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href="#Summary_of_analogy_between_magnetic_circuits_and_electrical_circuits"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Summary of analogy between magnetic circuits and electrical circuits</span> </div> </a> <ul id="toc-Summary_of_analogy_between_magnetic_circuits_and_electrical_circuits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gyrator" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gyrator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Gyrator</span> </div> </a> <ul id="toc-Gyrator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_voltage" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magnetic_voltage"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Magnetic voltage</span> </div> </a> <ul id="toc-Magnetic_voltage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_current" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magnetic_current"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Magnetic current</span> </div> </a> <ul id="toc-Magnetic_current-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_capacitance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magnetic_capacitance"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Magnetic capacitance</span> </div> </a> <ul id="toc-Magnetic_capacitance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_inductance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magnetic_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Magnetic inductance</span> </div> </a> <ul id="toc-Magnetic_inductance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Three_phase_transformer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three_phase_transformer"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Three phase transformer</span> </div> </a> <ul id="toc-Three_phase_transformer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformer_with_gap_and_leakage_flux" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformer_with_gap_and_leakage_flux"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Transformer with gap and leakage flux</span> </div> </a> <ul id="toc-Transformer_with_gap_and_leakage_flux-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Magnetic_impedance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magnetic_impedance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Magnetic impedance</span> </div> </a> <button aria-controls="toc-Magnetic_impedance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Magnetic impedance subsection</span> </button> <ul id="toc-Magnetic_impedance-sublist" class="vector-toc-list"> <li id="toc-Magnetic_complex_impedance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_complex_impedance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Magnetic complex impedance</span> </div> </a> <ul id="toc-Magnetic_complex_impedance-sublist" class="vector-toc-list"> <li id="toc-Magnetic_effective_resistance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Magnetic_effective_resistance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1.1</span> <span>Magnetic effective resistance</span> </div> </a> <ul id="toc-Magnetic_effective_resistance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_reactance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Magnetic_reactance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1.2</span> <span>Magnetic reactance</span> </div> </a> <ul id="toc-Magnetic_reactance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Limitations_of_the_analogy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limitations_of_the_analogy"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Limitations of the analogy</span> </div> </a> <ul id="toc-Limitations_of_the_analogy-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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Model for magnetic circuits</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">This article is about modeling the magnetic field in magnetic components such as transformers and chokes. For the circuit element, see <a href="/wiki/Gyrator" title="Gyrator">Gyrator</a>.</div> <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 dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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.sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/File:VFPt_Solenoid_correct2.svg" class="mw-file-description" title="Solenoid"><img alt="Solenoid" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/190px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="190" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/285px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/380px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Magnetism" title="Magnetism">Magnetism</a></li> <li><a href="/wiki/Optics" title="Optics">Optics</a></li> <li><a href="/wiki/History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li> <li><a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li> <li><a href="/wiki/List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li> <li><a href="/wiki/List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrostatics" title="Electrostatics">Electrostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Charge_density" title="Charge density">Charge density</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb law</a></li> <li><a href="/wiki/Electret" title="Electret">Electret</a></li> <li><a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a></li> <li><a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li> <li><a href="/wiki/Electric_field" title="Electric field">Electric field</a></li> <li><a href="/wiki/Electric_flux" title="Electric flux">Electric flux</a></li> <li><a href="/wiki/Electric_potential" title="Electric potential">Electric potential</a></li> <li><a href="/wiki/Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li> <li><a href="/wiki/Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li> <li><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss law</a></li> <li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Permittivity" title="Permittivity">Permittivity</a></li> <li><a href="/wiki/Polarization_density" title="Polarization density">Polarization</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Potential energy</a></li> <li><a href="/wiki/Static_electricity" title="Static electricity">Static electricity</a></li> <li><a href="/wiki/Triboelectric_effect" title="Triboelectric effect">Triboelectricity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetostatics" title="Magnetostatics">Magnetostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère law</a></li> <li><a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li> <li><a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss magnetic law</a></li> <li><a href="/wiki/Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></li> <li><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li> <li><a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></li> <li><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li> <li><a href="/wiki/Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li> <li><a href="/wiki/Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li> <li><a href="/wiki/Displacement_current" title="Displacement current">Displacement current</a></li> <li><a href="/wiki/Eddy_current" title="Eddy current">Eddy current</a></li> <li><a href="/wiki/Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li> <li><a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li> <li><a href="/wiki/Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li> <li><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li> <li><a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday law</a></li> <li><a href="/wiki/Jefimenko%27s_equations" title="Jefimenko's equations">Jefimenko equations</a></li> <li><a href="/wiki/Larmor_formula" title="Larmor formula">Larmor formula</a></li> <li><a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz law</a></li> <li><a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li> <li><a href="/wiki/London_equations" title="London equations">London equations</a></li> <li><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a></li> <li><a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li> <li><a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></li> <li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrical_network" title="Electrical network">Electrical network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/AC_motor" title="AC