# Practical Work of Three-Phased Induction Motor Modeling in a Classical Spreadsheet Software
## 4 Spreadsheet Modeling of the Induction Motor
### Model Proposed in the Spreadsheet  
At steady state, when the rotational speed does not vary anymore, the mechanical equilibrium is reached between the motor and its load. This equilibrium is expressed by the equation \(T_{u}=T_{load}\), where \(T_{u}\) is the useful torque furnished by the motor, and \(T_{load}\) is the resistant torque of the load when it is driven at the resulting \(\Omega_{r}\) rotational speed. When the induction motor is used normally, the operating point of the motor-load set is located in the \"linear\" part of the mechanical characteristic of the motor, as can be seen in Figure 7. On this part of the motor characteristic (where \(s\leq 10\%\)), by identifying the electromagnetic torque \(T_{em}\) and the useful torque \(T_{u}\), one often considers:  
\[T_{em}=K.s=\frac{K}{\Omega_{s}}\cdot(\Omega_{s}-\Omega_{r})=K-\frac{K}{\Omega_ {s}}\cdot\Omega_{r}. \tag{33}\]  
Many of the exercises given to students deal with this intersection point determination, often neglecting the mechanical losses. The most important point to visualize and understand is that the driven load acts just as much as the induction motor on the determination of the motor/load operating point. To model the induction motor at a given operating point, one must be aware of the power balance presented in Figure 4, where \(U\) stands for the RMS (Root Mean Square) voltage value between two terminals of the three-phase induction motor, \(V\) is the RMS value at the terminals of a winding, \(I\) is the RMS value of a line current, \(J\) is the RMS value in a winding, \(R\) is the resistance measured between two terminals of the three-phase induction motor, \(r\) is the resistance of a winding, \(T_{em}\) is the electromagnetic torque moment, \(T_{u}\) is the useful torque moment, and \(T_{p}\) is the torque due to losses.  
There are several types of mechanical load models for electric motors. Constant torque load is typical for applications such as conveyors or compressors. Linearly variable torque load increases linearly with speed, and it is often used to model fans and centrifugal pumps. Quadratic, or parabolic torque load is proportional to the square of the rotational speed and is also used to more precisely model fan and pump resistances in terms of speed. Hyperbolic torque also exists but is less common. It can be observed in some specific applications such as some braking systems, where the resistive torque decreases as speed increases. Additionally, in speed controller systems for electric motors, the resistive torque can be designed to decrease with increasing speed in order to maintain a constant speed. The model proposed in the supplementary file models a torque load up to the quadratic torque, whose shape is defined as follows:  
\[T_{load}=a_{0}+a_{1}\cdot\Omega+a_{2}\cdot\Omega^{2}. \tag{34}\]When slip is sufficiently small to warrant the validity of the linear hypothesis of the motor torque, the operating point is visible at the intersection of the two lines \(T_{em}(\Omega)\) and \(T_{load}(\Omega)\), which are, respectively, presented in Equations (33) and (34), and represented in red and blue in Figure 9. These models make it possible to predict the behavior of the motor according to the applied load and allow optimization of its operating point for different applications. An example of such an intersection is presented in Figure 9.  
The spreadsheet given alongside this paper on the MDPI website in the section \"Supplementary Material\" helped to initiate this work.