The influence of the semimagnetic stator wedges of different sizes on the electromagnetic characteristics and the behavior of the induction motor is investigated. The study will be carried out with both analytical calculations and FEM analysis. The analytical calculations will take into account the stator and rotor slots, as well as the iron core saturation in order to study the spatial and timedependent harmonic content of the airgap magnetic flux density and electromagnetic torque. The size of the wedge plays an important role as it determines the tooth tips saturation, the high harmonic content of the airgap magnetic flux density, and the electromagnetic characteristics of the induction motor.
It is well known that the existence of space and time harmonics in the electromagnetic variables of a symmetrical induction motor, fed by symmetrical threephase voltage system, depends on several causes. Mainly, it depends on the presence of stator and rotor slots and the magnetic saturation of the stator and rotor core, especially at the tooth tips [
Since the induction motor is the most widely used electrical machine in industrial applications, there is a continuous research concerning the optimization of the motor’s electromagnetic variables and behavior, as well as improved design aiming at the motor’s longer life cycle [
Many methods have been proposed during the years, aiming at the reduction of higher rank harmonics. Such methods are the skewing, the construction of cage rotors with totally closed slots [
Another method proposed for the reduction of the airgap magnetic flux density harmonic content is the use of semimagnetic wedges for the closure of the stator slot openings. The wedges lead the flux in the slot opening and produce a more uniform flux density distribution in the airgap but increase also the slot leakage flux. In [
In this work, the influence of semimagnetic wedges for the closure of the stator slot openings on the electromagnetic variables and the airgap magnetic flux density harmonic content, in induction motors, will be thoroughly investigated, using both analytical calculations and FE analysis. Particularly, the influence of different sized semimagnetic wedges closing the stator slot openings on the electromagnetic characteristics, harmonic content, iron core saturation, and the behavior of the induction motor will be studied.
The investigation will be carried out in three steps. In the first part, through analytical formulas the space and timedependent airgap magnetic flux density content will be shown. The airgap magnetic flux density and saturation related harmonics will be also presented. Finally, the harmonic index of the motor’s electromagnetic torque will be demonstrated.
In the second part, a specific induction motor will be simulated and studied under timeharmonic FE analysis, taking into account the nonlinear magnetic characteristic of the stator and rotor iron core. Four different models will be created and studied, one with open stator slots and three more with different sized semimagnetic wedges closing the stator slot openings. Generally, it is proposed that the value of the relative magnetic permeability of the wedge material should be between 5 and 12. For all cases, in this work the relative magnetic permeability of the wedge material is considered equal to 10. The timeharmonic FEA will show that one of the three wedged motors has the best behavior.
In the third part of this work, the motor with open stator slots and the wedged one presenting the optimum behavior, resulting from the Timeharmonic Analysis, will be simulated at nominal load under transient FE analysis. The transient analysis will lead to the torque and stator current waveforms. The current and electromagnetic torque’s harmonic contents will be shown with the application of the fast fourier transform (FFT).
The FEM analysis will reveal harmonic rank numbers which depend on the relative position between the stator and rotor and which cannot be predicted through analytical calculations. Furthermore, the electromagnetic variables of each model, such as electromagnetic torque, stator current, efficiency, and power factor, will be calculated and compared to each other.
In previous work [
From (
Considering that the motor examined has 36 stator slots and 28 rotor slots and 4 poles, (
Also, from (
From (
According to [
So, the influence of the rotor and stator MMF on the magnetic flux density, due to the saturation will be
From (
From (
In [
The induction motor used for this research is a 3phase, 4 kW, 400 V, with 28 rotor slots and 36 stator slots, 4pole induction squirrelcage motor. The stator phase resistance was measured in the laboratory through DC current injection. Also, for the analysis the nonlinear magnetic BH characteristic is taken into consideration, extracted from the manufacturer data sheets. Four models have been simulated and studied. The original model with open stator slots and three more with different sized semimagnetic wedges closing the stator slot openings. The wedge material has a linear BH characteristic and the relative magnetic permeability is considered equal to 10. For each of these three models, wedges of different size are applied. In Figure
The four simulated models: (a) original, (b) wA, (c) wB, and (d) wC.
The models have been simulated under timeharmonic analysis. For each model the electromagnetic torque, stator current amplitude, efficiency, and power factor for speed values between 0 and synchronous speed have been extracted. In Figures
Simulation results at starting and at 1470 rpm.
Model  Starting torque (Nm)  Starting current (A)  Torque at 1470 rpm (Nm)  Current at 1470 rpm (Nm) 

