{"id":62013,"date":"2021-11-16T14:01:42","date_gmt":"2021-11-16T12:01:42","guid":{"rendered":"https:\/\/wpcalc.com\/?p=62013"},"modified":"2026-09-30T18:04:45","modified_gmt":"2026-09-30T15:04:45","slug":"air-core-coil-inductance","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/air-core-coil-inductance\/","title":{"rendered":"Air Core Coil Inductance Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Flat Spiral Coil Inductance Tool<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"62013\" data-favorite-title=\"Air Core Coil Inductance\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/air-core-coil-inductance\/\" data-favorite-excerpt=\"Calculate the inductance of a flat spiral air core coil from its inner diameter, outer diameter, and number of turns. Enter the coil dimensions in millimeters and read the result in nanohenries.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-magnet\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/engineering\/air-core-coil-inductance\/\" method=\"POST\" class=\"calculator\" id=\"calculator-62013\" data-post=\"62013\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Outer diameter<\/label><div class=\"term\">d\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"outer_diameter\" id=\"field-1\" aria-describedby=\"calc-unit-1 calc-info-1\" step=\"0.01\" value=\"20\"\/><div class=\"unit\" id=\"calc-unit-1\">mm<\/div><small id=\"calc-info-1\">Outer diameter of the spiral coil<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Inner diameter<\/label><div class=\"term\">d\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"inner_diameter\" id=\"field-2\" aria-describedby=\"calc-unit-2 calc-info-2\" step=\"0.01\" value=\"10\"\/><div class=\"unit\" id=\"calc-unit-2\">mm<\/div><small id=\"calc-info-2\">Inner diameter of the spiral coil<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Number of turns<\/label><div class=\"term\">N<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"number_of_turns\" id=\"field-3\" aria-describedby=\"calc-info-3\" step=\"1\" value=\"5\"\/><small id=\"calc-info-3\">Total number of spiral turns<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"62013\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Air Core Flat Spiral Inductance<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"nH\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"62013\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Air Core Flat Spiral Inductance Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ L \\,(\\text{nH}) = \\frac{4.921 \\, N^2 \\left( d_1 + d_2 \\right)^2}{15 d_1 - 7 d_2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(L\\) \u2014 inductance of the coil (nH)<\/li>\n<li>\\(N\\) \u2014 total number of spiral turns<\/li>\n<li>\\(d_1\\) \u2014 inner diameter of the spiral (mm)<\/li>\n<li>\\(d_2\\) \u2014 outer diameter of the spiral (mm)<\/li>\n<\/ul>\n<p>This is a Wheeler-type empirical approximation for flat (planar) spiral coils wound on air. Both diameters are entered in millimeters, and the formula returns the inductance directly in nanohenries. The expression is only meaningful when the denominator \\(15d_1 - 7d_2\\) is positive.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Air Core Flat Spiral Inductance \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a flat spiral coil on a PCB has an inner diameter of 10 mm, an outer diameter of 20 mm, and 5 turns. What is its inductance?<\/p>\n<p>\\[ L = \\frac{4.921 \\times 5^2 \\times (10 + 20)^2}{15 \\times 10 - 7 \\times 20} = \\frac{4.921 \\times 25 \\times 900}{150 - 140} = \\frac{110\\,722.5}{10} \\approx 11\\,072.25 \\text{ nH} \\]<\/p>\n<p><strong>Result:<\/strong> \\(L \\approx 11\\,072\\) nH, which is about 11.07 \u00b5H.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter the <strong>Outer diameter<\/strong> (\\(d_2\\)) of the spiral in millimeters. This is the diameter at the widest point of the winding.<\/li>\n<li>Enter the <strong>Inner diameter<\/strong> (\\(d_1\\)) in millimeters \u2014 the diameter of the open center of the spiral.<\/li>\n<li>Enter the <strong>Number of turns<\/strong> (\\(N\\)) of the spiral winding.<\/li>\n<li>Read the result, shown as <strong>Air Core Flat Spiral Inductance<\/strong> in nanohenries (nH). Divide by 1000 if you need the value in microhenries (\u00b5H).<\/li>\n<\/ol>\n<p>The calculation runs instantly as you type, so you can quickly sweep different turn counts or diameters to hit a target inductance.<\/p>\n<h2>What is an air core flat spiral coil?<\/h2>\n<p>A flat spiral coil is a conductor wound in a single plane, like a clock spring, so that every turn lies flat instead of stacking in a cylinder. When there is no ferromagnetic core inside \u2014 just air, or the FR-4 of a printed circuit board \u2014 the coil is called an air core coil. Because the inductance comes purely from the geometry of the winding, this calculator needs only the two diameters and the number of turns to estimate it.