Update Dim. 12 octobre 2025

English/Index.md

@@ -15,6 +15,7 @@ Dream car: Ford Mustang (from the 60s)
 
 
 # IELTS
+Session: 24-25 April
 ## Writing Task 1 - Visual Information 
 This bar chart shows the percentage of the population living in urban areas, worldwide and on specific continents, from 1950 to 2030.
 
@@ -55,3 +56,38 @@ However, I would argue that applying a restriction to international travel would
 
 Overall, I would say that denying the impact of transportation would be disingenuous. I do however wholeheartedly believe that restricting international travel would not yield the results people might expect.
 
+
+# Speaking
+## **Internet**
+
+How often do you go online?
+
+What do you use the internet for?
+
+How do you get online?
+
+Do you have your own computer?
+
+## **Hobbies**
+
+Do you have a hobby?
+
+What equipment do you need for it?
+
+Do you think hobbies should be shared with other people?
+
+Did you have a hobby as a child?
+
+## **Music**
+
+Do you like music?
+
+What’s your favourite type of music?
+
+If you could learn a musical instrument, what would it be?
+
+Do you think music is important?
+
+
+Notes
+![[Pasted image 20251007151529.png]]

EnjeuxSocioEco/TD2.md

@@ -0,0 +1,83 @@
+![[2025-10 ESTIA TD Energie Climat ING1 A.pdf#page=2]]
+
+### 1) Les deux leviers pour baisser les émissions de l’UE
+
+- **Réduire les émissions**
+- **Augmenter les puis de carbone**
+
+---
+
+### 2) L’UE est-elle sur la bonne trajectoire ?
+
+Données du sujet : 1990 = **3,87 GtCO₂e** ; 2018 = **3,05 GtCO₂e**. Objectif 2030 : **–55 %** vs 1990, soit un niveau cible de **1,74 GtCO₂e**. Entre 1990 et 2018, la baisse observée est de **0,82 Gt** (≈ –21 %), bien en-deçà des –55 % requis pour 2030 ; il faut donc **accélérer fortement** la baisse. Conclusion : **non, la trajectoire 1990-2018 n’est pas suffisante pour atteindre –55 % en 2030.**
+
+---
+
+### 3) À rythme inchangé, quand atteindrait-on –55 % ?
+
+Baisse moyenne 1990→2018 : $((3{,}87-3{,}05),\text{Gt}/28\ \text{ans} \approx 0{,}0293,\text{Gt/an})$.  
+À partir de 2018, années nécessaires pour passer de 3,05 à 1,74 Gt : $((3{,}05-1{,}7415)/0{,}0293 \approx 44{,}7\ \text{ans})$.  
+**Échéance ≈ 2018 + 45 ≃ 2063.**
+
+---
+
+### 4) Effet de serre : texte complété
+
+L’effet de serre est un mécanisme **naturel** qui se produit en plusieurs étapes. Le soleil envoie de **l’énergie** sous forme de **rayonnements** à la Terre. Une fois réchauffée, la Terre rejette la chaleur vers **l’espace**. La chaleur est retenue par des gaz présents dans l’atmosphère, les **gaz à effet de serre**. On les appelle ainsi, car comme dans une serre de jardin, ils **emprisonnent** l’énergie et la renvoient vers la Terre. Sans cela, la température sur Terre serait de **–18 °C** au lieu de **15 °C**.
+
+---
+
+### 5) Conséquence de l’augmentation des GES
+
+**Renforcement de l’effet de serre** → **réchauffement climatique** global, avec effets associés : hausse du niveau de la mer, événements extrêmes plus fréquents/intenses, perturbations des écosystèmes et de l’agriculture, etc. ([IPCC](https://www.ipcc.ch/report/ar6/syr/downloads/report/IPCC_AR6_SYR_FullVolume.pdf?utm_source=chatgpt.com "CLIMATE CHANGE 2023"))
+
+---
+
+### 6) Pourquoi cibler surtout le CO₂ ? Quels autres leviers ?
+
+- Le **CO₂** est **le principal** contributeur anthropique au forçage, issu surtout de la combustion d’énergies fossiles ; il a une **longue durée de vie** et s’accumule (enjeu cumulatif). D’où la priorité à sa réduction. ([IPCC](https://www.ipcc.ch/report/ar6/wg3/chapter/chapter-2/?utm_source=chatgpt.com "Chapter 2: Emissions trends and drivers"))
+    
+- **Autres leviers** complémentaires : réduire **CH₄** (fuites de gaz, déchets, élevage), **N₂O** (engrais), **gaz fluorés**, **protéger/augmenter les puits** (forêts, sols), **CCS/CCUS**, et **sobriété/efficacité** transversales. ([IPCC](https://www.ipcc.ch/srccl/?utm_source=chatgpt.com "Special Report on Climate Change and Land"))
+    
+
+---
+
+### 7) D’après le document GIEC « Émissions de CO₂ et activités humaines depuis 1850 »
+
+**a.** Les émissions liées aux activités humaines **ont fortement augmenté** depuis 1850, passant de niveaux très faibles au XIXᵉ siècle à des niveaux records au XXIᵉ siècle (pics autour de ~2017-2019). ([Our World in Data](https://ourworldindata.org/co2-emissions?utm_source=chatgpt.com "CO₂ emissions"))  
+**b.** En **2017**, la part des **énergies fossiles + ciment** dans les émissions de CO₂ était d’environ **~89 %**, le reste provenant des changements d’usage des terres (~11 %). ([Carbon Brief](https://www.carbonbrief.org/analysis-global-co2-emissions-set-to-rise-2-percent-in-2017-following-three-year-plateau/?utm_source=chatgpt.com "Analysis: Global CO2 emissions set to rise 2% in 2017 after ..."))  
+**c.** L’usage des énergies fossiles **émet du CO₂** lors de leur combustion ; plus on consomme de charbon/pétrole/gaz, plus les émissions augmentent. **Stratégies** :
+
+- **Décarboner l’offre** (sortie des fossiles, renouvelables, nucléaire, CCUS) ;
+    
+- **Électrifier** les usages finaux et **améliorer l’efficacité** ;
+    
+- **Réduire la demande** (sobriété) ;
+    
+- **Limiter les émissions liées aux terres** (lutte contre la déforestation, gestion durable des sols). ([IPCC](https://www.ipcc.ch/report/ar6/wg3/chapter/chapter-6/?utm_source=chatgpt.com "Chapter 6: Energy systems"))
+    
+
+---
+
+## Sources
+
+- ESTIA / 2025-2026 / JB Jarin – **TD « Climat et GES » (énoncé et données)**. 2025.
+    
+- **IPCC AR6, WGIII – Chapter 2: Emissions trends and drivers** (version web). ([IPCC](https://www.ipcc.ch/report/ar6/wg3/chapter/chapter-2/?utm_source=chatgpt.com "Chapter 2: Emissions trends and drivers"))
+    
+- **IPCC AR6, WGIII – Chapter 2 (PDF)**. ([IPCC](https://www.ipcc.ch/report/ar6/wg3/downloads/report/IPCC_AR6_WGIII_Chapter02.pdf?utm_source=chatgpt.com "Emissions Trends and Drivers"))
+    
+- **IPCC AR6, WGIII – Chapter 6: Energy systems**. ([IPCC](https://www.ipcc.ch/report/ar6/wg3/chapter/chapter-6/?utm_source=chatgpt.com "Chapter 6: Energy systems"))
+    
+- **IPCC AR6 Synthesis Report (2023), Full Volume (PDF)**. ([IPCC](https://www.ipcc.ch/report/ar6/syr/downloads/report/IPCC_AR6_SYR_FullVolume.pdf?utm_source=chatgpt.com "CLIMATE CHANGE 2023"))
+    
+- **Our World in Data – CO₂ emissions** (séries longues, niveau global). ([Our World in Data](https://ourworldindata.org/co2-emissions?utm_source=chatgpt.com "CO₂ emissions"))
+    
+- **CarbonBrief (2017)** – _Global CO₂ emissions 2017: land-use changes ≈ 11 %_ (donc fossiles+ciment ≈ 89 %). ([Carbon Brief](https://www.carbonbrief.org/analysis-global-co2-emissions-set-to-rise-2-percent-in-2017-following-three-year-plateau/?utm_source=chatgpt.com "Analysis: Global CO2 emissions set to rise 2% in 2017 after ..."))
+    
+- **US EPA – Climate Change Indicators / GHG overview** (rappels sur sources et GES). ([Environmental Protection Agency](https://www.epa.gov/ghgemissions/global-greenhouse-gas-overview?utm_source=chatgpt.com "Global Greenhouse Gas Overview | US EPA"))
+    
+
+---
+
+Si tu veux, je peux te fournir la feuille de calcul détaillant les calculs des questions 2 et 3.

