{"id":62008,"date":"2021-11-16T13:50:59","date_gmt":"2021-11-16T11:50:59","guid":{"rendered":"https:\/\/wpcalc.com\/?p=62008"},"modified":"2026-09-30T18:38:07","modified_gmt":"2026-09-30T15:38:07","slug":"three-phase-power-by-two-watt-meters-method","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/three-phase-power-by-two-watt-meters-method\/","title":{"rendered":"Three-Phase Power by Two Wattmeters 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\">Two Wattmeters Power Measurement Tool<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"62008\" data-favorite-title=\"Three-Phase Power by Two Wattmeters\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/three-phase-power-by-two-watt-meters-method\/\" data-favorite-excerpt=\"Calculate the total active power in a three-phase three-wire system using the two-wattmeter method. Enter the two line voltages (V12, V13), line currents (I2, I3) and the phase displacement angle to get the result in watts.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-electrical\">\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\/three-phase-power-by-two-watt-meters-method\/\" method=\"POST\" class=\"calculator\" id=\"calculator-62008\" data-post=\"62008\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Voltage V\u2081\u2082<\/label><div class=\"term\">V\u2081\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"voltage_v\" id=\"field-1\" aria-describedby=\"calc-unit-1\" step=\"0.01\" value=\"400\"\/><div class=\"unit\" id=\"calc-unit-1\">V<\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Current I\u2082<\/label><div class=\"term\">I\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"current_i\" id=\"field-2\" aria-describedby=\"calc-unit-2\" step=\"0.01\" value=\"10\"\/><div class=\"unit\" id=\"calc-unit-2\">A<\/div><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Displacement angle<\/label><div class=\"term\">\u03b8<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"displacement_angle\" id=\"field-3\" aria-describedby=\"calc-unit-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"360\" value=\"30\"\/><div class=\"unit\" id=\"calc-unit-3\">\u00b0<\/div><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Voltage V\u2081\u2083<\/label><div class=\"term\">V\u2081\u2083<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"voltage_v\" id=\"field-4\" aria-describedby=\"calc-unit-4\" step=\"0.01\" value=\"400\"\/><div class=\"unit\" id=\"calc-unit-4\">V<\/div><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Current I\u2083<\/label><div class=\"term\">I\u2083<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"current_i\" id=\"field-5\" aria-describedby=\"calc-unit-5\" step=\"0.01\" value=\"10\"\/><div class=\"unit\" id=\"calc-unit-5\">A<\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"62008\">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>Three Phase Power<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"W\"><\/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=\"62008\" 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>Two Wattmeter Method Power 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\\[ P = V_{12} \\cdot I_2 \\cdot \\cos(30^\\circ + \\theta) + V_{13} \\cdot I_3 \\cdot \\cos(30^\\circ - \\theta) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(P\\) \u2014 total active power of the three-phase system (W)<\/li>\n<li>\\(V_{12}\\) \u2014 line voltage between lines 1 and 2 (V)<\/li>\n<li>\\(V_{13}\\) \u2014 line voltage between lines 1 and 3 (V)<\/li>\n<li>\\(I_2\\) \u2014 line current in line 2 (A)<\/li>\n<li>\\(I_3\\) \u2014 line current in line 3 (A)<\/li>\n<li>\\(\\theta\\) \u2014 phase displacement angle between voltage and current (degrees)<\/li>\n<\/ul>\n<p>The formula adds the readings of the two wattmeters: the first measures \\(V_{12} I_2 \\cos(30^\\circ + \\theta)\\) and the second measures \\(V_{13} I_3 \\cos(30^\\circ - \\theta)\\). Their sum gives the total active power, valid for balanced and unbalanced three-phase three-wire loads.<\/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>Two Wattmeters \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a 400 V three-phase system with line currents of 10 A in both lines 2 and 3, and a phase displacement angle of 30\u00b0.<\/p>\n<p>\\[ P = 400 \\cdot 10 \\cdot \\cos(30^\\circ + 30^\\circ) + 400 \\cdot 10 \\cdot \\cos(30^\\circ - 30^\\circ) \\]<\/p>\n<p>\\[ P = 4000 \\cdot \\cos 60^\\circ + 4000 \\cdot \\cos 0^\\circ = 4000 \\cdot 0.5 + 4000 \\cdot 1 \\]<\/p>\n<p><strong>Result:<\/strong> \\(P = 2000 + 4000 = 6000\\) W \u2014 the total three-phase active power is 6 kW. Note that the first wattmeter reads only 2000 W while the second reads 4000 W; only their sum equals the true total power.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the line voltage <strong>Voltage V\u2081\u2082<\/strong> \u2014 the voltage measured between lines 1 and 2 (default 400 V).<\/li>\n<li>Enter <strong>Current I\u2082<\/strong>, the current flowing in line 2 (default 10 A).<\/li>\n<li>Enter the <strong>Displacement angle<\/strong> \u03b8 between voltage and current in degrees (default 30\u00b0). Use 0\u00b0 for a purely resistive load.