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<title>Symmetrical components</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Symmetrical components</span></span>

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<p>In <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a>, the method of <b>symmetrical components</b> simplifies the analysis of a <a href="Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> power system exhibiting an <a href="Electrical_fault" title="Electrical fault">electrical fault</a> or other unbalanced condition.<sup id="cite_ref-FOOTNOTEAmbergRangel20201_1-0" class="reference"><a href="#cite_note-FOOTNOTEAmbergRangel20201-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The symmetrical components corresponding to an asymmetrical set of three phasors are:<sup id="cite_ref-FOOTNOTEAmbergRangel20201_1-1" class="reference"><a href="#cite_note-FOOTNOTEAmbergRangel20201-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<ul><li>Sequence 0 (also known as <b>zero sequence</b> or <b>homopolar</b>) is one-third the sum of the original three phasors.</li>

<li>Sequence 1 (<b>positive sequence</b>) is one-third the sum of the original three phasors rotated counterclockwise by 0°, 120°, and 240°.</li>

<li>Sequence 2 (<b>negative sequence</b>) is one-third the sum of the original three phasors rotated counterclockwise 0°, 240°, and 120°.</li></ul>

<p>The analysis of power system is much simpler in the domain of symmetrical components, because the resulting equations are mutually <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> if the power system itself is <a href="Balanced_system" class="mw-redirect" title="Balanced system">balanced</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In this case, each symmetrical component can be analyzed separately, similar to the <a href="Per-phase_analysis" class="mw-redirect" title="Per-phase analysis">per-phase analysis</a>.

</p><p>The <a href="Protective_relays" class="mw-redirect" title="Protective relays">protective relays</a> utilize the symmetric components for fault detection. For example, during the normal operation, the zero-sequence current is very small, so a high current value is a convenient and reliable indicator of a ground fault.<sup id="cite_ref-FOOTNOTEAnderson1998271–272_3-0" class="reference"><a href="#cite_note-FOOTNOTEAnderson1998271–272-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The basic idea dates back to 1895, when <a href="Galileo_Ferraris" title="Galileo Ferraris">Ferraris</a> et al. produced an analysis of a single-phase motor by splitting a field set inside it into two components revolving in the opposite directions. The concept now known as the positive and negative sequences was published by <a href="Ernst_Alexanderson" title="Ernst Alexanderson">Ernst Alexanderson</a> in 1913 in his work on phase balancers, and by L. G. Stokvis in 1912-1915 while investigating the voltage regulation. These works lacked the clear definition of a zero sequence.<sup id="cite_ref-FOOTNOTEEvans19335_4-0" class="reference"><a href="#cite_note-FOOTNOTEEvans19335-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>In 1918 <a href="Charles_Legeyt_Fortescue" class="mw-redirect" title="Charles Legeyt Fortescue">Charles Legeyt Fortescue</a> presented a paper<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> which demonstrated that any set of N unbalanced <a href="Phasor" title="Phasor">phasors</a> (that is, any such <i><a href="Polyphase_system" title="Polyphase system">polyphase</a></i> signal) could be expressed as the sum of N symmetrical sets of balanced phasors, for values of N that are prime. Only a single frequency component is represented by the phasors.

</p><p>In 1943 <a href="Edith_Clarke" title="Edith Clarke">Edith Clarke</a> published a textbook giving a method of use of symmetrical components for three-phase systems that greatly simplified calculations over the original Fortescue paper.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In a three-phase system, one set of phasors has the same <a href="Phase_sequence" class="mw-redirect" title="Phase sequence">phase sequence</a> as the system under study (positive sequence; say ABC), the second set has the reverse phase sequence (negative sequence; ACB), and in the third set the phasors A, B and C are in phase with each other (<a href="Zero_sequence" class="mw-redirect" title="Zero sequence">zero sequence</a>, the <a href="Common-mode_signal" title="Common-mode signal">common-mode signal</a>). Essentially, this method converts three unbalanced phases into three independent sources, which makes <a href="Fault_(power_engineering)" class="mw-redirect" title="Fault (power engineering)">asymmetric fault</a> analysis more tractable.