EnjeuxSocioEco/TD3.md

@@ -0,0 +1,86 @@
+![[2025-10 ESTIA TD Energie Climat ING1 A.pdf#page=4]]
+
+### 1) Puissance totale – Fruges vs. Gravelines
+
+- **Parc(s) éolien(s) de Fruges** : 70 éoliennes. La fiche technique de l’éolienne du parc indique **2,3 MW** nominaux →  
+    $(P_\text{Fruges} = 70 \times 2{,}3 = \mathbf{161\ MW})$.
+
+- **CNPE de Gravelines** : 6 réacteurs de **900 MW** chacun →  
+    $(P_\text{Gravelines} = 6 \times 900 = \mathbf{5,400\ MW})$.
+
+
+---
+
+### 2) Nombre d’éoliennes pour égaler la puissance de Gravelines
+
+- Avec des machines **2 MW** : $(5,400 / 2 = \mathbf{2,700})$ éoliennes.
+    
+- Avec des machines **15 MW** : $(5,400 / 15 = \mathbf{360})$ éoliennes.
+    
+
+---
+
+### 3) Nombre d’éoliennes pour égaler la **production annuelle** (CF 85 % vs 45 %)
+
+Hypothèses demandées : **nucléaire CF = 85 %**, **éolien offshore CF = 45 %**.  
+On égalise l’énergie annuelle :   
+$N = \frac{P_\text{nuke}\times CF_\text{nuke}}{P_\text{turbine}\times CF_\text{wind}}$  
+
+- **Turbine 2 MW** : $(N = \frac{5,400 \times 0{,}85}{2 \times 0{,}45} \approx \mathbf{5,100})$ éoliennes.
+    
+- **Turbine 15 MW** : $(N = \frac{5,400 \times 0{,}85}{15 \times 0{,}45} \approx \mathbf{680})$ éoliennes.
+    
+
+> Vérif énergie annuelle de Gravelines (ordre de grandeur) : (5{,}4\ \text{GW} \times 8,760\ \text{h} \times 0{,}85 \approx \mathbf{40,2\ TWh/an}). (cohérent avec les historiques publics) ([Wikipedia](https://en.wikipedia.org/wiki/Gravelines_Nuclear_Power_Station?utm_source=chatgpt.com "Gravelines Nuclear Power Station"))
+
+---
+
+### 4) Limites et proposition pour un remplacement 100 % éolien
+
+- **Limites** de l’équivalence par facteur de charge :
+    
+    - Le **CF éolien est subi** (dépend du vent, variable, corrélé régionalement), donc la **puissance garantie** et l’**adéquation instantanée** ne sont pas assurées.
+        
+    - Besoin de **réserves**, **stockage**, **flexibilité de la demande**, **renforcement réseau** et **pilotage** pour passer les creux de production.
+        
+    - Le **CF nucléaire est choisi** au sens où l’outil est **pilotable** : on peut décider quand produire (modulation/arrêts planifiés), indépendamment de la météo.
+        
+- **Feuille de route 100 % éolien** (idée de mix et d’infrastructures) :
+    
+    - **Surdimensionnement** de la capacité éolienne (overbuild) par rapport à la moyenne nécessaire.
+        
+    - **Stockages** multi-échelles : batteries (intra-jour), **STEP** et **hydrogène**/PtX (inter-journalier à saisonnier).
+        
+    - **Flexibilité** (effacement industriel, pilotage conso, tarification dynamique), **réseaux** (renforcement, interconnexions), et **backup** pilotable bas-carbone transitoire si nécessaire.
+        
+    - **Diversification géographique** et **technologique** (onshore + offshore, couplage avec solaire) pour lisser la variabilité.
+        
+ 
+---
+
+### 5) Comparaison d’empreinte CO₂ annuelle (à **100 % de CF** pour simplifier)
+
+À puissance installée identique (**5,4 GW**), l’énergie annuelle serait (5{,}4 \times 8,760 = \mathbf{47{,}3\ TWh}). Les émissions annuelles dépendent alors des **facteurs ACV** (gCO₂e/kWh) :
+
+- **Nucléaire (France/Europe, ACV)** : ~**5,5–6 gCO₂e/kWh** (médiane UNECE ≈ 5,5 ; base ADEME ≈ 6). → **≈ 0,26–0,28 MtCO₂e/an**.
+    
+- **Éolien onshore** : ~**11–16 gCO₂e/kWh** (IPCC/UNECE ; ADEME ~14,1). → **≈ 0,52–0,67 MtCO₂e/an**.
+    
+- **Éolien offshore** : ~**12–23 gCO₂e/kWh** (UNECE ; ADEME ~15,6). → **≈ 0,57–1,09 MtCO₂e/an**.
+    
+
+**Conclusion** : sur une base ACV, **nucléaire et éolien sont tous deux très bas-carbone**, l’ordre de grandeur annuel reste **< 1,1 MtCO₂e** pour 5,4 GW. **Le nucléaire ressort généralement un peu plus bas** que l’éolien (surtout offshore haut de fourchette), mais **les fourchettes se recoupent** selon hypothèses (mix amont, logistique, durée de vie, facteurs régionaux). Le différentiel CO₂ n’est **pas l’argument principal** pour trancher entre ces deux options ; la décision se joue plutôt sur **pilotabilité, coûts système, réseaux, acceptabilité, calendrier et risques**.
+
+---
+
+## Sources
+
+- ESTIA / 2025-2026 / JB Jarin — **TD 3 : Mix énergétique, pilotable & intermittent (nucléaire / éolien)** — Énoncé et données.
+    
+- **UNECE (2021)** — _Life Cycle Assessment of Electricity Generation Options_ (PDF) — valeurs ACV : nucléaire ≈ 5,5 gCO₂e/kWh ; éolien onshore < 16 ; offshore < 23.
+    
+- **ADEME (Base Carbone / Bilans-GES)** — facteurs ACV pour électricité nucléaire en France ≈ **6 gCO₂e/kWh**. ([sfen.org](https://www.sfen.org/rgn/les-emissions-carbone-du-nucleaire-francais-37g-de-co2-le-kwh/?utm_source=chatgpt.com "Les émissions carbone du nucléaire français : 4g de CO2 ..."))
+    
+- **Les Énergies Renouvelables (2025)** — synthèse ADEME : éolien **onshore ≈ 14,1** gCO₂e/kWh ; **offshore ≈ 15,6** gCO₂e/kWh. ([ECOinfos](https://www.les-energies-renouvelables.eu/conseils/bilan-carbone/bilan-carbone-emission-co2-source-energie-renouvelable/?utm_source=chatgpt.com "Emission de CO2 par source d'énergie renouvelable"))
+    
+- **Wikipedia / EDF / presse** — données de cadrage public sur **Gravelines** (6×900 MW ; puissance ~5,46 GW ; prod. annuelle typique). ([Wikipedia](https://en.wikipedia.org/wiki/Gravelines_Nuclear_Power_Station?utm_source=chatgpt.com "Gravelines Nuclear Power Station"))

Español/Index.md

@@ -1,3 +1,4 @@
+# Martes
 ![[Pasted image 20250923163517.png]]
 1. debajo de la meza
 2. sobre la meza (o encima de la meza)
@@ -58,7 +59,7 @@ significado:
 4. Boca de metro
 
 ---
-Martes, 30 de Septiembre 2025.
+# Martes, 30 de Septiembre 2025.
 
 Llamar la attencion:
 - Oye
@@ -108,3 +109,110 @@ Conjuga el verbo ir:
 11. ¿Vamonos nosotros al restaurante?
 12. Mis hijos van a cordoba
 13. Carlos va de vacaciones
+
+---
+# Martes, 7 de octubre 2025
+
+Correction du mail de Juanjo:
+```
+Hola Inés,  
+  
+Para llegar a Ponme otra, coges la linea 4 del metro hasta la parada "parlamento". Despues, vas todo recto al norte y coges la segunda a la derecha. Sigues la callercalle hasta la Plaza San Isabel y el bar se situa  a la en la izquierda.  
+  
+¡Hasta luego!
+```
+
+Quand il y a 2 verbes dans une phrase, un à l'infinitif et un conjugué, on met le pronom avant ou après
+```mermaid
+flowchart TD
+1["Voy a te ayudar"]
+2["Te voy a ayudar"]
+3["Voy a ayudarte"]
+1 --> 2 & 3
+```
+
+Toujours mettre **de** après después et antes.
+Después/Antes **DE** la salida
+
+Le mot **para** s'utilise pour parler d'une destination (destinatario) ou d'un but (finalidad)
+
+Le mot **por** s'utilise pour parler d'une cause (causa).
+
+![[Pasted image 20251007161106.png]]
+
+Este verano vamos ***a*** ir ***de*** vacaciones ***a*** las islas Canarias.
+Queremos ir ***en*** avión y después hacer excursiones ***en*** coche. 
+Nos vamos a quedar ***en*** un pequeño hotel ***en*** el este de la isla. 
+El hotel está cerca de una playa, y hay muchas otras playas ***a*** pocos kilómetros de distancia.
+
+![[Pasted image 20251007162909.png]]
+
+Carlos va a jugar al golf.
+El niño va a ver la tele.
+Carmen y Maria van a cenar al restaurante.
+Los padres van a pintar la casa.
+La abuela va a acostarse / La abuela se va a acostar.
+Miguel va a prepararse para salir / Miguel se va a preparar para salir.
+
+
+Verbos irregulares
+
+|          | Preferir   | Querer   |
+| -------- | ---------- | -------- |
+| Yo       | prefiero   | quiero   |
+| Tu       | prefieres  | quieres  |
+| El       | prefiere   | quiere   |
+| Nosotros | preferimos | queremos |
+| Vosotros | preferis   | quereis  |
+| Ellos    | prefieren  | quieren  |
+
+Ejercicio
+
+1. Las tiendas (cerrar) **cierran** a las 8.
+
+2. El partido de tenis (empezar) **empieza** a las 1:15
+
+3. ¿(encender, tú) **enciendes** la luz, por favor?
+
+4. (yo, preferir) **Prefiero** una cerveza.
+
+5. ¿En qué (pensar, tú) **piensas** ?
+
+6. ¿(querer, tú) **Quieres** un café?
+
+7. Carlos siempre (perder) **pierde** el autobús.
+
+8. ¿(tú, entender) **Entiendes** lo que dice el profesor?
+
+9. El director (querer) **quiere** este informe para finales de mes.
+
+10. Juan (despertar) **despierta** siempre a los niños.
+
+11. La clase (comenzar) **comienza** a las 10:30.
+
+12. (yo, sentir) **Siento** no poder ir a la fiesta.
+
+13. Ellos (sentarse) **se sientan** siempre ahí.
+
+14. (nosotros, querer) **Queremos** vender nuestro piso.
+
+15. Hoy (yo, sentirse) **me siento** muy bien.
+
+16. ¿Cuándo (empezar, tú) **empiezas** con las clases de inglés?
+
+
+Audio
+
+Escucha a dos personas y marca la información que se refiere a Marta, a Antonio o a los dos.
+
+|                                                 | Antonio | Marta |
+| ----------------------------------------------- | ------- | ----- |
+| 1. Tiene poco dinero.                        |         | x     |
+| 2. Quiere ir a una ciudad.                   | x       |       |
+| 3. Busca relax.<br>                          |         | x     |
+| 4. Quiere viajar en barco.<br>               | x       | x     |
+| 5. Quiere un viaje cultural.<br>             | x       |       |
+| 6. Tiene muchos días de vacaciones.<br>      |         | x     |
+| 7. Quiere conocer una cultura diferente.<br> | x       | x     |
+| 8. Busca naturaleza.<br>                     | x       | x     |
+| 9. Tiene pocos días de vacaciones.           | x       |       |

GénieÉlectrique/Exam2022/Exam2022.md

@@ -0,0 +1,48 @@
+![[Exam2022.pdf]]
+# Exercice 1: Générateur équivalent Norton Thevenin
+
+a) 
+![[Drawing 2025-10-09 15.10.26.excalidraw|1000]]
+
+Convertir les deux Thevenin de gauche en Norton:
+![[Drawing 2025-10-09 15.10.26.excalidraw 1|1000]]
+$i_1=\frac{E_{Th_1}}{R_{Th_1}}=\frac{10V}{2K\Omega}=0.005A=5mA$ 
+$i_2=\frac{E_{Th_2}}{R_{Th_2}}=\frac{6V}{3K\Omega}=0.002A=2mA$
+
+J'ai deux Norton en parallèles. Je vais simplifier.
+![[Drawing 2025-10-09 15.10.26.excalidraw 1 1]|1000]
+$\frac{1}{R_{eq}}=\frac{1}{2K\Omega}+\frac{1}{3K\Omega}=\frac{5}{6000\Omega}\implies R_{eq}=\frac{6000}{5}\Omega=1.2K\Omega$  
+
+Maintenant j'ai deux générateurs de Norton en série, je vais les convertir en thévenin.
+![[Drawing 2025-10-09 15.10.26.excalidraw 1 1 1|1000]]
+
+Puis faire l'équivalent.
+![[Drawing 2025-10-09 15.10.26.excalidraw 1 1 1 1|1000]]
+
+b)
+$i_{target}=\frac{E_{eq}}{R_{eq}}=\frac{9}{2000}=0.0045A=4.5mA$ 
+
+# Exercice 2:
+![[Pasted image 20251009162100.png]]
+a)
+![[Drawing 2025-10-09 16.35.15.excalidraw]]
+Loi des mailles
+$E=U_R+U_L=R_i+L\frac{di}{dt}$; $\frac{E}{R}=i+\frac{L}{R}\frac{di}{dt}$ donc $\tau=\frac{L}{R}$
+
+b)
+$i=\frac{E}{R}$
+
+c)
+
+# Exercice 3:
+![[IMG_7441.jpeg]]
+
+
+# Exercice 4:
+
+![[Pasted image 20251009171649.png]]
+$H=\frac{V_0}{V_S}$
+$V_0=\frac{Z_L}{Z_L+Z_R}V_S$ 
+
+b)
+j