<\/p>\n<p>Air core coils are popular at radio frequencies because they remove nearly all core-related problems: no hysteresis, no saturation, no ferrite losses, and a highly linear inductance over a wide current range. Flat spirals in particular are easy to manufacture \u2014 they can be etched directly as copper traces on a PCB \u2014 which is why they appear in NFC and RFID antennas, wireless charging receivers, RF tank circuits, and metal detector search coils.<\/p>\n<p>The inductance \\(L\\) tells you how much magnetic flux the coil links for a given current, and it scales with the square of the turn count: doubling \\(N\\) roughly quadruples \\(L\\). Larger coils (bigger diameters) also produce more inductance. Keep in mind that at high frequencies the coil also behaves as a resonant circuit due to its parasitic capacitance and winding resistance, so the geometric inductance calculated here is only part of the full picture. When a flat spiral is combined with other inductors in a circuit, use the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/series-inductance\/\">Series Inductance<\/a> or <a href=\"https:\/\/wpcalc.com\/en\/engineering\/parallel-inductance\/\">Parallel Inductance<\/a> calculator to find the total, and don't forget the contribution of connecting traces, which you can estimate with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/straight-wire-inductor\/\">Straight Wire Inductance<\/a> calculator.<\/p>\n<h2>Accuracy, limitations and practical tips<\/h2>\n<ul>\n<li>The formula is an empirical approximation. It assumes the turns are evenly spaced across the radial width and that the conductor is thin compared to the coil diameter. Real coils can deviate by several percent or more.<\/li>\n<li>The denominator must be positive: \\(15d_1 &gt; 7d_2\\). If your inner diameter is too small relative to the outer diameter, the result becomes meaningless \u2014 redesign the spiral so the winding has a reasonable radial width.<\/li>\n<li>Conductor width, thickness, and skin effect are not included in the model. At very high frequencies, skin effect and proximity effect raise the effective resistance even though the inductance changes only slightly.<\/li>\n<li>Nearby metal objects, ground planes, or shielding alter the magnetic field and shift the actual inductance away from the calculated value. Keep the coil at least one coil-diameter away from large conductive surfaces when possible.<\/li>\n<li>Above the coil's self-resonant frequency (SRF) it no longer behaves as an inductor at all, so always check that your operating frequency is well below the SRF of the physical coil.<\/li>\n<li>For critical RF designs, verify the result with an LCR meter or a vector network analyzer, and fine-tune the number of turns experimentally.<\/li>\n<\/ul>\n<h2>Quick reference values<\/h2>\n<p>The table below shows calculated inductances for coils with \\(d_1 = 10\\) mm and \\(d_2 = 20\\) mm at various turn counts, so you can sense how strongly \\(N\\) influences the result:<\/p>\n<table>\n<thead>\n<tr>\n<th>Turns \\(N\\)<\/th>\n<th>Inductance (nH)<\/th>\n<th>Inductance (\u00b5H)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>3,986.01<\/td>\n<td>3.99<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>11,072.25<\/td>\n<td>11.07<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>28,344.96<\/td>\n<td>28.34<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>44,289.00<\/td>\n<td>44.29<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Once you know the target inductance, you can plug it into an impedance-matching stage such as a <a href=\"https:\/\/wpcalc.com\/en\/engineering\/pi-network-impedance-matching\/\">PI Network Impedance Matching<\/a> network to complete your RF front-end design.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the inductance of a flat spiral air core coil from its inner diameter, outer diameter, and number of turns. Enter the coil dimensions in millimeters and read the result in nanohenries.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[93],"class_list":["post-62013","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-coils-transformers"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62013","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=62013"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62013\/revisions"}],"predecessor-version":[{"id":65944,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62013\/revisions\/65944"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=62013"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=62013"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=62013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}