Équations des télégraphistes

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<span>Afficher / masquer la sous-section Équations</span> </button> <ul id="toc-Équations-sublist" class="vector-toc-list"> <li id="toc-Formulation_de_base" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formulation_de_base"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Formulation de base</span> </div> </a> <ul id="toc-Formulation_de_base-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditions_initiales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditions_initiales"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Conditions initiales</span> </div> </a> <ul id="toc-Conditions_initiales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligne_sans_perte" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligne_sans_perte"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Ligne sans perte</span> </div> </a> <ul id="toc-Ligne_sans_perte-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cas_du_régime_sinusoïdal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cas_du_régime_sinusoïdal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Cas du régime sinusoïdal</span> </div> </a> <button aria-controls="toc-Cas_du_régime_sinusoïdal-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>Afficher / masquer la sous-section Cas du régime sinusoïdal</span> </button> <ul id="toc-Cas_du_régime_sinusoïdal-sublist" class="vector-toc-list"> <li id="toc-Ligne_sans_pertes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligne_sans_pertes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ligne sans pertes</span> </div> </a> <ul id="toc-Ligne_sans_pertes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vitesse_de_propagation_et_impédance_caractéristique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vitesse_de_propagation_et_impédance_caractéristique"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Vitesse de propagation et impédance caractéristique</span> </div> </a> <ul id="toc-Vitesse_de_propagation_et_impédance_caractéristique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligne_à_vide" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligne_à_vide"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Ligne à vide</span> </div> </a> <ul id="toc-Ligne_à_vide-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-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 vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Équations des télégraphistes</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 20 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-20" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">20 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%A7%D9%84%D8%AA%D9%84%D8%BA%D8%B1%D8%A7%D9%81" title="معادلات التلغراف – arabe" lang="ar" hreflang="ar" data-title="معادلات التلغراف" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BB%D0%B5%D0%B3%D1%80%D0%B0%D1%84%D0%BD%D0%B8_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Телеграфни уравнения – bulgare" lang="bg" hreflang="bg" data-title="Телеграфни уравнения" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equacions_del_tel%C3%A8graf" title="Equacions del telègraf – catalan" lang="ca" hreflang="ca" data-title="Equacions del telègraf" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Telegrafn%C3%AD_rovnice" title="Telegrafní rovnice – tchèque" lang="cs" hreflang="cs" data-title="Telegrafní rovnice" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Leitungsgleichung" title="Leitungsgleichung – allemand" lang="de" hreflang="de" data-title="Leitungsgleichung" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Telegrapher%27s_equations" title="Telegrapher's equations – anglais" lang="en" hreflang="en" data-title="Telegrapher's equations" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaciones_del_telegrafista" title="Ecuaciones del telegrafista – espagnol" lang="es" hreflang="es" data-title="Ecuaciones del telegrafista" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%AA%D9%84%DA%AF%D8%B1%D8%A7%D9%81%DB%8C" title="معادلات تلگرافی – persan" lang="fa" hreflang="fa" data-title="معادلات تلگرافی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%94%D7%98%D7%9C%D7%92%D7%A8%D7%A3" title="משוואות הטלגרף – hébreu" lang="he" hreflang="he" data-title="משוואות הטלגרף" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9F%E0%A5%87%E0%A4%B2%E0%A5%80%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AB_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="टेलीग्राफ समीकरण – hindi" lang="hi" hreflang="hi" data-title="टेलीग्राफ समीकरण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazioni_dei_telegrafisti" title="Equazioni dei telegrafisti – italien" lang="it" hreflang="it" data-title="Equazioni dei telegrafisti" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%BB%E4%BF%A1%E6%96%B9%E7%A8%8B%E5%BC%8F" title="電信方程式 – japonais" lang="ja" hreflang="ja" data-title="電信方程式" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EC%8B%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="전신 방정식 – coréen" lang="ko" hreflang="ko" data-title="전신 