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 15.10.26.excalidraw 1 1 1 1.md

@@ -0,0 +1,96 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+200 Ohms ^LPhWKBMv
+
+A ^jNY6AaqW
+
+B ^8kJhcOlg
+
+1800 Ohms ^Vg8ka0Jn
+
+9V ^OP48Ok2s
+
+%%
+## Drawing
+```compressed-json
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+
+0sxuAFYADgT+gGZ9gBYj+KSeeOr+o/5SmG5nbaPtAHYXi6Pqzb7+gcT+26QCgkdRbF7aTabC48T48Y77F6bQFSBCEZTSbjxbb7BKJd79F77RL7HjbeI8JGFSDWFbiVDVZHMKCkNgAawQAGE2Pg2KQKgBieIIIVCtaQTS4bCs5QsoSNLk8vkSZnWZhwXCBbJiiCDQj4fAAZVgqwkgg82qZLPZAHUQZJuHwqRBLWyEEaYCb0GbysjZeiOOFcmh4si2
+
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+```
+%%

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 15.10.26.excalidraw 1 1 1.md

@@ -0,0 +1,128 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+200 Ohms ^LPhWKBMv
+
+A ^jNY6AaqW
+
+B ^8kJhcOlg
+
+600 Ohms ^Vg8ka0Jn
+
+1.2 KOhms ^lIZhsuC0
+
+0.6V ^n9fPoOFh
+
+8.4V ^OP48Ok2s
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 15.10.26.excalidraw 1 1.md

@@ -0,0 +1,138 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+200 Ohms ^LPhWKBMv
+
+A ^jNY6AaqW
+
+B ^8kJhcOlg
+
+600 Ohms ^Vg8ka0Jn
+
+1mA ^b1zNOGHD
+
+ieq ^TTulWADF
+
+Req ^lIZhsuC0
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 15.10.26.excalidraw 1.md

@@ -0,0 +1,148 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+200 Ohms ^LPhWKBMv
+
+A ^jNY6AaqW
+
+B ^8kJhcOlg
+
+600 Ohms ^Vg8ka0Jn
+
+3 KOhms ^SKn2gV3d
+
+2 KOhms ^r1BkRYAm
+
+1mA ^b1zNOGHD
+
+i1 ^V7UxoibQ
+
+i2 ^TTulWADF
+
+R1 ^sQbpZJvu
+
+R2 ^lf1Zf0m6
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 15.10.26.excalidraw.md

@@ -0,0 +1,128 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+200 Ohms ^LPhWKBMv
+
+A ^jNY6AaqW
+
+B ^8kJhcOlg
+
+600 Ohms ^Vg8ka0Jn
+
+3 KOhms ^SKn2gV3d
+
+2 KOhms ^r1BkRYAm
+
+10V ^RumA6HTq
+
+6V ^owVBcwYX
+
+1mA ^b1zNOGHD
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/Exam2022/attachments/Drawing 2025-10-09 16.35.15.excalidraw.md

@@ -0,0 +1,84 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+UL ^dcNekxvT
+
+UR ^rRoNBtHQ
+
+E ^bumf53e2
+
+i ^bENrNZkM
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