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<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

<p>By expanding a <a href="One-line_diagram" class="mw-redirect" title="One-line diagram">one-line diagram</a> to show the positive sequence, negative sequence, and zero sequence impedances of <a href="Electrical_generator" class="mw-redirect" title="Electrical generator">generators</a>, <a href="Transformer" title="Transformer">transformers</a> and other devices including <a href="Overhead_power_line" title="Overhead power line">overhead lines</a> and <a href="Electrical_cable" title="Electrical cable">cables</a>, analysis of such unbalanced conditions as a single line to ground short-circuit fault is greatly simplified. The technique can also be extended to higher order phase systems.

</p><p>Physically, in a three phase system, a positive sequence set of currents produces a normal rotating field, a negative sequence set produces a field with the opposite rotation, and the zero sequence set produces a field that oscillates but does not rotate between phase windings. Since these effects can be detected physically with sequence filters, the mathematical tool became the basis for the design of <a href="Protective_relay" title="Protective relay">protective relays</a>, which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip <a href="Circuit_breaker" title="Circuit breaker">circuit breakers</a> or take other steps to protect electrical systems.

</p><p>The analytical technique was adopted and advanced by engineers at <a href="General_Electric" title="General Electric">General Electric</a> and <a href="Westinghouse_Electric_Corporation" title="Westinghouse Electric Corporation">Westinghouse</a>, and after <a href="World_War_II" title="World War II">World War II</a> it became an accepted method for asymmetric fault analysis.

</p><p>As shown in the figure to the above right, the three sets of symmetrical components (positive, negative, and zero sequence) add up to create the system of three unbalanced phases as pictured in the bottom of the diagram. The imbalance between phases arises because of the difference in magnitude and phase shift between the sets of vectors. Notice that the colors (red, blue, and yellow) of the separate sequence vectors correspond to three different phases (A, B, and C, for example). To arrive at the final plot, the sum of vectors of each phase is calculated. This resulting vector is the effective phasor representation of that particular phase. This process, repeated, produces the phasor for each of the three phases.

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<div class="mw-heading mw-heading2"><h2 id="The_three-phase_case">The three-phase case</h2></div>

<p>Symmetrical components are most commonly used for analysis of <a href="Three-phase_electric_power" title="Three-phase electric power">three-phase electrical power systems</a>. The voltage or current of a three-phase system at some point can be indicated by three phasors, called the three components of the voltage or the current.

</p><p>This article discusses voltage; however, the same considerations also apply to current. In a perfectly balanced three-phase power system, the voltage phasor components have equal magnitudes but are 120 degrees apart. In an unbalanced system, the magnitudes and phases of the voltage phasor components are different.

</p><p>Decomposing the voltage phasor components into a set of symmetrical components helps analyze the system as well as visualize any imbalances.

If the three voltage components are expressed as <a href="Phasor" title="Phasor">phasors</a> (which are complex numbers), a complex vector can be formed in which the three phase components are the components of the vector. A vector for three phase voltage components can be written as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{abc}={\begin{bmatrix}V_{a}\\V_{b}\\V_{c}\end{bmatrix}}}</annotation>

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</math></span><img src="./1c4a63f29d7b401e9ca1d495561f1df3f8dbbb6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.338ex; height:9.176ex;" alt="{\displaystyle \mathbf {v} _{abc}={\begin{bmatrix}V_{a}\\V_{b}\\V_{c}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>and decomposing the vector into three symmetrical components gives

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}V_{a}\\V_{b}\\V_{c}\end{bmatrix}}={\begin{bmatrix}V_{a,0}\\V_{b,0}\\V_{c,0}\end{bmatrix}}+{\begin{bmatrix}V_{a,1}\\V_{b,1}\\V_{c,1}\end{bmatrix}}+{\begin{bmatrix}V_{a,2}\\V_{b,2}\\V_{c,2}\end{bmatrix}}}</annotation>

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</math></span><img src="./79c84f1643345240ac8362470896ba76db586b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:37.854ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}V_{a}\\V_{b}\\V_{c}\end{bmatrix}}={\begin{bmatrix}V_{a,0}\\V_{b,0}\\V_{c,0}\end{bmatrix}}+{\begin{bmatrix}V_{a,1}\\V_{b,1}\\V_{c,1}\end{bmatrix}}+{\begin{bmatrix}V_{a,2}\\V_{b,2}\\V_{c,2}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components (NSC). The sequence components differ only by their phase angles, which are symmetrical and so are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {2}{3}}\pi }">

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {2}{3}}\pi }</annotation>

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</math></span><img src="./83f6376d76b329c4e308f11100070ab12bf5a3e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.445ex; height:3.176ex;" alt="{\displaystyle \scriptstyle {\frac {2}{3}}\pi }" loading="lazy"></span> radians or 120°.

