Analytical solution for no‐load induction machine speed calculation during direct start‐up

Abstract

Summary This paper deals with the investigation into mathematical modeling of induction machine speed‐time and time‐speed dependencies during no‐load direct start‐up. Namely, a precise analytical formula for induction machine speed calculation during no‐load direct start‐up is given and solved in the form of the Lambert W function. The results obtained by using developed analytical solution are compared with corresponding results obtained by using methods based on numerical time‐domain computation (NTDC) that have been realized in two different program packages. Furthermore, the results obtained by using developed analytical solution are compared with corresponding experimental results. The findings showed good agreement between the obtained results. All developed MATLAB programs for speed‐time and time‐speed curve determination are given in the paper. The Mathematica program developed for solving the Lambert W equation is also provided.

An invertible dependence of the speed and time of the induction machine during no-load direct start-up

“…The IM starting process is the basic scientific and practical problem of this machine with a long tradition of observation [3][4][5][6][7][8][9][10][11][12][13][14][15]. The most common starting method of this machine is the direct-on-line method [3,4,8,11], which can be realized with or without the reduced voltage technique. Very popular conventional methods are also the variable rotor resistance method [9], the star-delta method [10] and the method based on the usage of auto-transformer [11][12].…”

“…Namely, during the IM direct start-up, the machine current can be 6-7 times higher than the rated phase current, which can cause the activation of machine relays [12,22,23]. For that reason, the IM starting process duration is very important for proper machine protection-relay settings (for example, setting of the over-current protective devices) [3][4][5]22,23]. In addition, the high IM phase current can cause voltage sags (for example, in distribution power systems, in small power system areas, etc.…”

“…The research into the IM starting time calculation is presented in several papers [3,4,[28][29][30][31][32]. Namely, a simple mathematical equation for starting time calculation, based on machine speed, torque and mechanical parameters, is presented in [30].…”

“…On the other hand, the IM starting time and speed-time curves, can also be determined by using Numerical Time-Domain Computation (NTDC) methods [3,4,30]. NTDC methods assume describing the machine with a set of differential equations (differential equations for describing machine electrical circuits as well as differential equations for describing mechanical motion) and their realization or implementation in certain programme packages (for example PSpice, Matlab, Mathematica, etc.).…”

“…However, all programme packages solve differential equations by using numerical techniques (for example -Backward Euler method, Dormand-Prince method, Bogacki-Shampine method, etc.) [3,4,30], and therefore require the usage of special programme packages.…”