motor">AC motor</a></li> <li><a href="/wiki/DC_motor" title="DC motor">DC motor</a></li> <li><a href="/wiki/Electric_machine" title="Electric machine">Electric machine</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motor</a></li> <li><a class="mw-selflink selflink">Gyrator–capacitor</a></li> <li><a href="/wiki/Induction_motor" title="Induction motor">Induction motor</a></li> <li><a href="/wiki/Linear_motor" title="Linear motor">Linear motor</a></li> <li><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li> <li><a href="/wiki/Permeance" title="Permeance">Permeance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li> <li><a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li> <li><a href="/wiki/Rotor_(electric)" title="Rotor (electric)">Rotor</a></li> <li><a href="/wiki/Stator" title="Stator">Stator</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li> <li><a href="/wiki/Four-current" title="Four-current">Four-current</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li> <li><a href="/wiki/Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li> <li><a href="/wiki/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li> <li><a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li> <li><a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li><a href="/wiki/Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li> <li><a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Larmor</a></li> <li><a href="/wiki/Emil_Lenz" title="Emil Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Georg_Ohm" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li><a href="/wiki/William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li><a href="/wiki/George_Singer" title="George Singer">Singer</a></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electromagnetism" title="Template:Electromagnetism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electromagnetism" title="Template talk:Electromagnetism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electromagnetism" title="Special:EditPage/Template:Electromagnetism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_model_of_a_simple_transformer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Gyrator-Capacitor_model_of_a_simple_transformer.png/330px-Gyrator-Capacitor_model_of_a_simple_transformer.png" decoding="async" width="330" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Gyrator-Capacitor_model_of_a_simple_transformer.png/495px-Gyrator-Capacitor_model_of_a_simple_transformer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Gyrator-Capacitor_model_of_a_simple_transformer.png/660px-Gyrator-Capacitor_model_of_a_simple_transformer.png 2x" data-file-width="852" data-file-height="547" /></a><figcaption>A simple transformer and its gyrator-capacitor model. R is the reluctance of the physical magnetic circuit.</figcaption></figure> <p>The <b>gyrator–capacitor model</b><sup id="cite_ref-Hamill_1-0" class="reference"><a href="#cite_note-Hamill-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> - sometimes also the capacitor-permeance model<sup id="cite_ref-Lambert_2-0" class="reference"><a href="#cite_note-Lambert-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> - is a <a href="/wiki/Lumped_element_model" class="mw-redirect" title="Lumped element model">lumped-element model</a> for <a href="/wiki/Magnetic_circuit" title="Magnetic circuit">magnetic circuits</a>, that can be used in place of the more common <a href="/wiki/Resistance%E2%80%93reluctance_model" class="mw-redirect" title="Resistance–reluctance model">resistance–reluctance model</a>. The model makes <a href="/wiki/Permeance" title="Permeance">permeance</a> elements analogous to electrical <a href="/wiki/Capacitance" title="Capacitance">capacitance</a> (<i>see <a href="/wiki/Magnetic_capacitance" class="mw-redirect" title="Magnetic capacitance">magnetic capacitance</a> section</i>) rather than <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> (<i>see <a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">magnetic reluctance</a></i>). Windings are represented as <a href="/wiki/Gyrator" title="Gyrator">gyrators</a>, interfacing between the electrical circuit and the magnetic model. </p><p>The primary advantage of the gyrator–capacitor model compared to the magnetic reluctance model is that the model preserves the correct values of energy flow, storage and dissipation.<sup id="cite_ref-González_3-0" class="reference"><a href="#cite_note-González-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mohammad_4-0" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The gyrator–capacitor model is an example of a <a href="/wiki/Mechanical%E2%80%93electrical_analogies#Other_energy_domains" title="Mechanical–electrical analogies">group of analogies</a> that preserve energy flow across energy domains by making power conjugate pairs of variables in the various domains analogous. It fills the same role as the <a href="/wiki/Impedance_analogy" title="Impedance analogy">impedance analogy</a> for the mechanical domain. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Nomenclature">Nomenclature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=1" title="Edit section: Nomenclature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Magnetic circuit</i> may refer to either the physical magnetic circuit or the model magnetic circuit. <a href="/wiki/Lumped-element_model" title="Lumped-element model">Elements</a> and <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">dynamical variables</a> that are part of the model magnetic circuit have names that start with the adjective <i>magnetic</i>, although this convention is not strictly followed. Elements or dynamical variables in the model magnetic circuit may not have a one to one correspondence with components in the physical magnetic circuit. Symbols for elements and variables that are part of the model magnetic circuit may be written with a subscript of M. For example, <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 C_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b36d6b18ea6be45cbe07fe5fcee599d531ff486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.621ex; height:2.509ex;" alt="{\displaystyle C_{M}}"></span> would be a magnetic capacitor in the model circuit. </p><p>Electrical elements in an associated electrical circuit may be brought into the magnetic model for ease of analysis. Model elements in the magnetic circuit that represent electrical elements are typically the <a href="/wiki/Duality_(electrical_circuits)" title="Duality (electrical circuits)">electrical dual</a> of the electrical elements. This is because transducers between the electrical and magnetic domains in this model are usually represented by gyrators. A gyrator will transform an element into its dual. For example, a magnetic inductance may represent an electrical capacitance. </p> <div class="mw-heading mw-heading2"><h2 id="Summary_of_analogy_between_magnetic_circuits_and_electrical_circuits">Summary of analogy between magnetic circuits and electrical circuits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=2" title="Edit section: Summary of analogy between magnetic circuits and electrical circuits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory. </p> <table class="wikitable"> <caption>Analogy between magnetic circuits and electrical circuits used in the gyrator-capacitor approach </caption> <tbody><tr> <th colspan="3">Magnetic</th> <th></th> <th colspan="3">Electric </th></tr> <tr> <th>Name</th> <th>Symbol</th> <th>Units</th> <th></th> <th>Name</th> <th>Symbol</th> <th>Units </th></tr> <tr> <td><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a> (MMF)</td> <td><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 {\mathcal {F}}=\int \mathbf {H} \cdot d\mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot d\mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c48d8f3597c285e51854dbe8b0adab1310a3d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.334ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot d\mathbf {l} }"></span></td> <td><a href="/wiki/Ampere-turn" title="Ampere-turn">ampere-turn</a></td> <td></td> <td><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a> (EMF)</td> <td><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 {\mathcal {E}}=\int \mathbf {E} \cdot d\mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot d\mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b602f8975cd82ea20972196947e69be59254c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.384ex; height:5.676ex;" alt="{\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot d\mathbf {l} }"></span></td> <td><a href="/wiki/Volt" title="Volt">volt</a> </td></tr> <tr> <td><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></td> <td><i><b>H</b></i></td> <td>ampere/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a> = <p><a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a>/<a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a> </p> </td> <td></td> <td><a href="/wiki/Electric_field" title="Electric field">Electric field</a></td> <td><i><b>E</b></i></td> <td>volt/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a> = <p><a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a>/<a href="/wiki/Coulomb" title="Coulomb">coulomb</a> </p> </td></tr> <tr> <td><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></td> <td><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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span></td> <td><a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Electric charge</td> <td>Q</td> <td><a href="/wiki/Coulomb" title="Coulomb">coulomb</a> </td></tr> <tr> <td>Flux rate of change </td> <td><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 {\dot {\Phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\Phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe38dcf97ad0622c1216a0c361acb0c0e36b6e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.676ex;" alt="{\displaystyle {\dot {\Phi }}}"></span> </td> <td>weber/second = <a href="/wiki/Volt" title="Volt">volt</a> </td> <td> </td> <td><a href="/wiki/Electric_current" title="Electric current">Electric current</a> </td> <td><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> </td> <td>coulomb/second = <a href="/wiki/Ampere" title="Ampere">ampere</a> </td></tr> <tr> <td>Magnetic admittance </td> <td><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 Y_{M}(\omega )={\frac {{\dot {\Phi }}(\omega )}{{\mathcal {F}}(\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{M}(\omega )={\frac {{\dot {\Phi }}(\omega )}{{\mathcal {F}}(\omega )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28aeb332454e4dabe9d6152c9655eb28198238ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.681ex; height:6.843ex;" alt="{\displaystyle Y_{M}(\omega )={\frac {{\dot {\Phi }}(\omega )}{{\mathcal {F}}(\omega )}}}"></span> </td> <td><a href="/wiki/Ohm" title="Ohm">ohm</a> = 1/siemens </td> <td> </td> <td><a href="/wiki/Admittance" title="Admittance">Electric admittance</a> </td> <td><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 Y_{E}(\omega )={\frac {I(\omega )}{V(\omega )}}={\frac {1}{\operatorname {Z_{E}(\omega )} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{E}(\omega )={\frac {I(\omega )}{V(\omega )}}={\frac {1}{\operatorname {Z_{E}(\omega )} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5b8778b95a3216da94f20238c1038302689408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.032ex; height:6.509ex;" alt="{\displaystyle Y_{E}(\omega )={\frac {I(\omega )}{V(\omega )}}={\frac {1}{\operatorname {Z_{E}(\omega )} }}}"></span> </td> <td><a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a> = 1/ohm </td></tr> <tr> <td>Magnetic conductance </td> <td><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 G_{M}=\operatorname {Re} (Y_{M}(\omega ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{M}=\operatorname {Re} (Y_{M}(\omega ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee81e6061f0076bc796309fee785eb6896ae25f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.001ex; height:2.843ex;" alt="{\displaystyle G_{M}=\operatorname {Re} (Y_{M}(\omega ))}"></span> </td> <td>ohm = 1/siemens </td> <td> </td> <td><a href="/wiki/Electric_conductance" class="mw-redirect" title="Electric conductance">Electric conductance</a> </td> <td><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 G_{E}=\operatorname {Re} (Y_{E}(\omega ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{E}=\operatorname {Re} (Y_{E}(\omega ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18ffb232639ce7beabb4e45e4fd88464b74eac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.059ex; height:2.843ex;" alt="{\displaystyle G_{E}=\operatorname {Re} (Y_{E}(\omega ))}"></span> </td> <td>siemens = 1/ohm </td></tr> <tr> <td>Magnetic capacitance (<a href="/wiki/Permeance" title="Permeance">Permeance</a>)</td> <td><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 {\mathcal {P}}={\frac {\operatorname {Im} (Y_{M}(\omega ))}{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mi>ω</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}={\frac {\operatorname {Im} (Y_{M}(\omega ))}{\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1147e45f76f8184777b2601e349774013733c5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.788ex; height:5.676ex;" alt="{\displaystyle {\mathcal {P}}={\frac {\operatorname {Im} (Y_{M}(\omega ))}{\omega }}}"></span></td> <td><a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a></td> <td></td> <td>Electric <a href="/wiki/Capacitance" title="Capacitance">capacitance</a></td> <td><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 C={\frac {\operatorname {Im} (Y_{E}(\omega ))}{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mi>ω</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {\operatorname {Im} (Y_{E}(\omega ))}{\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5dc0c3fbcedc1f94a2574113ab403ad2cac383d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.379ex; height:5.676ex;" alt="{\displaystyle C={\frac {\operatorname {Im} (Y_{E}(\omega ))}{\omega }}}"></span> </td> <td><a href="/wiki/Farad" title="Farad">farad</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Gyrator">Gyrator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=3" title="Edit section: Gyrator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Gyrator_Element.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Gyrator-Capacitor_Model_Gyrator_Element.png/220px-Gyrator-Capacitor_Model_Gyrator_Element.png" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Gyrator-Capacitor_Model_Gyrator_Element.png/330px-Gyrator-Capacitor_Model_Gyrator_Element.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Gyrator-Capacitor_Model_Gyrator_Element.png/440px-Gyrator-Capacitor_Model_Gyrator_Element.png 2x" data-file-width="1200" data-file-height="1050" /></a><figcaption>Definition of Gyrator as used by Hamill in the gyrator-capacitor approach paper.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gyrator" title="Gyrator">Gyrator</a></div> <p>A <b><a href="/wiki/Gyrator" title="Gyrator">gyrator</a></b> is a <a href="/wiki/Two-port_network" title="Two-port network">two-port element</a> used in network analysis. The gyrator is the complement of the <a href="/wiki/Transformer" title="Transformer">transformer</a>; whereas in a transformer, a voltage on one port will transform to a proportional voltage on the other port, in a gyrator, a voltage on one port will transform to a current on the other port, and vice versa. </p><p>The role gyrators play in the gyrator–capacitor model is as <a href="/wiki/Transducer" title="Transducer">transducers</a> between the electrical energy domain and the magnetic energy domain. An emf in the electrical domain is analogous to an mmf in the magnetic domain, and a transducer doing such a conversion would be represented as a transformer. However, real electro-magnetic transducers usually behave as gyrators. A transducer from the magnetic domain to the electrical domain will obey <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, that is, a rate of change of magnetic flux (a magnetic current in this analogy) produces a proportional emf in the electrical domain. Similarly, a transducer from the electrical domain to the magnetic domain will obey <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's circuital law</a>, that is, an electric current will produce a mmf. </p><p>A winding of N turns is modeled by a gyrator with a gyration resistance of N ohms.