6bd7a4299","href":"https://static.observableusercontent.com/files/290e82c827f2476ea93affb76c8fae2e9042e659fdda86f2cc9a93182fc1286afdf0413103d229b830a71f346052d10870127926d9f7b8a862f040f6bd7a4299","name":"light-i.webp","type":"image/webp","lastModified":1743914503231.115,"size":1257050},{"id":"225698bdf807c48e575199f9140e3cc20a13e4b185747dc20f37495ab878a1ce7712cc7797bdbb77b41f45fff8bf075f9da358ac44ce36ece14091050cf40e67","href":"https://static.observableusercontent.com/files/225698bdf807c48e575199f9140e3cc20a13e4b185747dc20f37495ab878a1ce7712cc7797bdbb77b41f45fff8bf075f9da358ac44ce36ece14091050cf40e67","name":"lightbulb.jpg","type":"image/jpeg","lastModified":1743815465739.173,"size":132429},{"id":"e3525f218c3e8813539b6f93cdb44376a1791415d54bcdf1e93f9278d7b6a04481ccead73dcc2c69b80f6f11c8edc4696347d0966d10e23c880e24199916ff3b","href":"https://static.observableusercontent.com/files/e3525f218c3e8813539b6f93cdb44376a1791415d54bcdf1e93f9278d7b6a04481ccead73dcc2c69b80f6f11c8edc4696347d0966d10e23c880e24199916ff3b","name":"bb@1.csv","type":"text/csv","lastModified":1744082246161.396,"size":5915}] 2f:{"title":"Incandescent light bulb","cells":[{"id":0,"value":"# Incandescent light bulb\n\nIt used to be that flashlights had things called *light bulbs* inside. These have a thin metal wire inside which is heated by an electric current.\n\n $\"light$ \n\nHow does the current through a light bulb vary with voltage? Does it follow Ohm’s law? With a lab power supply we can make a few measurements.","pinned":false,"hidden":false,"mode":"md"},{"id":73,"value":"Plot.plot({\n width: 600,\n height: 400,\n x: {\n label: \"↑ Voltage (V)\"\n },\n y: {\n label: \"Current (I) →\",\n grid: true\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(calculated_data, {x: \"u\", y: \"i\", marker: \"dot\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":494,"value":"For the first few points, we indeed observe Ohm’s law — current and voltage are proportional to each other. But for the next points, the current goes up much more slowly. This is because the resistance of the filament rises with temperature. ^[a]","pinned":false,"hidden":false,"mode":"md"},{"id":89,"value":"Light bulbs emit black-body radiation. The Stefan–Boltzmann law states that the radiated power is proportional to the fourth power of the temperature. With a few assumptions (most importantly, the temperature at the nominal voltage of the bulb), we can thus estimate how hot the filament gets. ^[b]","pinned":false,"hidden":false,"mode":"md"},{"id":322,"value":"temperature_reference = 2700","pinned":true,"hidden":false,"mode":"ojs"},{"id":489,"value":"{\n var options = {\n width: 300,\n height: 200,\n x: {\n label: \"Voltage (V)\"\n },\n };\n var plt1 = Plot.plot({\n ...options,\n y: {\n label: \"↑ Resistance (R)\",\n grid: true\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(calculated_data, {x: \"u\", y: \"r\"})\n ]\n })\n var plt2 = Plot.plot({\n ...options,\n y: {\n label: \"↑ Power (W)\",\n grid: true\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(calculated_data, {x: \"u\", y: \"p\"})\n ]\n })\n return html`

${plt1}${plt2}