방정식" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Telegraafvergelijkingen" title="Telegraafvergelijkingen – néerlandais" lang="nl" hreflang="nl" data-title="Telegraafvergelijkingen" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnania_telegrafist%C3%B3w" title="Równania telegrafistów – polonais" lang="pl" hreflang="pl" data-title="Równania telegrafistów" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%B5es_do_telegrafista" title="Equações do telegrafista – portugais" lang="pt" hreflang="pt" data-title="Equações do telegrafista" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BB%D0%B5%D0%B3%D1%80%D0%B0%D1%84%D0%BD%D1%8B%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Телеграфные уравнения – russe" lang="ru" hreflang="ru" data-title="Телеграфные уравнения" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Telegrafski_ena%C4%8Dbi" title="Telegrafski enačbi – slovène" lang="sl" hreflang="sl" data-title="Telegrafski enačbi" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BB%D0%B5%D0%B3%D1%80%D0%B0%D1%84%D0%BD%D1%96_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F" title="Телеграфні рівняння – ukrainien" lang="uk" hreflang="uk" data-title="Телеграфні рівняння" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%94%B5%E6%8A%A5%E5%91%98%E6%96%B9%E7%A8%8B" 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</div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" 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">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><p>Les <b>équations des télégraphistes</b> sont un <a href="/wiki/Syst%C3%A8me_d%27%C3%A9quations" title="Système d'équations">système</a> de deux <a href="/wiki/%C3%89quations_aux_d%C3%A9riv%C3%A9es_partielles" class="mw-redirect" title="Équations aux dérivées partielles">équations aux dérivées partielles</a> qui décrivent l'évolution de la tension et du courant sur une ligne électrique en fonction de la distance et du temps. </p><p><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> a conçu dans les années 1880 le modèle des lignes électriques qui aboutit à ces équations. Il s'applique à toute ligne électrique et à toute fréquence et couvre les phénomènes de transmission et de réflexion sur une <a href="/wiki/Ligne_de_transmission" title="Ligne de transmission">ligne de transmission</a>, qu'elle serve au <a href="/wiki/T%C3%A9l%C3%A9graphe" title="Télégraphe">télégraphe</a>, au <a href="/wiki/T%C3%A9l%C3%A9phone" title="Téléphone">téléphone</a> ou à tout autre usage, ainsi qu'aux <a href="/wiki/Ligne_%C3%A0_haute_tension" title="Ligne à haute tension">lignes de distribution</a> du <a href="/wiki/R%C3%A9seau_%C3%A9lectrique" title="Réseau électrique">réseau électrique</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Équations"><span id=".C3.89quations"></span>Équations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=1" title="Modifier la section : Équations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=1" title="Modifier le code source de la section : Équations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Formulation_de_base">Formulation de base</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=2" title="Modifier la section : Formulation de base" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=2" title="Modifier le code source de la section : Formulation de base"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Transmission_line_element.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Transmission_line_element.svg/220px-Transmission_line_element.svg.png" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Transmission_line_element.svg/330px-Transmission_line_element.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Transmission_line_element.svg/440px-Transmission_line_element.svg.png 2x" data-file-width="310" data-file-height="159" /></a><figcaption>Schéma des composants élémentaires d'une ligne de transmission.</figcaption></figure> <p>Une portion <a href="/wiki/Calcul_infinit%C3%A9simal" title="Calcul infinitésimal">infinitésimale</a> de ligne électrique peut être représentée par un <a href="/wiki/Quadripole" class="mw-redirect" title="Quadripole">quadripole</a> où : </p> <ul><li>la <a href="/wiki/R%C3%A9sistance_(%C3%A9lectricit%C3%A9)" title="Résistance (électricité)">résistance</a> linéique (par <a href="/wiki/Unit%C3%A9_de_longueur" title="Unité de longueur">unité de longueur</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> du conducteur est représentée par une résistance <a href="/wiki/Montage_en_s%C3%A9rie" title="Montage en série">série</a> (exprimée en <a href="/wiki/Ohm_(unit%C3%A9)" title="Ohm (unité)">ohms</a> par unité de longueur) ;</li> <li>l'<a href="/wiki/Inductance" title="Inductance">inductance</a> linéique <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/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> est représentée par une <a href="/wiki/Bobine_(%C3%A9lectricit%C3%A9)" title="Bobine (électricité)">bobine</a> (<a href="/wiki/Henry_(unit%C3%A9)" title="Henry (unité)">henrys</a> par unité de longueur) ;</li> <li>la <a href="/wiki/Capacit%C3%A9_%C3%A9lectrique" title="Capacité électrique">capacité</a> linéique <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> entre les deux conducteurs est représentée par un <a href="/wiki/Condensateur_(%C3%A9lectricit%C3%A9)" class="mw-redirect" title="Condensateur (électricité)">condensateur</a> C shunt (<a href="/wiki/Farad" title="Farad">farads</a> par unité de longueur) ;</li> <li>la <a href="/wiki/Conductance_%C3%A9lectrique" title="Conductance électrique">conductance</a> linéique <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> du milieu <a href="/wiki/Di%C3%A9lectrique" title="Diélectrique">diélectrique</a> séparant les deux conducteurs (<a href="/wiki/Siemens_(unit%C3%A9)" title="Siemens (unité)">siemens</a> par unité de longueur). Le <a href="/wiki/Sch%C3%A9ma_%C3%A9lectrique" title="Schéma électrique">schéma électrique</a> du modèle représente cette conductance par une résistance <a href="/wiki/Parall%C3%A8le_(%C3%A9lectricit%C3%A9)" title="Parallèle (électricité)">parallèle</a> de valeur de <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 1/G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f35458f37b087cd3c1f7340da90785cbe3448c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.152ex; height:2.843ex;" alt="{\displaystyle 1/G}"></span> ohms.</li></ul> <p>La résistance et la conductance croissent avec la <a href="/wiki/Fr%C3%A9quence" title="Fréquence">fréquence</a> et l'inductance varie dans de moindres proportions, à cause de l'<a href="/wiki/Effet_de_peau" title="Effet de peau">effet de peau</a> et, dans les <a href="/wiki/Paire_torsad%C3%A9e" title="Paire torsadée">lignes bifilaires</a>, de l'effet de proximité<sup id="cite_ref-Fontolliet199969-70_1-0" class="reference"><a href="#cite_note-Fontolliet199969-70-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. </p><p>Soient <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 U(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e0def6913191df7b9f94a09eb2339a920c7496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle U(x,t)}"></span> la tension et <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(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce7ffaa3abb61b66f8c38b142ef3e164aa56047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.184ex; height:2.843ex;" alt="{\displaystyle I(x,t)}"></span> le courant en un point éloigné d'une distance <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> du début de la ligne à un instant <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, on peut écrire deux <a href="/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles" title="Équation aux dérivées partielles">équations aux dérivées partielles</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)-RI(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>R</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)-RI(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57b60d7d8b5ee6a11e57ccd2c85684dc7998ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.566ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)-RI(x,t)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)-GU(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)-GU(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5e72fba7b9496ec89b598d636456236db5e391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.581ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)-GU(x,t)}"></span></dd></dl> <p>De cette formulation, on peut tirer deux équations ne faisant chacune intervenir qu'une variable : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial U}{\partial t}}(x,t)+GRU(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mo>+</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>G</mi> <mi>R</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial U}{\partial t}}(x,t)+GRU(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1669b84fd7f5ea9c89ba806b6c631085a5bbe90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:63.111ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial U}{\partial t}}(x,t)+GRU(x,t)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial I}{\partial t}}(x,t)+GRI(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <mo>+</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>G</mi> <mi>R</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial I}{\partial t}}(x,t)+GRI(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d9b433f14bed5fe8935e1da031f4e7159f74b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:60.8ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)+(RC+GL){\frac {\partial I}{\partial t}}(x,t)+GRI(x,t)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Conditions_initiales">Conditions initiales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=3" title="Modifier la section : Conditions initiales" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=3" title="Modifier le code source de la section : Conditions initiales"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ces équations doivent être complétées par la définition de <a href="/wiki/Condition_initiale" title="Condition initiale">conditions initiales</a>. </p><p>On peut ainsi définir la tension à l'extrémité initiale de la ligne comme celle d'une source