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 LockIn 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 AB 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.)
 Directly measuring from Vs to ground gives 0.5 V, as expected. This is done to verify the calibration later on.
 Shorting the A and B wires to ground gives ~0 V and lets us measure the noise. On the spectrum analyzer it was ~400 nV/rHz at 100 Hz and rising slowly to 4 uV/rHz at 100 mHz. In this state, the sensitivity was 10 mV, so the overall gain was 1000. That gives an input referred level of ~0.4 nV/rHz at the input.

Hooking up now to AB: 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 nonstationary; 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 peakpeak voltage (1 V). So the bridge sensitivity is 50e9 * 1e9 = 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 ~50100x better than this. The next steps should be:
 Solder it all together on a PCB to have less air current sensitivity and decent contacts.
 Use a lownoise FET input. Since the impedance of the bridge is ~5 kOhms at this frequency, we are probably current noise limited.
 Estimate the oscillator amplitude noise sensitivity.
