Leakage inductance

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inductive_leakage_factor_and_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Inductive leakage factor and inductance</span> </div> </a> <ul id="toc-Inductive_leakage_factor_and_inductance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Refined_inductive_leakage_factor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Refined_inductive_leakage_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Refined inductive leakage factor</span> </div> </a> <ul id="toc-Refined_inductive_leakage_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> 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perturbation of imperfectly coupled transformers</div> <p><b>Leakage inductance</b> derives from the electrical property of an imperfectly coupled <a href="/wiki/Transformer" title="Transformer">transformer</a> whereby each <a href="/wiki/Electromagnetic_coil" title="Electromagnetic coil">winding</a> behaves as a <a href="/wiki/Self-inductance" class="mw-redirect" title="Self-inductance">self-inductance</a> in <a href="/wiki/Series_and_parallel_circuits" title="Series and parallel circuits">series</a> with the winding's respective <a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">ohmic resistance</a> constant. These four winding constants also interact with the transformer's <a href="/wiki/Mutual_inductance" class="mw-redirect" title="Mutual inductance">mutual inductance</a>. The winding leakage inductance is due to leakage flux not linking with all turns of each imperfectly coupled winding. </p><p>Leakage reactance is usually the most important element of a power system transformer due to <a href="/wiki/Power_factor" title="Power factor">power factor</a>, <a href="/wiki/Voltage_drop" title="Voltage drop">voltage drop</a>, <a href="/wiki/Reactive_power" class="mw-redirect" title="Reactive power">reactive power</a> consumption and <a href="/wiki/Fault_(power_engineering)" class="mw-redirect" title="Fault (power engineering)">fault current</a> considerations.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Saarbafi-9_2-0" class="reference"><a href="#cite_note-Saarbafi-9-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Leakage inductance depends on the geometry of the core and the windings. Voltage drop across the <a href="/wiki/Inductive_reactance" class="mw-redirect" title="Inductive reactance">leakage reactance</a> results in often undesirable supply regulation with varying transformer load. But it can also be useful for <a href="/wiki/Harmonics_(electrical_power)" title="Harmonics (electrical power)">harmonic</a> isolation (<a href="/wiki/Attenuating" class="mw-redirect" title="Attenuating">attenuating</a> higher frequencies) of some loads.<sup id="cite_ref-FOOTNOTEIrwin1997362_3-0" class="reference"><a href="#cite_note-FOOTNOTEIrwin1997362-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Leakage inductance applies to any imperfectly coupled magnetic circuit device including <a href="/wiki/Electric_motor" title="Electric motor">motors</a>.<sup id="cite_ref-Pyrhonen_4-0" class="reference"><a href="#cite_note-Pyrhonen-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Leakage_inductance_and_inductive_coupling_factor">Leakage inductance and inductive coupling factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=1" title="Edit section: Leakage inductance and inductive coupling factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Coupling_coefficient2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Coupling_coefficient2.gif/350px-Coupling_coefficient2.gif" decoding="async" width="350" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Coupling_coefficient2.gif/525px-Coupling_coefficient2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Coupling_coefficient2.gif/700px-Coupling_coefficient2.gif 2x" data-file-width="708" data-file-height="456" /></a><figcaption>Fig. 1 L<sub>P</sub><sup>σ</sup>and L<sub>S</sub><sup>σ</sup> are primary and secondary <b>leakage inductances</b> expressed in terms of <b>inductive coupling coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span></b> under open-circuited conditions.</figcaption></figure> <p>The magnetic circuit's flux that does not interlink both windings is the leakage flux corresponding to primary leakage inductance L<sub>P</sub><sup>σ</sup> and secondary leakage inductance L<sub>S</sub><sup>σ</sup>. Referring to Fig. 1, these leakage inductances are defined in terms of transformer winding <a href="/wiki/Open-circuit_test" title="Open-circuit test">open-circuit</a> inductances and associated <a href="/wiki/Coupling_coefficient_(inductors)" class="mw-redirect" title="Coupling coefficient (inductors)">coupling coefficient</a> or coupling factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span>.<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><sup id="cite_ref-18-1_6-0" class="reference"><a href="#cite_note-18-1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><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><p>The primary open-circuit self-inductance is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{oc}^{pri}=L_{P}=L_{M}+L_{P}^{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{oc}^{pri}=L_{P}=L_{M}+L_{P}^{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0476dd4a5152e63f5b65d96bb68b95b7f35c5bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.629ex; height:3.509ex;" alt="{\displaystyle L_{oc}^{pri}=L_{P}=L_{M}+L_{P}^{\sigma }}" /></span> ------ <b>(Eq. 1.1a)</b></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}^{\sigma }=L_{P}\cdot {(1-k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}^{\sigma }=L_{P}\cdot {(1-k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e185d5fe9283e28471df70c5f50ec35f58af62bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.9ex; height:3.009ex;" alt="{\displaystyle L_{P}^{\sigma }=L_{P}\cdot {(1-k)}}" /></span> ------ <b>(Eq. 1.1b)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M}=L_{P}\cdot {k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M}=L_{P}\cdot {k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04118db6e798a0a075b3d02d08d1453d905fe841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.58ex; height:2.509ex;" alt="{\displaystyle L_{M}=L_{P}\cdot {k}}" /></span> ------ <b>(Eq. 1.1c)</b></dd></dl> <p>and </p> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{oc}^{pri}=L_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{oc}^{pri}=L_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107c51d5164958b9fb6a63d043a4d64f0602ce72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.099ex; height:3.176ex;" alt="{\displaystyle L_{oc}^{pri}=L_{P}}" /></span> is primary self-inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}^{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}^{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9f7a39ebc0ae6f680e9716d0b8095af406bae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.049ex; height:2.843ex;" alt="{\displaystyle L_{P}^{\sigma }}" /></span> is primary leakage inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244dc7cd3cce17fdf55c7d23032474f7e2167805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.542ex; height:2.509ex;" alt="{\displaystyle L_{M}}" /></span> is magnetizing inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is inductive coupling coefficient</li></ul></dd></dl> <div style="padding:1em; margin:0 0 0 1em; width:500px; border:1px solid; background:ivory;"> <p><b>Measuring basic transformer inductances & coupling factor</b> </p><p>Transformer self-inductances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2579498b31fad37f12a5f29864f14be642ccdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.049ex; height:2.509ex;" alt="{\displaystyle L_{P}}" /></span> & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ef2d071fea0ddcd21d9b50e18b78a6138fd69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.875ex; height:2.509ex;" alt="{\displaystyle L_{S}}" /></span> and mutual inductance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> are, in additive and subtractive series connection of the two windings, given by,<sup id="cite_ref-Brenner1959-591_8-0" class="reference"><a href="#cite_note-Brenner1959-591-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><dl><dd><dl><dd>in additive connection,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ser}^{+}=L_{P}+L_{S}+2M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ser}^{+}=L_{P}+L_{S}+2M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53557cc78c88a14f214148e9d70d409ee2243c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.402ex; height:2.843ex;" alt="{\displaystyle L_{ser}^{+}=L_{P}+L_{S}+2M}" /></span>, and,</dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd>in subtractive connection,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ser}^{-}=L_{P}+L_{S}-2M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ser}^{-}=L_{P}+L_{S}-2M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e14d60cb3523f2e6a0e4a79d11fc06f7776a11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.402ex; height:2.843ex;" alt="{\displaystyle L_{ser}^{-}=L_{P}+L_{S}-2M}" /></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd>such that these transformer inductances can be determined from the following three equations:<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><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ser}^{+}-L_{ser}^{-}=4M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <mn>4</mn> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ser}^{+}-L_{ser}^{-}=4M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f904b88064be9db755f906e1700c0629f4aea50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.731ex; height:2.843ex;" alt="{\displaystyle L_{ser}^{+}-L_{ser}^{-}=4M}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ser}^{+}+L_{ser}^{-}=2\cdot (L_{P}+L_{S})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ser}^{+}+L_{ser}^{-}=2\cdot (L_{P}+L_{S})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e24745b0dc7d98e16cfe0e2f7292c6773b3b4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.542ex; height:3.009ex;" alt="{\displaystyle L_{ser}^{+}+L_{ser}^{-}=2\cdot (L_{P}+L_{S})}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}=a^{2}.L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}=a^{2}.L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5affe2fd8e192a30af608ddea132df674f128914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.341ex; height:3.009ex;" alt="{\displaystyle L_{P}=a^{2}.L_{S}}" /></span>.</dd></dl></dd></dl></dd></dl></dd></dl> <p>The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:<sup id="cite_ref-Voltech_11-0" class="reference"><a href="#cite_note-Voltech-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Rhombus_12-0" class="reference"><a href="#cite_note-Rhombus-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><dl><dd>Per <b>Eq. 2.7</b>, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{sc}^{pri}=L_{S}\cdot {(1-k^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{sc}^{pri}=L_{S}\cdot {(1-k^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf8fb87f7378289be903d02ca7830cda5fce042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.681ex; height:3.343ex;" alt="{\displaystyle L_{sc}^{pri}=L_{S}\cdot {(1-k^{2})}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{sc}^{sec}=L_{P}\cdot {(1-k^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{sc}^{sec}=L_{P}\cdot {(1-k^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c34a78ae3e1ba08a4c2989589f5978a5f16dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.969ex; height:3.176ex;" alt="{\displaystyle L_{sc}^{sec}=L_{P}\cdot {(1-k^{2})}}" /></span></dd></dl></dd> <dd>Such that <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\sqrt {1-{\frac {L_{sc}^{pri}}{L_{S}}}}}={\sqrt {1-{\frac {L_{sc}^{sec}}{L_{P}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\sqrt {1-{\frac {L_{sc}^{pri}}{L_{S}}}}}={\sqrt {1-{\frac {L_{sc}^{sec}}{L_{P}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e6d0b4a9dccbb830b997b70d103594ec2fdcdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.749ex; height:7.843ex;" alt="{\displaystyle k={\sqrt {1-{\frac {L_{sc}^{pri}}{L_{S}}}}}={\sqrt {1-{\frac {L_{sc}^{sec}}{L_{P}}}}}}" /></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>The Campbell bridge circuit can also be used to determine transformer self-inductances and mutual inductance using a variable standard mutual inductor pair for one of the bridge sides.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> </div> <p>It therefore follows that the open-circuit self-inductance and inductive coupling factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{oc}^{sec}=L_{S}=L_{M2}+L_{S}^{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{oc}^{sec}=L_{S}=L_{M2}+L_{S}^{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee2eb9c497ab1c26d4e3ebc6d37d05c7dab3b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.216ex; height:2.843ex;" alt="{\displaystyle L_{oc}^{sec}=L_{S}=L_{M2}+L_{S}^{\sigma }}" /></span> ------ <b>(Eq. 1.2)</b>, and,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mi>M</mi> <mo>|</mo> </mrow> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1c02d026d527bf8ff6af773a5cacda02547308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.394ex; height:7.009ex;" alt="{\displaystyle k={\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}}" /></span>, with 0 < <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> < 1 ------ <b>(Eq. 1.3)</b></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}^{\sigma }=L_{S}\cdot {(1-k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}^{\sigma }=L_{S}\cdot {(1-k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd165d9d6e2b3e80cdb8b3d2541e792bd4ae40b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.551ex; height:3.009ex;" alt="{\displaystyle L_{S}^{\sigma }=L_{S}\cdot {(1-k)}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M2}=L_{S}\cdot {k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M2}=L_{S}\cdot {k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a213f35172f7eff84c31d2cdf79c30c71abe9d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.228ex; height:2.509ex;" alt="{\displaystyle L_{M2}=L_{S}\cdot {k}}" /></span></dd></dl> <p>and </p> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is mutual inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{oc}^{sec}=L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{oc}^{sec}=L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05deb387df7f9ffd0c235f1dd87ce4ff3cba90e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.038ex; height:2.509ex;" alt="{\displaystyle L_{oc}^{sec}=L_{S}}" /></span> is secondary self-inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}^{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}^{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12dead2df6d53e4608962d06d32d875473898ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.875ex; height:2.843ex;" alt="{\displaystyle L_{S}^{\sigma }}" /></span> is secondary leakage inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M2}=L_{M}/a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M2}=L_{M}/a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8837178129acafbbf60348609ae5477e28769438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.451ex; height:3.176ex;" alt="{\displaystyle L_{M2}=L_{M}/a^{2}}" /></span> is magnetizing inductance referred to the secondary</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is inductive coupling coefficient</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv {\sqrt {\frac {L_{p}}{L_{s}}}}\approx N_{P}/N_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo>≈</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv {\sqrt {\frac {L_{p}}{L_{s}}}}\approx N_{P}/N_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7bc777684bc3efc64a848d6af95a5bb02beb25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.882ex; height:7.509ex;" alt="{\displaystyle a\equiv {\sqrt {\frac {L_{p}}{L_{s}}}}\approx N_{P}/N_{S}}" /></span><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> is the approximate turns ratio</li></ul></dd></dl> <p>The electric validity of the transformer diagram in Fig. 1 depends strictly on open-circuit conditions for the respective winding inductances considered. More generalized circuit conditions are as developed in the next two sections. </p> <div class="mw-heading mw-heading2"><h2 id="Inductive_leakage_factor_and_inductance">Inductive leakage factor and inductance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=2" title="Edit section: Inductive leakage factor and inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Inductance#Mutual_inductance" title="Inductance">Inductance § Mutual inductance</a></div> <p>A <a href="/wiki/Transformer#Real_transformer" title="Transformer">nonideal</a> linear two-winding transformer can be represented by two mutual inductance-coupled circuit loops linking the transformer's five <a href="/wiki/Impedance_(electrical)" class="mw-redirect" title="Impedance (electrical)">impedance</a> constants as shown in Fig. 2.<sup id="cite_ref-18-1_6-1" class="reference"><a href="#cite_note-18-1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18-5_17-0" class="reference"><a href="#cite_note-18-5-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ElecTut_19-0" class="reference"><a href="#cite_note-ElecTut-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Basic_transformer_circuits.