# Practical Work of Three-Phased Induction Motor Modeling in a Classical Spreadsheet Software
## 3 Equivalent Model of the Induction Motor
### Equivalent Model Under Load  
Under load, the motor operates under similar power conditions as shown previously (single voltage \(V\)), but the line current intensity is now \(i_{1}\), which is significantly higher than \(i_{0}\). For an absorbed power \(P_{1}\), the power factor is now equal to \(\cos\varphi_{1}\).  
One can interpret this new situation by noting that the total power \(P_{1}\) absorbed by the motor is the sum of the power transmitted to the rotor \(P_{TR}\) and the stator losses \(P_{S}\). By neglecting the losses due to the Joule stator effect and admitting that the stator iron losses \(P_{IS}\) are the same at no load and under load, one can write the following for a phase:  
\[\frac{P_{1}}{3}\approx\frac{P_{TR}}{3}+\frac{P_{0}}{3}. \tag{9}\]  
This relationship leads to a model associating a dipole \(D_{S}\), which absorbs a power \(\frac{P_{1}}{3}\) and accounts for stator phenomena and a dipole \(D_{r}\) which absorbs a power \(\frac{P_{TR}}{3}\) and accounts for the behavior of the rotor \"seen\" from a power phase. The Fresnel diagram in Figure 6 allows us to determine the intensity \(I_{2}\) of the current passing through \(D_{R}\) from the experimental values \(V\), \(I_{0}\), \(\varphi_{0}\), \(I_{1}\), and \(\varphi_{1}\).  
To determine the elements of the dipole \(D_{R}\), since the intensity \(I_{2}\) of the current lags behind the voltage \(V\) applied to the considered phase, \(D_{R}\) can be represented by a resistance \(r\) associated in series with an inductance \(l\). To ensure that the presence of \(D_{R}\) properly accounts for the additional power absorbed by the motor under load, the power dissipated in the resistance \(r\) must equal the power \(P_{tr}\) transmitted to the rotor:  
\[P_{tr}=3\cdot r\cdot I_{2}^{2}. \tag{10}\]  
On the other hand, by imposing on the model the following additional condition that the \(i_{2}\) intensity of the current, which passes through \(D_{r}\), is proportional to the actual \(i_{R}\) intensity of the current in a phase of the rotor (\(i_{2}=k\cdot i_{R}\), and then \(I_{2}=k\cdot I_{R}\)), then if \(R_{R}\) is the resistance of a rotor phase, the power \(P_{JR}\) actually dissipated by the Joule effect in the rotor is  
\[P_{JR}=3\cdot R_{R}\cdot I_{R}^{2}. \tag{11}\]  
Then, since \(P_{JR}=s\cdot P_{tr}\), meaning \(P_{tr}=P_{JR}/s\), one can have, by setting \(r=\frac{R_{R}}{kR}\), the equality \(r=R/s\), with the following identifications.  
\[P_{tr}=3\cdot r\cdot I_{2}^{2}=3\cdot r\cdot(k\cdot I_{R})^{2}=\frac{3\cdot R _{R}\cdot I_{R}^{2}}{s}=\frac{P_{JR}}{s}. \tag{12}\]  
To identify the values of the \(D_{r}\) equivalent dipole, a locked rotor test is performed. The rotor is blocked and supplied with reduced voltage (10 to 20% of the nominal voltage), such that the current drawn is close to the nominal current. Then, the intensity \(I_{1}\) of the load current, the active power \(P_{1}\) and the reactive power \(Q_{1}\) are measured. Since the motor is supplied with a very low voltage, iron losses are negligible. For the model in Figure 6, this amounts to neglecting the intensity of the current in the stator (\(i_{0}<<i_{1}\)). Indeed, with a low voltage value, the magnetic circuit is far from saturation, and iron losses are also low. Since \(s=1\), \(R/s=R\), \(P_{1}\approx 3\cdot R\cdot I_{1}^{2}\) and \(Q_{1}\approx 3\cdot L\omega\cdot I_{1}^{2}\), then  
\[R=\frac{P_{1}}{3\cdot I_{1}^{2}}, \tag{13}\]  
\[l\cdot\omega=\frac{Q_{1}}{3\cdot I_{1}^{2}}. \tag{14}\]  
The induction motor is then completely characterized with the four constants measured, thanks to the two tests performed, at no-load (\(s=0\)) and with locked rotor (\(s=1\)).