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<div class="mw-heading mw-heading3"><h3 id="A_matrix">A matrix</h3></div>

<p>Define a phasor rotation operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, which rotates a phasor vector counterclockwise by 120 degrees when multiplied by it:

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<annotation encoding="application/x-tex">{\displaystyle \alpha \equiv e^{{\frac {2}{3}}\pi i}}</annotation>

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</math></span><img src="./5bad0857ab53fcd5c0ce725cf2f1f889973d7141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.915ex; height:3.509ex;" alt="{\displaystyle \alpha \equiv e^{{\frac {2}{3}}\pi i}}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}=1}</annotation>

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</math></span><img src="./5b90aa7bafac6b90b700d04e233ad38c3a97f17b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.803ex; height:2.676ex;" alt="{\displaystyle \alpha ^{3}=1}" loading="lazy"></span> so that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{-1}=\alpha ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{-1}=\alpha ^{2}}</annotation>

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</p><p>The zero sequence components have equal magnitude and are in phase with each other, therefore:

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<annotation encoding="application/x-tex">{\displaystyle V_{0}\equiv V_{a,0}=V_{b,0}=V_{c,0}}</annotation>

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<p>and the other sequence components have the same magnitude, but their phase angles differ by 120°. If the original unbalanced set of voltage phasors have positive or <i>abc</i> phase sequence, then:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{1}&\equiv V_{a,1}=\alpha V_{b,1}=\alpha ^{2}V_{c,1}\end{aligned}}}</annotation>

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</math></span><img src="./c03c8d576aba99739b6205f0d324df4c17fb70e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.373ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}V_{1}&\equiv V_{a,1}=\alpha V_{b,1}=\alpha ^{2}V_{c,1}\end{aligned}}}" loading="lazy"></span>,</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{2}&\equiv V_{a,2}=\alpha ^{2}V_{b,2}=\alpha V_{c,2}\end{aligned}}}</annotation>

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<p>meaning that

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{b,1}=\alpha ^{2}V_{1}\end{aligned}}}</annotation>

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</math></span><img src="./e9525d1dc1ad6c6f33293f497708f52fcf88011d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.373ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}V_{b,1}=\alpha ^{2}V_{1}\end{aligned}}}" loading="lazy"></span>,</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{c,1}=\alpha V_{1}\end{aligned}}}</annotation>

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</math></span><img src="./0350a4a41faf936262e24600006513142a72148d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.326ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}V_{c,1}=\alpha V_{1}\end{aligned}}}" loading="lazy"></span>,</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{b,2}=\alpha V_{2}\end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{c,2}=\alpha ^{2}V_{2}\end{aligned}}}</annotation>

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</math></span><img src="./c4aec0b0d864551c77502643580664fd49869512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.38ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}V_{c,2}=\alpha ^{2}V_{2}\end{aligned}}}" loading="lazy"></span>.</dd></dl>

<p>Thus,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} _{abc}&={\begin{bmatrix}V_{0}\\V_{0}\\V_{0}\end{bmatrix}}+{\begin{bmatrix}V_{1}\\\alpha ^{2}V_{1}\\\alpha V_{1}\end{bmatrix}}+{\begin{bmatrix}V_{2}\\\alpha V_{2}\\\alpha ^{2}V_{2}\end{bmatrix}}\\&={\begin{bmatrix}1&1&1\\1&\alpha ^{2}&\alpha \\1&\alpha &\alpha ^{2}\end{bmatrix}}{\begin{bmatrix}V_{0}\\V_{1}\\V_{2}\end{bmatrix}}\\&={\textbf {A}}\mathbf {v} _{012}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./a0835f85d0c6aafd921e02b29d8acee79366a1a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:37.329ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {v} _{abc}&={\begin{bmatrix}V_{0}\\V_{0}\\V_{0}\end{bmatrix}}+{\begin{bmatrix}V_{1}\\\alpha ^{2}V_{1}\\\alpha V_{1}\end{bmatrix}}+{\begin{bmatrix}V_{2}\\\alpha V_{2}\\\alpha ^{2}V_{2}\end{bmatrix}}\\&={\begin{bmatrix}1&1&1\\1&\alpha ^{2}&\alpha \\1&\alpha &\alpha ^{2}\end{bmatrix}}{\begin{bmatrix}V_{0}\\V_{1}\\V_{2}\end{bmatrix}}\\&={\textbf {A}}\mathbf {v} _{012}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>where