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Basic_transformer_circuits.jpg/250px-Basic_transformer_circuits.jpg" decoding="async" width="250" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e1/Basic_transformer_circuits.jpg 1.5x" data-file-width="342" data-file-height="236" /></a><figcaption>Fig. 2 Nonideal transformer circuit diagram</figcaption></figure> <p>where </p> <dl><dd><ul><li>M is mutual inductance</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203847b56e637a9df11d99c36e336e87632dd566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle R_{P}}" /></span> & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5aad493910032da9a1e9ebd853fb6f5b6dceffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.056ex; height:2.509ex;" alt="{\displaystyle R_{S}}" /></span> are primary and secondary winding resistances</li> <li>Constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2579498b31fad37f12a5f29864f14be642ccdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.049ex; height:2.509ex;" alt="{\displaystyle L_{P}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ef2d071fea0ddcd21d9b50e18b78a6138fd69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.875ex; height:2.509ex;" alt="{\displaystyle L_{S}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203847b56e637a9df11d99c36e336e87632dd566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle R_{P}}" /></span> & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5aad493910032da9a1e9ebd853fb6f5b6dceffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.056ex; height:2.509ex;" alt="{\displaystyle R_{S}}" /></span> are measurable at the transformer's terminals</li> <li>Coupling factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is defined as</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow> <mo>|</mo> <mi>M</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0108efa0728a77e572c87ee7bca2a421d8fa529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.843ex; height:3.509ex;" alt="{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}" /></span>, where 0 < <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> < 1 ------ <b>(Eq. 2.1)</b></dd></dl></dd></dl></dd></dl> <p>The winding turns ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> is in practice given as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {L_{P}/L_{S}}}=N_{P}/N_{S}\approx v_{P}/v_{S}\approx i_{S}/i_{P}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>≈</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>≈</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {L_{P}/L_{S}}}=N_{P}/N_{S}\approx v_{P}/v_{S}\approx i_{S}/i_{P}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f37948ecbd4bba86fe394cf12838073f750046" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.844ex; height:4.843ex;" alt="{\displaystyle a={\sqrt {L_{P}/L_{S}}}=N_{P}/N_{S}\approx v_{P}/v_{S}\approx i_{S}/i_{P}=}" /></span> ------ <b>(Eq. 2.2)</b>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where </p> <dl><dd><ul><li>N<sub>P</sub> & N<sub>S</sub> are primary and secondary winding turns</li> <li>v<sub>P</sub> & v<sub>S</sub> and i<sub>P</sub> & i<sub>S</sub> are primary & secondary winding voltages & currents.</li></ul></dd></dl> <p>The nonideal transformer's mesh equations can be expressed by the following voltage and flux linkage equations,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{P}=R_{P}\cdot i_{P}+{\frac {d\Psi {_{P}}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{P}=R_{P}\cdot i_{P}+{\frac {d\Psi {_{P}}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa082207872a06d6d2f3b1a7125bcc09bb24a190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.038ex; height:5.509ex;" alt="{\displaystyle v_{P}=R_{P}\cdot i_{P}+{\frac {d\Psi {_{P}}}{dt}}}" /></span> ------ <b>(Eq. 2.3)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{S}=-R_{S}\cdot i_{S}-{\frac {d\Psi {_{S}}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{S}=-R_{S}\cdot i_{S}-{\frac {d\Psi {_{S}}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c161a2cce00bb4c5552767b30a8ceab4f73da6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.15ex; height:5.509ex;" alt="{\displaystyle v_{S}=-R_{S}\cdot i_{S}-{\frac {d\Psi {_{S}}}{dt}}}" /></span> ------ <b>(Eq. 2.4)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{P}=L_{P}\cdot i_{P}-M\cdot i_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>M</mi> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{P}=L_{P}\cdot i_{P}-M\cdot i_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46197b594db1b83a0a2082a053b4d2e269d66a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.427ex; height:2.509ex;" alt="{\displaystyle \Psi _{P}=L_{P}\cdot i_{P}-M\cdot i_{S}}" /></span> ------ <b>(Eq. 2.5)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{S}=L_{S}\cdot i_{S}-M\cdot i_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mi>M</mi> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{S}=L_{S}\cdot i_{S}-M\cdot i_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b9e1958412059f32e2910b2085361db30ec3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.079ex; height:2.509ex;" alt="{\displaystyle \Psi _{S}=L_{S}\cdot i_{S}-M\cdot i_{P}}" /></span> ------ <b>(Eq. 2.6)</b>,</dd></dl> <dl><dd>where</dd></dl> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" /></span> is flux linkage</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\Psi }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\Psi }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b06eb29a2fd18bd296f3dc2a4adab155af1084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.86ex; height:5.509ex;" alt="{\displaystyle {\frac {d\Psi }{dt}}}" /></span> is <a href="/wiki/Derivative" title="Derivative">derivative</a> of flux linkage with respect to time.</li></ul></dd></dl> <p>These equations can be developed to show that, neglecting associated winding resistances, the ratio of a winding circuit's inductances and currents with the other winding <a href="/wiki/Short-circuit_test" title="Short-circuit test">short-circuited</a> and at <a href="/wiki/Open-circuit_test" title="Open-circuit test">open-circuit test</a> is as follows,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}\approx {\frac {L_{sc}}{L_{oc}}}\approx {\frac {L_{sc}^{sec}}{L_{P}}}\approx {\frac {L_{sc}^{pri}}{L_{S}}}\approx {\frac {i_{oc}}{i_{sc}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}\approx {\frac {L_{sc}}{L_{oc}}}\approx {\frac {L_{sc}^{sec}}{L_{P}}}\approx {\frac {L_{sc}^{pri}}{L_{S}}}\approx {\frac {i_{oc}}{i_{sc}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547f34d575cd3cef8e3986be59c315f98d722dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.18ex; height:6.343ex;" alt="{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}\approx {\frac {L_{sc}}{L_{oc}}}\approx {\frac {L_{sc}^{sec}}{L_{P}}}\approx {\frac {L_{sc}^{pri}}{L_{S}}}\approx {\frac {i_{oc}}{i_{sc}}}}" /></span> ------ <b>(Eq. 2.7)</b>,</dd></dl> <dl><dd>where,</dd></dl> <dl><dd><ul><li>i<sub>oc</sub> & i<sub>sc</sub> are open-circuit and short-circuit currents</li> <li>L<sub>oc</sub> & L<sub>sc</sub> are open-circuit and short-circuit inductances.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> is the inductive leakage factor or Heyland factor<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{sc}^{pri}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>r</mi> <mi>i</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{sc}^{pri}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7468d20e53368eb450d393b7a37edf64baf50768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.951ex; height:3.176ex;" alt="{\displaystyle L_{sc}^{pri}}" /></span> & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{sc}^{sec}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{sc}^{sec}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d453ae32fa58dd301185491798c96f90ff2dcbfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.064ex; height:2.509ex;" alt="{\displaystyle L_{sc}^{sec}}" /></span> are primary and secondary short-circuited leakage inductances.</li></ul></dd></dl> <p>The transformer inductance can be characterized in terms of the three inductance constants as follows,<sup id="cite_ref-Hameyer,_p._27_26-0" class="reference"><a href="#cite_note-Hameyer,_p._27-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brenner_18-7_27-0" class="reference"><a href="#cite_note-Brenner_18-7-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M}=a{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M}=a{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41181b11fd28434d8c25ae928894b95c4dd61566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.312ex; height:2.509ex;" alt="{\displaystyle L_{M}=a{M}}" /></span> ------ <b>(Eq. 2.8)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}^{\sigma }=L_{P}-a{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}^{\sigma }=L_{P}-a{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96140f5147e24d17b22763b47390a9b321524cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.709ex; height:2.843ex;" alt="{\displaystyle L_{P}^{\sigma }=L_{P}-a{M}}" /></span> ------ <b>(Eq. 2.9)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}^{\sigma }=L_{S}-{M}/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}^{\sigma }=L_{S}-{M}/a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1449ebc0b206e6bd85fa2911f00d15baa215d0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.524ex; height:3.009ex;" alt="{\displaystyle L_{S}^{\sigma }=L_{S}-{M}/a}" /></span> ------ <b>(Eq. 2.10)</b> ,</dd></dl> <p>where, </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TREQCCTHeyland.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/TREQCCTHeyland.jpg/550px-TREQCCTHeyland.jpg" decoding="async" width="550" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/05/TREQCCTHeyland.jpg 1.5x" data-file-width="721" data-file-height="263" /></a><figcaption>Fig. 3 Nonideal transformer equivalent circuit</figcaption></figure> <dl><dd><ul><li>L<sub>M</sub> is magnetizing inductance, corresponding to magnetizing reactance X<sub>M</sub></li> <li>L<sub>P</sub><sup>σ</sup> & L<sub>S</sub><sup>σ</sup> are primary & secondary leakage inductances, corresponding to primary & secondary leakage reactances X<sub>P</sub><sup>σ</sup> & X<sub>S</sub><sup>σ</sup>.