GénieÉlectrique/Index.md

@@ -60,7 +60,7 @@ a((" ")) --> R1 --> b((" "))
 a((" ")) --> R2 --> b((" "))
 b --> bb((" "))
 ```
-En parallèle, c'est la somme des inverses ($R_{parallèle}=\sum{\frac{1}{R_i}}$)
+En parallèle, c'est la somme des inverses ($\frac{1}{R_{parallèle}}=\sum{\frac{1}{R_i}}$)
 
 2 Resistances identiques en parallèle ont pour somme la moitié.
 Ex:
@@ -107,5 +107,22 @@ N'importe quel circuit peut être résumé en un générateur de courant avec un
 
 $R_{Th}=R_{N}$
 
-Théorème de Millman
+Théorème de Millman $\rightarrow$ $\frac{\sum_{i\in{[0,n]}}\frac{V_i}{R_i}}{\sum_{i\in{[0,n]}}\frac{1}{R_i}}$
+
 -> Le théorème bulldozer
+
+
+# Chapitre 2. Analyse temporelle.
+![[Pasted image 20251006090339.png]]
+
+# Chapitre 3. Analyse fréquentielle.
+
+![[Pasted image 20251006093323.png]]![[Pasted image 20251006093350.png]]
+
+![[Pasted image 20251006093856.png]]
+![[Pasted image 20251006093952.png]]
+![[Pasted image 20251006094004.png]]
+
+Diagramme de Bode -> Représente le signal sous forme phase et amplitude.
+
+![[Pasted image 20251009085622.png]]

GénieÉlectrique/TD2/Index.md

@@ -1 +1,63 @@
-![[TD_FGE_2025.pdf]]
+![[TD_FGE_2025.pdf#page=7]]
+# Exercise 1. Equivalent Generator
+## a. ![[1.A.]]
+- On éteint les générateurs: ![[1.A.Eteint]]
+$R_{th}=R_{eq}=\frac{R_1R_2}{R_1+R_2}$
+
+- On fait un circuit ouvert après AB: ![[1.A.Ouvert]]
+$U_{AB}=\frac{E\times R_1}{R_1+R_2}=E_{th}$
+
+## b. ![[1.B.]]
+- On éteint les générateurs ![[1.B.Eteint]]
+$R_n=R_{eq}=R_1+R_2$
+
+- On fait un court-circuit en AB: ![[1.B.Ouvert]]
+$I_n=\frac{R_1}{R_1+R_2}I$
+
+## c. ![[1.C.]]
+- On éteint les générateurs: ![[1.C.Eteint]]
+$R_{eq}=R_{th}=\frac{5r}{2}$
+- On fait un circuit ouvert après AB: ![[1.C.Ouvert]]
+$U_{AB}=E_{th}=$
+D'après la loi des mailles:
+$U_{AB}=U_{eq}-e$
+Que vaut $U_{eq}$ ? ![[1.C.Ouvert.Ueq]]
+Loi des mailles:
+$ri-ri+e-2e=0$
+
+Superposition:
+En éteignant le générateur de $2e$, on obtient $U_1=\frac{e}{2}$
+En éteignant le générateur de $e$, on obtient $U_2=2e-\frac{2e}{2}=e$ 
+$U_{eq}=U_1+U_2=e+\frac{e}{2}=\frac{3e}{2}$ 
+
+## g. ![[1.G.]]
+Générateur dépendant -> Problème
+Calcul de $I_n$: Courant lorsque court-circuit: 
+![[1.G.CourtCircuit]]
+$I_n+I_1=I$
+$I_1=\frac{V}{R_1}$
+$I_n=I-\frac{V}{R_1}$
+
+$0=V+\lambda V-R_2I_n$ 
+$R_2I_n=V(1+\lambda)$
+$V=\frac{I_nR_2}{1+\lambda}\implies I_n=I-\frac{I_nR_2}{R_1(1+\lambda)}$
+$\implies \boxed{I_n=I\times\frac{R_1(1+\lambda)}{R_2+R_1(1+\lambda)}}$
+
+![[02a58d4198.pdf]]
+
+# Exercise 2: Power analysis
+![[Pasted image 20251003135122.png]]
+![[Drawing 2025-10-03 13.51.44.excalidraw]]
+
+# Exercise 3: Millman Theorem
+![[Pasted image 20251003135519.png]]
+![[Drawing 2025-10-03 13.56.01.excalidraw]]
+Pour trouver $R_{th}$, on éteint les générateurs et on calcule $R_{eq}=R_{th}$.
+$R_{th}=\frac{5R}{4}$.
+
+Pour trouver $E_{th}$, on met un circuit ouvert entre A et B:
+D'après Millmann, $U_{AB}=\frac{\sum{\frac{E_i}{R_i}}}{\sum{\frac{1}{R_i}}}=\frac{\frac{E_1}{R}+\frac{E_2}{R}+\frac{E_3}{R}+\frac{0}{R}}{\frac{1}{R}+\frac{1}{R}+\frac{1}{R}+\frac{1}{R}}=\frac{E_1+E_2+E_3}{4}=E_{th}$  
+
+![[Drawing 2025-10-03 14.07.09.excalidraw]]
+Diviseur de tension: $U_{ch}=\frac{R_{ch}}{R_{ch}+R_{th}}\times E_{th}=\frac{R}{R+\frac{5R}{4}}\times \frac{E_1+E_2+E_3}{4}=\frac{E_1+E_2+E_3}{9}$  
+

GénieÉlectrique/TD2/draws/1.A..md

@@ -0,0 +1,72 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+Eth ^jO5b802B
+
+Rth ^2uyUdtY9
+
+A ^OyoHlbN9
+
+B ^86RjNZ1A
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+```
+%%

GénieÉlectrique/TD2/draws/1.A.Eteint.md

@@ -0,0 +1,68 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+R2 ^U1vjJCPz
+
+R1 ^u32HhQwQ
+
+A ^RnM7aduJ
+
+B ^XnsbGU9B
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
+zQD1oSraEAp6a7dKNcVZVhCBGAwT+RpbFkiaPRMg969o10hAOQDBCbeEkbpkrdt7DDi6DAzRz7863i9Ll5y8KJgED7Jb8BJlUqRt+AMLggWxuwQBuwgA
+```
+%%

GénieÉlectrique/TD2/draws/1.A.Ouvert.md

@@ -0,0 +1,78 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^CPJvQkOH
+
+B ^t1QOg47A
+
+R2 ^6gJLpm6D
+
+R1 ^hdwbIKzo
+
+E ^n3CrLVL2
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD2/draws/1.B..md

@@ -0,0 +1,74 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+Rn ^i75oHoQp
+
+In ^68I92ijO
+
+A ^gEFFliQi
+
+B ^spKJtRCJ
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD2/draws/1.B.Eteint.md

@@ -0,0 +1,70 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+R2 ^1yKgfeMI
+
+R1 ^r8UB5GCU
+
+A ^NaoPTYNK
+
+B ^Gfyr39D4
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
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+
+NpdrZ1ggLgsWIgnAXQjkEQyQ9wBlaFUQJg+lcLaUXFWrbCmY8GNggylYyLQ6PoGqKyKQHeZY6Yw4/WE45rUwIQ8QwXHCiA3YKrggMsMwN0HFXAHgwQ3Ff46Q5SMsIQIwLvL/fgP/dcuvWEMEIk6Jp8kIAyAYGvZVW0ejZ0epeUJmAYN0JkDk4ieU/+gyBhIk8k6k/PIhOAONNqLqOEErVLS2EAA=
+```
+%%

GénieÉlectrique/TD2/draws/1.B.Ouvert.md

@@ -0,0 +1,96 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+R1 ^Xc7xtiqs
+
+In ^vr9UufFb
+
+A ^88cXLVoX
+
+B ^igiyhZIj
+
+R2 ^XMD8P6QJ
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
+fBRu4tD0BaHAwFqEyAETUWuRBlGLFQnoTMJ+AQWJBHADc4ukGocIJ1BOGNggAA==
+```
+%%

GénieÉlectrique/TD2/draws/1.C..md

@@ -0,0 +1,106 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^NwSZUS7g
+
+B ^8ef5hHpu
+
+e ^D4CTYvHY
+
+r ^NqrzqP4Y
+
+r ^GQAYKmof
+
+2r ^cIRrfnec
+
+2e ^9KXLA8iV
+
+e ^FbQ7QgwY
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD2/draws/1.C.Eteint.md

@@ -0,0 +1,100 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^OAnSGuXk
+
+B ^IieGrM5I
+
+r ^qHWQwtHs
+
+r ^v4qE4AYM
+
+2r ^DwLCKoG1
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+```
+%%

GénieÉlectrique/TD2/draws/1.C.Ouvert.Ueq.md

@@ -0,0 +1,95 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+r ^R47jNlSN
+
+r ^rZO2ofvP
+
+e ^7OipIK7s
+
+2e ^LYeY1VjP
+
+i ^HWEOcV7s
+
+ri ^xAb2jGkU
+
+## Embedded Files
+994ab975e81811be8219a079dc2b2096ec009408: $$U_{eq}$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD2/draws/1.C.Ouvert.md

@@ -0,0 +1,115 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^uKv7Ylmu
+
+B ^4RpUiNTT
+
+r ^tWbLRAeW
+
+r ^sTYHFhDY
+
+2r ^moGgLdTM
+
+2e ^Am7NHbbn
+
+e ^Ms9SzNpt
+
+i=0 ^wCeZlrVv
+
+## Embedded Files
+e1349fb6e62250567f75c748e6408aa92e51b67d: $$U_{AB}=E_{th}=?$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD2/draws/1.G..md

@@ -0,0 +1,105 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^x6oPPXYd
+
+B ^msYCM7Iq
+
+V ^KVYrrAlP
+
+R2 ^Pnbocb34
+
+R1 ^qyyPBV5x
+
+I ^NzRi4UuW
+
+## Embedded Files
+62e203f7c1ef225e71380d45a803cdc8a165e5d3: $$\lambda V$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD2/draws/1.G.CourtCircuit.md

@@ -0,0 +1,115 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^EoZSlIXb
+
+B ^y0LFRZMc
+
+V ^1JzfkbiQ
+
+R2 ^ojz6EWpY
+
+R1 ^2Mw6Q6np
+
+I ^kyS8URiZ
+
+In ^HQTUxM35
+
+## Embedded Files
+62e203f7c1ef225e71380d45a803cdc8a165e5d3: $$\lambda V$$
+
+%%
+## Drawing
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+```
+%%

GénieÉlectrique/TD2/draws/Drawing 2025-10-03 13.51.44.excalidraw.md

@@ -0,0 +1,99 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+8a649409eaf2056eb2e976f7d24d6382a0f878a7: $$I_1$$
+
+e2c089acc2a2571c0bae769e70c2453afa2b9a1e: $$I_2$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD2/draws/Drawing 2025-10-03 13.56.01.excalidraw.md

@@ -0,0 +1,136 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+A ^7qwiESYk
+
+B ^uecM9vk5
+
+R ^leCUteIX
+
+R ^TSV77M2j
+
+R ^9xYWNiMm
+
+R ^eCLif9Ks
+
+R ^HguGo8CV
+
+Rch ^SVDgTuPm
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD2/draws/Drawing 2025-10-03 14.07.09.excalidraw.md

@@ -0,0 +1,74 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+Rch ^KcK3m4pV
+
+Rth ^99lES8HE
+
+Eth ^m1umCxq0
+
+Uth ^bzrXMcOI
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
+DBQmEElKR7RMD80zOR8HZXQXZcGkb4lkv5hXRXxXd4f7WD+AjQTRwhXrOwQBOwgA
+```
+%%

GénieÉlectrique/TD3/Draws/Drawing 2025-10-03 16.08.29.excalidraw.md

@@ -0,0 +1,54 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD3/Index.md

@@ -0,0 +1,52 @@
+
+
+![[TD_FGE_2025.pdf#page=10]]
+![[b5364d1697.pdf#page=5]]
+# Exercise 1.
+
+Loi des mailles $\rightarrow E=U_R+UC$
+$E=R_i+U_C$
+$E=RC\frac{dU_C}{dt}+U_C$