</p>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{012}={\begin{bmatrix}V_{0}\\V_{1}\\V_{2}\end{bmatrix}},{\textbf {A}}={\begin{bmatrix}1&1&1\\1&\alpha ^{2}&\alpha \\1&\alpha &\alpha ^{2}\end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./33a707da776dbf8d65a4eec79ea2e7c2d15eadf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.365ex; height:9.509ex;" alt="{\displaystyle \mathbf {v} _{012}={\begin{bmatrix}V_{0}\\V_{1}\\V_{2}\end{bmatrix}},{\textbf {A}}={\begin{bmatrix}1&1&1\\1&\alpha ^{2}&\alpha \\1&\alpha &\alpha ^{2}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>If instead the original unbalanced set of voltage phasors have negative or <i>acb</i> phase sequence, the following matrix can be similarly derived:

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {A}}_{acb}={\begin{bmatrix}1&1&1\\1&\alpha &\alpha ^{2}\\1&\alpha ^{2}&\alpha \end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./b34d9c99cb25e71ef16c4786dff33bb134ce765c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:22.381ex; height:9.509ex;" alt="{\displaystyle {\textbf {A}}_{acb}={\begin{bmatrix}1&1&1\\1&\alpha &\alpha ^{2}\\1&\alpha ^{2}&\alpha \end{bmatrix}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Decomposition">Decomposition</h3></div>

<p>The sequence components are derived from the analysis equation

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{012}={\textbf {A}}^{-1}\mathbf {v} _{abc}}</annotation>

</semantics>

</math></span><img src="./b1c49b78cdacdf908ee47fbb451351bef5221d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.49ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} _{012}={\textbf {A}}^{-1}\mathbf {v} _{abc}}" loading="lazy"></span></dd></dl>

<p>where

</p>

<msup>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {A}}^{-1}={\frac {1}{3}}{\begin{bmatrix}1&1&1\\1&\alpha &\alpha ^{2}\\1&\alpha ^{2}&\alpha \end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./53ddbc63a55195f01e44956c3740909cd09bccb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.193ex; height:9.509ex;" alt="{\displaystyle {\textbf {A}}^{-1}={\frac {1}{3}}{\begin{bmatrix}1&1&1\\1&\alpha &\alpha ^{2}\\1&\alpha ^{2}&\alpha \end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>The above two equations tell how to derive symmetrical components corresponding to an asymmetrical set of three phasors:

</p>

<ul><li>Sequence 0 is one-third the sum of the original three phasors.</li>

<li>Sequence 1 is one-third the sum of the original three phasors rotated counterclockwise 0°, 120°, and 240°.</li>

<li>Sequence 2 is one-third the sum of the original three phasors rotated counterclockwise 0°, 240°, and 120°.</li></ul>

<p>Visually, if the original components are symmetrical, sequences 0 and 2 will each form a triangle, summing to zero, and sequence 1 components will sum to a straight line.

</p>

<div class="mw-heading mw-heading3"><h3 id="Intuition">Intuition</h3></div>

<p>The phasors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle V_{(ab)}=V_{(a)}-V_{(b)};\;V_{(bc)}=V_{(b)}-V_{(c)};\;V_{(ca)}=V_{(c)}-V_{(a)}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle V_{(ab)}=V_{(a)}-V_{(b)};\;V_{(bc)}=V_{(b)}-V_{(c)};\;V_{(ca)}=V_{(c)}-V_{(a)}}</annotation>

</semantics>

</math></span><img src="./8869117c5c4cacec960f0f6e2d2dbc1cf4ef2b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.754ex; height:2.343ex;" alt="{\displaystyle \scriptstyle V_{(ab)}=V_{(a)}-V_{(b)};\;V_{(bc)}=V_{(b)}-V_{(c)};\;V_{(ca)}=V_{(c)}-V_{(a)}}" loading="lazy"></span> form a closed triangle (e.g., outer voltages or line to line voltages). To find the synchronous and inverse components of the phases, take any side of the outer triangle and draw the two possible equilateral triangles sharing the selected side as base. These two equilateral triangles represent a synchronous and an inverse system.

</p><p>If the phasors V were a perfectly synchronous system, the vertex of the outer triangle not on the base line would be at the same position as the corresponding vertex of the equilateral triangle representing the synchronous system. Any amount of inverse component would mean a deviation from this position. The deviation is exactly 3 times the inverse phase component.