</li></ul></dd></dl> <p>The transformer can be expressed more conveniently as the <a href="/wiki/Equivalent_circuit" title="Equivalent circuit">equivalent circuit</a> in Fig. 3 with secondary constants referred (i.e., with prime superscript notation) to the primary,<sup id="cite_ref-Hameyer,_p._27_26-1" class="reference"><a href="#cite_note-Hameyer,_p._27-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brenner_18-7_27-1" class="reference"><a href="#cite_note-Brenner_18-7-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}^{\sigma \prime }=a^{2}L_{S}-aM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mi>a</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}^{\sigma \prime }=a^{2}L_{S}-aM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8108add5ffc8554c2c8a197f4daefa60adb4855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.978ex; height:3.343ex;" alt="{\displaystyle L_{S}^{\sigma \prime }=a^{2}L_{S}-aM}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{S}^{\prime }=a^{2}R_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{S}^{\prime }=a^{2}R_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a42b10610a1f43123c44e8983e13f5b56396e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.495ex; height:3.343ex;" alt="{\displaystyle R_{S}^{\prime }=a^{2}R_{S}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{S}^{\prime }=aV_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <mi>a</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{S}^{\prime }=aV_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3e29505fb77c9c25c059d46f539d86863bf3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.623ex; height:2.843ex;" alt="{\displaystyle V_{S}^{\prime }=aV_{S}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{S}^{\prime }=I_{S}/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{S}^{\prime }=I_{S}/a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3361ef987fb3daf888230056c5eac662b26593b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.122ex; height:3.009ex;" alt="{\displaystyle I_{S}^{\prime }=I_{S}/a}" /></span>.</dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TREQCCTHeyland-to-k.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/TREQCCTHeyland-to-k.jpg/550px-TREQCCTHeyland-to-k.jpg" decoding="async" width="550" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/24/TREQCCTHeyland-to-k.jpg 1.5x" data-file-width="721" data-file-height="279" /></a><figcaption>Fig. 4 Nonideal transformer equivalent circuit in terms of coupling coefficient k<sup id="cite_ref-Brenner_18-18_28-0" class="reference"><a href="#cite_note-Brenner_18-18-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=M/{\sqrt {L_{P}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=M/{\sqrt {L_{P}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a88bfd252998731fa85886d99bf7a2a96a6f0d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.163ex; height:3.509ex;" alt="{\displaystyle k=M/{\sqrt {L_{P}L_{S}}}}" /></span> ------ <b>(Eq. 2.11)</b></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {L_{P}/L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {L_{P}/L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272cda0886f23eacbdb3bf35b0145fb26ac866d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.739ex; height:4.843ex;" alt="{\displaystyle a={\sqrt {L_{P}/L_{S}}}}" /></span> ------ <b>(Eq. 2.12)</b>,</dd></dl> <p>we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aM={\sqrt {L_{P}/L_{S}}}\cdot k\cdot {\sqrt {L_{P}L_{S}}}=kL_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> <mo>⋅</mo> <mi>k</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mi>k</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aM={\sqrt {L_{P}/L_{S}}}\cdot k\cdot {\sqrt {L_{P}L_{S}}}=kL_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510c5643abf9ce7c5ca1303becea4d872be06111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.358ex; height:4.843ex;" alt="{\displaystyle aM={\sqrt {L_{P}/L_{S}}}\cdot k\cdot {\sqrt {L_{P}L_{S}}}=kL_{P}}" /></span> ------ <b>(Eq. 2.13)</b>,</dd></dl> <p><br /> which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,<sup id="cite_ref-Brenner_18-7_27-2" class="reference"><a href="#cite_note-Brenner_18-7-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:TREQCCTHeylandConverted.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/TREQCCTHeylandConverted.jpg/400px-TREQCCTHeylandConverted.jpg" decoding="async" width="400" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fb/TREQCCTHeylandConverted.jpg 1.5x" data-file-width="541" data-file-height="196" /></a><figcaption>Fig. 5 Simplified nonideal transformer equivalent circuit</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}^{\sigma }=L_{S}^{\sigma \prime }=L_{P}\cdot (1-k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}^{\sigma }=L_{S}^{\sigma \prime }=L_{P}\cdot (1-k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd51cb3615b2042fa8aa3988fce79be3bbba9866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.206ex; height:3.009ex;" alt="{\displaystyle L_{P}^{\sigma }=L_{S}^{\sigma \prime }=L_{P}\cdot (1-k)}" /></span> ------ <b>(Eq. 2.14 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }" /></span> Eq. 1.1b)</b></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M}=kL_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M}=kL_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27e3b9529f66b9e9ed57ec30514c30faa6627a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.901ex; height:2.509ex;" alt="{\displaystyle L_{M}=kL_{P}}" /></span> ------ <b>(Eq. 2.15 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }" /></span> Eq. 1.1c)</b>.</dd></dl> <p>The nonideal transformer in Fig. 4 can be shown as the simplified equivalent circuit in Fig. 5, with secondary constants referred to the primary and without ideal transformer isolation, where, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{M}=i_{P}-i_{S}^{'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{M}=i_{P}-i_{S}^{'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb6de2177e7ff4912ab1c5299c1a85b7cf7806f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.064ex; height:3.343ex;" alt="{\displaystyle i_{M}=i_{P}-i_{S}^{'}}" /></span> ------ <b>(Eq. 2.16)</b> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58214c1cff32defe4bfa755178e5133688f69805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.509ex;" alt="{\displaystyle i_{M}}" /></span> is magnetizing current excited by flux Φ<sub>M</sub> that links both primary and secondary windings</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c3f965fa98393eac42e8008bbc7cf43e7d7d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.269ex; height:2.509ex;" alt="{\displaystyle i_{P}}" /></span> is the primary current</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{S}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{S}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322c3eb749a2c116c648af4df3db9579da0af117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.095ex; height:2.843ex;" alt="{\displaystyle i_{S}'}" /></span> is the secondary current referred to the primary side of the transformer.</li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Refined_inductive_leakage_factor">Refined inductive leakage factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=3" title="Edit section: Refined inductive leakage factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="padding:1em; margin:0 0 0 1em; width:500px; border:1px solid; background:ivory;"> <p><b>Refined inductive leakage factor derivation</b> </p><p>a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow> <mo>|</mo> <mi>M</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0108efa0728a77e572c87ee7bca2a421d8fa529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.843ex; height:3.509ex;" alt="{\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}" /></span> --------------------- <b>(Eq. 2.1)</b>:</dd></dl> <p>b. Per Eq. 2.7 & <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-42">IEC IEV 131-12-42</a> Inductive leakage factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-k^{2}=1-{\frac {M^{2}}{L_{P}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-k^{2}=1-{\frac {M^{2}}{L_{P}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fef06a8404c6b23673286dec48ec3595bddecb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.558ex; height:6.176ex;" alt="{\displaystyle \sigma =1-k^{2}=1-{\frac {M^{2}}{L_{P}L_{S}}}}" /></span> ------ <b>(Eq. 2.7)</b> & <b>(Eq. 3.7a)</b></dd></dl> <p>c. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {M^{2}}{L_{P}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {M^{2}}{L_{P}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32904eb83eb243c0e2bdad9698bb941c4f9eadc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.761ex; height:6.176ex;" alt="{\displaystyle {\frac {M^{2}}{L_{P}L_{S}}}}" /></span> multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a^{2}}{a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a^{2}}{a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564bc6db92c0d47a9144f69183ef2d3a1b278e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.12ex; height:6.009ex;" alt="{\displaystyle {\frac {a^{2}}{a^{2}}}}" /></span> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e40c0d9bf239e84b8dc22be37c473eeabc7593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.476ex; height:6.343ex;" alt="{\displaystyle \sigma =1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}}" /></span> ----------------- <b>(Eq. 3.7b)</b></dd></dl> <p>d. Per Eq. 2-8 & knowing that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}L_{S}=L_{S}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}L_{S}=L_{S}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b0559236ce2a1ada5f172afac7eec8905cc6fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.133ex; height:3.343ex;" alt="{\displaystyle a^{2}L_{S}=L_{S}^{\prime }}" /></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1cfe97b82f599a6da67ab9f26038e8c6e4dda35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.192ex; height:7.009ex;" alt="{\displaystyle \sigma =1-{\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}" /></span> ---------------------- <b>(Eq. 3.7c)</b></dd></dl> <p>e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d80e3d4a35a5480a104a3ca319a5b7be7143ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:6.761ex; height:7.009ex;" alt="{\displaystyle {\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}" /></span> multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L_{M}.L_{M}}{L_{M}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>.