The transfer function for the capacitor bank is
$$T(s)=\frac{ks}{md}$$
and the Transfer function for the mechanical system is
$$T(s)=\frac{1}{ms^2+bs+k}$$
When we take the inverse Laplace transform of these equations multiplied together we get the equations
$$\begin{matrix}
\dot{v}(t)=\dot{v}(t)\\
\ddot{v}(t)=-\frac{k}{m}v(t)-\frac{b}{m}\dot{v}+\frac{v_{in}}{md}F
\end{matrix}$$
With our states being $v(t),\dot{v}(t)$ the state-space model of the accelerometer becomes
$$\begin{bmatrix}
\dot{v}(t)\\ \ddot{v}(t)
\end{bmatrix}=
\begin{bmatrix}
0&1\\
-\frac{k}{m}&-\frac{b}{m}\\
\end{bmatrix}
\begin{bmatrix}
v(t)\\ \dot{v}(t)
\end{bmatrix}+
\begin{bmatrix}
0\\ \frac{v_{in}}{md}
\end{bmatrix}F$$
$$y(t)=
\begin{bmatrix}
-\frac{k}{m}C&-\frac{b}{m}C
\end{bmatrix}
\begin{bmatrix}
v(t)\\ \dot{v}(t)
\end{bmatrix}+C\frac{v_{in}}{md}F$$
- candyAccelerometerMode
[[Capacitive Accelerometer]]
General General Accelerometer State-Space Model
[[State-Space Model]]