Circular differential microphone arrays (CDMAs) have been extensively studied in speech and audio applications for their steering flexibility, potential to achieve frequency-invariant directivity patterns, and high directivity factors (DFs). However, CDMAs suffer from both white noise amplification and deep nulls in the DF and in the white noise gain (WNG) due to spatial aliasing, which considerably restricts their use in practical systems. The minimum-norm filter can improve the WNG by using more microphones than required for a given differential array order; but this filter increases the array aperture (radius), which exacerbates the spatial aliasing problem and worsens the nulls problem in the DF. Through theoretical analysis, this research finds that the nulls of the CDMAs are caused by the zeros in the denominators of the filters' coefficients, i.e., the zeros of the Bessel function. To deal with both the white noise amplification and deep nulls problems, this paper develops an approach that combines different rings of microphones together with appropriate radii. The resulting robust concentric circular differential microphone arrays (CCDMAs) can mitigate both problems. Simulation results justify the superiority of the robust CCDMA approach over the traditional CDMAs and robust CDMAs.