The effect of the radial and tangential electromagnetic stresses on the vibrations of synchronous machines

The main effect due to rotor eccentricity in electrical machines is an imbalance of the electromagnetic forces acting upon rotor and stator surfaces, with a consequent development of a net radial force, known in the literature as Unbalanced Magnetic Pull. The authors of previous researches have generally calculated this force by means of the expression of the radial component of the Maxwell stress tensor, neglecting the tangential one. In this paper different approaches will be considered in order to evaluate the electromagnetic stresses acting in synchronous machines: the circuit approach, the field approach and a novel approach based on the electromechanical features of the machine. From the Maxwell theory, many authors maintain that the air-gap magnetic field in rotating electrical machines can not be purely radial, otherwise the phenomenon of the torque generation could not be explained. In this paper, the generation of torque is explained by the Lorentz forces, which act tangentially at the air-gap just considering the field as radial. These tangential forces can be associated to a tangential stress, which in turn can be expressed as the Maxwell tangential tensor and therefore correlated to an equivalent tangential component of the magnetic field. A definition of the equivalent tangential flux density and a relationship between the tangential and the radial stresses will be found. Therefore, the values of these parameters will be estimated in case of machines with salient-poles (hydrogenerators) and smooth-poles (turbo-generators). Finally, the effect of rotor eccentricity on open circuit and in load condition will be evaluated, by modifying the expressions of the forces due to the unbalanced magnetic pull obtained by the authors in a previous research. The results point out that, in steady-state conditions, the tangential stress can be actually disregarded for both types of synchronous machines, even if its effect is less negligible for the smooth-poles machines than for the salient-poles ones, due to the constructive characteristics of these machines.