Dise帽o y simulaci贸n de un aceler贸metro MEMS
Analizando el siguiente modelo:
\[L_f = L_{fvo} + L_x\]
Capacitancia
\[A = L_{fov} tc\]
\[C = \frac{Q}{V} = \frac{\varepsilon A}{d}\]
Donde:
$\varepsilon$: Permitividad
\[C_1 = \frac{\varepsilon A}{d+x}\quad,\quad C_2 = \frac{\varepsilon A}{d-x}\]
\[\begin{aligned}
\Delta C &= C_1 - C_2 = \varepsilon A \left(\frac{1}{d +x} - \frac{1}{d-x}\right)\\
&= \varepsilon A\left(\frac{-2x}{d^2 - x^2}\right)
\end{aligned}\]
Desarrollando:
\[\tag{1}
\Delta C x^2 - 2\varepsilon A x - \Delta C d^2 = 0\]
Resolviendo para $(1)$:
\[\boxed{x = \frac{\varepsilon A}{\Delta C} \pm \sqrt{\left(\varepsilon \frac{A}{\Delta C}\right)^2 + d^2}}\]
Fuerza el茅ctrica
\[F_e = \frac{1 \varepsilon A V^2}{2 d^2} = ma\]
\[\boxed{m = \frac{1 \varepsilon A V^2}{2 a d^2}}\]
Resorte
\[Fr = kx = ma\]
\[\boxed{k = \frac{ma}{x}}\]