## 2 Theory  
### Impedance at Different Slip Rates  
The most commonly used models in induction machine analysis can be divided into two categories: the transient model and the steady state model. As the transient time of the induction machine is very short, the transience caused by the inductor will quickly fall to the negligible range in the early stage of start-up, so the characteristics of voltage and current will be dominated by the steady-state impedance. The equivalent circuit of the induction machine under the steady state is shown in Figure 1, where \(R_{s}\) is the stator resistance; \(R_{r}\) is the rotor resistance referred to the stator side; \(X_{m}\) is the excitation reactance; \(X_{s}\) is the stator equivalent reactance; \(X_{r}\) is the rotor reactance referred to the stator side; \(Z\) is the input impedance; \(R\) is the input resistance; \(X\) is the input reactance; and \(s\) is the slip rate. The input impedance of the induction machine can be expressed as  
\[R+jX=R_{s}+jX_{s}+\frac{jX_{m}(R_{r}/s+jX_{r})}{jX_{m}+(R_{r}/s+jX_{r})}. \tag{1}\]  
Dividing the resistance and reactance in the impedance, the input resistance and input reactance are respectively:  
\[R(s)=R_{s}+\frac{X_{m}^{2}R_{r}/s}{\left(R_{r}/s\right)^{2}+\left(X_{m}+X_{r} \right)^{2}}, \tag{2}\]  
\[X(s)=X_{s}+X_{m}-\frac{X_{m}^{2}(X_{m}+X_{r})}{\left(R_{r}/s\right)^{2}+\left( X_{m}+X_{r}\right)^{2}}. \tag{3}\]  
In Figure 1, only the resistance is affected by the slip rate. Due to the change in the resistance, both the input resistance and the input reactance are affected and become time-varying impedance. When the rotor speed changes from a static to synchronous state, the slip rate changes from 1 to 0, and the resistance and reactance changes are shown in Figure 2. The values of the curves will vary with the capacity, however, the trend of the curve will remain the same. Therefore, the vertical axis does not mark the scale. In Figure 2, the reactance is a monotonically decreasing curve. At the beginning of the start-up, the reactance is flat and there is no change, but the resistance increases at the beginning of the start-up, with a maximum reading near the synchronous speed [23]. The reactance value and the resistance value will intersect near the maximum resistance value, and when the slip rate continues to decrease, the resistance drops rapidly, so when the slip rate is zero, the resistance drops to nearly zero. Expressed in polynomial fractions, the input resistance and input reactance can be expressed as  
\[R=\frac{R_{s}+sX_{m}^{2}/R_{r}+s^{2}R_{s}(X_{m}+X_{r})^{2}/R_{r}^{2}}{1+s^{2}( X_{m}+X_{r})^{2}/R_{r}^{2}}, \tag{4}\]  
\[X=\frac{(X_{m}+X_{s})}{1+s^{2}(X_{m}+X_{r})^{2}/R_{r}^{2}}+\frac{s^{2}\Big{[} \big{(}X_{m}+X_{r}\big{)}^{2}(X_{m}+X_{s})-X_{m}^{2}(X_{m}+X_{r})\Big{]}/R_{r} ^{2}}{1+s^{2}(X_{m}+X_{r})^{2}/R_{r}^{2}}. \tag{5}\]  
Figure 1: Steady-state equivalent circuit.  
Therefore, the input impedance can be expressed as a polynomial fraction.  
\[R+jX=\frac{(\beta_{0}+\beta_{1}s+\beta_{2}s^{2})+j(\beta_{3}+\beta_{4}s^{2})}{1+ \alpha_{2}s^{2}}. \tag{6}\]  
Comparing Equations (4)-(6), the relationship between the polynomial fractional coefficients and the induction machine parameters is  
\[\alpha_{2}=(X_{m}+X_{r})^{2}/R_{r}^{2}, \tag{7}\]  
\[\beta_{0}=R_{s}, \tag{8}\]  
\[\beta_{1}=X_{m}^{2}/R_{r}, \tag{9}\]  
\[\beta_{2}=R_{s}(X_{m}+X_{r})^{2}/R_{r}^{2}, \tag{10}\]  
\[\beta_{3}=X_{m}+X_{s}, \tag{11}\]  
\[\beta_{4}=(X_{m}+X_{r})^{2}(X_{m}+X_{s})-X_{m}^{2}(X_{m}+X_{r}). \tag{12}\]