<\/li>\n<li>Enter <strong>Voltage V\u2081\u2083<\/strong>, the voltage between lines 1 and 3 (default 400 V).<\/li>\n<li>Enter <strong>Current I\u2083<\/strong>, the current in line 3 (default 10 A).<\/li>\n<li>Read the <strong>Three Phase Power<\/strong> result in watts.<\/li>\n<\/ol>\n<h2>What Is the Two Wattmeter Method?<\/h2>\n<p>The two wattmeter method (Aron connection) is the standard technique for measuring active power in three-phase three-wire systems. Two wattmeters are connected with their current coils in two of the three lines and their voltage coils between those lines and the third line. The total active power of the system is the <em>algebraic sum<\/em> of the two readings, regardless of whether the load is balanced or unbalanced, star or delta connected.<\/p>\n<p>Each wattmeter measures the product of a line voltage and a line current, multiplied by the cosine of the angle between them. Because those angles are \\((30^\\circ + \\theta)\\) and \\((30^\\circ - \\theta)\\) for a balanced load, the individual readings depend heavily on the power factor \u2014 one wattmeter may even read zero at \u03b8 = 60\u00b0 or negative at \u03b8 &gt; 60\u00b0. The sum, however, always equals the true total active power \\(P = \\sqrt{3}\\,V_L I_L \\cos\\theta\\) for a balanced system.<\/p>\n<p>This method matters in practice because it needs only two metering points instead of three, works without access to a neutral conductor, and remains accurate under unbalanced load conditions. It is used for commissioning tests, load studies and energy monitoring in industrial installations. To fully analyse a circuit, pair this result with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/real-reactive-power\/\">Real and Reactive Power calculator<\/a>, since the two wattmeter readings can also yield the reactive power and power factor of the system.<\/p>\n<h2>Interpreting the Result<\/h2>\n<p>A positive total power confirms that net power flows into the load. If one of the wattmeter terms becomes negative (which happens when the displacement angle exceeds 60\u00b0, i.e. power factor below 0.5), it must be subtracted rather than added \u2014 in the physical measurement this corresponds to reversing a wattmeter connection. The calculator handles the signs automatically through the cosine terms.<\/p>\n<p>For a balanced system with \\(V_{12} = V_{13} = V_L\\) and \\(I_2 = I_3 = I_L\\), you can sanity-check the result against \\(P = \\sqrt{3}\\,V_L I_L \\cos\\theta\\). With the default inputs, \\(\\sqrt{3} \\cdot 400 \\cdot 10 \\cdot \\cos 30^\\circ \\approx 6000\\) W, which matches the calculator output exactly.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Three-wire systems only.<\/strong> The method is valid for three-phase three-wire circuits (no neutral). In four-wire systems with neutral current, a third wattmeter is required.<\/li>\n<li><strong>Angle convention.<\/strong> \u03b8 is the phase displacement between the line voltage and the line current. For a purely resistive load \u03b8 = 0\u00b0, for an inductive load \u03b8 &gt; 0\u00b0 (current lagging).<\/li>\n<li><strong>Unbalanced loads.<\/strong> The formula stays valid, but you must use the actual measured voltages and currents of each line \u2014 the symmetric case \\(V_{12} = V_{13}\\), \\(I_2 = I_3\\) only applies to balanced loads.<\/li>\n<li><strong>Distortion.<\/strong> The result assumes sinusoidal voltages and currents. With significant harmonic distortion, additional measurement techniques are needed.<\/li>\n<\/ul>\n<h2>Practical Tips<\/h2>\n<p>When performing the actual measurement, make sure both wattmeters share the same reference line (line 1 in this configuration) and that polarity markings are respected; otherwise the two readings will not sum correctly. If the conductors are undersized, voltage drop will distort the readings \u2014 check suitability with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/wire-gauge\/\">Wire Gauge calculator<\/a>. For single-phase circuits, a different approach is needed; see the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/single-phase-transformer\/\">Single Phase Transformer calculator<\/a> for related single-phase calculations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the total active power in a three-phase three-wire system using the two-wattmeter method. Enter the two line voltages (V12, V13), line currents (I2, I3) and the phase displacement angle to get the result in watts.<\/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":[21],"class_list":["post-62008","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-electrical"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62008","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=62008"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62008\/revisions"}],"predecessor-version":[{"id":66022,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/62008\/revisions\/66022"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=62008"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=62008"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=62008"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}