+
+La solution est une courbe de la forme $U_C(t)=A_e^{-t/RC}+B$ 
+
+Conditions aux limites:
+À $t=0$, $U_C(t)=0$ par lecture du circuit.
+$U_C(t)=A_e^{-0/RC}+B=A+B$
+$\implies 0=A+B\implies A=-B$
+
+À $t\rightarrow ∞$, $U_C(t)=E_0$ par lecture du circuit.
+$U_C(∞)=B$ par la forme de la solution.
+Finalement, $U_C(t)=-E_0e^{-t/RC}+E_0=E_0(1-e^{-t/RC})$ 
+
+$V=E-U_C(t)$
+$V(t)=E_0e^{-t/RC}$
+
+# Exercise 2.
+![[Pasted image 20251003152305.png]]
+
+
+$\tau=RC=10E^3\times100E^{-6}=1s$ 
+![[IMG_7370.jpeg]]
+
+# Exercise 3: Second order DC circuits
+![[Pasted image 20251003160147.png]]
+
+$\implies I=CL\frac{d^2i_1}{dt^2}+i_1$ 
+
+Puisque $0=LC\frac{d^2U}{dt^2}+U+°\times\frac{dU}{dt}$ 
+$\implies$ Solution: $U(t)=A\cos{(\omega_0t)}+B\sin{(\omega_0t)}$ 
+À $t=0$, $U(t)=0$ donc $A=0$.
+
+Donc $U(t)=B\sin{(\omega_0t)}$
+d'où: $\frac{du}{dt}=B\omega_0\cos{(\omega_0t)}$
+Or, $\frac{dU}{dt}=\frac{i_C}{C}$ et $i=i_C+i_L$
+Donc $\frac{dU}{dt}=\frac{i-i_L}{C}$
+
+À $t=0$:
+$\frac{dU}{dt}=B\omega_0=\frac{I}{C}$
+Parceque $I_L$ est continu:
+$I_L(0^+)=I_L(0^-)=0$
+$\implies B=\frac{I}{i\omega_0}=\frac{I}{\frac{C}{\sqrt{LC}}}=\frac{I\sqrt{LC}}{C}$ 
+
+Finalement, $U(t)=\frac{I\sqrt{LC}}{C}\sin{(\frac{1}{\sqrt{LC}}t)}$ 

GénieÉlectrique/TD4/Index.md

@@ -0,0 +1,200 @@
+![[TD_FGE_2025.pdf#page=15]]
+
+# Exercise 1. Fresnel vector
+![[Pasted image 20251006094239.png]]
+a) polar form (modulus and argument)
+$\overrightarrow{v_1}=(15A,\frac{7\pi}{6})=15e^{j(\phi)}=15e^{j210}$ 
+
+b) rectangular form (a+jb)
+$\overrightarrow{v_1}=(\begin{matrix}15A\cos{(\frac{7\pi}{6})} \\ 15A\sin{(\frac{7\pi}{6})}\end{matrix})=(\begin{matrix}-12.99A \\ -7.5A\end{matrix})$ 
+$\overrightarrow{v_1}=a+jb$ avec $a=15\cos{210}$ et $b=15\sin{210}$
+$\overrightarrow{v_1}=-12.99 - 7.5j$
+
+
+# Exercise 2:
+![[Pasted image 20251006100509.png]]
+a) 
+$\overrightarrow{V_1}=63+j16=\sqrt{63^2+16^2}e^{j\arctan{(\frac{16}{63})}}=65e^{j0.249}$ 
+$\overrightarrow{V_2}=-5-j12=\sqrt{(-5)^2+(-12)^2}e^{j\arctan{(\frac{-12}{-5})}}=13e^{j(1.176+\pi)}$ 
+
+$\overrightarrow{V_1}\wedge\overrightarrow{V_2}=65\times13\times e^{j(0.249+\pi+1.176)}$
+
+b)
+$\overrightarrow{V_1}/\overrightarrow{V_2}=\frac{65}{13}e^{j(0.249-\pi-1.176)}$
+
+# Exercise 3:
+![[Pasted image 20251006101747.png]]
+## Figure 2: ![[Drawing 2025-10-06 10.49.08.excalidraw]]
+a)
+$U=\underline{z}_1i_1\implies i_1=\frac{\underline{U}}{\underline{z_1}}=\frac{\underline{U}}{R}$ 
+$U=\underline{z}_2i_2\implies i_2=\frac{\underline{U}}{\underline{z_2}}=\frac{\underline{U}}{jL\omega}$ 
+
+$||\underline{i_1}||=||\frac{\underline{U}}{R}||=\frac{||\underline{U}||}{||R||}=5A$ 
+$||\underline{i_2}||=||\frac{\underline{U}}{\underline{Z_2}}||=\frac{||\underline{U}||}{||\underline{Z_2}||}=2A$ 
+
+b)
+$\underline{i}=\underline{i_1}+\underline{i_2}$ 
+$i_2=\frac{\underline{U}}{jL\omega}=\frac{-j\underline{U}}{L\omega}$
+![[Drawing 2025-10-06 10.55.18.excalidraw]]
+
+$||\underline{i}||=\sqrt{||\underline{i_1}||^2+||\underline{i_2}||^2}=\sqrt{29}\approx5.3851648071A$
+$\phi=\arctan{(\frac{||\underline{i_2}||}{||\underline{i_1}||})}=\arctan{(\frac{-2}{5})}\approx-0.3805063771 rad$ 
+
+c)
+$\frac{1}{Z_{eq}}=\frac{1}{Z_R}+\frac{1}{Z_L}\implies Z_{eq}=\frac{\underline{Z_L}\underline{Z_R}}{\underline{Z_L}+\underline{Z_R}}$ 
+$\frac{1}{Z_{eq}}=\frac{1}{R}+\frac{1}{jL\omega}=\frac{1}{R}-j\frac{1}{L\omega}$
+ou
+$Z_{eq}=\frac{jRL\omega}{R+jL\omega}=\frac{(R-jL\omega)(jRL\omega)}{(R+jL\omega)(R-jL\omega)}=\frac{RL^2\omega^2+jR^2L\omega^2}{R^2+L^2\omega^2}=\frac{RL^2\omega^2}{R^2+L^2\omega^2}+j\frac{R^2L\omega}{R^2+L^2\omega^2}$ 
+$||Z_{eq}||=\frac{||R||||Z_L||}{||R+Z_L||}=\frac{1000}{\sqrt{20^2+50^2}}\approx\frac{1000}{53.8516480713}=18.5695338177\Omega$ 
+
+d)
+$\underline{U}=\underline{Z_{eq}}\underline{I}$
+$||\underline{U}||=||\underline{Z_{eq}}||||\underline{I}||\approx18.5695338177\times5.3851648071=99.9999999993V$ 
+
+## Figure 3: ![[Drawing 2025-10-06 11.17.18.excalidraw|1000]]
+a)
+$U=\underline{z}_1i_1\implies i_1=\frac{\underline{U}}{\underline{z_1}}=\frac{\underline{U}}{R}$ 
+$U=\underline{z_2}i_2\implies i_2=\frac{\underline{U}}{\underline{z_2}}=\frac{\underline{U}}{\frac{1}{jC\omega}}$ 
+
+$||\underline{i_1}||=||\frac{\underline{U}}{R}||=\frac{||\underline{U}||}{||R||}=\frac{120}{20}=6A$ 
+$||\underline{i_2}||=||\frac{\underline{U}}{\underline{Z_2}}||=\frac{||\underline{U}||}{||\underline{Z_2}||}=\frac{120}{30}=4A$ 
+
+b)
+$\underline{i}=\underline{i_1}+\underline{i_2}$ 
+$i_2=\frac{\underline{U}}{\frac{1}{jC\omega}}=\underline{U}jC\omega$ 
+
+![[Drawing 2025-10-06 11.25.11.excalidraw]]
+$||\underline{i}||=\sqrt{||\underline{i_1}||^2+||\underline{i_2}||^2}=\sqrt{52}\approx7.2111025509A$
+$\phi=\arctan{(\frac{||\underline{i_2}||}{||\underline{i_1}||})}=\arctan{(\frac{4}{6})}\approx0.5880026035 rad$ 
+
+c)
+$\frac{1}{Z_{eq}}=\frac{1}{Z_R}+\frac{1}{Z_C}\implies Z_{eq}=\frac{\underline{Z_C}\underline{Z_R}}{\underline{Z_C}+\underline{Z_R}}$ 
+$\frac{1}{Z_{eq}}=\frac{1}{R}+\frac{1}{\frac{1}{jC\omega}}=\frac{1}{R}+jC\omega$ 
+ou
+$||Z_{eq}||=\frac{||R||||Z_C||}{||R+Z_C||}=\frac{20\times30}{\sqrt{20^2+30^2}}\approx\frac{600}{36.0555127546}=16.6410058868\Omega$ 
+
+d)
+$\underline{U}=\underline{Z_{eq}}\underline{I}$
+$||\underline{U}||=||\underline{Z_{eq}}||||\underline{I}||\approx16.6410058868\times7.2111025509=119.9999999998V$ 
+
+## Figure 4: ![[Drawing 2025-10-06 11.38.13.excalidraw|1000]]
+a)
+$U=\underline{z}_1i_1\implies i_1=\frac{\underline{U}}{\underline{z_1}}=\frac{\underline{U}}{\frac{1}{jC\omega}}$ 
+$U=\underline{z}_2i_2\implies i_2=\frac{\underline{U}}{\underline{z_2}}=\frac{\underline{U}}{jL\omega}$ 
+
+$||\underline{i_1}||=||\frac{\underline{U}}{\underline{Z_1}}||=\frac{||\underline{U}||}{||\underline{Z_1}||}=\frac{100}{20}=5A$ 
+$||\underline{i_2}||=||\frac{\underline{U}}{\underline{Z_2}}||=\frac{||\underline{U}||}{||\underline{Z_2}||}=\frac{100}{30}=3\frac{1}{3}A$ 
+
+b)
+$\underline{i}=\underline{i_1}+\underline{i_2}$ 
+$i_1=\frac{\underline{U}}{\frac{1}{jC\omega}}=\underline{U}jC\omega$ 
+$i_2=\frac{\underline{U}}{jL\omega}=\frac{-j\underline{U}}{L\omega}$ 
+
+![[Drawing 2025-10-06 11.43.56.excalidraw]]
+$||\underline{i}||=||\underline{i_1}||-||\underline{i_2}||=5-3\frac{1}{3}=1\frac{2}{3}A$
+$\phi=180°$ ou $\pi rad$ 
+
+c)
+$\frac{1}{Z_{eq}}=\frac{1}{Z_L}+\frac{1}{Z_C}\implies Z_{eq}=\frac{\underline{Z_C}\underline{Z_L}}{\underline{Z_C}+\underline{Z_L}}$ 
+$\frac{1}{Z_{eq}}=\frac{1}{jL\omega}+\frac{1}{\frac{1}{jC\omega}}=-j\frac{1}{L\omega}+jC\omega$ 
+ou
+$||Z_{eq}||=\frac{||Z_L||||Z_C||}{||Z_L+Z_C||}=\frac{20\times30}{||20-30||}\approx\frac{600}{10}=60\Omega$ 
+
+d)
+$\underline{U}=\underline{Z_{eq}}\underline{I}$
+$||\underline{U}||=||\underline{Z_{eq}}||||\underline{I}||\approx60\times1\frac{2}{3}=100V$ 
+
+## Figure 5:![[Drawing 2025-10-06 12.02.59.excalidraw|1000]]
+a)
+$U=\underline{z}_1i_1\implies i_1=\frac{\underline{U}}{\underline{z_1}}=\frac{\underline{U}}{\frac{1}{jC\omega}}$ 
+$U=\underline{z}_2i_2\implies i_2=\frac{\underline{U}}{\underline{z_2}}=\frac{\underline{U}}{jL\omega}$ 
+$U=\underline{z_3}i_3\implies i_3=\frac{\underline{U}}{\underline{z_3}}=\frac{\underline{U}}{R}$ 
+
+$||\underline{i_1}||=||\frac{\underline{U}}{\underline{Z_1}}||=\frac{||\underline{U}||}{||\underline{Z_1}||}=\frac{100}{30}=3\frac{1}{3}A$ 
+$||\underline{i_2}||=||\frac{\underline{U}}{\underline{Z_2}}||=\frac{||\underline{U}||}{||\underline{Z_2}||}=\frac{100}{60}=1\frac{2}{3}A$ 
+$||\underline{i_3}||=||\frac{\underline{U}}{R}||=\frac{||\underline{U}||}{||R||}=\frac{100}{20}=5A$ 
+
+b)
+$\underline{i}=\underline{i_1}+\underline{i_2}+i_3$ 
+$i_1=\frac{\underline{U}}{\frac{1}{jC\omega}}=\underline{U}jC\omega$ 
+$i_2=\frac{\underline{U}}{jL\omega}=\frac{-j\underline{U}}{L\omega}$ 
+
+![[Drawing 2025-10-06 12.09.17.excalidraw]]
+
+$||\underline{i}||=\sqrt{(||\underline{i_1}+\underline{i_2}||)^2+||i_3||^2}=\sqrt{(3\frac{1}{3}-1\frac{2}{3})^2+5^2}=\sqrt{27.7777777778}\approx5.2704627669A$ 
+$\phi=\arctan{(\frac{||\underline{i_{1+2}}||}{||\underline{i_3}||})}=\arctan{(\frac{1\frac{2}{3}}{5})}\approx0.3217505544 rad$ 
+
+c)
+$\frac{1}{Z_{eq}}=\frac{1}{Z_L}+\frac{1}{Z_C}+\frac{1}{Z_R}=\frac{Z_RZ_C+Z_CZ_L+Z_RZ_L}{Z_RZ_LZ_C}$ 