<sup id="cite_ref-Hamill_1-2" class="reference"><a href="#cite_note-Hamill-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 100">: 100 </span></sup> </p><p>Transducers that are not based on magnetic induction may not be represented by a gyrator. For instance, a <a href="/wiki/Hall_effect_sensor" title="Hall effect sensor">Hall effect sensor</a> is modelled by a transformer. </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_voltage">Magnetic voltage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=4" title="Edit section: Magnetic voltage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Magnetic voltage</b>, <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_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39c47348c7acfafbf415f12afaaff6e0a0900ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.803ex; height:2.009ex;" alt="{\displaystyle v_{m}}"></span>, is an alternate name for <i><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a></i> (mmf), <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 {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> (<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a>: <a href="/wiki/Ampere" title="Ampere">A</a> or <a href="/wiki/Amp-turn" class="mw-redirect" title="Amp-turn">amp-turn</a>), which is analogous to electrical <a href="/wiki/Voltage" title="Voltage">voltage</a> in an electric circuit.<sup id="cite_ref-Mohammad_4-1" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 42">: 42 </span></sup><sup id="cite_ref-González_3-1" class="reference"><a href="#cite_note-González-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 5">: 5 </span></sup> Not all authors use the term <i>magnetic voltage</i>. The magnetomotive force applied to an element between point A and point B is equal to the line integral through the component of the magnetic field strength, <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 \mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f017b876ed763037d8818ec5dfbbdc6703e0f683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{m}={\mathcal {F}}=-\int _{A}^{B}\mathbf {H} \cdot d\mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{m}={\mathcal {F}}=-\int _{A}^{B}\mathbf {H} \cdot d\mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354a431525a8a54f22abdf357dd59c45d3acf8fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.181ex; height:6.176ex;" alt="{\displaystyle v_{m}={\mathcal {F}}=-\int _{A}^{B}\mathbf {H} \cdot d\mathbf {l} }"></span> The <a href="/wiki/Resistance%E2%80%93reluctance_model" class="mw-redirect" title="Resistance–reluctance model">resistance–reluctance model</a> uses the same equivalence between magnetic voltage and magnetomotive force. </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_current">Magnetic current</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=5" title="Edit section: Magnetic current"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Magnetic_current" title="Magnetic current">Magnetic current</a>, an electromagnetic field quantity.</div> <p><b>Magnetic current</b>, <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/e48a29710ab6618c7e08679ee9cd06ba1264ef0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle i_{m}}"></span>, is an alternate name for the <i>time rate of change of flux</i>, <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 {\dot {\Phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\Phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe38dcf97ad0622c1216a0c361acb0c0e36b6e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.676ex;" alt="{\displaystyle {\dot {\Phi }}}"></span> (<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a>: <a href="/wiki/Weber_(unit)" title="Weber (unit)">Wb</a>/sec or <a href="/wiki/Volts" class="mw-redirect" title="Volts">volts</a>), which is analogous to electrical current in an electric circuit.<sup id="cite_ref-Lambert_2-1" class="reference"><a href="#cite_note-Lambert-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 2429">: 2429 </span></sup><sup id="cite_ref-Mohammad_4-2" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> In the physical circuit, <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 {\dot {\Phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\Phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe38dcf97ad0622c1216a0c361acb0c0e36b6e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.676ex;" alt="{\displaystyle {\dot {\Phi }}}"></span>, is <a href="/wiki/Magnetic_current#Magnetic_displacement_current" title="Magnetic current">magnetic displacement current</a>.<sup id="cite_ref-Mohammad_4-3" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> The magnetic current flowing through an element of cross section, <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, is the area integral of the magnetic flux density <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 \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.901ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} }"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{m}={\dot {\Phi }}={\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{m}={\dot {\Phi }}={\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f89c8ce81ed7d553bccb448e7c97217c32a649" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.885ex; height:5.843ex;" alt="{\displaystyle i_{m}={\dot {\Phi }}={\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S} }"></span> The resistance–reluctance model uses a different equivalence, taking magnetic current to be an alternate name for flux, <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span>. This difference in the definition of magnetic current is the fundamental difference between the gyrator-capacitor model and the resistance–reluctance model. The definition of magnetic current and magnetic voltage imply the definitions of the other magnetic elements.<sup id="cite_ref-Mohammad_4-4" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 35">: 35 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_capacitance">Magnetic capacitance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=6" title="Edit section: Magnetic capacitance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Magnetic_capacitance"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Permeance_Element.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gyrator-Capacitor_Model_Permeance_Element.png/260px-Gyrator-Capacitor_Model_Permeance_Element.png" decoding="async" width="260" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gyrator-Capacitor_Model_Permeance_Element.png/390px-Gyrator-Capacitor_Model_Permeance_Element.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Gyrator-Capacitor_Model_Permeance_Element.png/520px-Gyrator-Capacitor_Model_Permeance_Element.png 2x" data-file-width="4200" data-file-height="2400" /></a><figcaption>Permeance of a rectangular prism element</figcaption></figure> <p><b>Magnetic capacitance</b> is an alternate name for <a href="/wiki/Permeance" title="Permeance">permeance</a>, (<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a>: <a href="/wiki/Henry_(unit)" title="Henry (unit)">H</a>). It is represented by a capacitance in the model magnetic circuit. Some authors use <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 C_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05edb806fdb6ed21a51764a20bd34eb64a67bc39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.401ex; height:2.509ex;" alt="{\displaystyle C_{\mathrm {M} }}"></span> to denote magnetic capacitance while others use <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and refer to the capacitance as a permeance. Permeance of an element is an <a href="/wiki/Extensive_property" class="mw-redirect" title="Extensive property">extensive property</a> defined as the magnetic flux, <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 \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span>, through the cross sectional surface of the element divided by the <a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a>, <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 {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span>, across the element'<sup id="cite_ref-González_3-2" class="reference"><a href="#cite_note-González-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 6">: 6 </span></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\mathrm {M} }=P={\frac {\int \mathbf {B} \cdot d\mathbf {S} }{\int \mathbf {H} \cdot d\mathbf {l} }}={\frac {\Phi }{\mathcal {F}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\mathrm {M} }=P={\frac {\int \mathbf {B} \cdot d\mathbf {S} }{\int \mathbf {H} \cdot d\mathbf {l} }}={\frac {\Phi }{\mathcal {F}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef83582cda97c4020bcba43813d144b40a9f6906" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.127ex; height:6.843ex;" alt="{\displaystyle C_{\mathrm {M} }=P={\frac {\int \mathbf {B} \cdot d\mathbf {S} }{\int \mathbf {H} \cdot d\mathbf {l} }}={\frac {\Phi }{\mathcal {F}}}}"></span> </p><p>For a bar of uniform cross-section, magnetic capacitance is given by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\mathrm {M} }=P=\mu _{\mathrm {r} }\mu _{0}{\frac {S}{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>l</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\mathrm {M} }=P=\mu _{\mathrm {r} }\mu _{0}{\frac {S}{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/487ac1e0a3bdfc312563002b03c85b7f8f2870e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.413ex; height:5.509ex;" alt="{\displaystyle C_{\mathrm {M} }=P=\mu _{\mathrm {r} }\mu _{0}{\frac {S}{l}}}"></span> where: </p> <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 \mu _{\mathrm {r} }\mu _{0}=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {r} }\mu _{0}=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c17451693ac055a610c6bf2688956d5e9ea76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.235ex; height:2.176ex;" alt="{\displaystyle \mu _{\mathrm {r} }\mu _{0}=\mu }"></span> is the <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">magnetic permeability</a>,</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is the element cross-section, and</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is the element length.