An electrical system includes a cycloidal electric machine having a stator and a balanced rotor. The rotor is eccentrically positioned radially within the stator, such that the rotor moves with two degrees of freedom (2DOF). The 2DOF motion includes rotating motion about the rotor axis and orbiting motion about the stator axis. A rotor constraint mechanism constrains the motion of the rotor, such that the rotor is able to generate torque on a coupled load. Part of the rotor constraint mechanism may be integrally formed with the rotor. A coupling mechanism may be coupled to the rotor and configured to translate the 2DOF into 1DOF, i.e., rotation without orbital motion. The rotor may include mutually-coupled rotors. At least one counterweight may be connected to the rotor, e.g., externally or within the airgap. Balancing of the balanced rotor may be optionally provided via the multiple rotors and/or the counterweight(s).