Force-sensing capacitor

Typical force sensitive capacitors are examples of parallel plate capacitors. For small deflections, there is a linear relationship between applied force and change in capacitance, which can be shown as follows:

The capacitance, ${\displaystyle C}$, equals ${\displaystyle \varepsilon A/d}$, where ${\displaystyle \varepsilon }$ is permeability, ${\displaystyle A}$ is the area of the sensor and ${\displaystyle d}$ is the distance between parallel plates. If the material is linearly elastic (so follows Hooks Law), then the displacement, due to an applied force ${\displaystyle F}$, is ${\displaystyle x=F/k}$, where ${\displaystyle k}$ is the spring constant. Combining these equations gives the capacitance after an applied force as:

${\displaystyle C=\varepsilon A/(d_{nominal}-F/k)}$, where ${\displaystyle d_{nominal}}$ is the separation between parallel plates when no force is applied.

This can be rearranged to:

${\displaystyle C=(\varepsilon Ad_{nominal}+\varepsilon AF/k)/(d_{nominal}^{2}-F^{2}/k^{2})}$

Assuming that ${\displaystyle d_{nominal}^{2}>>F^{2}/k^{2}}$, which is true for small deformations where ${\displaystyle d_{nominal}>>x}$, we can simplify this to:

C ${\displaystyle \simeq (\varepsilon Ad_{nominal}+\varepsilon AF/k)/(d_{nominal}^{2})}$

It follows that:

C ${\displaystyle \simeq C_{nominal}+\varepsilon AF/kd_{nominal}^{2}}$

C ${\displaystyle \simeq C_{nominal}+BF}$ where ${\displaystyle B=\epsilon A/kd^{2}}$, which is constant for a given sensor.

We can express the change in capacitance ${\displaystyle \Delta C}$ as:

${\displaystyle \Delta C=BF}$

SingleTact makes force-sensitive capacitors using moulded silicon between two layers of polymide to construct a 0.35 mm thick sensor, with force ranges from 1 N to 450 N.[3] The 8mm SingleTact has a nominal capacitance of 75 pF, which increases by 2.2 pF when the rated force is applied.[3]

Force-sensing capacitors can be used to create low-profile force-sensitive buttons. They have been used in medical imaging to map pressures in the esophagus[4][5]and to image breast[6][7] and prostate cancer.[8]