From ec92d95c56b8dbfbc7a37224b5b952b22493b838 Mon Sep 17 00:00:00 2001 From: Vincent Date: Fri, 8 Nov 2024 08:00:32 +0100 Subject: [PATCH] Bonjour Valentin --- Physique1A/Physique_1A_05_C_OH.markdown | 19 ++++++++++++++++++- Physique1A/img/04_C/RLC_series.jpg | Bin 0 -> 16764 bytes assets/manim/RLC_resolution.mp4 | Bin 0 -> 42453 bytes 3 files changed, 18 insertions(+), 1 deletion(-) create mode 100644 Physique1A/img/04_C/RLC_series.jpg create mode 100644 assets/manim/RLC_resolution.mp4 diff --git a/Physique1A/Physique_1A_05_C_OH.markdown b/Physique1A/Physique_1A_05_C_OH.markdown index 7a61327..b735dc0 100644 --- a/Physique1A/Physique_1A_05_C_OH.markdown +++ b/Physique1A/Physique_1A_05_C_OH.markdown @@ -184,6 +184,8 @@ $$y(t) = A \exp(-r_1t) + B \exp(-r_2t)$$ où $r_1$ et $r_2$ sont les deux racines du polynôme caractéristique. (En général pour un problème physique, on aura deux racines négatives pour un problème amorti. Sinon la solution tend vers l'infini...) + + $A$ et $B$ sont déterminés par les conditions initiales du problème. On appelle cela le régime **apériodique**. Le système revient vers un état de repos sans oscillations. @@ -198,4 +200,19 @@ $A$ et $B$ sont déterminés par les conditions initiales du problème. Il s'agit du cas limite pour lequel il n'y a pas d'oscillations. -**$\Delta < 0$, les racines sont complexes :** \ No newline at end of file +**$\Delta < 0$, les racines sont complexes :** + +Dans ce cas, $r_{1,2} = -\dfrac{1}{2 \tau} \pm j\sqrt{{\omega_0}^2-\dfrac{1}{4 \tau^2}}$ + +**Remarque :** En physique, et tout particulièrement en électronique, on a tendance à écrire le nombre complexe $j$ qui correspond à $j^2 = -1$. Le but est de ne pas le confondre avec l'intensité électrique, par exemple. + +Afin de simplifier l'écriture de $r_{1,2}$, on va souvent poser : $\alpha = -\dfrac{1}{2 \tau}$ et $\omega = \sqrt{{\omega_0}^2-\dfrac{1}{4 \tau^2}}$ pour écrire finalement : +$r_{1,2} = \alpha \pm j\omega$ + +Ce qui permettra d'écrire la solution : + +$$\boxed{y(t) = A e^{-\alpha t} \cos(\omega t + \varphi)}$$ + +On appelle ce régime un **régime pseudo-périodique**. Le système va revenir vers son état de repos en oscillant autour de la position d'équilibre. Ces oscillations sont caractérisées par la **pseudo-période** $\omega$. + + \ No newline at end of file diff --git a/Physique1A/img/04_C/RLC_series.jpg b/Physique1A/img/04_C/RLC_series.jpg new file mode 100644 index 0000000000000000000000000000000000000000..185f2fd6d30aa974715e64c09502ff9ebbb5f876 GIT binary patch literal 16764 zcmeHu2|Sc--}WW@zVFKvipZ8VVOp#SX(8*BB_SlqV45jH_7I99YlM`2pX^DsoT5cL@8vp?S8hUWFdjtOEL9-7WBLgE7GYc#Df=Vu69}O+-K04aHQG>6(3qB9f zanti0P&vuKYk8GX(wFb({e&zgsZ(#8`Oo*`q*a}6g)p-S2nq>{96Tf=dst3QT|-mr znD%LX14AQYlQS1CUb3>jY-8)};(E>Py1R$pZT~v~@W7zZ2Vvn64JFEYg>Cq=liY?1A{}uBco%V$7e9u*|~4u=NA_7>pwO& z30uVNoxOR{0JMLd*1yf{SM%Zq=e3WHj+TybZ(cO}0>Fcon~wf~3IoqcOUA3dypl)n zGx42D$a>q%ETwuL$M1BjpG81gZRQ|;Z)$&;**`b2kUurEf1B99&ubiDqon~CkCq#N z0^|$XQL@1Q2M6st9gE$I4n!Zqy#YOL&WQ13D)4aA^O(Qr@rUZHKIQ2b71_#oCUS&h zV)%P+GI*M{XI5-hw=@Rpmk|?Y#9_@rR+Wf4uhdA1V9!RiFK-S?CSx9Tao-SA?GOx2 z5z@lJLXlDt9aGg*pws7}wT5k6RzA_ZWukcpks@&PtCwqA>VBpnQj76K99s7}$-du( 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