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]]