To get a feel for the Capacitive Bridge problems, we setup a simple bridge using fixed (1 nF) caps on a breadboard. We used an SR830 Lock-In amplifier to drive it and readout the noise.
We measured the cap values with an LCR meter. They were all within a few % of 0.99 nF.
With a 0.5 V drive to the top of the bridge, the A-B voltage was ~2 mV as expected from the matching of the capacitors.
(** Note about the gain in the SR830: In order to find the magnitude of the input referred signal, one has to divide by G. G = (10 V)/ Sensitivity. 'Sensitivity' is the setting on the front panel.)
Hooking up now to A-B: the signal is ~10x larger than the 'dark' noise everywhere. 2 uV/rHz @ 100 Hz, 10 uV/rHz @ 10 Hz, 50 uV/rHz @ 1 Hz. The spectrum is very non-stationary; changing by factors of several up and down between averages. Probably a problem with the cheapo contacts in the breadboard + wind. The gain in this state was still 1000. So at 1 Hz, its 50 nV/rHz referred to the input.
To convert into units of capacitance fluctuation, we multiply by the capacitance of the capacitors (1 nF) and divide out by the peak-peak voltage (1 V). So the bridge sensitivity is 50e-9 * 1e-9 = 5 x 10^-17 F/rHz.
If we assume that we will have a capacitive displacement transducer giving 1 nF capacitance change for a 0.1 mm displacement, this bridge would have a sensitivity of 5 x 10^-12 m/rHz @ 1 Hz. We would like to do ~50-100x better than this. The next steps should be: