From 41e65d200ace53398d466f23ef91f88695b1eb05 Mon Sep 17 00:00:00 2001
From: Benjamin Chausse <benjamin@chausse.xyz>
Date: Tue, 3 Dec 2024 18:06:25 -0500
Subject: Up to bertha decharge

---
 annexe/main.tex | 21 +++++++++++++++------
 1 file changed, 15 insertions(+), 6 deletions(-)

(limited to 'annexe/main.tex')

diff --git a/annexe/main.tex b/annexe/main.tex
index 9b4a50f..e574faf 100644
--- a/annexe/main.tex
+++ b/annexe/main.tex
@@ -28,7 +28,6 @@
 \newcommand{\vlt}{V_L(t)}
 \newcommand{\vlz}{V_L(0)}
 
-\newcommand{\e}{\mathrm{e}}
 \newcommand{\al}{\alpha}
 \newcommand{\la}{\lambda}
 \newcommand{\wz}{\omega_{0}}
@@ -92,11 +91,21 @@
 	\subsection{Décharge}
 	\begin{gather*}
 		\begin{align}
-			0 & = \vlt + V_C(t) + V_R(t)                                                           \\
-			0 & = \vlt+\frac{1}{C}\int I_(t)\dt + R_I(t)                                           \\
-			0 & = \ddt{2}\left[ \vlt + \frac{1}{LC}\iint\vlt\dt^2 + \frac{R}{L}\ddt{} \vlt \right] \\
-			0 & = \ddt{2}\vlt+ \frac{R}{L}\ddt{}\vlt +\frac{1}{LC}\vlt                             \\
-			0 & = \ddt{2}+2\al\la A\e
+			0         & = \vlt + V_C(t) + V_R(t)                                                           \\
+			0         & = \vlt+\frac{1}{C}\int I_(t)\dt + R_I(t)                                           \\
+			0         & = \ddt{2}\left[ \vlt + \frac{1}{LC}\iint\vlt\dt^2 + \frac{R}{L}\ddt{} \vlt \right] \\
+			0         & = \ddt{2}\vlt+ \frac{R}{L}\ddt{}\vlt +\frac{1}{LC}\vlt                             \\
+			0         & = \ddt{2}+\vlt+2\al\ddt{}\vlt+\ws                                                  \\
+			\nonumber\text{Posons que: }
+			          &
+			\vlt=Ae^{\la t} \Rightarrow
+			\ddt{}\vlt=\la Ae^{\la t} \Rightarrow
+			\ddt{2}\vlt=\la^{2}Ae^{\la t}                                                                  \\
+			0         & = \la^2Ae^{\la t} +2\al Ae^{\la t} + \ws Ae^{\la t}                                \\
+			0         & = Ae^{\la t} \left(\la^2 +2\al\la+\ws \right)                                      \\
+			0         & = \la^2 +2\al\la+\ws                                                               \\
+			\la_{1,2} & = \frac{-2\al\pm\sqrt{(-2\al)^2-4\ws}}{2(-2\al)}                                   \\
+			\la{1,2}  & = -\al\pm\sqrt{\al^2-\ws}
 		\end{align}
 	\end{gather*}
 \end{appendix}
-- 
cgit v1.2.3