sinusoïdale </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(0,t)=U_{0}\sin(2\pi ft)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(0,t)=U_{0}\sin(2\pi ft)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe7d8737a36f9e0e2508087f91da66ee5612401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.033ex; height:2.843ex;" alt="{\displaystyle U(0,t)=U_{0}\sin(2\pi ft)}"></span></dd></dl> <p>et définir une relation entre courant et tension à l'autre extrémité de la ligne située à une distance <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> </p> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(l,t)=R'I(l,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>R</mi> <mo>′</mo> </msup> <mi>I</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(l,t)=R'I(l,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec19c42101f64a3dab8ba4ab43db2151468232d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.254ex; height:3.009ex;" alt="{\displaystyle U(l,t)=R'I(l,t)}"></span> pour une ligne chargée par une résistance <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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cc152440f75fd8f842f4225a7484bb431b3343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle R'}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(l,t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(l,t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b12b5a1a53ffceb5260a4687d63c937d969fa8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.809ex; height:2.843ex;" alt="{\displaystyle I(l,t)=0}"></span> pour une ligne à vide, <abbr class="abbr" title="et cetera">etc.</abbr></li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ligne_sans_perte">Ligne sans perte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=4" title="Modifier la section : Ligne sans perte" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=4" title="Modifier le code source de la section : Ligne sans perte"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans beaucoup de cas, on peut négliger les pertes résistives. On pose alors <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/b7e38203eea12f5613721e2fc5e185b8eef9e800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=0}"></span> et <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2141d3c1164708ef8f436aa25143234d38904f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.088ex; height:2.176ex;" alt="{\displaystyle G=0}"></span>. Les équations s'écrivent : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147c756383aaa6aaa5b3e45440abb776ae969d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.777ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial U}{\partial x}}(x,t)=-L{\frac {\partial I}{\partial t}}(x,t)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1862d1872908819dc7d65777ecbc49661ba420d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.119ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial I}{\partial x}}(x,t)=-C{\frac {\partial U}{\partial t}}(x,t)}"></span></dd></dl> <p>On peut les combiner pour former deux équations de propagation, qui sont des <a href="/wiki/%C3%89quation_des_ondes" title="Équation des ondes">équations de d'Alembert</a> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4be026bdfd9abf1a63b3b655c5e78989c75f0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.505ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd8fad1cea9370a7383ad76280507b80584575e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.416ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}I}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}I}{\partial t^{2}}}(x,t)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Cas_du_régime_sinusoïdal"><span id="Cas_du_r.C3.A9gime_sinuso.C3.AFdal"></span>Cas du régime sinusoïdal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=5" title="Modifier la section : Cas du régime sinusoïdal" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=5" title="Modifier le code source de la section : Cas du régime sinusoïdal"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On considère une tension sinusoïdale complexe <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> de pulsation <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> et de <a href="/wiki/Vecteur_d%27onde" title="Vecteur d'onde">vecteur d'onde</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 {\overrightarrow {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d116a52f51b869325e137116547379f35353c922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {k}}}"></span> se propageant selon l'axe <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> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{0}e^{j(\omega t+{\overrightarrow {k}}.{\overrightarrow {x}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msup> <mo>,</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo>→</mo> </mover> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{0}e^{j(\omega t+{\overrightarrow {k}}.{\overrightarrow {x}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4404adf9a98816825bce1d16550995f2a9a7224d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.398ex; height:4.343ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{0}e^{j(\omega t+{\overrightarrow {k}}.