From 8647653ce2a8f798cb5780d22838a175626daf43 Mon Sep 17 00:00:00 2001 From: Vincent Date: Wed, 23 Oct 2024 17:23:28 +0200 Subject: [PATCH] PhyA1TD4 --- Physique1A/Physique_1A_04_C_RT.markdown | 216 ++++++++++++++++++- Physique1A/Physique_1A_04_EX.markdown | 69 +++++- Physique1A/img/04_C/RC_application.png | Bin 0 -> 17566 bytes Physique1A/img/04_C/RC_application_annot.png | Bin 0 -> 30423 bytes Physique1A/img/04_C/RC_series_dis.png | Bin 0 -> 16301 bytes Physique1A/img/04_EX/bob_04.png | Bin 0 -> 18923 bytes Physique1A/img/04_EX/bob_05.png | Bin 0 -> 12972 bytes Physique1A/img/04_EX/bob_06.png | Bin 0 -> 20598 bytes Physique1A/img/04_EX/cap_03.png | Bin 0 -> 14023 bytes Physique1A/img/04_EX/cap_05.png | Bin 0 -> 18018 bytes Physique1A/img/04_EX/cap_06.png | Bin 0 -> 14331 bytes assets/manim/RC_series_voltage_dis.mp4 | Bin 0 -> 193824 bytes 12 files changed, 283 insertions(+), 2 deletions(-) create mode 100644 Physique1A/img/04_C/RC_application.png create mode 100644 Physique1A/img/04_C/RC_application_annot.png create mode 100644 Physique1A/img/04_C/RC_series_dis.png create mode 100644 Physique1A/img/04_EX/bob_04.png create mode 100644 Physique1A/img/04_EX/bob_05.png create mode 100644 Physique1A/img/04_EX/bob_06.png create mode 100644 Physique1A/img/04_EX/cap_03.png create mode 100644 Physique1A/img/04_EX/cap_05.png create mode 100644 Physique1A/img/04_EX/cap_06.png create mode 100644 assets/manim/RC_series_voltage_dis.mp4 diff --git a/Physique1A/Physique_1A_04_C_RT.markdown b/Physique1A/Physique_1A_04_C_RT.markdown index 7ebb771..07e934e 100644 --- a/Physique1A/Physique_1A_04_C_RT.markdown +++ b/Physique1A/Physique_1A_04_C_RT.markdown @@ -171,4 +171,218 @@ Une fois le calcul de la tension dans le condensateur effectué, il est assez si On obtient alors : $i(t) = \dfrac{E}{R} e^{-\frac{t}{RC}}$ - \ No newline at end of file + + +**Remarque importante :** + +L'intensité dans le condensateur est nulle à $t<0$. Il y a donc une discontinuité de l'intensité dans le condensateur au moment ou la charge débute ! + +### C. Décharge du condensateur : + +On considère désormais que le condensateur est chargé, et qu'on éteint le générateur à $t=0$. + +On peut modéliser cette expérience par le circuit ci-dessous : + +![](./img/04_C/RC_series_dis.png) + +- À $t < 0$, on considère que le condensateur est chargé, et que l'interrupteur est ouvert $\Rightarrow u_C = E$ et $i=0$. +- À $t=0$, on ferme l'interrupteur (le condensateur va donc commencer à se décharger) + +On cherche maintenant à déterminer le comportement du circuit pour $t\geq0$ + +**Équations caractéristiques du circuit** + +- Loi des mailles : $(1) : 0 = u_C(t) + u_R(t)$ +- Loi d'Ohm : $(2) : u_R(t) = R.i(t)$ +- Loi du condensateur : $(3) : i(t) = C \dfrac{du_C}{dt}$ + +On va maintenant combiner ces trois équations afin d'obtenir une équation unique qui ne contient que $u_C(t)$. + +**Équation différentielle en $u_C(t)$** + +$(3) : i(t) = C \dfrac{du_C}{dt}$ + +On élimine $i(t)$ en utilisant $(2) : i(t) = \dfrac{u_R(t)}{R}$ + +$\Leftrightarrow \dfrac{u_R(t)}{R} = C \dfrac{du_C}{dt}$ + +On élimine $u_R(t)$ en utilisant $(1) : u_R(t) = - u_C(t)$ + +$\Leftrightarrow \dfrac{- u_C(t)}{R} = C \dfrac{du_C}{dt}$ + +Il ne reste qu'à remettre en forme l'équation : + +$\Leftrightarrow \boxed{\dfrac{du_C}{dt} +\dfrac{u_C(t)}{RC} = 0}$ + +**Résolution de l'équation différentielle** + +1. Solution particulière $u_{C,p}(t)$: + +On cherche la solution en régime permanent. On sait que lorsque la charge du condensateur sera terminée, on aura $\frac{du_C}{dt} = 0$. L'équation devient alors : +$$\dfrac{1}{RC} u_{C,p}(t) = 0$$ +$$\Leftrightarrow \boxed {u_{C,p}(t) = 0}$$ + +2. Solution homogène $u_{C,h}(t)$: + +*Remarque : Ici l'équation sans second membre est donc la même que l'équation de base !* + +On écrit l'équation dite "sans second membre", c'est à dire qu'on écrit l'équation sans les termes qui ne dépendent pas de $u_C(t)$ : +$$\dfrac{du_{C,h}}{dt} + \dfrac{1}{RC} u_{C,h}(t) = 0$$ + +**Théorème :** L'équation $y'(t) + a.y(t) = 0$ a pour solution : $y(t) = \alpha e^{-at}$ où $\alpha$ est une constante à déterminer. + +Donc ici, on peut en déduire que $\boxed{u_{C,h}(t) = \alpha e^{-\frac{t}{RC}}}$ + +3. Conclusion (et détermination de la constante $\alpha$) : + +La solution complète de l'équation différentielle est : + +$$u_C(t) = u_{C,p}(t) + u_{C,h}(t) = 0 + \alpha e^{-\frac{t}{RC}}$$ + +Il faut déterminer $\alpha$ afin de terminer la résolution. Pour ce faire, on doit utiliser une condition intiale qui donne la valeur de $u_C(t)$ à un temps donné. Ici, on sait que $u_C(t=0) = E$. + +Alors, +$$u_C(t=0) = E$$ +$$\Leftrightarrow \alpha e^{-\frac{0}{RC}} = E$$ +$$\Leftrightarrow \alpha = E$$ + +Et on peut donc finalement écrire que : + +$$\boxed{u_{C}(t) = Ee^{-\frac{t}{RC}}}$$ + +**Tracé de la courbe :** + + + + +### D. Exercice d'application : + +On considère le circuit suivant : + +![](./img/04_C/RC_application.png) + +- À $t < 0$, on considère que le condensateur est déchargé, et que l'interrupteur est ouvert. +- À $t=0$, on ferme l'interrupteur. + +1. Que vaut $u_C(t=0)$? +2. Déterminer $u_C(t)$ pour $t>0$. +3. Une fois le régime permanent atteint dans la situation précédente, on rouvre l'interrupteur. On choisit comme nouvelle origine des temps cet instant. Donner la nouvelle expression de $u_C(t)$. + +### SOLUTION : + +Il est de bon usage de commencer par annoter tout le circuit : + +![](./img/04_C/RC_application_annot.png) + +1. Le condensateur étant déchargé à $t=0$, on peut dire que $u_C(t=0)=0$. + +2. On fait tout d'abord l'inventaire des équations caractéristiques du circuit : + +- $(1)$ Loi des mailles : $E - u_{R_1}(t) - u_{R_2}(t) = 0$ +- $(2)$ Loi des mailles : $u_{R_2}(t) = u_C(t)$ +- $(3)$ Loi des noeuds : $i(t) = i_C(t) + i_R(t)$ +- $(4)$ Loi d'Ohm : $u_{R_1}(t) = R_1 . i(t)$ +- $(5)$ Loi d'Ohm : $u_{R_2}(t) = R_2 . i_R(t)$ +- $(6)$ Loi du condensateur : $i_{C}(t) = C \dfrac{du_C}{dt}$ + +$$(6) : i_{C}(t) = C \dfrac{du_C}{dt}$$ + +On remplace $i_C(t)$ en utilisant $(3) : i_C(t) = i(t) - i_R(t)$ + +$$\Leftrightarrow i(t) - i_R(t) = C \dfrac{du_C}{dt}$$ + +On remplace $i(t)$ et $i_R(t)$ en utilisant $(4, 5) : i(t) = \dfrac{u_{R_1}(t)}{R_1}, i_R(t)= \dfrac{u_{R_2}(t)}{R_2}$ + +$$\Leftrightarrow \dfrac{u_{R_1}(t)}{R_1} - \dfrac{u_{R_2}(t)}{R_2} = C \dfrac{du_C}{dt}$$ + +On remplace $u_{R_1}(t)$ et $u_{R_2}(t)$ en utilisant $(1, 2) : u_{R_2}(t) = u_{C}(t), u_{R_1}(t)= E - u_{R_2}(t) = E - u_{C}(t)$ + +$$\Leftrightarrow \dfrac{E - u_{C}(t)}{R_1} - \dfrac{u_{C}(t)}{R_2} = C \dfrac{du_C}{dt}$$ + +Il ne reste plus qu'à remettre l'équation en forme : + +$$\Leftrightarrow C \dfrac{du_C}{dt} = \dfrac{E}{R_1} - \dfrac{u_{C}(t)}{R_1} - \dfrac{u_{C}(t)}{R_2} $$ + +$$\Leftrightarrow C \dfrac{du_C}{dt} + \dfrac{u_{C}(t)}{R_1} + \dfrac{u_{C}(t)}{R_2}= \dfrac{E}{R_1}$$ + +$$\Leftrightarrow C \dfrac{du_C}{dt} + R_2\dfrac{u_{C}(t)}{R_1R_2} + R_1\dfrac{u_{C}(t)}{R_1R_2}= \dfrac{E}{R_1}$$ + +$$\Leftrightarrow C \dfrac{du_C}{dt} + \dfrac{R_1+R_2}{R_1R_2}u_C(t)= \dfrac{E}{R_1}$$ + +$$\Leftrightarrow \boxed{\dfrac{du_C}{dt} + \dfrac{R_1+R_2}{R_1R_2C}u_C(t)= \dfrac{E}{R_1C}}$$ + +On peut alors résoudre l'équation différentielle : + +- Solution particulière : $\dfrac{du_C}{dt} = 0$ + +$$\Leftrightarrow \dfrac{R_1+R_2}{R_1R_2C}u_{C,p}(t)= \dfrac{E}{R_1C}$$ + +$$\Leftrightarrow u_{C,p}(t)= \dfrac{E}{\cancel{R_1C}}.\dfrac{\cancel{R_1}R_2\cancel{C}}{R_1+R_2}$$ + +$$\Leftrightarrow \boxed{u_{C,p}(t)= \dfrac{ER_2}{R_1+R_2}}$$ + +- Solution homogène : + +$$\dfrac{du_{C,h}}{dt} + \dfrac{R_1+R_2}{R_1R_2C}u_{C,h}(t)= 0$$ + +D'après le théorème, on a alors : $\boxed{u_{C,h}(t) = \alpha \exp\left(\dfrac{-(R_1+R_2)t}{R_1R_2C}\right)}$ + +- Conclusion : + +$$u_C(t) = u_{C,p}(t) + u_{C,h}(t) = \dfrac{ER_2}{R_1+R_2} + \alpha \exp\left(\dfrac{-(R_1+R_2)t}{R_1R_2C}\right)$$ + +On détermine $\alpha$ grâce à la condition initiale $u_C(t=0)=0$ : + +$$u_C(t=0) = 0$$ + +$$\Leftrightarrow \dfrac{ER_2}{R_1+R_2} + \alpha \exp\left(\dfrac{-(R_1+R_2).0}{R_1R_2C}\right) = 0$$ + +$$\Leftrightarrow \dfrac{ER_2}{R_1+R_2} + \alpha .1 = 0$$ + +$$\Leftrightarrow \alpha = - \dfrac{ER_2}{R_1+R_2}$$ + +Et on en déduit finalement : + +$$\boxed{u_C(t) = \dfrac{ER_2}{R_1+R_2}\left(1 - \exp\left(\dfrac{-(R_1+R_2).t}{R_1R_2C}\right)\right)}$$ + +Dans ces cas là, il peut être utile de poser $\tau = \dfrac{R_1R_2C}{R_1+R_2}$ afin de simplifier l'écriture de l'équation : + +$$\boxed{u_C(t) = \dfrac{ER_2}{R_1+R_2}\left(1 - \exp\left(\dfrac{-t}{\tau}\right)\right)}$$ + +3. Lorsqu'on rouvre l'interrupteur : +- Le circuit redevient un simple circuit RC série +- La condition initiale devient le régime permanent de l'étude précédente, donc $u_C(t=0) = \frac{ER_2}{R_1+R_2}$ + +L'équation différentielle est alors la même que celle du cours : + +$\boxed{\dfrac{du_C}{dt} +\dfrac{u_C(t)}{R_1C} = \dfrac{E}{R_1C}}$ + +**Résolution :** + +- Solution particulière : + +$$\dfrac{1}{R_1C} u_{C,p}(t) = \dfrac{E}{R_1C}$$ +$$\Leftrightarrow \boxed {u_{C,p}(t) = E}$$ + +- Solution homogène : + +$$\dfrac{du_{C,h}}{dt} + \dfrac{1}{R_1C} u_{C,h}(t) = 0$$ + +$$\Leftrightarrow \boxed{u_{C,h}(t) = \alpha e^{-\frac{t}{R_1C}}}$$ + +- Conclusion (et détermination de la constante $\alpha$) : + +La solution complète de l'équation différentielle est : + +$$u_C(t) = u_{C,p}(t) + u_{C,h}(t) = E + \alpha e^{-\frac{t}{R_1C}}$$ + +Or, +$$u_C(t=0) = \dfrac{ER_2}{R_1+R_2} $$ +$$\Leftrightarrow E + \alpha e^{-\frac{0}{R_1C}} = \dfrac{ER_2}{R_1+R_2}$$ +$$\Leftrightarrow \alpha = \dfrac{ER_2}{R_1+R_2} - E$$ +$$\Leftrightarrow \alpha = \dfrac{ER_2}{R_1+R_2} - E\dfrac{R_1 + R_2}{R_1+R_2}$$ +$$\Leftrightarrow \alpha = -\dfrac{ER_1}{R_1+R_2}$$ + +Et on peut donc finalement écrire que : + +$$\boxed{u_{C}(t) = E -\dfrac{ER_1}{R_1+R_2}e^{-\frac{t}{R_1C}}}$$ \ No newline at end of file diff --git a/Physique1A/Physique_1A_04_EX.markdown b/Physique1A/Physique_1A_04_EX.markdown index 171c353..9079a67 100644 --- a/Physique1A/Physique_1A_04_EX.markdown +++ b/Physique1A/Physique_1A_04_EX.markdown @@ -8,4 +8,71 @@ id: Phy1_04_EX # Chapitre 4 : Régimes Transitoires - EXERCICES -![](./img/03_EX/Ex8_IMDEA.png) \ No newline at end of file +## Exercice 1 : + +![](./img/04_EX/cap_03.png) + +On considère le montage ci-dessus. Initialement, le condensateur est déchargé, le générateur est éteint, et l'interrupteur est ouvert. + +1. L'interrupteur reste ouvert, et on allume le générateur. Déterminer $u(t)$, et préciser la constante de temps de ce circuit. +2. On suppose que le régime permanent de la question précédente est atteint depuis longtemps, et on ferme l'interrupteur. On choisit cet instant comme nouvelle origine des temps. Donner l'équation différentielle alors vérifiée par $u(t)$, et la résoudre. +3. Tracer la courbe de l'évolution de $u_C(t)$ sur les deux phases. +4. Effectuer un bilan de puissance sur les deux phases. + +## Exercice 2 : + +![](./img/04_EX/cap_05.png) + +On considère le montage ci-dessus. Initialement, les deux condensateurs sont déchargés, et les interrupteurs $K_1$ et $K_2$ sont ouverts. + +1. L'interrupteur $K_2$ restant ouvert, on ferme $K_1$ pendant un temps assez long pour les tensions sur les condensateurs $C_1$ et $C_2$ deviennent constantes en fonction du temps. Déterminer les valeurs de ces tensions lorsqu'elles sont devenues constantes. +2. On ferme l'interrupteur $K_2$. Déterminer les nouvelles valeurs de la tension dans chacun des condensateurs lorsqu'elles sont devenues constantes. Quel est le temps nécessaire pour que cela se produise ? + +## Exercice 3 : + +On relie à une résistance $R$ deux condensateurs de capacité $C_1$ et $C_2$ initialement chargés. À $t=0$, on ferme l'interrupteur. Pour $t<0$, on a $u_1(t<0) = U_{01}$ et $u_2(t<0) = U_{02} E/8R$. 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