</li></ul> <p>For <a href="/wiki/Phasor_analysis" class="mw-redirect" title="Phasor analysis">phasor analysis</a>, the magnetic permeability<sup id="cite_ref-Arkadiew_6-0" class="reference"><a href="#cite_note-Arkadiew-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and the permeance are complex values.<sup id="cite_ref-Arkadiew_6-1" class="reference"><a href="#cite_note-Arkadiew-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Popov_7-0" class="reference"><a href="#cite_note-Popov-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Permeance is the reciprocal of <a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">reluctance</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_inductance">Magnetic inductance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=7" title="Edit section: Magnetic inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Magnetic_induction_(disambiguation)" class="mw-redirect mw-disambig" title="Magnetic induction (disambiguation)">magnetic induction</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Magnetic_Inductance.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Magnetic_Inductance.png/330px-Magnetic_Inductance.png" decoding="async" width="330" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Magnetic_Inductance.png/495px-Magnetic_Inductance.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Magnetic_Inductance.png/660px-Magnetic_Inductance.png 2x" data-file-width="4350" data-file-height="2700" /></a><figcaption>Circuit equivalence between a magnetic inductance and an electric capacitance.</figcaption></figure> <p>In the context of the gyrator-capacitor model of a magnetic circuit, <b>magnetic inductance</b> <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_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6fa203c8846ba711f0c81fe162789853f42c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.322ex; height:2.509ex;" alt="{\displaystyle L_{\mathrm {M} }}"></span>(<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a>: <a href="/wiki/Farad" title="Farad">F</a>) is the analogy to inductance in an electrical circuit. </p><p>For phasor analysis the magnetic inductive reactance is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mathrm {L} }=\omega L_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {L} }=\omega L_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deda4d3638ab578f3e7621f1165bf62a3eedef0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.455ex; height:2.509ex;" alt="{\displaystyle x_{\mathrm {L} }=\omega L_{\mathrm {M} }}"></span> where: </p> <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_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6fa203c8846ba711f0c81fe162789853f42c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.322ex; height:2.509ex;" alt="{\displaystyle L_{\mathrm {M} }}"></span> is the magnetic 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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the magnetic circuit</li></ul> <p>In the complex form it is a positive imaginary number: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle jx_{\mathrm {L} }=j\omega L_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle jx_{\mathrm {L} }=j\omega L_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65220ce451211a047340c8386162de8cdc534290" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.398ex; height:2.509ex;" alt="{\displaystyle jx_{\mathrm {L} }=j\omega L_{\mathrm {M} }}"></span> </p><p>The magnetic potential energy sustained by magnetic inductance varies with the frequency of oscillations in electric fields. The average power in a given period is equal to zero. Due to its dependence on frequency, magnetic inductance is mainly observable in magnetic circuits which operate at <a href="/wiki/Very_high_frequency" title="Very high frequency">VHF</a> and/or <a href="/wiki/Ultra_high_frequency" title="Ultra high frequency">UHF</a> frequencies.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="reference Mohammad says magnetic inductance will not considered a magnetic element on page 43 (April 2020)">citation needed</span></a></i>]</sup> </p><p>The notion of magnetic inductance is employed in analysis and computation of circuit behavior in the gyrator–capacitor model in a way analogous to <a href="/wiki/Inductance" title="Inductance">inductance</a> in electrical circuits. </p><p>A magnetic inductor can represent an electrical capacitor.<sup id="cite_ref-Mohammad_4-5" class="reference"><a href="#cite_note-Mohammad-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 43">: 43 </span></sup> A shunt capacitance in the electrical circuit, such as intra-winding capacitance can be represented as a series inductance in the magnetic circuit. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Three_phase_transformer">Three phase transformer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=9" title="Edit section: Three phase transformer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png/330px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png" decoding="async" width="330" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png/495px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png/660px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer.png 2x" data-file-width="3900" data-file-height="2700" /></a><figcaption>Three phase transformer with windings and permeance elements.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png/330px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png" decoding="async" width="330" height="428" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png/495px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png/660px-Gyrator-Capacitor_Model_Example_Three_Phase_Transformer_Schematic.png 2x" data-file-width="3300" data-file-height="4275" /></a><figcaption>Schematic using gyrator-capacitor model for transformer windings and capacitors for permeance elements</figcaption></figure> <p>This example shows a <a href="/wiki/Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> <a href="/wiki/Transformer" title="Transformer">transformer</a> modeled by the gyrator-capacitor approach. The transformer in this example has three primary windings and three secondary windings. The magnetic circuit is split into seven reluctance or permeance elements. Each winding is modeled by a gyrator. The gyration resistance of each gyrator is equal to the number of turns on the associated winding. Each permeance element is modeled by a capacitor. The value of each capacitor in <a href="/wiki/Farads" class="mw-redirect" title="Farads">farads</a> is the same as the inductance of the associated permeance in <a href="/wiki/Henry_(unit)" title="Henry (unit)">henrys</a>. </p><p>N<sub>1</sub>, N<sub>2</sub>, and N<sub>3</sub> are the number of turns in the three primary windings. N<sub>4</sub>, N<sub>5</sub>, and N<sub>6</sub> are the number of turns in the three secondary windings. Φ<sub>1</sub>, Φ<sub>2</sub>, and Φ<sub>3</sub> are the fluxes in the three vertical elements. <a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a> in each permeance element in <a href="/wiki/Weber_(unit)" title="Weber (unit)">webers</a> is numerically equal to the charge in the associated capacitance in <a href="/wiki/Coulomb" title="Coulomb">coulombs</a>. The energy in each permeance element is the same as the energy in the associated capacitor. </p><p>The schematic shows a three phase generator and a three phase load in addition to the schematic of the transformer model. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Transformer_with_gap_and_leakage_flux">Transformer with gap and leakage flux</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=10" title="Edit section: Transformer with gap and leakage flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png/330px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png" decoding="async" width="330" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png/495px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png/660px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux.png 2x" data-file-width="2850" data-file-height="2513" /></a><figcaption>Transformer with gap and leakage flux.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png/330px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png" decoding="async" width="330" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png/495px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png/660px-Gyrator-Capacitor_Model_Example_Transformer_with_Gap_and_Leakage_Flux_Schematic.png 2x" data-file-width="3300" data-file-height="2550" /></a><figcaption>Gyrator-capacitor model of a transformer with a gap and leakage flux.</figcaption></figure> <p>The gyrator-capacitor approach can accommodate <a href="/wiki/Leakage_inductance" title="Leakage inductance">leakage inductance</a> and air gaps in the magnetic circuit. Gaps and leakage flux have a permeance which can be added to the equivalent circuit as capacitors. The permeance of the gap is computed in the same way as the substantive elements, except a relative permeability of unity is used. The permeance of the leakage flux may be difficult to compute due to complex geometry. It may be computed from other considerations such as measurements or specifications. </p><p>C<sub>PL</sub> and C<sub>SL</sub> represent the primary and secondary leakage inductance respectively. C<sub>GAP</sub> represents the air gap permeance. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Magnetic_impedance">Magnetic impedance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=11" title="Edit section: Magnetic impedance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Magnetic_complex_impedance">Magnetic complex impedance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=12" title="Edit section: Magnetic complex impedance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Magnetic_impedance.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Magnetic_impedance.png/330px-Magnetic_impedance.png" decoding="async" width="330" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Magnetic_impedance.png/495px-Magnetic_impedance.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Magnetic_impedance.png/660px-Magnetic_impedance.png 2x" data-file-width="4350" data-file-height="2700" /></a><figcaption>Circuit equivalence between a magnetic impedance and an electric admittance.</figcaption></figure> <p><b>Magnetic complex impedance</b>, also called full magnetic resistance, is the <a href="/wiki/Quotient" title="Quotient">quotient</a> of a complex sinusoidal magnetic tension (<a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a>, <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 {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span>) on a passive <a href="/wiki/Magnetic_circuit" title="Magnetic circuit">magnetic circuit</a> and the resulting complex sinusoidal magnetic current (<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 {\dot {\Phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\Phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe38dcf97ad0622c1216a0c361acb0c0e36b6e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.676ex;" alt="{\displaystyle {\dot {\Phi }}}"></span>) in the circuit. Magnetic impedance is analogous to <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a>. </p><p>Magnetic complex impedance (<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a>: <a href="/wiki/Siemens_(unit)" title="Siemens (unit)">S</a>) is determined by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{M}={\frac {\mathcal {F}}{\dot {\Phi }}}=z_{M}e^{j\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Φ</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{M}={\frac {\mathcal {F}}{\dot {\Phi }}}=z_{M}e^{j\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1724062f8e27c687830afe139364cedfef1b85d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.519ex; height:5.843ex;" alt="{\displaystyle Z_{M}={\frac {\mathcal {F}}{\dot {\Phi }}}=z_{M}e^{j\phi }}"></span> where <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 z_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6066282e2acf6716ce4d51ce375cc0463963ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.04ex; height:2.009ex;" alt="{\displaystyle z_{M}}"></span> is the modulus of <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 Z_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ee772fe8af0a46a79e98373509267247bf0421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.547ex; height:2.509ex;" alt="{\displaystyle Z_{M}}"></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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is its phase. The <a href="/wiki/Argument" title="Argument">argument</a> of a complex magnetic impedance is equal to the difference of the phases of the magnetic tension and magnetic current. Complex magnetic impedance can be presented in following form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{M}=z_{M}e^{j\phi }=z_{M}\cos \phi +jz_{M}\sin \phi =r_{M}+jx_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ϕ</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>+</mo> <mi>j</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mi>j</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{M}=z_{M}e^{j\phi }=z_{M}\cos \phi +jz_{M}\sin \phi =r_{M}+jx_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ea6d8debbc1f8c3977ee429d0b33ef81c9d471" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:49.115ex; height:3.009ex;" alt="{\displaystyle Z_{M}=z_{M}e^{j\phi }=z_{M}\cos \phi +jz_{M}\sin \phi =r_{M}+jx_{M}}"></span> where <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_{M}=z_{M}\cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{M}=z_{M}\cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26483075fd7109299058cad51be3adae87d385df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.417ex; height:2.509ex;" alt="{\displaystyle r_{M}=z_{M}\cos \phi }"></span> is the real part of the complex magnetic impedance, called the effective magnetic resistance, 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 x_{M}=z_{M}\sin \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{M}=z_{M}\sin \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2b27c514f3ed15aa990a85ce6f350ffcb45458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.443ex; height:2.509ex;" alt="{\displaystyle x_{M}=z_{M}\sin \phi }"></span> is the imaginary part of the complex magnetic impedance, called the reactive magnetic resistance. The magnetic impedance is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{M}={\sqrt {r_{M}^{2}+x_{M}^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{M}={\sqrt {r_{M}^{2}+x_{M}^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d96b4f0ef9d42c30e4b195d124a85c32533051" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.246ex; height:4.843ex;" alt="{\displaystyle z_{M}={\sqrt {r_{M}^{2}+x_{M}^{2}}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\arctan {\frac {x_{M}}{r_{M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\arctan {\frac {x_{M}}{r_{M}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0011a26d85eb7bf0f3cf22c304f909ec39fe42d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.462ex; height:5.009ex;" alt="{\displaystyle \phi =\arctan {\frac {x_{M}}{r_{M}}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Magnetic_effective_resistance">Magnetic effective resistance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=13" title="Edit section: Magnetic effective resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Magnetic effective resistance</b> is the <a href="/wiki/Real_analysis" title="Real analysis">real</a> component of complex magnetic impedance. This causes a magnetic circuit to lose magnetic potential energy.