{\overrightarrow {x}})}}"></span></dd></dl> <p>La dérivée partielle de <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> par rapport au temps est alors : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial t}(x,t)=j\omega U(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial t}(x,t)=j\omega U(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5cf5890aa653411c63c6b1485a5883f75ecf303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.134ex; height:5.509ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial t}(x,t)=j\omega U(x,t)}"></span></dd></dl> <p>De même, la dérivée partielle de <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>par rapport à <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> est : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial x}(x,t)=jk_{x}{\text{ }}U(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>j</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial x}(x,t)=jk_{x}{\text{ }}U(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca42753daf01647c49f11920305cfcfdcc79d7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.652ex; height:5.509ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\partial U \over \partial x}(x,t)=jk_{x}{\text{ }}U(x,t)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ligne_sans_pertes">Ligne sans pertes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=6" title="Modifier la section : Ligne sans pertes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=6" title="Modifier le code source de la section : Ligne sans pertes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans la ligne sans pertes : <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/b7e38203eea12f5613721e2fc5e185b8eef9e800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=0}"></span> et <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2141d3c1164708ef8f436aa25143234d38904f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.088ex; height:2.176ex;" alt="{\displaystyle G=0}"></span>, la première <a href="/wiki/%C3%89quation_des_ondes" title="Équation des ondes">équation des ondes</a> est </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4be026bdfd9abf1a63b3b655c5e78989c75f0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.505ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}U}{\partial x^{2}}}(x,t)=LC{\frac {\partial ^{2}U}{\partial t^{2}}}(x,t)}"></span></dd></dl> <p>La solution générale de cette équation de propagation est la somme d'une onde de tension <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 U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"></span> se propageant dans les <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> croissants et d'une autre <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 U_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaec4ad37894dbbbf185dad9af9dcb365ab8916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.561ex; height:2.509ex;" alt="{\displaystyle U_{r}}"></span> se propageant dans le sens des <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> décroissants : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} ^{2},U(x,t)=U_{i}(x,t)+U_{r}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} ^{2},U(x,t)=U_{i}(x,t)+U_{r}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeffde18cc61c335887e8d749a9d3b2982779e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.619ex; height:3.176ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} ^{2},U(x,t)=U_{i}(x,t)+U_{r}(x,t)}"></span></dd></dl> <p>avec </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\begin{cases}U_{i}(x,t)=U_{i}e^{j(\omega t-k_{i}x)}\\U_{r}(x,t)=U_{r}e^{j(\omega t+k_{r}x)}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\begin{cases}U_{i}(x,t)=U_{i}e^{j(\omega t-k_{i}x)}\\U_{r}(x,t)=U_{r}e^{j(\omega t+k_{r}x)}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a319372b77462b19c1278666eed5ac3528567e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.345ex; height:6.176ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},{\begin{cases}U_{i}(x,t)=U_{i}e^{j(\omega t-k_{i}x)}\\U_{r}(x,t)=U_{r}e^{j(\omega t+k_{r}x)}\end{cases}}}"></span></dd></dl> <p>Les indices <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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> et <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> renvoient aux ondes « incidente » et « réfléchie ». </p><p>En injectant l'expression des tensions <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 U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"></span> et <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 U_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaec4ad37894dbbbf185dad9af9dcb365ab8916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.561ex; height:2.509ex;" alt="{\displaystyle U_{r}}"></span> dans cette équation, on obtient : </p> <dl><dd>pour <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=\lbrace i,r\rbrace :k_{l}^{2}U_{j}(x,t)=-\omega ^{2}LC{\text{ }}U_{l}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>,</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>:</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=\lbrace i,r\rbrace :k_{l}^{2}U_{j}(x,t)=-\omega ^{2}LC{\text{ }}U_{l}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb66b448cc4ae7dcbf6ff181d322881f79f0572d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.373ex; height:3.343ex;" alt="{\displaystyle l=\lbrace i,r\rbrace :k_{l}^{2}U_{j}(x,t)=-\omega ^{2}LC{\text{ }}U_{l}(x,t)}"></span></dd></dl> <p>En simplifiant par <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 U_{j}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{j}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34858baacfc17a9dbfbacea375c5b063d4e92c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.51ex; height:3.009ex;" alt="{\displaystyle U_{j}(x,t)}"></span>, on obtient la relation de dispersion suivante : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{l}^{2}=-\omega ^{2}LC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{l}^{2}=-\omega ^{2}LC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ab4dfd933e84dcfc4466db21b185c93a3220a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.021ex; height:3.343ex;" alt="{\displaystyle k_{l}^{2}=-\omega ^{2}LC}"></span></dd></dl> <p>soit </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{l}=j\omega {\sqrt {LC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{l}=j\omega {\sqrt {LC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61144962a45a83102ff160091e5c87c6e0cdc5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.721ex; height:3.009ex;" alt="{\displaystyle k_{l}=j\omega {\sqrt {LC}}}"></span></dd></dl> <p>Donc pour les ondes incidente et réfléchie, le <a href="/wiki/Nombre_d%27onde" title="Nombre d'onde">nombre d'onde</a> est imaginaire pur. </p><p>La constante d'<a href="/wiki/Att%C3%A9nuation" title="Atténuation">atténuation</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 \alpha =\Re (k_{l})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\Re (k_{l})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b7cc18d24be89f2b5db1532d5be61140c78d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.253ex; height:2.843ex;" alt="{\displaystyle \alpha =\Re (k_{l})}"></span> est nulle et la constante de propagation <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 \beta =\Im (k_{l})=\omega {\sqrt {LC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi mathvariant="normal">ℑ</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\Im (k_{l})=\omega {\sqrt {LC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7315dbaa7aca209a6b508de1634d143f258b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.291ex; height:3.176ex;" alt="{\displaystyle \beta =\Im (k_{l})=\omega {\sqrt {LC}}}"></span> et en posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\omega {\sqrt {LC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\omega {\sqrt {LC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1ae96760d9da725b356e11e2eaee6d56e2d04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.041ex; height:3.009ex;" alt="{\displaystyle k=\omega {\sqrt {LC}}}"></span>, l'expression de la tension <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> est : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{i}e^{j(\omega t-kx)}+U_{r}e^{j(\omega t+kx)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msup> <mo>,</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{i}e^{j(\omega t-kx)}+U_{r}e^{j(\omega t+kx)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e455352e3b42d9696850d20b7509f0be7fdc5eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.497ex; height:3.343ex;" alt="{\displaystyle \forall (x,t)\in \mathbb {R} \times \mathbb {R} ^{\text{+}},U(x,t)=U_{i}e^{j(\omega t-kx)}+U_{r}e^{j(\omega t+kx)}}"></span></dd></dl> <p>Puisque la constante d'atténuation <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> est nulle, la propagation se fait bien sans pertes. </p> <div class="mw-heading mw-heading3"><h3 id="Vitesse_de_propagation_et_impédance_caractéristique"><span id="Vitesse_de_propagation_et_imp.C3.A9dance_caract.C3.A9ristique"></span>Vitesse de propagation et impédance caractéristique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=7" title="Modifier la section : Vitesse de propagation et impédance caractéristique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=7" title="Modifier le code source de la section : Vitesse de propagation et impédance caractéristique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Puisque <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 U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"></span> se propage dans le sens des x croissants et <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 U_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaec4ad37894dbbbf185dad9af9dcb365ab8916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.561ex; height:2.509ex;" alt="{\displaystyle U_{r}}"></span>, dans le sens des x décroissants alors leur expression est de la forme </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v_{i}}{\Bigr )}\\U_{r}(x,t)=U_{r}{\Bigl (}t+{x \over v_{r}}{\Bigr )}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v_{i}}{\Bigr )}\\U_{r}(x,t)=U_{r}{\Bigl (}t+{x \over v_{r}}{\Bigr )}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3986c454ab5c24e45c2dae77107b4246cbf89cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.906ex; height:9.343ex;" alt="{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v_{i}}{\Bigr )}\\U_{r}(x,t)=U_{r}{\Bigl (}t+{x \over v_{r}}{\Bigr )}\end{cases}}}"></span>avec <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 {\begin{cases}v_{i},{\text{la vitesse de propagation de l'onde incidente}}\\v_{r},{\text{la vitesse de propagation de l'onde réfléchie}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>la vitesse de propagation de l'onde incidente</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>la vitesse de propagation de l'onde réfléchie</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}v_{i},{\text{la vitesse de propagation de l'onde incidente}}\\v_{r},{\text{la vitesse de propagation de l'onde réfléchie}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c034e2fabe582ae3be2db2ae4ed419985764b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.584ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}v_{i},{\text{la vitesse de propagation de l'onde incidente}}\\v_{r},{\text{la vitesse de propagation de l'onde réfléchie}}\end{cases}}}"></span></dd></dl> <p>La vitesse de propagation est la vitesse à laquelle se déplace la phase de l'onde. </p><p>D'après l'étude précédente sur les solutions de l'équation des télégraphistes en <a href="/wiki/R%C3%A9gime_sinuso%C3%AFdal" title="Régime sinusoïdal">régime sinusoïdal</a> sans pertes, on a : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v}{\Bigr )}=U_{i0}\ e^{j(\omega t-kx)}=U_{i0}\ e^{j\omega {\Bigl (}t-{k \over \omega }x{\Bigr )}}\\U_{r}(x,t)=U_{i}{\Bigl (}t+{x \over v}{\Bigr )}=U_{r0}\ e^{j(\omega t+kx)}=U_{r0}\ e^{j\omega {\Bigl (}t+{k \over \omega }x{\Bigr )}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>v</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>ω</mi> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>v</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>ω</mi> </mfrac> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v}{\Bigr )}=U_{i0}\ e^{j(\omega t-kx)}=U_{i0}\ e^{j\omega {\Bigl (}t-{k \over \omega }x{\Bigr )}}\\U_{r}(x,t)=U_{i}{\Bigl (}t+{x \over v}{\Bigr )}=U_{r0}\ e^{j(\omega t+kx)}=U_{r0}\ e^{j\omega {\Bigl (}t+{k \over \omega }x{\Bigr )}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918c106ee7ed279606f4eec60dab6046ee9f50e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:56.111ex; height:12.509ex;" alt="{\displaystyle {\begin{cases}U_{i}(x,t)=U_{i}{\Bigl (}t-{x \over v}{\Bigr )}=U_{i0}\ e^{j(\omega t-kx)}=U_{i0}\ e^{j\omega {\Bigl (}t-{k \over \omega }x{\Bigr )}}\\U_{r}(x,t)=U_{i}{\Bigl (}t+{x \over v}{\Bigr )}=U_{r0}\ e^{j(\omega t+kx)}=U_{r0}\ e^{j\omega {\Bigl (}t+{k \over \omega }x{\Bigr )}}\end{cases}}}"></span></dd></dl> <p>Par identification, on a : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=v_{r}={\omega \over k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=v_{r}={\omega \over k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ed8e962dd795f38b668ad3368e291af7ca429e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.508ex; height:4.843ex;" alt="{\displaystyle v_{i}=v_{r}={\omega \over k}}"></span></dd></dl> <p>La vitesse de propagation de l'onde incidente et de l'onde réfléchie vaut donc <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={\frac {\omega }{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {\omega }{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe492407aa055a244a167ec1fbf7ed0b3a5d0ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.508ex; height:4.843ex;" alt="{\displaystyle v={\frac {\omega }{k}}}"></span> et comme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{l}=j\omega {\sqrt {LC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{l}=j\omega {\sqrt {LC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61144962a45a83102ff160091e5c87c6e0cdc5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.721ex; height:3.009ex;" alt="{\displaystyle k_{l}=j\omega {\sqrt {LC}}}"></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 v={\frac {1}{\sqrt {LC}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {1}{\sqrt {LC}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5340020701eb09a7b59fe195561f79e44611b650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.347ex; height:6.176ex;" alt="{\displaystyle v={\frac {1}{\sqrt {LC}}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ligne_à_vide"><span id="Ligne_.C3.A0_vide"></span>Ligne à vide</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=8" title="Modifier la section : Ligne à vide" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=8" title="Modifier le code source de la section : Ligne à vide"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone"><span class="mw-valign-text-top noviewer" typeof="mw:File"><a href="/wiki/Fichier:Fairytale_warning.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/17px-Fairytale_warning.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/26px-Fairytale_warning.