diff options
Diffstat (limited to 'annexe')
| -rw-r--r-- | annexe/main.tex | 33 |
1 files changed, 22 insertions, 11 deletions
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} |