Five Means of Approximation of Critical Values of the Damping Coefficient in the No-Current Model of a Synchronous Electric Motor
摘要
Under certain assumptions, the operation of a synchronous electric motor and operation of a phase-locked loop system can be described by a second-order differential equation, which includes two dimensionless parameters and does not include electric currents. This equation takes a significant place in Leonov’s nonlocal reduction method, which provides conditions when the global asymptotic stability of this equation alone entails the global asymptotic stability of a multidimensional phase system. F. Tricomi established that there exists a critical value for the linear damping coefficient in this equation, which is a continuous function of another parameter and separates the case when the given equation is globally stable from cases without global stability. The critical values have no explicit representation. This has prompted a number of mathematicians to derive analytical upper and lower bounds to the critical values. We earlier carried out computer analysis of the equation under consideration, and we proposed linear and sinusoidal approximations of the critical values and calculated their errors. In the present work, we consider another means of the sinusoidal approximation and two means of the parabolic approximations, and the maximum moduli of their absolute and relative errors are calculated. These considerations show that the linear approximation allows the calculation of critical values with absolute and relative errors on the order of 10–2, the parabolic approximation ensures their calculation with absolute and relative errors on the order of 10–3, and the sinusoidal approximation provides errors on the order of 10–5.