<sup id="cite_ref-Pohl_8-0" class="reference"><a href="#cite_note-Pohl-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Küpfmüller_9-0" class="reference"><a href="#cite_note-Küpfmüller-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Active power in a magnetic circuit equals the product of magnetic effective resistance <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_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab9070000e25d7e098d94cb284a8888af704447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.788ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {M} }}"></span> and magnetic current squared <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_{\mathrm {M} }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {M} }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20cafeda3fe88a3db999de1672f26d815b2e683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.762ex; height:3.176ex;" alt="{\displaystyle I_{\mathrm {M} }^{2}}"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=r_{\mathrm {M} }I_{\mathrm {M} }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=r_{\mathrm {M} }I_{\mathrm {M} }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b7997b58d6bba022086f88f34142c16e27d92c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.394ex; height:3.176ex;" alt="{\displaystyle P=r_{\mathrm {M} }I_{\mathrm {M} }^{2}}"></span> </p><p>The magnetic effective resistance on a complex plane appears as the side of the resistance triangle for magnetic circuit of an alternating current. The effective magnetic resistance is bounding with the effective magnetic conductance <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 g_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261bc91c21e4571819c5642c8447fd3cced2506f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.848ex; height:2.009ex;" alt="{\displaystyle g_{\mathrm {M} }}"></span> by the expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mathrm {M} }={\frac {r_{\mathrm {M} }}{z_{\mathrm {M} }^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\mathrm {M} }={\frac {r_{\mathrm {M} }}{z_{\mathrm {M} }^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4d1424cc1f995b20e549f5af19bc96082f93d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.603ex; height:5.843ex;" alt="{\displaystyle g_{\mathrm {M} }={\frac {r_{\mathrm {M} }}{z_{\mathrm {M} }^{2}}}}"></span> where <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 z_{\mathrm {M} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{\mathrm {M} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a89c8caaec4a8f6dd9050d0cb1c5109f879ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.009ex;" alt="{\displaystyle z_{\mathrm {M} }}"></span> is the full magnetic impedance of a magnetic circuit. </p> <div class="mw-heading mw-heading4"><h4 id="Magnetic_reactance">Magnetic reactance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=14" title="Edit section: Magnetic reactance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Magnetic complex reluctance</a></div> <p><b>Magnetic reactance</b> is the parameter of a passive magnetic circuit, or an element of the circuit, which is equal to the square root of the difference of squares of the magnetic complex impedance and magnetic effective resistance to a magnetic current, taken with the sign plus, if the magnetic current lags behind the magnetic tension in phase, and with the sign minus, if the magnetic current leads the magnetic tension in phase. </p><p>Magnetic reactance <sup id="cite_ref-Pohl_8-1" class="reference"><a href="#cite_note-Pohl-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Popov_7-1" class="reference"><a href="#cite_note-Popov-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Küpfmüller_9-1" class="reference"><a href="#cite_note-Küpfmüller-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> is the component of magnetic complex impedance of the <a href="/wiki/Alternating_current" title="Alternating current">alternating current</a> circuit, which produces the phase shift between a magnetic current and magnetic tension in the circuit. It is measured in units of <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 {\tfrac {1}{\Omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Ω</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\Omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c128d22dd003867dc45aae5c7741b46d49f2aa23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.023ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{\Omega }}}"></span> and is denoted 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> (or <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>). It may be inductive <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_{L}=\omega L_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{L}=\omega L_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16de5e6b3df97a57730f19818a2cb482920d90f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.767ex; height:2.509ex;" alt="{\displaystyle x_{L}=\omega L_{M}}"></span> or capacitive <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_{C}={\tfrac {1}{\omega C_{M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>ω</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{C}={\tfrac {1}{\omega C_{M}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceedf6a95d326ea542899ad1521ab24e3c2f4c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.509ex; height:4.009ex;" alt="{\displaystyle x_{C}={\tfrac {1}{\omega C_{M}}}}"></span>, where <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of a magnetic current, <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 the <a href="/wiki/Magnetic_inductivity" class="mw-redirect" title="Magnetic inductivity">magnetic inductiance</a> of a circuit, <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 C_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b36d6b18ea6be45cbe07fe5fcee599d531ff486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.621ex; height:2.509ex;" alt="{\displaystyle C_{M}}"></span> is the magnetic capacitance of a circuit. The magnetic reactance of an undeveloped circuit with the inductance and the capacitance which are connected in series, is equal: <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="{\textstyle x=x_{L}-x_{C}=\omega L_{M}-{\frac {1}{\omega C_{M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ω</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=x_{L}-x_{C}=\omega L_{M}-{\frac {1}{\omega C_{M}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7c4a08d90b2088804b7e2f46e589b9329b2342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:28.287ex; height:4.009ex;" alt="{\textstyle x=x_{L}-x_{C}=\omega L_{M}-{\frac {1}{\omega C_{M}}}}"></span>. If <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_{L}=x_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{L}=x_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5e6ba47265d1d2ae87db7900166804a5d75c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.591ex; height:2.009ex;" alt="{\displaystyle x_{L}=x_{C}}"></span>, then the net reactance <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> and <a href="/wiki/Resonance" title="Resonance">resonance</a> takes place in the circuit. In the general case <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="{\textstyle x={\sqrt {z^{2}-r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x={\sqrt {z^{2}-r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52bd9ef5aaa2fb6de7331c1f6cd9cc0488239c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.84ex; height:3.509ex;" alt="{\textstyle x={\sqrt {z^{2}-r^{2}}}}"></span>. When an energy loss is absent (<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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></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 x=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c798ab79074bda962dc350b24e4b742d3a9b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.676ex;" alt="{\displaystyle x=z}"></span>. The angle of the phase shift in a magnetic circuit <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="{\textstyle \phi =\arctan {\frac {x}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi =\arctan {\frac {x}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711b9899c8e87271bcf27ced894bbd58f2ce408f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.113ex; height:3.009ex;" alt="{\textstyle \phi =\arctan {\frac {x}{r}}}"></span>. On a complex plane, the magnetic reactance appears as the side of the resistance triangle for circuit of an alternating current. </p> <div class="mw-heading mw-heading2"><h2 id="Limitations_of_the_analogy">Limitations of the analogy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrator%E2%80%93capacitor_model&action=edit&section=15" title="Edit section: Limitations of the analogy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The limitations of this analogy between magnetic circuits and electric circuits include the following; </p> <ul><li>The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic circuits not all of the magnetic field is confined to the magnetic circuit because magnetic permeability also exists outside materials (see <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>). Thus, there may be significant "<a href="/wiki/Leakage_flux" class="mw-redirect" title="Leakage flux">leakage flux</a>" in the space outside the magnetic cores. If the leakage flux is small compared to the main circuit, it can often be represented as additional elements. In extreme cases, a lumped-element model may not be appropriate at all, and <a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">field theory</a> is used instead.