</p><p>The synchronous component is in the same manner 3 times the deviation from the "inverse equilateral triangle". The directions of these components are correct for the relevant phase. It seems counter intuitive that this works for all three phases regardless of the side chosen but that is the beauty of this illustration. The graphic is from <a href="Napoleon's_Theorem" class="mw-redirect" title="Napoleon's Theorem">Napoleon's Theorem</a>, which matches a graphical calculation technique that sometimes appears in older references books.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Poly-phase_case">Poly-phase case</h2></div>

<p>The idea can be expanded to <i>N</i> phases: an asymmetrical set of <i>N</i> <a href="Phasor" title="Phasor">phasors</a> can be expressed as a <a href="Linear_combination" title="Linear combination">linear combination</a> of <i>N</i> symmetrical sets of phasors by means of a <a href="Complex_number" title="Complex number">complex</a> <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Fortescue's theorem (symmetrical components) is based on the <a href="Superposition_theorem" class="mw-redirect" title="Superposition theorem">superposition principle</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> so it is applicable to linear power systems only, or to linear approximations of non-linear power systems.

</p><p>It can be seen that the transformation matrix A above is a <a href="DFT_matrix" title="DFT matrix">DFT matrix</a>, and as such, symmetrical components can be calculated for any poly-phase system.

</p>

<div class="mw-heading mw-heading2"><h2 id="Contribution_of_harmonics_to_symmetrical_components_in_3-phase_power_systems">Contribution of harmonics to symmetrical components in 3-phase power systems</h2></div>

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<p><a href="Harmonics_(electrical_power)" title="Harmonics (electrical power)">Harmonics</a> often occur in power systems as a consequence of non-linear loads. Each order of harmonics contributes to different sequence components. The fundamental and harmonics of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 3n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle 3n+1}</annotation>

</semantics>

</math></span><img src="./0acd79fb81983b062a9a528e01adf34ce04e6fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.909ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 3n+1}" loading="lazy"></span> will contribute to the positive sequence component. Harmonics of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 3n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle 3n-1}</annotation>

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</math></span><img src="./5600e7a0b68d8c5087b8686df00add9b10dff8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.909ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 3n-1}" loading="lazy"></span> will contribute to the negative sequence. Harmonics of order <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 3n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle 3n}</annotation>

</semantics>

</math></span><img src="./c51dcd453638b5dcf83021eead788465ec32267c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 3n}" loading="lazy"></span> contribute to the zero sequence.

</p><p>Note that the rules above are only applicable if the phase values (or distortion) in each phase are exactly the same. Please further note that even harmonics are not common in power systems.

</p>

<div class="mw-heading mw-heading2"><h2 id="Consequence_of_the_zero_sequence_component_in_power_systems">Consequence of the zero sequence component in power systems</h2></div>

<p>The zero sequence represents the component of the unbalanced phasors that is equal in magnitude and phase. Because they are in phase, zero sequence currents flowing through an n-phase network will sum to n times the magnitude of the individual zero sequence currents components. Under normal operating conditions this sum is small enough to be negligible. However, during large zero sequence events such as lightning strikes, this nonzero sum of currents can lead to a larger current flowing through the neutral conductor than the individual phase conductors. Because neutral conductors are typically not larger than individual phase conductors, and are often smaller than these conductors, a large zero sequence component can lead to overheating of neutral conductors and to fires.

</p><p>One way to prevent large zero sequence currents is to use a delta connection, which appears as an open circuit to zero sequence currents. For this reason, most transmission, and much sub-transmission is implemented using delta. Much distribution is also implemented using delta, although "old work" distribution systems have occasionally been "wyed-up" (converted from <a href="Delta-wye_transformer" title="Delta-wye transformer">delta</a> to <a href="Delta-wye_transformer" title="Delta-wye transformer">wye</a>) so as to increase the line's capacity at a low converted cost, but at the expense of a higher central station protective relay cost.

