Fields with frequencies below megahertz are challenging for Rydberg-atom-based measurements, due to the low-frequency electric field screening effect that is caused by the alkali-metal atoms adsorbed on the inner surface of the container. In this paper, we investigate on electric fields measurements in the ULF, VLF and LF bands in a Cs vapor cell with built-in parallel electrodes. With optimization of the applied DC field, we achieve high-sensitive detection of the electric field at frequencies of 1kHz, 10kHz and 100kHz based on Rydberg-atom sensor, with the minimum electric field strength down to 18.0μV/cm, 6.9μV/cm and 3.0μV/cm, respectively. The corresponding sensitivity is 5.7 μV/cm/{\sqrt{Hz}}, 2.2μV/cm/{\sqrt{Hz}} and 0.95μV/cm/{\sqrt{Hz}} for ULF, VLF and LF fields, which is better than 1-cm dipole antenna. Besides, the linear dynamic range of Rydberg-atom sensor is over 50 dB. This work presents the potential to enable more applications that utilize atomic sensing technology in ULF, VLF and LF fields.

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