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/34px-Fairytale_warning.png 2x" data-file-width="64" data-file-height="64" /></a></span></div><div class="bandeau-cell">Cette section est vide, insuffisamment détaillée ou incomplète. <a href="/wiki/Sp%C3%A9cial:EditPage/%C3%89quations_des_t%C3%A9l%C3%A9graphistes" title="Spécial:EditPage/Équations des télégraphistes">Votre aide</a> est la bienvenue ! <a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">Comment faire ?</a></div></div> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=9" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=9" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Démoulin2014"><span class="ouvrage" id="Bernard_Démoulin2014">Bernard <span class="nom_auteur">Démoulin</span>, <cite class="italique">Éléments sur la théorie des lignes de transmission : d1322</cite>, <a href="/wiki/Techniques_de_l%27ing%C3%A9nieur" class="mw-redirect" title="Techniques de l'ingénieur">Techniques de l'ingénieur</a>, <time>2014</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://www.techniques-ingenieur.fr/base-documentaire/energies-th4/outils-d-analyse-en-electronique-de-puissance-et-metrologie-42278210/elements-sur-la-theorie-des-lignes-de-transmission-d1322/">présentation en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+sur+la+th%C3%A9orie+des+lignes+de+transmission&rft.pub=Techniques+de+l%27ing%C3%A9nieur&rft.stitle=d1322&rft.aulast=D%C3%A9moulin&rft.aufirst=Bernard&rft.date=2014&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89quations+des+t%C3%A9l%C3%A9graphistes"></span></span></span>.</li> <li><span class="ouvrage" id="Fontolliet1999"><span class="ouvrage" id="Pierre-Gérard_Fontolliet1999">Pierre-Gérard <span class="nom_auteur">Fontolliet</span>, <cite class="italique">Systèmes de télécommunications : Traité d'électricité, volume XVIII</cite>, Lausanne, Presses polytechniques et universitaires romandes, <time>1999</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z5_3AAKZJBsC">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 71<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Syst%C3%A8mes+de+t%C3%A9l%C3%A9communications&rft.place=Lausanne&rft.pub=Presses+polytechniques+et+universitaires+romandes&rft.stitle=Trait%C3%A9+d%27%C3%A9lectricit%C3%A9%2C+volume+XVIII&rft.aulast=Fontolliet&rft.aufirst=Pierre-G%C3%A9rard&rft.date=1999&rft.pages=71&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89quations+des+t%C3%A9l%C3%A9graphistes"></span></span></span>.</li></ul> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Fontolliet199969-70-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-Fontolliet199969-70_1-0">↑</a> </span><span class="reference-text"><a href="#Fontolliet1999">Fontolliet 1999</a>, <abbr class="abbr" title="page(s)">p.</abbr> 69-70. </span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Commission_électrotechnique_internationale"><a href="/wiki/Commission_%C3%A9lectrotechnique_internationale" title="Commission électrotechnique internationale">Commission électrotechnique internationale</a>, <cite style="font-style:normal">« Théorie des circuits : éléments de circuit et leurs caractéristiques »</cite>, dans <cite class="italique">IEC 60050 Vocabulaire électrotechnique international année=2002</cite> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-86">lire en ligne</a>)</small>, <abbr class="abbr" title="page(s)">p.</abbr> 131-12-86 : Ligne de transmission<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=IEC+60050+Vocabulaire+%C3%A9lectrotechnique+international+ann%C3%A9e%3D2002&rft.atitle=Th%C3%A9orie+des+circuits+%3A+%C3%A9l%C3%A9ments+de+circuit+et+leurs+caract%C3%A9ristiques&rft.aucorp=%5B%5BCommission+%C3%A9lectrotechnique+internationale%5D%5D&rft.pages=131-12-86+%3A+Ligne+de+transmission&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89quations+des+t%C3%A9l%C3%A9graphistes"></span></span>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&veaction=edit&section=10" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quations_des_t%C3%A9l%C3%A9graphistes&action=edit&section=10" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ligne_de_transmission" title="Ligne de transmission">Ligne de transmission</a></li> <li><a href="/wiki/Angle_de_transport" title="Angle de transport">Angle de transport</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:%C3%89lectricit%C3%A9_et_%C3%A9lectronique" title="Portail de l’électricité et de l’électronique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/24px-Octicons-circuit-board.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/36px-Octicons-circuit-board.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Octicons-circuit-board.svg/48px-Octicons-circuit-board.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:%C3%89lectricit%C3%A9_et_%C3%A9lectronique" title="Portail:Électricité et électronique">Portail de l’électricité et de l’électronique</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:T%C3%A9l%C3%A9communications" title="Portail des télécommunications"><img alt="icône décorative" 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