`\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":649,"value":"viewof temp_graph_selection = Inputs.radio([\"ut\", \"tr\"], {label: \"Show\", value: \"ut\",\n format: x => ({\"ut\": \"Voltage / temperature\", \"tr\": \"Temperature / resistance\"}[x])})","pinned":false,"hidden":false,"mode":"ojs"},{"id":104,"value":" temp_graph_selection == \"ut\" ?\n Plot.plot({\n insetLeft: 22, // make room for color bar\n x: {\n label: \"Voltage (V) →\",\n grid: true,\n },\n y: {\n label: \"↑ Temperature (K)\",\n grid: true,\n },\n marks: [\n Plot.lineY(calculated_data, {x: \"u\", y: \"t\"}),\n // color band\n Plot.rect(bb_table, {y1: o => o.temp - 0.5 * dt, y2: o => o.temp + 0.5 * dt, x1: 0, x2: 0, insetLeft: -20, insetRight: 2, insetTop: -0.5, fill: \"css\", strokeWidth: 0})\n ]\n })\n :\n Plot.plot({\n insetBottom: 22, // make room for color bar\n x: {\n label: \"Temperature (K) →\",\n grid: true,\n },\n y: {\n label: \"↑ Resistance (Ω)\",\n grid: true,\n },\n marks: [\n Plot.lineY(calculated_data, {x: \"t\", y: \"r\"}),\n // color band\n Plot.rect(bb_table, {x1: o => o.temp - 0.5 * dt, x2: o => o.temp + 0.5 * dt, y1: 0, y2: 0, insetBottom: -20, insetTop: 2, insetLeft: -0.5, fill: \"css\", strokeWidth: 0})\n ]\n })\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":356,"value":"## Does a light bulb always start?\n\nThis seems straightforward, and in real life most electricity sources have approximately constant voltage. Such sources deliver more power to lower resistances (P = U² / R). So at low temperature we see low resistance, and high current — an *inrush current*. This will quickly heat up the filament.\n\n![light-u.webp](${await FileAttachment(\"light-u.webp\").url()})\n\nWith a lab power supply, you can measure in constant current mode. For such sources, power decreases with lower resistance (P = I² × R). So, due to low initial resistance, the filament may not heat up at all.\n\nBut this is not the case here. Our measured current always went up with increasing voltage. We can visualize this by plotting the required power required to maintain a given resistance, and compare it with the power which would be supplied by the current source. ^[c]","pinned":false,"hidden":false,"mode":"md"},{"id":377,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":558,"value":"viewof graph1_options = Inputs.form({\n select: Inputs.radio([\"cc\", \"cv\"], {label: \"Select\", value: \"cc\", format: x => ({cc: \"Constant current\", cv: \"Constant voltage\"}[x])}),\n index: Inputs.range([2, calculated_data.length], {label: \"Data point\", step: 1, value: Math.floor(calculated_data.length / 2)}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":445,"value":"So, even with a constant current supply, the bulb always starts. There is one intersection point, and delivered power is always greater than the required power to the left of that point. It sometimes is only by a small margin, and these cases are easily visible: it takes about a second for the temperature to settle.\n\n![light-i.webp](${await FileAttachment(\"light-i.webp\").url()})","pinned":false,"hidden":false,"mode":"md"},{"id":462,"value":"If you’re old enough to remember flashlights with incandescent light bulbs, you may remember seeing this effect when your batteries were flat. Flat batteries tend to have high internal resistance, and thus will not deliver that initial inrush current.","pinned":false,"hidden":false,"mode":"md"},{"id":312,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":306,"value":"### Source data","pinned":false,"hidden":false,"mode":"md"},{"id":573,"value":"bb_table = FileAttachment(\"bb@1.csv\").csv({typed: true})","pinned":true,"hidden":false,"mode":"ojs"},{"id":20,"value":"measured_u_i = [\n {u: 0.000, i: 0.000},\n {u: 0.008, i: 0.010},\n {u: 0.016, i: 0.020},\n {u: 0.025, i: 0.030},\n {u: 0.033, i: 0.040},\n {u: 0.042, i: 0.050},\n {u: 0.052, i: 0.060},\n {u: 0.062, i: 0.070},\n {u: 0.073, i: 0.080},\n {u: 0.085, i: 0.090},\n {u: 0.100, i: 0.100},\n {u: 0.118, i: 0.110},\n {u: 0.138, i: 0.120},\n {u: 0.169, i: 0.130},\n {u: 0.192, i: 0.135},\n {u: 0.218, i: 0.140},\n {u: 0.255, i: 0.145},\n {u: 0.321, i: 0.150},\n {u: 0.435, i: 0.155},\n {u: 0.553, i: 0.160},\n {u: 0.637, i: 0.165},\n {u: 0.715, i: 0.170},\n {u: 0.786, i: 0.175},\n {u: 0.855, i: 0.180},\n {u: 0.923, i: 0.185},\n {u: 0.995, i: 0.190},\n {u: 1.072, i: 0.195},\n {u: 1.148, i: 0.200},\n {u: 1.219, i: 0.205},\n {u: 1.293, i: 0.210},\n {u: 1.365, i: 0.215},\n {u: 1.438, i: 0.220},\n {u: 1.501, i: 0.225},\n {u: 1.583, i: 0.230},\n {u: 1.664, i: 0.235},\n {u: 1.743, i: 0.240},\n {u: 1.824, i: 0.245},\n {u: 1.916, i: 0.250},\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"### Derived data","pinned":false,"hidden":false,"mode":"md"},{"id":341,"value":"viewof table = Inputs.table(calculated_data, {format: {\n \"u\": u => u.toFixed(3) + \" V\",\n \"i\": i => i.toFixed(3) + \" A\",\n \"p\": p => p.toFixed(3) + \" W\",\n \"r\": r => r.toFixed(3) + \" Ω\",\n \"t\": t => t.toFixed(0) + \" K\",\n \"lm_w\": e => e.toFixed(2) + \" lm/W\",\n \"css_bb\": css => html` ${css}`\n}, align: {\"css_bb\": \"right\"}})","pinned":false,"hidden":false,"mode":"ojs"},{"id":123,"value":"// power at 293K, i.e. 20°C or 68°F. This represents the power we receive from and radiate to the environment\n// The exact choise here doesn’t matter, for any reasonable room temperature this power level is really low.\npower_0 = Math.pow(293 / temperature_reference, 4) * last_power","pinned":true,"hidden":false,"mode":"ojs"},{"id":331,"value":"function power_to_temperature(p) {\n return Math.pow(p / last_power + power_0, 0.25) * temperature_reference;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":22,"value":"calculated_data = measured_u_i.map(o => {\n var p = o.u * o.i;\n var t = power_to_temperature(p);\n var lm_w = 0;\n var css = \"black\";\n if (t > bb_table[0].temp && t < bb_table.at(-1).temp)\n {\n var ix = Math.floor((t - 200) / dt);\n var bb1 = bb_table[ix]\n var bb2 = bb_table[ix + 1]\n var mix = (t - bb1.temp) / (bb2.temp - bb1.temp);\n lm_w = (1 - mix) * bb1.lm_w + mix * bb2.lm_w;\n css = bb1.css;\n }\n return {...o, p: p, r: o.u / o.i, t: t, lm_w: lm_w, css_bb: css};\n })\n","pinned":true,"hidden":false,"mode":"ojs"},{"id":265,"value":"last_power = measured_u_i.at(-1).u * measured_u_i.at(-1).i","pinned":true,"hidden":false,"mode":"ojs"},{"id":272,"value":"dt = bb_table[1].temp - bb_table[0].temp","pinned":true,"hidden":false,"mode":"ojs"},{"id":146,"value":"### CSS","pinned":false,"hidden":false,"mode":"md"},{"id":148,"value":"function sRGB([r, g, b])\n{\n // both sRGB and Display-P3 use this transfer function.\n var a = 0.055;\n r = r < .0031308 ? r / 12.92 : (1 + a) * Math.pow(r, 1/2.4) - a;\n g = g < .0031308 ? g / 12.92 : (1 + a) * Math.pow(g, 1/2.4) - a;\n b = b < .0031308 ? b / 12.92 : (1 + a) * Math.pow(b, 1/2.4) - a;\n return [r, g, b]\n}","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}