+$Z_{eq}=\frac{Z_RZ_LZ_C}{Z_RZ_C+Z_CZ_L+Z_RZ_L}$
+ou
+$||Z_{eq}||=\frac{||Z_R||||Z_L||||Z_C||}{||Z_RZ_C||+||Z_CZ_L||+||Z_RZ_L||}$ 
+
+# Exercise 4: Application
+![[Pasted image 20251006122351.png]]
+
+a)
+![[Drawing 2025-10-06 12.24.01.excalidraw]]
+$Z_{eq}=30+\frac{80}{j}=30-j80$
+$Z_C=\frac{1}{jC\omega}=\frac{80}{j}$
+
+b) ![[Drawing 2025-10-06 12.28.41.excalidraw]]
+$Z_{eq}=(r,\theta)=(\sqrt{30^2+80^2}, \arctan{(\frac{-80}{30})})=(85.4400374532, -1.2120256565 rad)$ 
+$V=85.44e^{-1.21j}$ 
+
+# Exercise 5: Application, Study of a real inductor 
+The series model of a real inductor is defined by its series resistance $R_S$ and its series inductance $L_S$. It is possible to define a parallel model equivalent to the series model as shown in Figure 6.
+![[Pasted image 20251008135058.png]]
+
+a)
+Modèle en série $\implies Z_{eq_1}=L_S+R_S=R_S+jL_S\omega$ 
+Modèle parallèle $\implies \frac{1}{Z_{eq_2}}=\frac{1}{L_P}+\frac{1}{R_P}$ $\implies Z_{eq_2}=\frac{R_PjL_P\omega}{R_P+jL_P\omega}$
+
+Multiplication par le conjugué
+$\frac{jR_PL_P\omega(R_P-jL_P\omega)}{(R_P+jL_P\omega)(R_P-jL_P\omega)}=\frac{R_PL_P^2\omega^2+jR_P^2L_P\omega}{R_P^2+L_P^2\omega^2}$
+ 
+Or, $Z_{eq_1}=Z_{eq_2}\implies\begin{cases}R_S=\frac{R_PL_P^2\omega^2}{R_P^2+L_P^2\omega^2} \\ L_S\omega=\frac{R_P^2L_P\omega}{R_P^2+L_P^2\omega^2}\end{cases}$ 
+
+$R_S+jL_S\omega=\frac{R_P^2L_P\omega}{R_P^2+L_P^2\omega^2}+j\frac{R_PL_P^2\omega^2}{R_P^2+L_P^2\omega^2}$ 
+
+b)
+$U(t)=U_2\cos{(\omega t)}$
+$\underline{U}=U_2e^{j\omega t}$ 
+$\underline{I}=\frac{\underline{U}}{\underline{Z_{eq}}}=\frac{U_2e^{j\omega t}}{\underline{Z_{eq}}}$ 
+
+c)
+![[Drawing 2025-10-08 14.39.38.excalidraw|1000]]
+
+$V_{total}^2=V_{L_S}^2+(V_0+V_{R_S})^2$ 
+$V_{total}^2=V_{L_S}^2+V_0^2+2V_0V_{R_S}+V_{R_S}^2$ 
+
+$V_{bobine}^2=V_{L_S}^2+V_{R_S}^2$ 
+
+D'après Pythagore:
+$$
+\begin{cases}
+V_{bobine}^2=V_{L_S}^2+V_{R_S}^2 \\
+V_{total}^2=V_{L_S}^2+(V_0+V_{R_S})^2 \\
+V_0=R_0I
+\end{cases}
+\implies
+\begin{cases}
+V_{bobine}^2=L_S^2\omega^2I^2+R_S^2I^2 \\
+V_{total}^2=L_S^2\omega^2I^2+(R_S+R_0)^2I^2 \\
+V_0=R_0I
+\end{cases}
+$$
+![[IMG_7432.jpeg|400]]
+
+$$
+\begin{cases}
+L_S^2=(V_{bobine}^2-\frac{R_S^2V_0^2}{R_0^2})\times\frac{R_0^2}{V_0^2\omega^2} \\
+R_S=\frac{V_{tot}^2-V_{bobine}^2-V_0^2}{2V_0}\times R_0
+\end{cases}
+$$
+
+d)
+Application Numérique: $\begin{matrix}R_S=4.93\Omega \\ L_S=32mH\end{matrix}$ 
+
+e)
+Au lieu de mettre $R_0$ en série pour mesurer les tensions, on peut le mettre en parallèle pour mesurer les courants.

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 10.49.08.excalidraw.md

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+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
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+
+%%
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+hACGUMJsEAA=
+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 10.55.18.excalidraw.md

@@ -0,0 +1,77 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+U ^Kn15tU1P
+
+i1 ^ffSESVe2
+
+i2 ^cDjedbwX
+
+i1+i2=i ^ia5sDJST
+
+## Embedded Files
+42ee155d1d029d6bce99644c5e742c89bee942b2: $$\phi$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 11.17.18.excalidraw.md

@@ -0,0 +1,118 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+z1 ^EowiJ97o
+
+120V ^4cH6Vc2o
+
+i1 ^f4UjqGOJ
+
+i2 ^ZuhZPq89
+
+20Ohm ^Ty07DFA4
+
+30Ohm ^TLXDc5jm
+
+U ^otL86C55
+
+z2 ^60xhEJfu
+
+i ^rJpW4k64
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 11.25.11.excalidraw.md

@@ -0,0 +1,83 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+U ^rE2xVULQ
+
+i1 ^R0buD7IG
+
+i2 ^VpTNriPF
+
+i1+i2=i ^uPh71FIl
+
+## Embedded Files
+42ee155d1d029d6bce99644c5e742c89bee942b2: $$\phi$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 11.38.13.excalidraw.md

@@ -0,0 +1,126 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+100V ^bx8y19kl
+
+i1 ^ts395ueC
+
+i2 ^6bkWwky2
+
+20Ohm ^OmupJgvl
+
+U ^7EpTWyO7
+
+z2 ^seZWaoSi
+
+i ^L6idugZg
+
+z1 ^LqTadgge
+
+30Ohm ^MhxsJawN
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 11.43.56.excalidraw.md

@@ -0,0 +1,77 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+U ^WfMsGdeJ
+
+i1 ^xqG4OZ8H
+
+i2 ^o1TS0Wht
+
+i1+i2=i ^VxBLTjnh
+
+## Embedded Files
+42ee155d1d029d6bce99644c5e742c89bee942b2: $$\phi$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
+z/2CA+aJYvx/DDcOQhLxL3AUJSVRA6LBrDY/xWjJVZVMM2awERapVtZI1pAHIpAc4eG+rdr9wDrTrBLRL3ktuoLdgDQCA2AeQxo/xcAuL+L/xPrJL6jUoAajABqfz+AALZJQ1YQwQwbkiGuttOaPT5JHNzOMb3pBgxomQmb8lwx4SfE3QmbCbSbqkcjYA9G0RwQdYrYIArYQAA==
+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 12.02.59.excalidraw.md

@@ -0,0 +1,144 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+100V ^p1eRjU8M
+
+i1 ^YXbvr1C1
+
+i2 ^wKdzvhnJ
+
+30Ohm ^4PzJD6Sf
+
+U ^7hNtRzAx
+
+z2 ^QVPbMOT1
+
+i ^T3rxvThY
+
+z1 ^LR5RpCOg
+
+60Ohm ^fCro86jg
+
+z3 ^qwnpSfwX
+
+20Ohm ^csUo0fdU
+
+i3 ^8FtfpeP0
+
+%%
+## Drawing
+```compressed-json
+N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATLZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHALRY8RMpWdx8Q1TdIEfARcZgRmBShcZQUebQBGAFZtAGYaOiCEfQQOKGZuAG1wMFAwMogSbggAdgAtfQAWAAkAawAFCgBxDh5agBkANQANADk+gEEAIQHMdLLIWEQqoKI5
+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 12.09.17.excalidraw.md

@@ -0,0 +1,85 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+U ^OmHpSpZT
+
+i1 ^2y3lUdg4
+
+i2 ^kdicXDhQ
+
+i1+i2+i3=i ^r7j9c9O7
+
+i3 ^wtUlGdR0
+
+## Embedded Files
+42ee155d1d029d6bce99644c5e742c89bee942b2: $$\phi$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 12.24.01.excalidraw.md

@@ -0,0 +1,57 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+9673c2680c9c0e69946c77027116ca5b82a54c31: $$30\Omega$$
+
+a827b999a096f2876c3babb0da8f13e2fbff4f01: $$80\Omega$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-06 12.28.41.excalidraw.md

@@ -0,0 +1,59 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+-80 ^LwU2aGJA
+
+30 ^5VbKPYJT
+
+## Embedded Files
+feb9fea6595245059dfddd6901e8d08536b7323c: $$\theta$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+```
+%%

GénieÉlectrique/TD4/attachments/Drawing 2025-10-08 14.39.38.excalidraw.md

@@ -0,0 +1,183 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+R0 ^8Fs0OQLO
+
+RS ^QnSgsORY
+
+LS ^0KpOtU2S
+
+## Embedded Files
+8c89c325f01932b1ea6b5ab190bb4a36212df99a: $$\color{#9C36B5}V_{L_S}$$
+
+def7c29d8031f630dd7913235d2b34a448557e4d: $$\color{#1971C2}V_0$$
+
+55be3bc14ff58d958e0f93b4eecf51e1c8983580: $$\color{#F08C02}i$$
+
+0eec7588357b84a6e134895882b123d8e6e5d2bb: $$\color{#2F9E44}V_{R_S}$$
+
+8dd0451c5433788bb390deb0dde0ec8097b27edb: $$\color{#343a40}V_{bobine}$$
+
+0eabec84a6e0457fd947222bde0bab265c32161d: $$\color{#e03131}V_{total}$$
+
+57e2ddc4017248083fb3142a71ae60d36cd18a0a: $$\color{#e03131}V_{total}$$
+
+ee2202e4bbc6d1e3a4c88ed96592596aad3f5646: $$\color{#343a40}V_{bobine}$$
+
+04825015f524ba0d3a5e67e9b1256cbbd8c8f3fb: $$\color{#1971C2}V_0$$
+
+58b2bf57d8f215dcd06bccdf6d9fec94638cd0b9: $$\color{#F08C02}i$$
+
+354c0f551db2bba8f82c773ee794b8dfdb7bf06e: $$\color{#e03131}V_{total}$$
+
+46ab13516241fab23c86b96295c8541d8738770f: $$\color{#9C36B5}V_{L_S}$$
+
+98e0632e71aa6be16fdeaca9e3071665868bf70d: $$\color{#343a40}V_{bobine}$$
+
+4ed5148bc3056edbd335824c2c65e2ea4e78be79: $$\color{#2F9E44}V_{R_S}$$
+
+a0a5873f27c023f5acfbfd112bf261257e856e17: $$\color{#1971C2}V_0$$
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

GénieÉlectrique/TD5/Index.md

@@ -0,0 +1,67 @@
+![[TD_FGE_2025.pdf#page=20]]
+
+# Exercise 1. Composed Transfer Function
+![[Pasted image 20251009091732.png]]![[Pasted image 20251009091748.png]]
+
+1. 
+Étude asymptotique de $\underline{G}=1+j\tau_i\omega$:
+$G_{dB}=||20log(\underline{G})||=20log(\sqrt{1^2+\tau^2\omega^2})$ 
+$\phi=arg(\underline{G})=\arctan(\frac{\tau\omega}{1})$ 
+
+Étude en basses fréquences:
+Pour $\omega\rightarrow∞$:
+$G_{dB}\approx20log(1)=0$
+$\phi\approx\arctan{(0)}=0$
+
+Étude en hautes fréquences:
+Pour $\omega>>1$
+$G_{dB}\approx20log(\tau\omega)=20log(\tau)+20log(\omega)$
+$\phi\approx\arctan{(∞)}=\frac{\pi}{2}$
+
+On cherche le croisement des deux asymptotes.
+On cherche donc $20log(\tau\omega)=0$, ce qui implique que $\tau\omega=1$ et par conséquent que $\omega=\frac{1}{\tau}$.