</p>

<ul><li><a href="Symmetry" title="Symmetry">Symmetry</a></li>

<li><a href="Direct-quadrature-zero_transformation" title="Direct-quadrature-zero transformation">Direct-quadrature-zero transformation</a></li>

<li><a href="Alpha%E2%80%93beta_transformation" title="Alpha–beta transformation">Alpha–beta transformation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<dl><dt>Notes</dt></dl>

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<li id="cite_note-FOOTNOTEAmbergRangel20201-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAmbergRangel20201_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAmbergRangel20201_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAmbergRangel2020">Amberg & Rangel 2020</a>, p. 1.</span>

</li>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFBlackburn1993" class="citation book cs1">Blackburn, J. Lewis (1993-06-07). <i>Symmetrical Components for Power Systems Engineering</i> (1st ed.). New York: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-8767-7</bdi>.</cite></span>

</li>

<li id="cite_note-FOOTNOTEAnderson1998271–272-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAnderson1998271–272_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAnderson1998">Anderson 1998</a>, pp. 271–272.</span>

</li>

<li id="cite_note-FOOTNOTEEvans19335-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEvans19335_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEvans1933">Evans 1933</a>, p. 5.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Charles L. Fortescue, "<a rel="nofollow" class="external text" href="http://www.energyscienceforum.com/files/fortescue/methodofsymmetrical.pdf">Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks</a>". Presented at the 34th annual convention of the AIEE (American Institute of Electrical Engineers) in Atlantic City, N.J. on 28 June 1918. Published in: <i>AIEE Transactions</i>, vol. 37, part II, pages 1027–1140 (1918). For a brief history of the early years of symmetrical component theory, see: J. Lewis Blackburn, <i>Symmetrical Components for Power Engineering</i> (Boca Raton, Florida: CRC Press, 1993), pages 3–4.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Gabriele Kass-Simon, Patricia Farnes, Deborah Nash (ed), <i>Women of Science: Righting the Record</i>, Indiana University Press, 1993, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0253208130</bdi>. pages 164-168</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFWagnerEvans1933" class="citation book cs1">Wagner, C. F.; Evans, R. D. (1933). <i>Symmetrical Components</i>. New York and London: McGraw Hill. p. 265.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFHadjsaïdSabonnadière2013" class="citation book cs1">Hadjsaïd, Nouredine; Sabonnadière, Jean-Claude (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bpEeycYeWJIC&pg=PT244"><i>Power Systems and Restructuring</i></a>. John Wiley & Sons. p. 244. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118599921</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFMathisPauli1999" class="citation book cs1">Mathis, Wolfgang; Pauli, Rainer (1999). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/047134608X.W2507"><i>Network Theorems</i></a>. Wiley Online Library. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F047134608X.W2507">10.1002/047134608X.W2507</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>047134608X</bdi>. <q>[…] the results of Fortescue […] are proven by the superposition theorem, and for this reason, a direct generalization to nonlinear networks is impossible.</q></cite></span>

</li>

</ol></div></div>

<dl><dt>Bibliography</dt></dl>

<ul><li><cite id="CITEREFAmbergRangel2020" class="citation web cs1">Amberg, Ariana; Rangel, Alex (2020). <a rel="nofollow" class="external text" href="https://selinc.com/api/download/100688">"Tutorial on Symmetrical Components: Part 2"</a>. Schweitzer Engineering Laboratories, Inc<span class="reference-accessdate">. Retrieved <span class="nowrap">22 June</span> 2025</span>.</cite></li>

<li><cite id="CITEREFAnderson1998" class="citation book cs1">Anderson, Paul M. (1998-12-09). <a rel="nofollow" class="external text" href="https://dn790002.ca.archive.org/0/items/POWERSYSTEMPROTECTIONP.M.Anderson/POWER%20SYSTEM%20PROTECTION%2C%20P.%20M.%20Anderson.pdf"><i>Power System Protection</i></a> <span class="cs1-format">(PDF)</span>. New York: Wiley-IEEE Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7803-3427-2</bdi>.</cite></li>

<li>J. Lewis Blackburn <i>Symmetrical Components for Power Systems Engineering</i>, Marcel Dekker, New York (1993). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-8767-6</bdi></li>

<li>William D. Stevenson, Jr. <i>Elements of Power System Analysis Third Edition</i>, <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, New York (1975). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-061285-4</bdi>.</li>

<li><cite id="CITEREFEvans1933" class="citation book cs1">Evans, R. D. (1933). <a rel="nofollow" class="external text" href="https://archive.org/details/dli.ernet.288563"><i>Symmetrical Components</i></a>. New York: Mcgraw-Hill.</cite></li>

<li>Westinghouse Corporation, <i>Applied Protective Relaying</i>, 1976, Westinghouse Corporation, no ISBN, Library of Congress card no. 76-8060 - a standard reference on electromechanical protective relays</li></ul>

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