From 12255158025a0660e8cd19e3bc7c52692267470f Mon Sep 17 00:00:00 2001 From: Vincent Date: Wed, 30 Oct 2024 13:49:12 +0100 Subject: [PATCH] TD OH --- Physique1A/Physique_1A_05_C_OH.markdown | 19 +++++-- Physique1A/Physique_1A_05_EX.markdown | 68 ++++++++++++++++++++++++ Physique1A/img/05_EX/LL.png | Bin 0 -> 14300 bytes Physique1A/img/05_EX/RCRC.png | Bin 0 -> 18313 bytes Physique1A/img/05_EX/RLC_parallel.png | Bin 0 -> 27253 bytes Physique1A/img/05_EX/RLC_series.png | Bin 0 -> 17111 bytes Physique1A/img/05_EX/RLRC.png | Bin 0 -> 24195 bytes 7 files changed, 84 insertions(+), 3 deletions(-) create mode 100644 Physique1A/Physique_1A_05_EX.markdown create mode 100644 Physique1A/img/05_EX/LL.png create mode 100644 Physique1A/img/05_EX/RCRC.png create mode 100644 Physique1A/img/05_EX/RLC_parallel.png create mode 100644 Physique1A/img/05_EX/RLC_series.png create mode 100644 Physique1A/img/05_EX/RLRC.png diff --git a/Physique1A/Physique_1A_05_C_OH.markdown b/Physique1A/Physique_1A_05_C_OH.markdown index 4156a1f..130352b 100644 --- a/Physique1A/Physique_1A_05_C_OH.markdown +++ b/Physique1A/Physique_1A_05_C_OH.markdown @@ -12,11 +12,17 @@ id: Phy1_05_C ![](./img/05_C/LC_series.png) -On s'intéresse au circuit représenté ci-dessus. À $t=0$, on ferme l'interrupteur. On s'intéresse à la tension $u_C(t)$ pour $t>0$. +On s'intéresse au circuit représenté ci-dessus. On considère qu'à $t<0$, le condensateur est chargé, et $u_C(t<0) = U_0$. + +À $t=0$, on ferme l'interrupteur. On s'intéresse à la tension $u_C(t)$ pour $t>0$. ### A. Étude qualitative -On peut se demander comment va évoluer la tension du condensateur. En effet, dans un premier temps, lorsqu'on ferme l'interrupteur, la bobine va s'opposer à l'établissement du courant dans le circuit. Puis, progressivement, le courant va s'établir, et les charges vont s'accumuler sur le condensateur. +On peut se demander comment va évoluer la tension du condensateur. En effet, dans un premier temps, lorsqu'on ferme l'interrupteur, le condensateur va avoir tendance à se décharger. Cependant, la bobine va s'opposer à l'établissement du courant, qui va donc augmenter de manière progressive et non brutale. + +Lorsque la tension du condensateur devient nulle, le mouvement de charges devrait alors s'interrompre. Nous savons cependant que l'intensité électrique est continue dans une bobine : la bobine continue à alimenter le mouvement des charges, et va donc forcer la charge du condensateur dans le sens opposé. + +Intuitivement, on devrait donc voir apparaître une oscillation de la tension du condensateur entre une valeur positive et une valeur négative. ### B. Étude quantitative @@ -53,4 +59,11 @@ $\Leftrightarrow \boxed{\dfrac{d^2u_C}{dt^2} + \dfrac{1}{LC} u_C(t) = 0}$ - Il s'agit d'une équation différentielle linéaire du **second ordre** à coefficients réels constants. - On ne va pas pouvoir appliquer la même méthode de résolution que pour les équations différentielles du premier ordre, car notre théorème pour la solution homogène ne s'applique plus ! Pas de panique, on va en inventer un autre :) -- On dit que cette équation différentielle est caractéristique d'un oscillateur harmonique \ No newline at end of file +- On remarque que les fonctions qui vérifient cette équation sont les fonctions trigonométriques ! +- On appelle ce genre de système **un oscillateur harmonique**. + +## 2. L'oscillateur harmonique + +> Un oscillateur harmonique est un oscillateur idéal dont l'évolution au cours du temps est décrite par une fonction sinusoïdale, dont la fréquence ne dépend que des caractéristiques du système et dont l'amplitude est constante. (*Wikipédia*) + +Il faut comprendre dans le mot idéal, un oscillateur sans frottements. \ No newline at end of file diff --git a/Physique1A/Physique_1A_05_EX.markdown b/Physique1A/Physique_1A_05_EX.markdown new file mode 100644 index 0000000..a592ac3 --- /dev/null +++ b/Physique1A/Physique_1A_05_EX.markdown @@ -0,0 +1,68 @@ +--- +layout: default +title: "Physique A1 - Chapitre 5 - Exercices" +date: 2024-07-01 12:19:36 +0200 +categories: exercises +id: Phy1_05_EX +--- + +# Chapitre 5 : Oscillateurs Électriques - EXERCICES + +## Exercice 1 : Deux bobines en série + +![](./img/05_EX/LL.png) + +Le circuit schématisé ci-dessus comporte deux bobines d'inductance $L_1$ et $L_2$, ainsi qu'un condensateur de capacité $C$. Le condensateur est initialement chargé, et $u(t<0) = U_0$. À $t=0$, on ferme l'interrupteur. + +1. Justifier que $u(t=0^+) = U_0$. +2. Montrer que $\dfrac{du}{dt}(t=0^+) = 0$ +3. Déterminer l'équation différentielle vérifiée par $u(t)$. +4. Résoudre, qualifier le régime suivi par $u(t)$, et préciser ses caractéristiques. Tracer l'allure de $u(t)$. + +## Exercice 2 : RLC série + +![](./img/05_EX/RLC_series.png) + +On s'intéresse à l'évolution de la tension $u_C(t)$ aux bornes du condensateur dans le circuit ci-dessus. Avant la fermeture de l'interrupteur à $t=O$, on suppose que le condensateur est déchargé. + +1. Que vaut la tension $u_C(t)$ en régime permanent (après avoir fermé l'interrupteur) ? +2. Établir l'équation différentielle vérifiée par $u_C(t)$. Donner l'expression des grandeurs caractéristiques de cette équation (pulsation propre et facteur d'amortissement, ou facteur de qualité). +3. Que valent les conditions initiales $u_C(t=0)$ et $\dfrac{du_C}{dt}(t=0)$ ? +4. On prend pour valeurs numériques : $E=10V, R = 1k\Omega, L = 10mH, C = 1\mu F$ + a. Qualifier le régime transitoire dans ce cas. + b. Résoudre l'équation différentielle pour $u_C(t)$. + c. Tracer $u_C(t)$. +5. Mêmes questions avec : $E=10V, R = 0.1k\Omega, L = 1mH, C = 1 nF$ +6. Mêmes questions avec : $E=10V, R = 2k\Omega, L = 1mH, C = 1 nF$ + +## Exercice 3 : Double RC + +![](./img/05_EX/RCRC.png) + +Le circuit ci-dessus comporte deux résistances $R$ identiques, et deux condensateurs de capacité $C$ identiques, et initialement déchargés. À $t=0$, on ferme l'interrupteur. + +1. Justifier que $u(t=0^+) = 0$. +2. Montrer que $\dfrac{du}{dt}(t=0^+) = 0$ +3. Que vaut la tension $u(t)$ en régime permanent ? +4. Montrer que l'équation différentielle vérifiée par $u(t)$ est : +$$\dfrac{E}{\tau^2} = \dfrac{d^2u}{dt^2} + \dfrac{3}{\tau} \dfrac{du}{dt} + \dfrac{u(t)}{\tau^2}$$ +et donner l'expression de $\tau$. +6. Résoudre, qualifier le régime suivi par $u(t)$, et préciser ses caractéristiques. Tracer l'allure de $u(t)$. + +## Exercice 4 : RL-RC + +![](./img/05_EX/RLRC.png) + +Dans le circuit ci-dessus, les dipôles $RL$ et $RC$ ont la même constante de temps, et $E = 10V$. Pour $t<0$, le condensateur est déchargé. On ferme l'interrupteur à $t=0$. + +Déterminer l'équation différentielle vérifiée par $u(t)$. Qualifier le régime de $u(t)$, puis résoudre. + +## Exercice 5 : RLC parallèle + +![](./img/05_EX/RLC_parallel.png) + +Dans le circuit ci-dessus, on ferme l'interrupteur à $t=0$. On considère qu'avant cela, le condensateur est déchargé. On s'intéresse à l'évolution de la tension $u(t)$. + +1. Déterminer les valeurs de $u,i,i_1,i_2,i_3$ à $t=0^+$, puis en régime permanent. +2. Établir l'équation différentielle vérifiée par $u(t)$. Identifier les grandeurs caractéristiques et donner leur expression. +3. 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