+Alors, $G_{dB}(\frac{1}{\tau})20log(\sqrt{2})\approx3$
+$\phi(\frac{1}{\tau})=\arctan{(1)}=\frac{\pi}{4}$
+
+![[Drawing 2025-10-09 09.27.54.excalidraw]]
+
+2. 
+$G_2=\frac{1}{G}$, donc
+$G_{dB_2}=20log(1)-20log(|\underline{G}|)=-G_{dB}$
+
+$\phi_2=-\phi$
+
+![[Drawing 2025-10-09 09.53.13.excalidraw]]
+
+3. 
+$G_{tot}=\frac{(1+j\tau_1\omega)(1+j\tau_2\omega)}{(1+j\tau_3\omega)(1+j\tau_4\omega)}=\frac{G_1G_2}{G_3G_4}$
+
+# Exercise 2: .Bandpass filter 2nd order
+
+![[Pasted image 20251009105118.png]]
+
+1. 
+![[Pasted image 20251009105236.png]]
+
+On cherche $\underline{V_S}$.
+
+$Z_{eq}$ correspond à $L$ et $C$, donc $\frac{1}{Z_{eq}}=\frac{1}{Z_L}+\frac{1}{Z_C}=jC\omega-j\frac{1}{L\omega}$ 
+Par le pont diviseur, $\underline{V_S}=\frac{Z_{eq}}{R+Z_{eq}}V_e=\frac{1}{\frac{R}{Z_{eq}+1}}V_e$
+$\implies H=\frac{V_S}{V_e}=\frac{1}{1+jR(C\omega-\frac{1}{L\omega})}$ 
+Je cherche $\alpha$ et $\beta$ tel que $C\omega-\frac{1}{L\omega}=\alpha(\frac{\omega}{\beta}-\frac{\beta}{\omega})$ 
+$\frac{\alpha}{\beta}=C$ et $\alpha\beta=\frac{1}{L}\implies\begin{cases} \alpha=\beta C\implies\alpha=C\sqrt{\frac{1}{LC}}=\sqrt{\frac{C}{L}} \\ \beta^2C=\frac{1}{L}\implies\beta=\sqrt{\frac{1}{LC}} \end{cases}$
+
+
+Donc $H(j\omega)=\frac{1}{1+jR\sqrt{\frac{C}{L}}(\frac{\omega}{\sqrt{\frac{1}{LC}}}-\frac{\sqrt{\frac{1}{LC}}}{\omega})}$ 
+
+# Exercise 3:
+![[Pasted image 20251009115819.png]]
+## Part I: Impedance analyses
+1. 
+$Z_{eq}=\frac{RjL\omega}{R+jL\omega}+\frac{\frac{R}{jC\omega}}{R+\frac{1}{jC\omega}}=$ 
+![[IMG_7440.jpeg]]
+
+2. 
+
+
+

GénieÉlectrique/TD5/attachments/Drawing 2025-10-09 09.27.54.excalidraw.md

@@ -0,0 +1,77 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+383d65aa2a52a07a31727ced72b99a4a026f9067: $$G_{dB}$$
+
+acd1800c42346192175acf7183643ef2a6dff614: $$\phi$$
+
+359c7b9d494f03fc07d1931f1ca8e042626674dd: $$\color{red}\frac{1}{\tau}$$
+
+4f89de5548103229895bfdabab304099d2883258: $$\color{red}\frac{1}{\tau}$$
+
+%%
+## Drawing
+```compressed-json
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+```
+%%

GénieÉlectrique/TD5/attachments/Drawing 2025-10-09 09.53.13.excalidraw.md

@@ -0,0 +1,77 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+383d65aa2a52a07a31727ced72b99a4a026f9067: $$G_{dB}$$
+
+acd1800c42346192175acf7183643ef2a6dff614: $$\phi$$
+
+359c7b9d494f03fc07d1931f1ca8e042626674dd: $$\color{red}\frac{1}{\tau}$$
+
+4f89de5548103229895bfdabab304099d2883258: $$\color{red}\frac{1}{\tau}$$
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

ModélisationEnMeca/Drawing 2025-10-01 14.27.18.excalidraw.md

@@ -0,0 +1,141 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+6a262ccbae353e8cc05b0faf6160875fddb93275: $$\overrightarrow{e_y}$$
+
+7fdb0c24b187f7d76522a9c4d485a7425759d17e: $$I$$
+
+1e089744421e86f53df500b70e37ec1831d9c1f5: $$\overrightarrow{e_x}$$
+
+6aa0cfeefb27ef8536ae8116afe5b050994abc34: $$\overrightarrow{e_r}$$
+
+685c1767b0522fdc5f185927bd8de5104266e352: $$\color{red}i$$
+
+521dc4d5162a4746ca7ba548126ef74c6fa207ff: $$\color{red}e$$
+
+bcb7b78cd00504331b57a5710f16ed64fb98eaf7: $$J$$
+
+ca3832c943862ce2128ea4860529dcd5a1553467: $$\color{blue}A$$
+
+d52cb46dd0d13af67043b52511647cec78d2c75f: $$bi$$
+
+bab93bdeb4c8624eeda96c4eab791df06a61e16c: $$\color{red}bi$$
+
+49585cb74ff6727096eea73a3eb3afdeb50fb3fc: $$C$$
+
+a953b785a61514b2de4dcb91ff99f6691405e474: $$\overrightarrow{e_\theta}$$
+
+58cd790aed46987ebcfd2457d056c93d1bac0343: $$O$$
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

ModélisationEnMeca/Drawing 2025-10-02 09.54.32.excalidraw.md

@@ -0,0 +1,133 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+09a164dcad95a326d155dcba78f412fcdf60a9ef: $$\color{red}\overrightarrow{j}$$
+
+092dd01314566fc0cf41be707360e9bb8b56e0cf: $$\color{red}\overrightarrow{i}$$
+
+95747095a5e08af685fdf95159a25b3e4e62cc18: $$\color{red}\overrightarrow{F_{N1}}$$
+
+06ceb09334ec479a6d0575deb1653daa2af1ba52: $$\color{red}\overrightarrow{F_{N2}}$$
+
+96545cdb9e24cc91564b055942f45c57256f412e: $$\color{orange}\overrightarrow{F_{T1}}$$
+
+a842e24e0aa6b190dc49313dd3bca13d31c14df7: $$\color{orange}\overrightarrow{F_{T2}}$$
+
+4ea08391950e45a3369e4b21af22ce969cda61a7: $$\color{purple}\overrightarrow{p}$$
+
+5a009ab579b55495433ed0aedaab35936dc45921: $$\alpha$$
+
+efc680781f725614cb59983ceb8a2e9010330d10: $$\color{blue}I$$
+
+b28f47b6021dea66e6ac89e80c3a163ce3df26a1: $$\color{blue}J$$
+
+fcd48c9f21ed956313606096cc8d14d4bdb2aa45: $$\color{blue}H$$
+
+88be5f7d962298e8d6f162f82b78765a4063ce1b: $$\color{blue}G$$
+
+%%
+## Drawing
+```compressed-json
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+
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+```
+%%

ModélisationEnMeca/Drawing 2025-10-02 10.15.02.excalidraw.md

@@ -0,0 +1,89 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+b060c72e4479d8d99037b3bfe7a59fc8ac66af63: $$\color{red}\theta$$
+
+434e92c3fc83d47bfbc23cc011f37feb3dfd4c49: $$\pi$$
+
+3c821b85ce03f5a1e4e80af9b7b226532ea9f70f: $$\alpha$$
+
+3466c5e409ca3caa2f68e17dcd0fc445fb0090c0: $$\alpha$$
+
+26f6a29fa97ffac5b09861594a652d8744b4c619: $$\frac{\pi}{2}-\alpha$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+```
+%%

ModélisationEnMeca/Drawing 2025-10-02 11.54.47.excalidraw.md

@@ -0,0 +1,57 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+edcc6bc64370a2abcfa36ee0836eeea684899aa8: $$\overrightarrow{F_T}$$
+
+09b32b2f343cfc9984dcaa57a763e8edd191131b: $$\overrightarrow{F_N}$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+```
+%%

ModélisationEnMeca/Drawing 2025-10-02 12.14.37.excalidraw.md

@@ -0,0 +1,119 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+G ^a64nsYrJ
+
+## Embedded Files
+b98018b905b5ee15ac7c32c42a3719f3f745b3c5: $$\overrightarrow{x_0}$$
+
+a0a29d6e61781ef232c4f2cb1e7bec347ab88142: $$\overrightarrow{y_0}$$
+
+a42d190bde77f8aadf045842fdf4730a99db50c8: $$\overrightarrow{x_1}$$
+
+c88aa21145c5cc413113eaeabfaa71659633ac55: $$\overrightarrow{y_1}$$
+
+a0080036c4c28a770da1e2e61015dcfb12e8a681: $$O_0$$
+
+b25418d5c33c03057113385a6b94b00113e21229: $$\theta(+)$$
+
+d1b127a8fe03cea322af5e9441038a3e1e79b147: $$\color{red}\overrightarrow{p}$$
+
+c671c06e0c71806be6ad8bce161837c754d4f40b: $$\color{#9C36B5}\overrightarrow{R_y}$$
+
+5f2fbdbc02c52a70a2975d035f9941fe71855286: $$\color{#9C36B5}\overrightarrow{R_x}$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+```
+%%

ModélisationEnMeca/Index.md

@@ -80,6 +80,8 @@ $\overrightarrow{AB}=b(\cos{\alpha_2}\overrightarrow{x}+\sin{\alpha_2}\overright
 ![[MG.pdf#page=11]]
 
 # Cinématique
+Formule Babar = ![[Pasted image 20251001143754.png]]
+
 $\overrightarrow{V}_{M\in S_s/R_0}=\frac{d}{dt}\overrightarrow{O_0M}(t)|_{B_0}$   
 
 ![[TD-Ven. 26 septembre 2025|1000]]
@@ -103,3 +105,236 @@ $\Omega_{1/0}=\dot{\theta} \overrightarrow{z}$
 $xyzxyz$
 Pour calculer le produit vectoriel de $x$ et $y$, on prend le suivant ($z$)
 Vers la droite, c'est positif ($x • y = z$) et vers la gauche c'est négatif ($y • x = -z$)
+
+
+
+$\overrightarrow{v}_{O_1}\in{1/0}=\frac{d}{dt}\overrightarrow{O_0O_1}|_O$ 
+$\overrightarrow{O_0O_1}=\overrightarrow{O_0I}+\overrightarrow{IO_1}=X\overrightarrow{x_0}+R\overrightarrow{y_0}$ 
+$\frac{d}{dt}\times X\overrightarrow{x_0}+R\overrightarrow{y_0}|_O=\dot{X}\overrightarrow{x_0}\times\frac{d\overrightarrow{x_0}}{dt}|_O+\dot{R}\overrightarrow{y_0}+R\frac{d\overrightarrow{y_0}}{dt}|_O=\dot{X}\overrightarrow{x_0}$ 
+$\overrightarrow{V_{I\in{1/0}}}=\overrightarrow{V_{O_1\in{1/0}}}+\overrightarrow{IO_1}\overrightarrow{\Omega_{1/0}}$  
+	$=\dot{X}\overrightarrow{x_0}+R\dot{\theta}\overrightarrow{x_0}=(\dot{X}+R\dot{\theta})\overrightarrow{x_0}$
+
+Roulement à bille
+![[Drawing 2025-10-01 14.27.18.excalidraw]]
+Repère fix ($0$) $\rightarrow (\overrightarrow{e_x}, \overrightarrow{e_y})$ 
+Repère se déplaçant avec la bille $\rightarrow (\overrightarrow{e_\theta}, \overrightarrow{e_r})$ 
+Vitesse de la bague intérieure par rapport à $0$: $\Omega_{i/0}=\dot{\theta}_i\overrightarrow{e_z}$ 
+Vitesse de la bague extérieure par rapport à $0$: $\Omega_{e/0}=\dot{\theta}_e\overrightarrow{e_z}$ 
+
+Calculer la vitesse du point $I\in{\color{red}i}$ Par rapport à $O$:
