  We describe the design principles, fabrication, and characterization of a
precision AC resonant capacitance bridge (RCB) sensor, based on a resonant
differential planar printed circuit board transformer with a solid (ungapped)
MnZn ferrite core, demonstrating a short-term sensitivity at 293 K of 0.225 +/-
0.005 aF/rtHz, at around 120 kHz resonance frequency and 1 Hz Fourier
measurement frequency. At 120 K the RCB short term noise sensitivity is 0.118
+/- 0.005 aF/rtHz. We compare the ungapped configuration to five different
RCBs; three with a core gap of 65 um and two with a core gap of 130 um. Their
average room temperature short term noise sensitivities are 0.30 +/- 0.01
aF/rtHz and 0.45 +/- 0.01 aF/rtHz, while the cryogenic operation of these
transformers at 120 K resulted in averaged sensitivities of 0.23 +/- 0.01
aF/rtHz and of 0.36 +/- 0.01 aF/rtHz respectively. Multi-hour room temperature
runs, with one core of each of the three gap types, proved the stability of
their long-term sensitivities of 0.234 +/- 0.005 aF/rtHz, 0.338 +/- 0.009
aF/rtHz, and 0.435 +/- 0.010 aF/rtHz for the ungapped (40-hour duration) and
the 65 um and 130 um (28-hour duration) cores respectively. At 0.1 mHz, a
critical frequency for space gravitational wave detectors, the respective
sensitivities are 0.25 +/- 0.02 aF/rtHz, 0.35 +/- 0.02 aF/rtHz, and 0.53 +/-
0.07 aF/rtHz. Measurements with the ungapped transformer configuration for
temperatures from 325 K to 349 K further validate the dependence of the noise
model on the temperature and permeability. The performance of our RCB with an
ungapped core matches the calculated performance value and shows an improvement
in signal-to-noise of two or more compared with capacitance bridges developed
for similar applications. A further factor of about two noise reduction is
achieved by cooling to 120 K.
