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| author | Benjamin Chausse <benjamin@chausse.xyz> | 2024-12-03 18:06:25 -0500 |
|---|---|---|
| committer | Benjamin Chausse <benjamin@chausse.xyz> | 2024-12-03 18:06:25 -0500 |
| commit | 41e65d200ace53398d466f23ef91f88695b1eb05 (patch) | |
| tree | 4208be6402e4ad10193dcb795295cea7a13c5ebb | |
| parent | 663b8f8b7560508afbac568412cfda9ea57130c5 (diff) | |
Up to bertha decharge
| -rw-r--r-- | annexe/main.tex | 21 |
1 files changed, 15 insertions, 6 deletions
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} |