## 3 Capacitive Sensing with Electronic Detection
### Electronic Detection System
#### 3.3.4 Sensitivity and Noise Considerations  
The charge amplifier (see amplifiers on Figure 12) is built of a high gain, single-ended, folded cascade low-noise operational amplifier with feedback impedance formed by parallel connection of \(C_{f}\) and \(R_{f}\). The time constant is selected in such a way that the _S/N_ at the output is optimal. The resistance \(R_{f}\) must be very high, _i.e._, in a range of several tens of \(\mathrm{G}\Omega\) in order to implement low pole frequency at small capacitor \(C_{f}\)[29]. One charge amplifier processes charges from two differential sensors that are driven by excitation signals (\(V_{\mathrm{S1}p}\), \(V_{\mathrm{S1}n}\)) and (\(V_{\mathrm{S2}p}\), \(V_{\mathrm{S2}n}\)) with different frequencies.  
The decisive parameter regarding the detection sensitivity of a measurement system is the signal to noise ratio at the output of the charge amplifier (2) in 1 Hz bandwidth around the corresponding spectral line. The details of those calculations are given in Supplementary file. The _S/N_ ratio is proportional to the excitation signal amplitude \(V_{Sx}\) and the ratio \(\delta C_{x}\)/\(C_{f}\) and inversely proportional to noise of the amplifier \(V_{\mathrm{ndop}}\) and the parasitic capacitance gain ratio given by \(\left(1+\sum C_{\mathrm{VG}}\big{/}C_{f}\right)\). These considerations instruct us about the optimum characteristics of the sensor-electronic pair:  
\[\left(\frac{S}{N}\right)\cong\frac{V_{\mathrm{Cho}}}{\sqrt{2}\cdot V_{\mathrm{ ndCho}}}=\frac{V_{Sx}\frac{\delta C_{x}}{\sqrt{2}C_{f}}}{\sqrt{3}\left[V_{ \mathrm{ndop}}\left(1+\frac{\sum C_{\mathrm{VG}}}{C_{f}}\right)\right]} \tag{2}\]  
The minimum detectable capacitance difference of sensor x can be calculated using Equation (3), taking into consideration that the signal power must be at least three times as large as the noise power in the specified band. The charge amplifier is designed with the following parameters: \(V_{\mathrm{ndop}}\left(\omega_{x}\right)\leq 7.5\cdot 10^{-9}\)\(\mathrm{V}\cdot\mathrm{S}^{\mathrm{1}\mathrm{2}}\), \(V_{Sx}=5\,\mathrm{V}\), \(C_{f}=2\,\mathrm{pF}\) and \(\sum C_{\mathrm{VG}}=2\,\mathrm{pF}\). The reduction of the _S/N_ ratio caused by analogue and digital mixers is negligible, because each mixing contributes approximately \(\sqrt{2}\), if designed properly [30, 31]:  
\[\delta C\cong 3\sqrt{2}\,\frac{V_{\mathrm{ndop}}}{V_{Sx}}\left(C_{f}+\sum C_{ \mathrm{VG}}\right)\cong 4.5\cdot 10^{-20}\left[\frac{F}{\sqrt{\mathrm{Hz}}}\right] \tag{3}\]  
For a comb capacitive sensor with the separation between two comb fingers of \(d_{0}\cong 1\,\mathrm{\SIUnitSymbolMicro m}\), and the initial capacitance of approximately \(C_{0}=0.5\,\mathrm{pF}\), it is possible to estimate the minimum detectable distance change in plates \(\delta_{\mathrm{m}}\) between two fingers due to the adsorption according to Equation (4) and it is approximately 80 fm that is well below the thickness of one layer of adsorbed target molecules estimated to 0.5 nm [32] for the TNT. This fact can be used to measure the density of target molecules in the atmosphere around the sensor. The fact that \(\delta_{\mathrm{m}}\) is much smaller than the size of the target molecule makes the measurements possible even if the comb capacitor surface is not covered completely and the density of target molecules is smaller than the number of target molecules at vapour pressure:  
\[\delta_{=}\cong\frac{d_{0}\cdot\delta C}{\left(C_{0}+\delta C\right)}\cong 8 \cdot 10^{-14}\ m \tag{4}\]