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ADS1284: Low frequency noise peaks in shorted-input noise spectrum

Part Number: ADS1284
Other Parts Discussed in Thread: LM27762, REF6250

We are evaluating ADS1284 and during our tests, on our prototype board, we have noticed that while we achieve very similar SNR (RMS noise) numbers to the datasheet, we are seeing a few low-frequency (~3 Hz, ~6 Hz) monochromatic noise peaks in the shorted-input data spectrum.

We do not think it is the power supply or reference, but with the noise being monochromatic, the power system would be my first guess.  We run 4 ADS1284's all off the same +/- 2.5V analogue supplies and use the REF6250 reference. The +/- 2.5V supplies are generated from a 3.3V SMPS ---> LM27762 ---> +/- 2.5V ---> (REF6250) ---> ADS1284