</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L_{M}.L_{M}}{L_{M}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04de5919c58b61af6860f2412e7a26fc841f5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.954ex; height:6.343ex;" alt="{\displaystyle {\frac {L_{M}.L_{M}}{L_{M}^{2}}}}" /></span> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{\prime }}{L_{M}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{\prime }}{L_{M}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc86fdfac37810989bfffd5e447f4edd2dd5f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:17.344ex; height:8.009ex;" alt="{\displaystyle \sigma =1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{\prime }}{L_{M}}}}}}" /></span> ------------------ <b>(Eq. 3.7d)</b></dd></dl> <p>f. Per Eq. 3.5 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 1.1b & Eq. 2.14 and Eq. 3.6 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 1.1b & Eq. 2.14: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f320e6ef137036722af31c6208cd4ac08acb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.305ex; height:6.009ex;" alt="{\displaystyle \sigma =1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}" /></span> --- <b>(Eq.3.7e)</b></dd></dl> <p>All equations in this article assume steady-state constant-frequency waveform conditions the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> values of which are dimensionless, fixed, finite & positive but less than 1. </p> </div> <p>Referring to the flux diagram in Fig. 6, the following equations hold:<sup id="cite_ref-Erickson_29-0" class="reference"><a href="#cite_note-Erickson-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kim1963-3_30-0" class="reference"><a href="#cite_note-Kim1963-3-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Main_%26_leakage_inductances.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Main_%26_leakage_inductances.jpg/190px-Main_%26_leakage_inductances.jpg" decoding="async" width="190" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3d/Main_%26_leakage_inductances.jpg 1.5x" data-file-width="229" data-file-height="330" /></a><figcaption>Fig. 6 Magnetizing and leakage flux in a magnetic circuit<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Erickson_29-1" class="reference"><a href="#cite_note-Erickson-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kim1963-4_32-0" class="reference"><a href="#cite_note-Kim1963-4-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <dl><dd>σ<sub>P</sub> = Φ<sub>P</sub><sup>σ</sup>/Φ<sub>M</sub> = L<sub>P</sub><sup>σ</sup>/L<sub>M</sub><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.1 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 2.7)</b></dd></dl> <p>In the same way, </p> <dl><dd>σ<sub>S</sub> = Φ<sub>S</sub><sup>σ'</sup>/Φ<sub>M</sub> = L<sub>S</sub><sup>σ'</sup>/L<sub>M</sub><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.2 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 2.7)</b></dd></dl> <p>And therefore, </p> <dl><dd>Φ<sub>P</sub> = Φ<sub>M</sub> + Φ<sub>P</sub><sup>σ</sup> = Φ<sub>M</sub> + σ<sub>P</sub>Φ<sub>M</sub> = (1 + σ<sub>P</sub>)Φ<sub>M</sub><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.3)</b></dd></dl> <dl><dd>Φ<sub>S</sub><sup>'</sup> = Φ<sub>M</sub> + Φ<sub>S</sub><sup>σ'</sup> = Φ<sub>M</sub> + σ<sub>S</sub>Φ<sub>M</sub> = (1 + σ<sub>S</sub>)Φ<sub>M</sub><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.4)</b></dd></dl> <dl><dd>L<sub>P</sub> = L<sub>M</sub> + L<sub>P</sub><sup>σ</sup> = L<sub>M</sub> + σ<sub>P</sub>L<sub>M</sub> = (1 + σ<sub>P</sub>)L<sub>M</sub><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.5 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 1.1b & Eq. 2.14)</b></dd></dl> <dl><dd>L<sub>S</sub><sup>'</sup> = L<sub>M</sub> + L<sub>S</sub><sup>σ'</sup> = L<sub>M</sub> + σ<sub>S</sub>L<sub>M</sub> = (1 + σ<sub>S</sub>)L<sub>M</sub><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> ------ <b>(Eq. 3.6 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /></span> Eq. 1.1b & Eq. 2.14)</b>,</dd></dl> <p>where </p> <dl><dd><ul><li>σ<sub>P</sub> & σ<sub>S</sub> are, respectively, primary leakage factor & secondary leakage factor</li></ul></dd></dl> <dl><dd><ul><li>Φ<sub>M</sub> & L<sub>M</sub> are, respectively, mutual flux & magnetizing inductance</li></ul></dd></dl> <dl><dd><ul><li>Φ<sub>P</sub><sup>σ</sup> & L<sub>P</sub><sup>σ</sup> are, respectively, primary leakage flux & primary leakage inductance</li></ul></dd></dl> <dl><dd><ul><li>Φ<sub>S</sub><sup>σ'</sup> & L<sub>S</sub><sup>σ'</sup> are, respectively, secondary leakage flux & secondary leakage inductance both referred to the primary.</li></ul></dd></dl> <p>The leakage ratio σ can thus be refined in terms of the interrelationship of above winding-specific inductance and Inductive leakage factor equations as follows:<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}{^{'}}}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{'}}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}{^{'}}}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{'}}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b85fa6df88f9097ac409a62ba57bea058baee320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:86.953ex; height:9.176ex;" alt="{\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}{^{'}}}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{'}}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}" /></span> ------ <b>(Eq. 3.7a to 3.7e)</b>.</dd></dl> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Leakage inductance can be an undesirable property, as it causes the voltage to change with loading. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Kvglr.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Kvglr.jpg/220px-Kvglr.jpg" decoding="async" width="220" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Kvglr.jpg/330px-Kvglr.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Kvglr.jpg/440px-Kvglr.jpg 2x" data-file-width="639" data-file-height="790" /></a><figcaption>High leakage transformer</figcaption></figure> <p>In many cases it is useful. Leakage inductance has the useful effect of limiting the current flows in a transformer (and load) without itself dissipating power (excepting the usual non-ideal transformer losses). Transformers are generally designed to have a specific value of leakage inductance such that the leakage reactance created by this inductance is a specific value at the desired frequency of operation. In this case, actually working useful parameter is not the leakage inductance value but the <a href="/wiki/Short-circuit_inductance" title="Short-circuit inductance">short-circuit inductance</a> value. </p><p>Commercial and distribution transformers rated up to say 2,500 kVA are usually designed with short-circuit impedances of between about 3% and 6% and with a corresponding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd34222b597f30c8f61770e00bdf191a02729e32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:2.843ex;" alt="{\displaystyle X/R}" /></span> ratio (winding reactance/winding resistance ratio) of between about 3 and 6, which defines the percent secondary voltage variation between no-load and full load. Thus for purely resistive loads, such transformers' full-to-no-load <a href="/wiki/Voltage_regulation" title="Voltage regulation">voltage regulation</a> will be between about 1% and 2%. </p><p>High leakage reactance transformers are used for some negative resistance applications, such as neon signs, where a voltage amplification (transformer action) is required as well as current limiting. In this case the leakage reactance is usually 100% of full load impedance, so even if the transformer is shorted out it will not be damaged. Without the leakage inductance, the negative resistance characteristic of these gas discharge lamps would cause them to conduct excessive current and be destroyed. </p><p>Transformers with variable leakage inductance are used to control the current in <a href="/wiki/Arc_welding" title="Arc welding">arc welding</a> sets. In these cases, the leakage inductance limits the <a href="/wiki/Electric_current" title="Electric current">current</a> flow to the desired magnitude. Transformer leakage reactance has a large role in limiting circuit fault current within the maximum allowable value in the power system.