From 486cb2ce4bd4693dc45c1d2c5dfe72c0dca8d6d9 Mon Sep 17 00:00:00 2001
From: Sarah Gosselin <sarah@gosselin.xyz>
Date: Tue, 3 Dec 2024 18:54:22 -0500
Subject: leaving Kaapeh

---
 annexe/main.tex | 33 ++++++++++++++++++++++-----------
 1 file changed, 22 insertions(+), 11 deletions(-)

(limited to 'annexe/main.tex')

diff --git a/annexe/main.tex b/annexe/main.tex
index 5e270fb..650bc8e 100644
--- a/annexe/main.tex
+++ b/annexe/main.tex
@@ -5,12 +5,14 @@
 \usepackage{booktabs}
 \usepackage{bookmark}
 \usepackage{subcaption}
-\usepackage[american]{circuitikz}
+% \usepackage[american]{circuitikz}
 % \usepackage{showframe}
 \usepackage{float}
 \usepackage{multicol}
 \usepackage{siunitx}
 \usepackage{amsmath}
+\usepackage{mathtools}
+\usepackage{cancel}
 \usepackage[dvipsnames]{xcolor}
 \usepackage[T1]{fontenc}
 \usepackage{csquotes}
@@ -32,6 +34,9 @@
 \newcommand{\vst}{V_S(t)}
 \newcommand{\vlz}{V_L(0)}
 
+\newcommand{\ilt}{I_L(t)}
+\newcommand{\ict}{I_C(t)}
+
 \newcommand{\al}{\alpha}
 \newcommand{\la}{\lambda}
 \newcommand{\wz}{\omega_{0}}
@@ -73,11 +78,11 @@
 	Les fonction suivantes seront utilisées et référencé tout au long de l'annexe.
 
 	\begin{gather}
-		V_C(t) = \frac{1}{C}\int I_C(t) \dt \label{eq:vct}\\
-		I_C(t) = C\frac{\text{d}}{\dt}V_C(t) \label{eq:ict}\\
+		\vct = \frac{1}{C}\int\ict \dt \label{eq:vct}\\
+		\ict = C\frac{\text{d}}{\dt}\vct \label{eq:ict}\\
 		\nonumber\\
-		V_L(t) = L\frac{\text{d}}{\dt}I_L(t) \label{eq:vlt}\\
-		I_L(t) = \frac{1}{L}\int V_L(t)\dt \label{eq:ilt}\\
+		\vlt = L\frac{\text{d}}{\dt}\ilt \label{eq:vlt}\\
+		\ilt = \frac{1}{L}\int \vlt\dt \label{eq:ilt}\\
 		\nonumber\\
 		V(t) = RI(t) \label{eq:vri}\\
 		\nonumber\\
@@ -96,12 +101,18 @@
 
 
 	\begin{gather}
-		V_S = V_C(t) + V_R(t) + V_L(t) \\
-		V_S = \frac{1}{C}\int I(t) \dt + RI(t) + V_L \\
-		V_S = \frac{1}{CL} \iint \vlt \dt\ \dt + \frac{R}{L}\int\vlt\dt + V_L\\
-		\ddt{2}V_S = \ddt{2}\vlt + \frac{R}{L}\ddt{}\vlt + \frac{1}{CL}\vlt\\
-		\nonumber\\
-		\nonumber \text{On pose la forme de la solution homogène: } V_{L_h} = Ae^{\la t}\\
+		\vst = \vct + \vrt + \vlt \\
+		\vst = \frac{1}{C}\int I(t) \dt + RI(t) + \vlt \\
+		\vst = \frac{1}{CL} \iint \vlt \dt\ \dt + \frac{R}{L}\int\vlt\dt + \vlt\\
+		\nonumber \text{On substitue les termes $R, L, C$ par les \cref{eq:alpha,eq:omega_0}}\\
+		\ddt{2}V_S = \ddt{2}\vlt + 2\al\ddt{}\vlt + \omega^2\vlt\\
+		\nonumber \text{On pose la forme de la solution homogène: } V_{L_h} = Ae^{\la t} = 0\\
+		0 = \la^2Ae^{\la t} + 2\al\la Ae^{\la t} + \omega_0^2Ae^{\la t}\\
+		0 = \cancelto{0}{Ae^{\la t}}\left(\la^2 + 2\al\la + \omega_0\right)\\
+		\la_{1,2} = \frac{-2\al \pm \sqrt{(2\al)^2 - 4\omega_0^2}}{2}\\
+		\la_{1,2} = -\al\pm\sqrt{\al^2 - \omega_0^2}\\
+		\nonumber\text{Puisque le discriminant est négatif, on le multiplie par $-1$ et on sort $j$.}\\
+		% \la_{1,2} = -\al\pm j\sqrt{\}
 	\end{gather}
 
 	\subsection{Décharge}
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