We consider the operation principles of the sensor of an external electric field on the bases of microring resonators that consist of bent-strip waveguides with vertical or horizontal slots filled with nematic liquid crystal. The mode-field distribution and dispersion parameters of the bent-slot waveguides are calculated by using the algorithm based on the method of lines. The influence of the waveguide and microresonator structure (slot width, position and orientation, microresonator radius, etc.) on the sensor sensitivity is analyzed.