<sup id="cite_ref-Saarbafi-9_2-1" class="reference"><a href="#cite_note-Saarbafi-9-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In addition, the leakage inductance of a HF-transformer can replace a series <a href="/wiki/Inductor" title="Inductor">inductor</a> in a <a href="/wiki/Resonant_converter" title="Resonant converter">resonant converter</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In contrast, connecting a conventional <a href="/wiki/Transformer" title="Transformer">transformer</a> and an inductor in series results in the same electric behavior as of a leakage transformer, but this can be advantageous to reduce the <a href="/wiki/Eddy_current" title="Eddy current">eddy current losses</a> in the transformer windings caused by the stray field. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Blocked_rotor_test" title="Blocked rotor test">Blocked rotor test</a></li> <li><a href="/wiki/Circle_diagram" title="Circle diagram">Circle diagram</a></li> <li><a href="/wiki/Inductance#Mutual_inductance" title="Inductance">Mutual inductance</a></li> <li><a href="/wiki/Induction_motor#Steinmetz_equivalent_circuit" title="Induction motor">Steinmetz equivalent circuit</a></li> <li><a href="/wiki/Short-circuit_inductance" title="Short-circuit inductance">Short-circuit inductance</a></li> <li><a href="/wiki/Short-circuit_test" title="Short-circuit test">Short-circuit test</a></li> <li><a href="/wiki/Voltage_regulation" title="Voltage regulation">Voltage regulation</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Equality is approached when the leakage inductances are small.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFKim1963">Kim 1963</a>, p. 1</span> </li> <li id="cite_note-Saarbafi-9-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Saarbafi-9_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Saarbafi-9_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSaarbafiMclean2014">Saarbafi & Mclean 2014</a>, AESO Transformer Modelling Guide, p. 9 of 304</span> </li> <li id="cite_note-FOOTNOTEIrwin1997362-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEIrwin1997362_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFIrwin1997">Irwin 1997</a>, p. 362.</span> </li> <li id="cite_note-Pyrhonen-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pyrhonen_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPyrhönenJokinenHrabovcová2008">Pyrhönen, Jokinen & Hrabovcová 2008</a>, Chapter 4 Flux Leakage</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">The terms inductive coupling factor and inductive leakage factor are in this article as defined in <a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">International Electrotechnical Commission</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160619074202/http://www.electropedia.org/iev/iev.nsf/d253fda6386f3a52c1257af700281ce6?OpenForm">Electropedia</a>'s <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-41">IEV-131-12-41, Inductive coupling factor</a> and <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-42">IEV-131-12-42, Inductive leakage factor</a>.</span> </li> <li id="cite_note-18-1-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-18-1_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-18-1_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, §18-1 Mutual Inductance, pp. 587-591</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">IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-41">IEV 131-12-41 <b>Inductive coupling factor</b></a></span> </li> <li id="cite_note-Brenner1959-591-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brenner1959-591_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, §18-1 Mutual Inductance - Series connection of Mutual Inductance, pp. 591-592</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">Brenner & Javid 1959, pp. 591-592, Fig. 18-6</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Harris 1952, p. 723, fig. 43</span> </li> <li id="cite_note-Voltech-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Voltech_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVoltech2016">Voltech 2016</a>, Measuring Leakage Inductance</span> </li> <li id="cite_note-Rhombus-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rhombus_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRhombus_Industries1998">Rhombus Industries 1998</a>, Testing Inductance</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">This measured <a href="/wiki/Short-circuit_inductance" title="Short-circuit inductance">short-circuit inductance</a> value is often referred to as the leakage inductance. See for example are, <a rel="nofollow" class="external text" href="http://www.voltech.com/Articles/104-105%20Leakage%20Inductance/104-105.pdf">Measuring Leakage Inductance</a>, <a rel="nofollow" class="external text" href="http://www.rhombus-ind.com/app-note/l-leak.pdf">Testing Inductance</a>. The formal leakage inductance is given by <b>(Eq. 2.14)</b>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Harris 1952, p. 723, fig. 42</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Khurana 2015, p. 254, fig. 7.33</span> </li> <li id="cite_note-18-5-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-18-5_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, §18-5 The Linear Transformer, pp. 595-596</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 24</span> </li> <li id="cite_note-ElecTut-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-ElecTut_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSingh2016">Singh 2016</a>, Mutual Inductance</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, §18-6 The Ideal Transformer, pp. 597-600: Eq. 2.2 holds exactly for an ideal transformer where, at the limit, as self-inductances approach an infinite value ( <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2579498b31fad37f12a5f29864f14be642ccdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.049ex; height:2.509ex;" alt="{\displaystyle L_{P}}" /></span> → ∞ & <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ef2d071fea0ddcd21d9b50e18b78a6138fd69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.875ex; height:2.509ex;" alt="{\displaystyle L_{S}}" /></span> → ∞ ), the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{P}/L_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{P}/L_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27616c765ea87111b0ddfa957fd4ca9634607587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.087ex; height:2.843ex;" alt="{\displaystyle L_{P}/L_{S}}" /></span> approaches a finite value.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 24, eq. 3-1 thru eq. 3-4</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 25, eq. 3-13</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFKnowlton1949">Knowlton 1949</a>, pp. §8–67, p. 802: Knowlton describes <b>The Leakage Factor</b> as "The total flux which passes through the yoke and enters the pole = Φ<sub>m</sub> = Φ<sub>a</sub> + Φ<sub>e</sub> and the ratio Φ<sub>m</sub>/Φ<sub>a</sub> is called the leakage factor and is greater than 1." This factor is evidently different from the inductive leakage factor described in this Leakage inductance article.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-42">IEV ref. 131-12-42: "<b>Inductive leakage factor</b></a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">IEC 60050 (Publication date: 1990-10). Section 221-04: Magnetic bodies, <a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=221-04-12">IEV ref. 221-04-12: "<b>Magnetic leakage factor</b> - the ratio of the total magnetic flux to the useful magnetic flux of a magnetic circuit."</a> This factor is also different from the inductive leakage factor described in this Leakage inductance article.</span> </li> <li id="cite_note-Hameyer,_p._27-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hameyer,_p._27_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hameyer,_p._27_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 27</span> </li> <li id="cite_note-Brenner_18-7-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-Brenner_18-7_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Brenner_18-7_27-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Brenner_18-7_27-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, §18-7 Equivalent Circuit for the nonideal transformer, pp. 600-602 & fig. 18-18</span> </li> <li id="cite_note-Brenner_18-18-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brenner_18-18_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrennerJavid1959">Brenner & Javid 1959</a>, p. 602, "Fig. 18-18 In this equivalent circuit of a (nonideal) transformer the elements are physically realizable and the isolationg property of the transformer has been retained."</span> </li> <li id="cite_note-Erickson-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-Erickson_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Erickson_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEricksonMaksimovic2001">Erickson & Maksimovic 2001</a>, Chapter 12 Basic Magnetic Theory, §12.2.3. Leakage inductances</span> </li> <li id="cite_note-Kim1963-3-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kim1963-3_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKim1963">Kim 1963</a>, pp. 3-12, Magnetice Leakage in Transformers; pp. 13-19, Leakage Reactance in Transformers.