</li> <li>Magnetic circuits are <a href="/wiki/Nonlinear_element" class="mw-redirect" title="Nonlinear element">nonlinear</a>; the permeance in a magnetic circuit is not constant, unlike capacitance in an electrical circuit, but varies depending on the magnetic field. At high magnetic fluxes the <a href="/wiki/Ferromagnetic_materials" class="mw-redirect" title="Ferromagnetic materials">ferromagnetic materials</a> used for the cores of magnetic circuits <a href="/wiki/Saturation_(magnetic)" title="Saturation (magnetic)">saturate</a>, limiting further increase of the magnetic flux, so above this level the permeance decreases rapidly. In addition, the flux in ferromagnetic materials is subject to <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a>; it depends not just on the instantaneous MMF but also on the history of MMF. After the source of the magnetic flux is turned off, <a href="/wiki/Remanent_magnetism" class="mw-redirect" title="Remanent magnetism">remanent magnetism</a> is left in ferromagnetic materials, creating flux with no MMF.</li></ul> <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=Gyrator%E2%80%93capacitor_model&action=edit&section=16" title="Edit section: References"><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-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"> Hamill parenthetically includes "(per turn)" on page 97. <sup id="cite_ref-Hamill_1-1" class="reference"><a href="#cite_note-Hamill-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Hamill-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hamill_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hamill_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hamill_1-2"><sup><i><b>c</b></i></sup></a></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 q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHamill1993" class="citation journal cs1">Hamill, D.C. (1993). "Lumped equivalent circuits of magnetic components: the gyrator-capacitor approach". <i>IEEE Transactions on Power Electronics</i>. <b>8</b> (2): <span class="nowrap">97–</span>103. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993ITPE....8...97H">1993ITPE....8...97H</a>. <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%2F63.223957">10.1109/63.223957</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Power+Electronics&rft.atitle=Lumped+equivalent+circuits+of+magnetic+components%3A+the+gyrator-capacitor+approach&rft.volume=8&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E97-%3C%2Fspan%3E103&rft.date=1993&rft_id=info%3Adoi%2F10.1109%2F63.223957&rft_id=info%3Abibcode%2F1993ITPE....8...97H&rft.aulast=Hamill&rft.aufirst=D.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrator%E2%80%93capacitor+model" class="Z3988"></span></span> </li> <li id="cite_note-Lambert-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lambert_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lambert_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLambertMahseredjianMartı´nez-DuróSirois2015" class="citation journal cs1">Lambert, M.; Mahseredjian, J.; Martı´nez-Duró, M.; Sirois, F. (2015). "Magnetic Circuits Within Electric Circuits: Critical Review of Existing Methods and New Mutator Implementations". <i>IEEE Transactions on Power Delivery</i>. <b>30</b> (6): <span class="nowrap">2427–</span>2434. <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%2FTPWRD.2015.2391231">10.1109/TPWRD.2015.2391231</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:38890643">38890643</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Power+Delivery&rft.atitle=Magnetic+Circuits+Within+Electric+Circuits%3A+Critical+Review+of+Existing+Methods+and+New+Mutator+Implementations&rft.volume=30&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2427-%3C%2Fspan%3E2434&rft.date=2015&rft_id=info%3Adoi%2F10.1109%2FTPWRD.2015.2391231&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A38890643%23id-name%3DS2CID&rft.aulast=Lambert&rft.aufirst=M.&rft.au=Mahseredjian%2C+J.&rft.au=Mart%C4%B1%C2%B4nez-Dur%C3%B3%2C+M.&rft.au=Sirois%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrator%E2%80%93capacitor+model" class="Z3988"></span></span> </li> <li id="cite_note-González-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-González_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-González_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-González_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGonzálezEhsani2018" class="citation journal cs1">González, Guadalupe G.; Ehsani, Mehrdad (2018-03-12). <a rel="nofollow" class="external text" href="https://doi.org/10.35840%2F2631-5068%2F6512">"Power-Invariant Magnetic System Modeling"</a>. <i>International Journal of Magnetics and Electromagnetism</i>. <b>4</b> (1): <span class="nowrap">1–</span>9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.35840%2F2631-5068%2F6512">10.35840/2631-5068/6512</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1969.1%2FETD-TAMU-2011-08-9730">1969.1/ETD-TAMU-2011-08-9730</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2631-5068">2631-5068</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Magnetics+and+Electromagnetism&rft.atitle=Power-Invariant+Magnetic+System+Modeling&rft.volume=4&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E9&rft.date=2018-03-12&rft_id=info%3Ahdl%2F1969.1%2FETD-TAMU-2011-08-9730&rft.issn=2631-5068&rft_id=info%3Adoi%2F10.35840%2F2631-5068%2F6512&rft.aulast=Gonz%C3%A1lez&rft.aufirst=Guadalupe+G.&rft.au=Ehsani%2C+Mehrdad&rft_id=https%3A%2F%2Fdoi.org%2F10.35840%252F2631-5068%252F6512&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrator%E2%80%93capacitor+model" class="Z3988"></span></span> </li> <li id="cite_note-Mohammad-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mohammad_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mohammad_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Mohammad_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Mohammad_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Mohammad_4-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Mohammad_4-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohammad2014" class="citation thesis cs1">Mohammad, Muneer (2014-04-22). <a rel="nofollow" class="external text" href="https://oaktrust.library.tamu.edu/handle/1969.1/152720"><i>An Investigation of Multi-Domain Energy Dynamics</i></a> (PhD thesis).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=An+Investigation+of+Multi-Domain+Energy+Dynamics&rft.degree=PhD&rft.date=2014-04-22&rft.aulast=Mohammad&rft.aufirst=Muneer&rft_id=https%3A%2F%2Foaktrust.library.tamu.edu%2Fhandle%2F1969.1%2F152720&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrator%E2%80%93capacitor+model" class="Z3988"></span></span> </li> <li id="cite_note-Arkadiew-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Arkadiew_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Arkadiew_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Arkadiew W. <i>Eine Theorie des elektromagnetischen Feldes in den ferromagnetischen Metallen</i>. – Phys. Zs., H. 14, No 19, 1913, S. 928-934.</span> </li> <li id="cite_note-Popov-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Popov_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Popov_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopov,_V._P.1985" class="citation book cs1 cs1-prop-foreign-lang-source">Popov, V. P. (1985). <i>The Principles of Theory of Circuits</i> (in Russian). 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