# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 2 Transient Process Modeling in Induction Motor  
### Coordinate Conversion  
To simplify the mathematical description of an induction motor (IM), coordinate system transformations and the space vector concept are often employed [67]. The goal of such transformations is to obtain equations with constant coefficients, making analysis and control easer. This simplification is achieved using the two-axis theory, which considers not the actual magnetic flux in the air gap but its components along two mutually perpendicular axes--longitudinal and transverse (\(d\)-axis and \(q\)-axis) [68]. Coordinate transformations are widely used in technical applications to simplify complex systems by converting coordinates from one system to another. This process is known as coordinate conversion. According to mathematical principles, the number of variables remains unchanged after a coordinate transformation. Typically, a system of equations with time-varying coefficients (e.g., in the ABC phase coordinate system) is transformed into a system with constant coefficients in the \(dq0\) coordinate system. In general, the transformation involves converting an equation system with varying coefficients in the phase coordinate system (A, B, C or 1A, 1B, 1C) into an equivalent system expressed in the \(dq0\) coordinate system. This transformation simplifies the mathematical representation and facilitates the analysis and control of electrical machines [69]. An example of such a transformation is the conversion of the stator current space vector \(\hat{I}\) from the phase coordinates (IA, 1B, 1C) into the \(dq0\) coordinate system (Figure 2). If the phase currents contain a zero-sequence component, such that we define the zero-sequence component as: \((i_{1A}+i_{1B}+i_{1C})/3=i_{0}\). This zero-sequence current \(i_{0}\) may vary according to any complex principle and should not be confused with the alternating zero-sequence space vector current \(\hat{I}\). The concept of zero-sequence current is introduced as part of the coordinate transformation methodology to provide a complete theoretical framework. However, in the context of the four models developed and analyzed in this study, zero-sequence current is not present due to the assumption of a balanced three-phase system. Under balanced conditions, the sum of the phase currents is zero, and no zero-sequence current flows. This assumption simplifies the analysis and allows the focus to remain on the primary components of transient and starting dynamics. The coordinate axis 0 is perpendicular to the plane of the figure. In this case, the space vector \(\dot{I}\) in the ABC coordinate system can be expressed as follows: \(i^{\prime}_{1A}=i_{1A}-i_{0}\), \(i^{\prime}_{1B}=i_{1B}-i_{0}\), \(i^{\prime}_{1C}=i_{1C}-i_{0}\), such that \(i^{\prime}_{1A}+i^{\prime}_{1B}+i^{\prime}_{1C}=0\). Transforming the vector projections onto the phase axes provides the phase winding currents \(i_{1A}\), \(i_{1B}\), \(i_{1C}\), while projections onto the rotor axes yield the currents \(i_{d}\) and \(i_{q}\). In the general case, when the stator windings are connected in wye with a neutral wire, it is possible for \(i_{1A}+i_{1B}+i_{1C}\
eq 0\). In this situation, we account for the zero-sequence component as: \((i_{1A}+i_{1B}+i_{1C})/3=i_{0}\), where \(i_{0}\) is the stator current's zero-sequence component. \[i_{1A} =i^{\prime}_{1A}+i_{0} \tag{1}\] \[i_{1B} =i^{\prime}_{1B}+i_{0}\] (2) \[i_{1C} =i^{\prime}_{1C}+i_{0}. \tag{3}\]  
It should be noted that the currents marked with \"prime\" (e.g., \(i^{\prime}_{1A}\)) no longer contain the zero-sequence component.