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 29, Fig. 26</span> </li> <li id="cite_note-Kim1963-4-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kim1963-4_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKim1963">Kim 1963</a>, p. 4, Fig. 1, Magnetic field due to current in the inner winding of a core-type transformer; Fig. 2, Magnetic field due to current in the outer winding of Fig. 1</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 28, eq. 3-31</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 28, eq. 3-32</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 29, eq. 3-33</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><a href="#CITEREFKim1963">Kim 1963</a>, p. 10, eq. 12</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 29, eq. 3-34</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><a href="#CITEREFKim1963">Kim 1963</a>, p. 10, eq. 13</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 29, eq. 3-35</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, pp. 29, eq. 3-36</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><a href="#CITEREFHameyer2001">Hameyer 2001</a>, p. 29, eq. 3-37</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation 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.cs1-maint{color:#18911f}}</style><cite class="citation conference cs1"><a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/9265771"><i>11kW, 70kHz LLC Converter Design for 98% Efficiency</i></a>. 2020 IEEE 21st Workshop on Control and Modeling for Power Electronics. November 2020. pp. <span class="nowrap">1–</span>8. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FCOMPEL49091.2020.9265771">10.1109/COMPEL49091.2020.9265771</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:227278364">227278364</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=11kW%2C+70kHz+LLC+Converter+Design+for+98%25+Efficiency&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E8&rft.date=2020-11&rft_id=info%3Adoi%2F10.1109%2FCOMPEL49091.2020.9265771&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A227278364%23id-name%3DS2CID&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F9265771&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=8" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrennerJavid1959" class="citation conference cs1">Brenner, Egon; Javid, Mansour (1959). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6D0jAAAAMAAJ">"Chapter 18 – Circuits with Magnetic Coupling"</a>. <i>Analysis of Electric Circuits</i>. McGraw-Hill. pp. esp. 586–617.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Chapter+18+%E2%80%93+Circuits+with+Magnetic+Coupling&rft.btitle=Analysis+of+Electric+Circuits&rft.pages=esp.+586-617&rft.pub=McGraw-Hill&rft.date=1959&rft.aulast=Brenner&rft.aufirst=Egon&rft.au=Javid%2C+Mansour&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6D0jAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDidenkoSirotin2012" class="citation conference cs1">Didenko, V.; Sirotin, D. 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Busan, Republic of Korea.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Accurate+Measurement+of+Resistance+and+Inductance+of+Transformer+Windings&rft.btitle=XX+IMEKO+World+Congress+%E2%80%93+Metrology+for+Green+Growth&rft.place=Busan%2C+Republic+of+Korea&rft.date=2012-09-09%2F2012-09-14&rft.aulast=Didenko&rft.aufirst=V.&rft.au=Sirotin%2C+D.&rft_id=http%3A%2F%2Fwww.imeko.org%2Fpublications%2Fwc-2012%2FIMEKO-WC-2012-TC4-O24.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEricksonMaksimovic2001" class="citation conference cs1">Erickson, Robert W.; Maksimovic, Dragan (2001). <a rel="nofollow" class="external text" href="https://ieee.li/pdf/introduction_to_power_electronics/chapter_12.pdf">"Chapter 12: Basic Magnetics Theory (Instructor slides only for book)"</a> <span class="cs1-format">(PDF)</span>. <i>Fundamentals of Power Electronics</i> (2nd ed.). Boulder: University of Colorado (slides) / Springer (book). pp. 72 slides. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-7270-7" title="Special:BookSources/978-0-7923-7270-7"><bdi>978-0-7923-7270-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Chapter+12%3A+Basic+Magnetics+Theory+%28Instructor+slides+only+for+book%29&rft.btitle=Fundamentals+of+Power+Electronics&rft.place=Boulder&rft.pages=72+slides&rft.edition=2nd&rft.pub=University+of+Colorado+%28slides%29+%2F+Springer+%28book%29&rft.date=2001&rft.isbn=978-0-7923-7270-7&rft.aulast=Erickson&rft.aufirst=Robert+W.&rft.au=Maksimovic%2C+Dragan&rft_id=https%3A%2F%2Fieee.li%2Fpdf%2Fintroduction_to_power_electronics%2Fchapter_12.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150427022958/http://www.electropedia.org/">"Electropedia: The World's Online Electrotechnical Vocabulary"</a>. IEC 60050 (Publication date: 1990-10). Archived from <a rel="nofollow" class="external text" href="http://www.electropedia.org/">the original</a> on 2015-04-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Electropedia%3A+The+World%27s+Online+Electrotechnical+Vocabulary&rft.pub=IEC+60050+%28Publication+date%3A+1990-10%29&rft_id=http%3A%2F%2Fwww.electropedia.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHameyer2001" class="citation book cs1">Hameyer, Kay (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130210003139/http://materialy.itc.pw.edu.pl/zpnis/electric_machines_I/ForStudents/Script_EMIHanneberger.pdf"><i>Electrical Machines I: Basics, Design, Function, Operation</i></a> <span class="cs1-format">(PDF)</span>. RWTH Aachen University Institute of Electrical Machines. Archived from <a rel="nofollow" class="external text" href="http://materialy.itc.pw.edu.pl/zpnis/electric_machines_I/ForStudents/Script_EMIHanneberger.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-02-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electrical+Machines+I%3A+Basics%2C+Design%2C+Function%2C+Operation&rft.pub=RWTH+Aachen+University+Institute+of+Electrical+Machines&rft.date=2001&rft.aulast=Hameyer&rft.aufirst=Kay&rft_id=http%3A%2F%2Fmaterialy.itc.pw.edu.pl%2Fzpnis%2Felectric_machines_I%2FForStudents%2FScript_EMIHanneberger.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarris1952" class="citation book cs1">Harris, Forest K. (1952). <i>Electrical Measurements</i> (5th printing (1962) ed.). New York, London: John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electrical+Measurements&rft.place=New+York%2C+London&rft.edition=5th+printing+%281962%29&rft.pub=John+Wiley+%26+Sons&rft.date=1952&rft.aulast=Harris&rft.aufirst=Forest+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeyland1894" class="citation journal cs1">Heyland, A. (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". <i>ETZ</i>. <b>15</b>: <span class="nowrap">561–</span>564.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ETZ&rft.atitle=A+Graphical+Method+for+the+Prediction+of+Power+Transformers+and+Polyphase+Motors&rft.volume=15&rft.pages=%3Cspan+class%3D%22nowrap%22%3E561-%3C%2Fspan%3E564&rft.date=1894&rft.aulast=Heyland&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeyland1906" class="citation book cs1">Heyland, A. (1906). <a rel="nofollow" class="external text" href="https://archive.org/details/graphicaltreatme00heylrich"><i>A Graphical Treatment of the Induction Motor</i></a>. Translated by George Herbert Rowe; Rudolf Emil Hellmund. McGraw-Hill. pp. 48 pages.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Graphical+Treatment+of+the+Induction+Motor&rft.pages=48+pages&rft.pub=McGraw-Hill&rft.date=1906&rft.aulast=Heyland&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgraphicaltreatme00heylrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIrwin1997" class="citation book cs1">Irwin, J. D. (1997). <i>The Industrial Electronics Handbook</i>. A CRC handbook. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-8343-4" title="Special:BookSources/978-0-8493-8343-4"><bdi>978-0-8493-8343-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Industrial+Electronics+Handbook&rft.series=A+CRC+handbook&rft.pub=Taylor+%26+Francis&rft.date=1997&rft.isbn=978-0-8493-8343-4&rft.aulast=Irwin&rft.aufirst=J.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhurana2015" class="citation book cs1">Khurana, Rohit (2015). <i>Electronic Instrumentation and Measurement</i>. Vikas Publishing House. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789325990203" title="Special:BookSources/9789325990203"><bdi>9789325990203</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Instrumentation+and+Measurement&rft.pub=Vikas+Publishing+House&rft.date=2015&rft.isbn=9789325990203&rft.aulast=Khurana&rft.aufirst=Rohit&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKim1963" class="citation book cs1">Kim, Joong Chung (1963). <a rel="nofollow" class="external text" href="https://ir.library.oregonstate.edu/downloads/kp78gj731"><i>The Determination of Transformer Leakage Reactance by Using an Inpulse Driving Function</i></a>. University of Oregon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Determination+of+Transformer+Leakage+Reactance+by+Using+an+Inpulse+Driving+Function&rft.pub=University+of+Oregon&rft.date=1963&rft.aulast=Kim&rft.aufirst=Joong+Chung&rft_id=https%3A%2F%2Fir.library.oregonstate.edu%2Fdownloads%2Fkp78gj731&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKnowlton1949" class="citation book cs1">Knowlton, A.E., ed. (1949). <i>Standard Handbook for Electrical Engineers</i> (8th ed.). McGraw-Hill. p. 802, § 8–67: The Leakage Factor.