Positive and negative damping in synchronous machines

The straight mathematical analysis of the damping problem on the synchronous machine leads to very complicated results. From these results, the conception of the phenomena occurring in the machine is very difficult and in addition the formulas obtained are so long that the calculation becomes tedious and errors can easily be made. In this paper, an attempt has been made to find a simple physical explanation for the positive and negative damping torques of the synchronous machine and on this basis to derive simpler formulas for the calculation of the damping torques. It has been found that the torque of the synchronous machine, as a double-fed machine, consists of three components, two dependent on the speed of the machine and the third dependent on the angle between the rotating field and the rotor. When the synchronous machine oscillates, the change of the first two components produces positive and negative damping torques of the synchronous machine. The change of the third component produces the synchronizing torque. The positive damping torque depends mainly on the primary voltage and secondary resistance; the negative damping torque depends mainly on the excitation and the primary resistance. This knowledge made it possible to derive a simple and very accurate formula for the positive damping torque as well as a simple approximate formula for negative damping torque. On the basis of these formulas, the influence of the two axes of the machine on the damping can be easily found. A comparison between the results obtained by using the simple formulas and the long accurate formulas as well as a comparison between calculation and test on a small machine have been carried through.