**4. Conclusion**

244 Induction Motors – Modelling and Control

**3.7. Discussions** 

and 26.

larger.

**Table 6.** Comparison of radial force at slip=0.05.

the low speed range, that is, high torque region.

the stator magnet-motive force and the rotor slot permeance.

frequency produce by the eccentricity of the rotor

and a randomized switching frequency PWM are taken into account.

Frequency 100Hz 400Hz *f*c 2 *f*<sup>c</sup> Line 0.815 0.0467 -- -- PWM 0.813 0.0455 0.00560 0.00263 P PWM 0.821 0.0455 0.00480 0.00171 F PWM 0.815 0.0454 0.00516 0.00142

The steady state characteristics of the induction motor can be calculated by the 2D FEM considering the modified conductivity in the rotor slot by (12) and the leakage inductance of stator coil end by (13). However, for the motor with one-slot skewing there is some error in

In the section 3.3, it is shown that the space variation of the radial force is approximately flat in the teeth and becomes a small value at the positions where the rotor slot exists. The radial force at one tooth is bigger than that at the other two teeth, when the stator winding is distributed in three slots as shown in Fig. 14. It is also shown that the radial force has a fundamental frequency of 2 times the line frequency of 50Hz, because this motor is two pole pairs machine. Moreover, some frequencies of the radial force are obtained by considering

In the section 3.4, when the motor is driven by the PWM inverter, the current and torque contain the component of the carrier frequency as well known. As a result, the waveform of the radial force also includes small noise. The Fourier analysis shows that the fundamental component is almost same as that driven by line source, and that the amplitude around 5 kHz, that is, carrier frequency is bigger than that of the line source as shown in Figs. 22

In the section 3.5, the measurement of vibration velocity shows that there are vibration at around natural frequency and some frequencies corresponding to the radial force. However, there are the other frequencies corresponding to the harmonics of line frequency and the

In the section 3.6, the effect of the randomized PWM inverter on the radial force is calculated. Two types of random PWM method, namely, a randomized pulse position PWM

However, the time spectrum shown in Figs 29 and 30 are approximately the same as that under line source, except for the reduction of radial forces at two times carrier frequency. The calculation was carried out for about five cycles of the fundamental frequency. If a very long period is calculated, the reduction of radial force at the carrier frequency becomes The natural frequencies of the motor can be estimated by considering the equivalent Young's modulus of the stator windings. For example, the lowest measured and calculated natural frequencies are 1,325 and 1,369 Hz for the stator core only, and are 637 and 587 Hz for the stator core with winding. They agree well with each other.

The steady state characteristics of the induction motor can be calculated by the 2D FEM considering the modified conductivity in the rotor slot and the leakage inductance of stator coil end. Using this simulation model, the radial force of the induction motor fed by the line source has been analyzed. It is shown that the frequencies are explained by the product of the fundamental stator MMF and the rotor slot permeance, and that the radial force is different at each tooth because of the distributed stator winding.

When the motor is driven by the PWM inverter, the fundamental component of radial force is almost same as that driven by line source and the amplitude around the carrier frequency is bigger than that of the line source. Moreover, the effect of the randomized PWM inverter on the radial force is calculated. The radial forces at two times carrier frequency can be reduced by using the randomized pulse position PWM or by the randomized switching frequency PWM.