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Standard+Handbook+for+Electrical+Engineers&rft.pages=802%2C+%C2%A7-8-67%3A+The+Leakage+Factor&rft.edition=8th&rft.pub=McGraw-Hill&rft.date=1949&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMIT-Press1977" class="citation conference cs1">MIT-Press (1977). "Self- and Mutual Inductances". <i>Magnetic circuits and transformers a first course for power and communication engineers</i>. Cambridge, Mass.: MIT-Press. pp. <span class="nowrap">433–</span>466. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-31082-6" title="Special:BookSources/978-0-262-31082-6"><bdi>978-0-262-31082-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Self-+and+Mutual+Inductances&rft.btitle=Magnetic+circuits+and+transformers+a+first+course+for+power+and+communication+engineers.&rft.place=Cambridge%2C+Mass.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E433-%3C%2Fspan%3E466&rft.pub=MIT-Press&rft.date=1977&rft.isbn=978-0-262-31082-6&rft.au=MIT-Press&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPyrhönenJokinenHrabovcová2008" class="citation book cs1">Pyrhönen, J.; Jokinen, T.; Hrabovcová, V. (2008). <a rel="nofollow" class="external text" href="https://wiley.com/en-us/Design+of+Rotating+Electrical+Machines%2C+2nd+Edition-p-9781118581575"><i>Design of Rotating Electrical Machines</i></a>. p. Chapter 4 Flux Leakage.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+of+Rotating+Electrical+Machines&rft.pages=Chapter+4+Flux+Leakage&rft.date=2008&rft.aulast=Pyrh%C3%B6nen&rft.aufirst=J.&rft.au=Jokinen%2C+T.&rft.au=Hrabovcov%C3%A1%2C+V.&rft_id=https%3A%2F%2Fwiley.com%2Fen-us%2FDesign%2Bof%2BRotating%2BElectrical%2BMachines%252C%2B2nd%2BEdition-p-9781118581575&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRhombus_Industries1998" class="citation web cs1"><a rel="nofollow" class="external text" href="http://rhombus-ind.com/app-note/l-leak.pdf">"Mutual Inductance"</a> <span class="cs1-format">(PDF)</span>. Rhombus Industries Inc. 1998<span class="reference-accessdate">. Retrieved <span class="nowrap">4 August</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mutual+Inductance&rft.pub=Rhombus+Industries+Inc.&rft.date=1998&rft_id=http%3A%2F%2Frhombus-ind.com%2Fapp-note%2Fl-leak.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSaarbafiMclean2014" class="citation web cs1">Saarbafi, Karim; Mclean, Pamela (2014). <a rel="nofollow" class="external text" href="https://www.aeso.ca/assets/linkfiles/4040.002-Rev02-Transformer-Modelling-Guide.pdf">"AESO Transformer Modelling Guide"</a> <span class="cs1-format">(PDF)</span>. Calgary: AESO - Alberta Electric System Operator (prepared by Teshmont Consultants LP). pp. 304 pages<span class="reference-accessdate">. Retrieved <span class="nowrap">August 6,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=AESO+Transformer+Modelling+Guide&rft.place=Calgary&rft.pages=304+pages&rft.pub=AESO+-+Alberta+Electric+System+Operator+%28prepared+by+Teshmont+Consultants+LP%29&rft.date=2014&rft.aulast=Saarbafi&rft.aufirst=Karim&rft.au=Mclean%2C+Pamela&rft_id=https%3A%2F%2Fwww.aeso.ca%2Fassets%2Flinkfiles%2F4040.002-Rev02-Transformer-Modelling-Guide.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSingh2016" class="citation web cs1">Singh, Mahendra (2016). <a rel="nofollow" class="external text" href="http://www.electronics-tutorials.ws/inductor/mutual-inductance.html">"Mutual Inductance"</a>. Electronics Tutorials<span class="reference-accessdate">. Retrieved <span class="nowrap">6 January</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mutual+Inductance&rft.pub=Electronics+Tutorials&rft.date=2016&rft.aulast=Singh&rft.aufirst=Mahendra&rft_id=http%3A%2F%2Fwww.electronics-tutorials.ws%2Finductor%2Fmutual-inductance.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVoltech2016" class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.voltech.com/support/technical-articles/measuring-leakage-inductance/">"Measuring Leakage Inductance"</a>. Voltech Instruments. 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">5 August</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Measuring+Leakage+Inductance&rft.pub=Voltech+Instruments&rft.date=2016&rft_id=https%3A%2F%2Fwww.voltech.com%2Fsupport%2Ftechnical-articles%2Fmeasuring-leakage-inductance%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALeakage+inductance" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leakage_inductance&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>IEC <a rel="nofollow" class="external text" href="http://www.electropedia.org">Electropedia</a> links: </p> <style data-mw-deduplicate="TemplateStyles:r1216972533">.mw-parser-output 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href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-21">Ideal voltage source</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-19">Inductance</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-23">Ideal current source</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-30">Coupling</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-33">Inductive coupling</a></li></ul> </td> <td class="col-break col-break-2"> <ul><li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-41">Inductive coupling factor</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-42">Inductive leakage factor</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-78">Ideal transformer</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=221-04-12">Magnetic leakage factor</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-35">Self-inductance</a></li> <li><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-36">Mutual inductance</a></li></ul> <p>  </p> </td></tr></tbody></table></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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transformer</a></li> <li><a href="/wiki/Distribution_transformer" title="Distribution transformer">Distribution transformer</a> <ul><li><a href="/wiki/Pad-mounted_transformer" title="Pad-mounted transformer">Pad-mounted transformer</a></li></ul></li> <li><a href="/wiki/Delta-wye_transformer" title="Delta-wye transformer">Delta-wye transformer</a></li> <li><a href="/wiki/Energy_efficient_transformer" title="Energy efficient transformer">Energy efficient transformer</a> <ul><li><a href="/wiki/Amorphous_metal_transformer" title="Amorphous metal transformer">Amorphous metal transformer</a></li></ul></li> <li><a href="/wiki/Flyback_transformer" title="Flyback transformer">Flyback transformer</a></li> <li><a href="/wiki/Grounding_transformer" title="Grounding transformer">Grounding transformer</a></li> <li><a href="/wiki/Instrument_transformer" title="Instrument transformer">Instrument transformer</a> <ul><li><a href="/wiki/Current_transformer" title="Current transformer">Current transformer</a></li> <li><a href="/wiki/Voltage_transformer" title="Voltage transformer">Voltage transformer</a></li></ul></li> <li><a href="/wiki/Isolation_transformer" title="Isolation transformer">Isolation transformer</a> <ul><li><a href="/wiki/Austin_transformer" title="Austin transformer">Austin transformer</a></li></ul></li> <li><a href="/wiki/Linear_variable_differential_transformer" title="Linear variable differential transformer">Linear variable differential transformer</a></li> <li><a href="/wiki/Parametric_transformer" title="Parametric transformer">Parametric transformer</a></li> <li><a href="/wiki/Planar_transformer" title="Planar transformer">Planar transformer</a></li> <li><a href="/wiki/Rotary_transformer" title="Rotary transformer">Rotary transformer</a></li> <li><a href="/wiki/Rotary_variable_differential_transformer" title="Rotary variable differential transformer">Rotary variable differential transformer</a></li> <li><a href="/wiki/Scott-T_transformer" title="Scott-T transformer">Scott-T transformer</a></li> <li><a href="/wiki/Solid-state_transformer" title="Solid-state transformer">Solid-state transformer</a></li> <li><a href="/wiki/Trigger_transformer" title="Trigger transformer">Trigger transformer</a></li> <li><a href="/wiki/Variable-frequency_transformer" title="Variable-frequency transformer">Variable-frequency transformer</a></li> <li><a href="/wiki/Zigzag_transformer" title="Zigzag transformer">Zigzag transformer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coils</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hybrid_coil" class="mw-redirect" title="Hybrid coil">Hybrid coil</a></li> <li><a href="/wiki/Induction_coil" title="Induction coil">Induction coil</a></li> <li><a href="/wiki/Oudin_coil" title="Oudin coil">Oudin coil</a></li> <li><a href="/wiki/Polyphase_coil" title="Polyphase coil">Polyphase coil</a></li> <li><a href="/wiki/Repeating_coil" title="Repeating coil">Repeating coil</a></li> <li><a href="/wiki/Tesla_coil" title="Tesla coil">Tesla coil</a></li> <li><a href="/wiki/Trembler_coil" title="Trembler coil">Trembler coil</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Manufacturers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ABB" title="ABB">ABB</a></li> <li><a href="/wiki/General_Electric" title="General Electric">General Electric</a></li> <li><a href="/wiki/Mitsubishi_Electric" title="Mitsubishi Electric">Mitsubishi Electric</a></li> <li><a href="/wiki/ProlecGE" title="ProlecGE">ProlecGE</a></li> <li><a href="/wiki/Schneider_Electric" title="Schneider Electric">Schneider Electric</a></li> <li><a href="/wiki/Siemens" title="Siemens">Siemens</a></li> <li><a href="/wiki/TBEA" title="TBEA">TBEA</a></li> <li><a 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Leakage inductance - Wikipedia