Bridge Sensitivity Formula
For a differential push-pull capacitive displacement sensor used in an AC bridge, where the output voltage is independent of the supply frequency, the sensitivity ($S$) is often characterized by the following relationship:
$ S = \frac{V_s}{2 C_0} $
This formula assumes an optimized bridge design or operating condition that nullifies frequency dependence. Here:
- $V_s$ represents the supply voltage.
- $C_0$ denotes the nominal capacitance of the sensor.
Sensor Parameters
The given parameters are:
- Supply Voltage, $V_s = 1\text{ V}$
- Nominal Capacitance, $C_0 = 0.01 \text{ }\mu F$
First, convert the nominal capacitance from microfarads ($ \mu F $) to Farads ($ F $):
$ C_0 = 0.01 \text{ }\mu F = 0.01 \times 10^{-6} \text{ F} = 1 \times 10^{-8} \text{ F} $
Sensitivity Calculation
Substitute the values of $ V_s $ and $ C_0 $ into the sensitivity formula:
$ S = \frac{1\text{ V}}{2 \times (1 \times 10^{-8} \text{ F})} = \frac{1}{2 \times 10^{-8}} \frac{\text{V}}{\text{F}} = 0.5 \times 10^8 \frac{\text{V}}{\text{F}} $
Unit Conversion to mV/pF
The options are provided in millivolts per picofarad (mV/pF). Convert the calculated sensitivity:
- $1 \text{ V} = 1000 \text{ mV}$
- $1 \text{ F} = 10^{12} \text{ pF}$
Apply these conversion factors:
$ S = 0.5 \times 10^8 \frac{\text{V}}{\text{F}} \times \frac{1000 \text{ mV}}{10^{12} \text{ pF}} = 0.5 \times 10^8 \times 10^3 \times 10^{-12} \frac{\text{mV}}{\text{pF}} $
$ S = 0.5 \times 10^{(8+3-12)} \frac{\text{mV}}{\text{pF}} = 0.5 \times 10^{-1} \frac{\text{mV}}{\text{pF}} $
$ S = 0.05 \text{ mV/pF} $
This calculated sensitivity matches one of the provided options.
Important Questions from Capacitive Transducers
The dynamic characteristics of capacitive transducers are similar to those of
- What device would you use to measure the output of a thermocouple ?
An air filled parallel plate electrostatic actuator is shown in the figure. The area of each capacitor plate is $100 \mu m \times 100 \mu m$. The distance between the plates $d_0 = 1 \mu m$ when both the capacitor charge and spring restoring force are zero as shown in Figure (a). A linear spring of constant $k=0.01 N/m$ is connected to the movable plate. When charge is supplied to the capacitor using a current source, the top plate moves as shown in Figure (b). The magnitude of minimum charge (Q) required to momentarily close the gap between the plates is _________ $\times 10^{-14} C$ (rounded off to two decimal places).
Note: Assume a full range of motion is possible for the top plate and there is no fringe capacitance. The permittivity of free space is $\epsilon_0 =8.85\times 10^{-12} F/m$ and relative permittivity of air ($\epsilon_r$) is 1.
A capacitive motion transducer circuit is shown. The gap $d$ between the parallel plates of the capacitor is varied as $d(t)=10^{-3}[1+0.1\sin(1000\pi t)]$ m. If the value of the capacitance is 2pF at $t = 0$ ms, the output voltage $V_o$ at $t = 2$ ms is
A parallel plate capacitive displacement sensor has a plate area of $2\text{ cm}^2$. The air gap between the plates is decreased by $0.1\text{ mm}$ from an initial value of $0.5\text{ mm}$. The percentage change in the capacitance value is ______ %.
(Assume permittivity as $\epsilon_0 = 8.854 \times 10^{-12}\text{ F/m}$)
