## 1 Introduction
### [PERSON]'s [PERSON] have described two invertible expressions for the speed-time curve representation of IM during direct startup. The following expression was derived in [20]:  
\[t=\frac{J\omega_{s}^{2}}{U_{T}^{2}R_{2}}\left(\frac{\left(R_{T}^{2}+\left(X_{T}+ X_{2}\right)^{2}\right)}{2}\left(1-s^{2}\right)+2R_{T}R_{2}(1-s)-R_{2}^{2}\log(s)\right) \tag{2}\]  
where \(s\) is the machine's slip, \(R_{T}\) and \(X_{T}\) and \(U_{T}\) represent the Thevenin equivalent resistance and reactance and voltage of the IM's equivalent circuit, respectively. On the other side, in [24], [PERSON] has proposed the following expression to represent the IM's speed-time characteristics during direct startup. \[t=\frac{J\omega_{s}}{2M_{br}}\left(\frac{1-s^{2}}{2s_{br}}-s_{br}\log(s)\right) \tag{3}\]  
where \(M_{br}\) is the maximum machine torque, and \(s_{br}\) is its corresponding slip. Details about the parameters given in Equations (2) and (3) are presented in Appendix A. However, it should be noted that for both derived expressions, the bearing losses were ignored. Proposed Expressions for Speed-Time Characteristics Representation, with Bearing Losses 2.3.1. The First Expression  
The basic mechanical equation that represents rotation of any IM is given based on [PERSON]'s second law of motion as follows:  
\[J\frac{d\omega}{dt}=M_{em}-M_{friction} \tag{4}\]  
The electromagnetic torque (\(M_{em}\)) can be expressed in terms of the Kloss equation. In addition, the torque due to friction (\(M_{friction}\)) can be represented using the speed of rotation. The time increment (\(dt\)) can be calculated as follows:  
\[dt=\frac{J\omega_{s}s^{2}+J\omega_{s}s_{br}^{2}}{B\omega_{s}s^{3}-B\omega_{s}s ^{2}+\left(2M_{br}s_{br}+B\omega_{s}s_{br}^{2}\right)s-B\omega_{s}s_{br}^{2}}ds \tag{5}\]  
Therefore, the total startup time up to realize a certain speed is obtained as follows:  
\[t=\int\limits_{1}^{s}\frac{J\omega_{s}s^{2}+J\omega_{s}s_{br}^{2}}{B\omega_{s }s^{3}-B\omega_{s}s^{2}+\left(2M_{br}s_{br}+B\omega_{s}s_{br}^{2}\right)s-B \omega_{s}s_{br}^{2}}ds \tag{6}\]  
Equation (6) can be written in a more general mathematical form as follows:  
\[t=\int\limits_{1}^{s}\frac{\alpha_{1}s^{2}+\alpha_{2}s+\alpha_{3}}{\alpha_{4 }s^{3}+\alpha_{5}s^{2}+\alpha_{6}s+\alpha_{7}}ds \tag{7}\]  
so that:  
\[ \begin{array}{l}\alpha_{1}=J\omega_{s},\\ \alpha_{2}=0,\\ \alpha_{3}=J\omega_{s}s_{br}^{2},\\ \alpha_{4}=B\omega_{s},\\ \alpha_{5}=-\alpha_{4},\\ \alpha_{6}=2M_{br}s_{br}+B\omega_{s}s_{br}^{2},\text{and}\\ \alpha_{7}=-B\omega_{s}s_{br}^{2}\end{array}  \tag{8}\]The previous integral expression can be easily solved for any IM with known appropriate parameters. Firstly, one can formulate the solution expression as follows, in which the unknown coefficients \(k_{1}\), \(k_{2}\), \(k_{3}\), \(s_{1}\), \(r_{1}\), \(r_{2}\), and \(r_{3}\) should be determined. \[\frac{\alpha_{1}s^{2}+\alpha_{2}s+\alpha_{3}}{\alpha_{4}s^{3}+\alpha_{5}s^{2}+ \alpha_{6}s+\alpha_{7}}=\frac{k_{1}}{s-s_{1}}+\frac{k_{2}s+k_{3}}{r_{1}s^{2}+r _{2}s+r_{3}} \tag{9}\]  
The coefficient \(s_{1}\) represents a real pole of the equation; thus:  
\[\alpha_{4}s^{3}+\alpha_{5}s^{2}+\alpha_{6}s+\alpha_{7}=0 \tag{10}\]  
The coefficients \(r_{1}\), \(r_{2}\), and \(r_{3}\) are determined by dividing the polynomial \(\alpha_{4}s^{3}+\alpha_{5}s^{2}+\alpha_{6}s+\alpha_{7}\) with the polynomial \(s-s_{1}\). The other coefficients can be defined as follows:  
\[k_{1}\Big{(}r_{1}s^{2}+r_{2}s+r_{3}\Big{)}+(s-s_{1})(k_{2}s+k_{3})=\alpha_{1}s ^{2}+\alpha_{2}s+\alpha_{3} \tag{11}\]  
Then, a system of equations is formed as follows:  
\[\left[ \begin{array}{c}k_{1}\\ k_{2}\\ k_{3}\end{array} \right]=\left[ \begin{array}{ccc}r_{1}&1&0\\ r_{2}&x_{1}&1\\ r_{3}&0&-x_{1}\end{array} \right]^{-1}\left[ \begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \alpha_{3}\end{array} \right] \tag{12}\]  
Hence, the time expression is obtained by solving the following integrals.