## 1 Introduction
CVSRT reflects a duration from the launch of voltage sags to the instant when an IM's speed becomes equal to the rated or other predefined value. CVSRT can be determined in several manners [18]. First, it can be measured by time-domain dynamic simulation that requires a complete set of IM parameters and an appropriate program package and mathematical equations set. This approach is quite monotonous since the simulation should be repeated until the correct CVSRT is found. Second, CVRST can be measured using finite element methods (FEMs). However, these methods require many machine data, including complete machine geometric data. Therefore, it is not a practical and hand-useful method, particularly for older machines [7; 8; 9]. Third, CVRST can be determined using an analytical solution of the speed-time characteristics curve. These approaches are beneficial as they give simple expressions [13; 23], and can be calculated using different program packages (such as Matlab, Maple, and others). In the literature, [PERSON], in [9], has presented an analytical formula for starting time calculation of medium- and high-voltage induction motors under conventional starter methods. However, to control the solution using his suggested expression, [PERSON] has used some additional parameters, and therefore the developed formulation is not entirely mathematically correct. Likewise, [PERSON] has introduced different solutions for this problem in [8; 20; 24]. He developed an invertible speed-time characteristics curve during machine startup using Kloss expression for the machine's torque representation [24]. In [20], invertible expressions were derived using the machine torque's full expression based on the Thevenin theorem and the machine's single-cage equivalent circuit. An analytical solution for IM acceleration and maximum acceleration value was presented in [24]. However, the expressions given did not include the bearing losses in the suggested models; therefore,IMs' speed-time characteristics modeling during no-load direct startups were not accurate enough. The IM starting time and speed-time curves can also be determined using numerical time-domain computation (NTDC) approaches [13]. NTDC methods assume representing the machine with a set of differential equations for electrical machine circuits, and mechanical motion. However, to describe the IM dynamic's behavior, numerical integration techniques should be used. NTDC can be implemented in specific program packages (for instance, PSpice, Matlab, Mathematica, etc.). However, these methods are slow, time-consuming, and require the usage of special software. This paper aims to present a novel mathematical expression for speed-time characteristics representation during DOL machine starting, including the bearing losses to accurately formulate IMs' speed-time characteristics modeling during no-load direct startups. In this regard, two expressions of the speed-time characteristics of IMs during no-load direct startup are derived. The first expression was derived based on the Kloss equation used for representing the torque, and the second expression was derived based on the torque expression determined from the Thevenin equivalent circuit of the machine. The goal is also to compare the speed-time characteristics obtained by using the proposed expressions with the results obtained using the expressions known in the literature. The derived expressions' accuracy was validated using laboratory measurement and computer simulation approaches. The results obtained show that the proposed formulations enable accurate machine modeling under the worst condition--during direct startup. It should be mentioned that the proposed method did not address saturation of the magnetic circuit and current displacement in the rotor bars, as the IEEE standard T-equivalent scheme for IM machine parameters estimation was used. These phenomena significantly impact torque-time and current-time curves, which may change the engine startup time.