cos 

Original  83.02  65.86  26.07  7.13  82.67  84.34 
wA  73.70  61.85  25.36  6.96  84.34  82.41 
wB  70.60  60.70  24.59  6.76  85.70  81.05 
wC  63.68  57.99  23.94  6.59  86.80  79.85 
Electromagnetic torque versus speed for the four simulated motors.
Stator current amplitude versus speed for the four simulated motors.
The use of semimagnetic wedges decreases the electromagnetic torque and the stator current amplitude, for every speed. In this work, the wedge material is considered linear due to its low relative magnetic permeability. So, as a result, the starting torque of the motors, where wedges are applied, is lower than the original model’s, due to the increased leakage flux.
At 1470 rpm, the stator current and as a consequence the electromagnetic torque of the models with wedges decrease with the increment of the wedge size, compared to the original. The same observation stands for the power factor due to the increase of the leakage flux. Nevertheless, the efficiency of wA, wB, and wC models is about 1.7%, 3% and 4.1% greater than the original’s, respectively. The above results will be explained later in this paper, through the study of the airgap magnetic flux density harmonic content occurring from the analytical calculations and FEM analysis.
In Figure
Normalized amplitude of the airgap magnetic flux density at 1470 rpm for the cases: (a) original model, (b) wA model, (c) wB model and (d) wC model.
Furthermore, as mentioned previously through the analytical calculations (
Moreover, from (
The harmonic content of the simulated models, presented in Figure
For each case, the Total Harmonic Distortionamplitude ratio (THD) was also calculated until rank number 50, at nominal speed for better comparison between the simulated models. The results are presented in Table
THDamplitude ratio at nominal speed.
Model  THD (%) 

Original  49.23 
wA model  48.35 
wB model  40.41 
wC model  41.03 
The wB model is characterized by the greatest reduction of the airgap magnetic flux density harmonic content. Also, it presents greater starting electromagnetic torque and slightly increased starting stator current than the wC model. Moreover, it proves to have greater efficiency than wA model and greater power factor than wC model at nominal speed. On the other hand, its power factor is reduced compared to wA model and its efficiency is less than that of wC model. From all of the above, the wB model presents an average satisfying behavior at starting and at nominal speed at the same time, compared to the wA and wC models.
In this paragraph, two models, the original and wB, will be simulated and studied under transient Rotating Machine (RM) analysis at the same load 30Nm. Same as in timeharmonic analysis, shown in the previous paragraph, the analysis will take into account the nonlinear magnetic BH characteristic of the stator and rotor iron core.
In Figures
The line current waveforms for: (a) the original and (b) the wb models.
The torque waveforms for (a) the original and (b) the wb models.
The frequency spectrums of the line current and torque signals are presented in Figure
Line current rotor slotrelated harmonics amplitudes.
Harmonic (Hz)  Open slots (dB)  Harmonic (Hz)  Wedged (dB) 

20  −50,86  22,31  −57,18 
80  −50,86  77,69  −58,15 
333,8  −52,26  332,3  −59,85 
633,8  −21,69  632,3  −27,81 
603,8  −62,91  604,6  −75,56 
663,1  −62,76  660  −75,47 
Torque rotor slotrelated harmonics amplitudes.
Harmonic (Hz)  Open slots (dB)  Harmonic (Hz)  Wedged (dB) 

30  −51,63  27,69  −56,76 
283,1  −52,71  282,3  −56,56 
331,5  −72,39  327,7  −95,95 
553,8  −64,25  554,6  −73,02 
613,1  −64,34  610  −72,93 
Frequency spectrum of the original (blue) and wb (red) models of the (a) line current and (b) torque.
In this work, the influence of different sized semimagnetic wedges on the airgap magnetic field harmonic content and the electromagnetic characteristics of the induction motor was investigated through analytical equations and FEM analysis. The results indicate that the use of semimagnetic wedges leads to the decrease of the airgap magnetic field harmonic content, which is also influenced by the wedge geometry. The wB model presented the least THD of its airgap magnetic flux density at nominal speed, as well as an average satisfying behavior at starting and nominal speed at the same time. For the above reasons, the wB model was selected to be studied and compared to the original one, under transient FEA operating at nominal load. The timedependent harmonic content of both the electromagnetic torque and the stator current has been significantly reduced in the case of the wedged model compared to the one with open slots. This behavior leads to slightly lower output power at nominal speed and lower power factor, but, on the other hand improvement of the motor’s efficiency and lower stator current at nominal speed. Finally, this work’s results state that the geometry of the wedge is crucial, since it determines the electromagnetic variables and behaviour of the induction motor.
Geometrical angle
Component of the airgap relative specific permeance varying with the angle
Stator slot pitch
Stator angular frequency
Rotor angular frequency
Airgap relative permeance
Airgap relative permeance saturation component
Mechanical angular frequency
Total, stator and rotor airgap magnetic flux density radial components
Stator frequency
Airgap length
Carter’s coefficient
Synchronous speed
Number of pole pairs
Number of stator slots per phase and pole
Slip
Stator and rotor slot numbers, respectively
Stator and rotor MMF, respectively
Stator and rotor MMF amplitude, respectively.
This work has been supported by the research program: “K. Karatheodori 2010,” of the Research Committee of the University of Patras, Greece.