Five means of approximation of critical values of the damping coefficient in the no-current model of the synchronous electric motor

Under some assumptions, operation of a synchronous electric motor and operation of a phase locking 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 ODE 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, and that induced a number of mathematicians to derive analytical estimates for the critical values from above and below. The authors of the present paper had carried out the computer analysis of the equation under consideration, and in the course of it they proposed a linear and sinusoidal approximations of the critical values and calculated their errors. In the present paper, another one means of the sinusoidal approximation and two means of the parabolic approximations are considered, and maximum modulus of their absolute and relative errors are calculated. These computations show that the linear approximation provides calculation of critical values with absolute and relative errors of the order of $10^{-2}$, the parabolic approximation ensures their calculation with absolute and relative errors of the order of $10^{-3}$, and the sinusoidal approximation provides the errors of the order of $10^{-5}$.