+$\overrightarrow{V_{I\in{i/0}}}=\overrightarrow{V_{0\in{i/0}}}+\overrightarrow{IO_1}\overrightarrow{\Omega_{i/0}}$ 
+$=\overrightarrow{0}-r_i\overrightarrow{e_r}\wedge\dot{\theta_i}\overrightarrow{e_z}$ 
+$=-r_i\dot{\theta_i}\overrightarrow{e_r}\wedge\overrightarrow{e_z}=r_i\dot{\theta_i}\overrightarrow{e_\theta}$  
+
+$\overrightarrow{V_{J\in{e/0}}}=r_e\dot{\theta_e}\overrightarrow{e_\theta}$
+
+$\overrightarrow{V_{C\in{bi/0}}}=\overrightarrow{V_{I\in{bi/0}}}+\overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}}$ 
+$\overrightarrow{V_{C\in{bi/0}}}=\overrightarrow{V_{J\in{bi/0}}}+\overrightarrow{CJ}\wedge\overrightarrow{\Omega_{bi/0}}=\overrightarrow{V_{J\in{bi/0}}}-\overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}}$ 
+
+$2\overrightarrow{V_{C\in{bi/0}}}=\overrightarrow{V_{I\in{bi/0}}}+\overrightarrow{V_{J\in{bi/0}}}$ 
+
+$I \in bi \quad I \in i$
+
+pas de glissement en $I$ :
+
+$\overrightarrow{V_{I \in bi/i}} = \overrightarrow{V_{I \in bi/0}} + \overrightarrow{V_{I \in 0/i}} = \overrightarrow{0}$
+
+$\overrightarrow{V_{I \in bi/0}} = - \overrightarrow{V_{I \in 0/i}} = \overrightarrow{V_{I \in ci/0}} = a \, \dot{\theta}_i \, \overrightarrow{e_\theta}$
+
+$\overrightarrow{V_{J \in bi/0}} = \overrightarrow{V_{J \in ee/0}} = R_e \, \dot{\theta}_e \, \overrightarrow{e_\theta}$
+
+$2 \, \overrightarrow{V_{C \in bi/0}} = \overrightarrow{V_{I \in bi/0}} + \overrightarrow{V_{J \in bi/0}}$
+
+$\overrightarrow{V_{C \in bi/0}} = \tfrac{1}{2} \big( n_i \, \dot{\theta}_i + n_e \, \dot{\theta}_e \big) \, \overrightarrow{e_\theta}$
+
+$\overrightarrow{V_{I\in{bi/0}}}-\overrightarrow{V_{J\in{bi/0}}}+\overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}} + \overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}}$ 
+
+Or, $\overrightarrow{A}\wedge\overrightarrow{X}=\overrightarrow{B}$ $\implies\boxed{\overrightarrow{X}=\overrightarrow{B}\wedge\frac{\overrightarrow{A}}{\overrightarrow{A}•\overrightarrow{A}}+\lambda\overrightarrow{A}}$ 
+
+$2\overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}}=\overrightarrow{V_{J\in{bi/0}}}-\overrightarrow{V_{I\in{bi/0}}}$ 
+$2\overrightarrow{CI}\wedge\overrightarrow{\Omega_{bi/0}}=R_e \, \dot{\theta}_e \, \overrightarrow{e_\theta}-R_i \, \dot{\theta}_i \, \overrightarrow{e_\theta}$ 
+$$
+\overrightarrow{\Omega_{bi/0}}=
+(r_e \, \dot{\theta}_e 
+-r_i \, \dot{\theta}_i)\overrightarrow{e_\theta}
+\wedge
+\frac
+{
+	2\overrightarrow{CI}
+}
+{
+	2\overrightarrow{CI}•2\overrightarrow{CI}
+}
++\lambda2\overrightarrow{CI}
+$$
+$$
+\overrightarrow{\Omega_{bi/0}}=
+(r_e \, \dot{\theta}_e 
+-r_i \, \dot{\theta}_i)\overrightarrow{e_\theta}
+\wedge
+\frac
+{
+	-(r_e-r_i)\overrightarrow{e_r}
+}
+{
+	(r_e-r_i)^2
+}
++\lambda(r_i-r_e)\overrightarrow{e_r}
+$$
+$$
+\overrightarrow{\Omega_{bi/0}}=
+\frac
+{
+-(r_e \, \dot{\theta}_e 
+-r_i \, \dot{\theta}_i)
+}
+{
+r_i-r_e
+}
+\overrightarrow{e_\theta}
+\wedge
+\overrightarrow{e_r}
++\lambda(r_i-r_e)\overrightarrow{e_r}
+$$
+$$
+\overrightarrow{\Omega_{bi/0}}=
+\frac
+{
+r_i \, \dot{\theta}_i
+-r_e \, \dot{\theta}_e 
+}
+{
+r_i-r_e
+}
+\overrightarrow{e_z}
++\lambda(r_i-r_e)\overrightarrow{e_r}
+$$
+
+# Statique
+À savoir:
+$\cos(-\alpha)=\cos(\alpha)$, et $\sin(-\alpha)=-\sin(\alpha)$
+
+![[Drawing 2025-10-02 09.54.32.excalidraw|1000]]
+
+Avec $\color{#AC46C5}\overrightarrow{p}$  le poids, $\color{#8888FF}I$ et $\color{#8888FF}J$ les points de contact entre le chariot et la pente. 
+Avec $m$ la masse du chariot, et $g$ la constante gravitationnelle.
+
+$\overrightarrow{IJ}=a\overrightarrow{i}$ 
+$\overrightarrow{HG}=h\overrightarrow{j}$
+$\overrightarrow{IG}=\overrightarrow{IH}+\overrightarrow{HG}=\frac{a}{2}\overrightarrow{i}+h\overrightarrow{j}$ 
+
+$\color{red}\overrightarrow{F_{N1}}\color{white}=F_{N1}\color{red}\overrightarrow{j}$ 
+$\color{orange}\overrightarrow{F_{T1}}\color{white}=F_{T1}\color{red}\overrightarrow{i}$
+$\color{red}\overrightarrow{F_{N2}}\color{white}=F_{N2}\color{red}\overrightarrow{j}$
+$\color{orange}\overrightarrow{F_{T2}}\color{white}=F_{T2}\color{red}\overrightarrow{i}$
+
+![[Drawing 2025-10-02 10.15.02.excalidraw|250]]
+$\color{red}\theta\color{white}=-\alpha+\frac{3\pi}{2}-2\pi=-(\alpha+\frac{\pi}{2})$
+$\cos{\color{red}\theta}=\cos{(-(\alpha+\frac{\pi}{2}))}=\cos{(\alpha+\frac{\pi}{2})}=-\sin{\alpha}$
+$\sin{\color{red}\theta}=\sin{(-(\alpha+\frac{\pi}{2}))}=-\sin{(\alpha+\frac{\pi}{2})}=-\cos{\alpha}$
+
+$\color{#AC46C5}\overrightarrow{p}\color{white}=mg(\cos{\color{red}\theta}\color{red}\overrightarrow{i}\color{white}+\sin{\color{red}\theta}\color{red}\overrightarrow{j}\color{white})$ 
+$\color{#AC46C5}\overrightarrow{p}\color{white}=mg(-\sin{\alpha}\color{red}\overrightarrow{i}\color{white}-\cos{\alpha}\color{red}\overrightarrow{j}\color{white})$
+$\color{#AC46C5}\overrightarrow{p}\color{white}=-mg(\sin{\alpha}\color{red}\overrightarrow{i}\color{white}+\cos{\alpha}\color{red}\overrightarrow{j}\color{white})$
+
+$$
+_I\cases{F_{N_1}\overrightarrow{j}+F_{T_1}\overrightarrow{i} \\ \overrightarrow{0}} 
++ _J\cases{
+	F_{N_2}\overrightarrow{j}+F_{T_2}\overrightarrow{i} \\ 
+	\overrightarrow{0}
+}
++ _G\cases{
+	-mg(\sin{\alpha}\color{red}\overrightarrow{i}\color{white}+\cos{\alpha}\color{red}\overrightarrow{j}\color{white}) \\ 
+	\overrightarrow{0}
+}
+=\cases{\overrightarrow{0} \\ \overrightarrow{0}}
+$$
+
+Comment transporter un torseur:
+$$
+_I\cases{
+\overrightarrow{R} \\
+\overrightarrow{O}=\overrightarrow{M_{\overrightarrow{R}/I}}
+}$$ $$
+_M\cases{
+	\overrightarrow{R} \\
+	\overrightarrow{M_{\overrightarrow{R}/M}} = \overrightarrow{M_{\overrightarrow{R}/I}} +\overrightarrow{MI}\wedge\overrightarrow{R}
+}
+$$
+$$
+_I\cases{
+	\overrightarrow{F_1} \\
+	\overrightarrow{M_{\overrightarrow{F_1}/I}}
+}
++_J\cases{
+	\overrightarrow{F_2} \\
+	\overrightarrow{M_{\overrightarrow{F_2}/J}}
+}
++_G\cases{
+	\overrightarrow{p} \\
+	\overrightarrow{0}
+}
+=_I\cases{
+	\overrightarrow{F_1} \\
+	\overrightarrow{M_{\overrightarrow{F_1}/I}}
+}
++_I\cases{
+	\overrightarrow{F_2} \\
+	\overrightarrow{M_{\overrightarrow{F_2}/I}}
+}
++_I\cases{
+	\overrightarrow{p} \\
+	\overrightarrow{M_{\overrightarrow{p}/I}}
+}
+$$
+Avec $\overrightarrow{M_{\overrightarrow{F_2}/I}} = \overrightarrow{M_{\overrightarrow{F_2}/J}} + \overrightarrow{IJ}\wedge\overrightarrow{F_2}$ , avec $\overrightarrow{M_{\overrightarrow{F_2}/J}}=\overrightarrow{0}$. Donc,
+$\overrightarrow{M_{\overrightarrow{F_2}/I}} = \overrightarrow{IJ}\wedge(F_{N_2}\overrightarrow{j}+F_{T_2}\overrightarrow{i})=a\overrightarrow{i}\wedge(F_{N_2}\overrightarrow{j}+F_{T_2}\overrightarrow{i})=aF_{N2}k$ (car $i\wedge j \rightarrow k$) 
+
+Avec $\overrightarrow{M_{\overrightarrow{p}/I}} =\overrightarrow{M_{\overrightarrow{p}/G}} + \overrightarrow{IG}\wedge\overrightarrow{p}=0 + (\frac{a}{2}\overrightarrow{i}+h\overrightarrow{j})\wedge(-mg(\sin{\alpha}\overrightarrow{i}+\cos{\alpha}\overrightarrow{j}))=mg(-\frac{a}{2}\cos{\alpha}+h\sin{\alpha})T_2$ 
+
+Quand il y a frottement, On a ![[Drawing 2025-10-02 11.54.47.excalidraw]] $||\overrightarrow{F_T}||=\lambda_f||\overrightarrow{F_N}||$ 
+$||\overrightarrow{F_T}||≤\lambda_a||\overrightarrow{F_N}||$
+
+$F_{N2}=0$
+$-\frac{a}{2}\cos{\alpha_c}+h\sin{\alpha_c}=0$
+$h\sin{\alpha_c}=\frac{a}{2}\cos{\alpha_c}$
+$\tan{\alpha_c}=\frac{a}{2h}$
+$\boxed{\alpha=\tan^{-1}{(\frac{a}{2h})}}$ 
+
+
+![[Drawing 2025-10-02 12.14.37.excalidraw]]
+
+Les translations bloquées génèrent des forces,
+les rotations bloquées génèrent des moments.
+
+$\overrightarrow{p}=mg\overrightarrow{x_0}$ 
+$\overrightarrow{R_x}=R_x\overrightarrow{x_0}$ 
+$\overrightarrow{R_y}=R_y\overrightarrow{y_0}$ 
+
+Torseur statique $\rightarrow _G\cases{mg\overrightarrow{x_0}\\\overrightarrow{0}}+_{O_0}\cases{R_x\overrightarrow{x_0}+R_y\overrightarrow{y_0}\\\overrightarrow{0}}$ 
+
+
+```functionplot
+---
+title: Hope over time
+xLabel: Time (days)
+yLabel: Hope (percentage)
+bounds: [0,30,0,1]
+disableZoom: true
+grid: true
+---
+hope(x)=0.85(1-x/30)^0.1
+```
+
+```functionplot
+---
+title: Understanting over time
+xLabel: Time (days)
+yLabel: Understanting (percentage)
+bounds: [0,30,-5,1]
+disableZoom: true
+grid: true
+---
+hope(x)=0.65(1-(x/26)^100)
+```
+