**3. Radial magnetic force**

230 Induction Motors – Modelling and Control

**Figure 5.** Natural vibration modes for stator core only.

winding.

Mode Measured [Hz] Calculated [Hz] 2 1,325 1,369 2 1,337 1,425 3 3,425 3,446 3 3,875 3,926

**Table 3.** Comparison of the calculated natural frequencies with the measured ones for the stator core only.

(a) 1,369Hz (b) 1,425Hz

Next, we calculate the natural frequencies of the stator with winding, where the space factor of winding is chosen to be 0.43 by considering the enameled wires. Three lowest natural frequencies and the natural vibration modes are shown in Table 4 and Fig. 6. The natural

(c) 3,446Hz (d) 3,926Hz

Mode Measured [Hz] Calculated [Hz] 2 637 587.0 3 1,770 1,544.6 4 2,694 2,739.0 **Table 4.** Comparison of the calculated natural frequencies with the measured ones for the stator with

frequencies of 587, 1,545 and 2,739Hz have mode 2, 3 and 4, respectively.

### **3.1. Analysis method of radial magnetic force**

The simulation of the electromagnetic force is implemented by using a 2D non-linear finite element method considering the rotor current coupled with voltage equations. As we consider the force and vibration at a steady state, the rotating speed is assumed to be constant. Then, the government equations are as follows,

**Figure 7.** Relationship between the smallest natural frequency and space factor.

$$\frac{\partial}{\partial \mathbf{x}} (\nu \frac{\partial A}{\partial \mathbf{x}}) + \frac{\partial}{\partial y} (\nu \frac{\partial A}{\partial y}) = -\sum\_{k} J\_{0k} + \sigma \frac{\partial A}{\partial \ t} \tag{6}$$

(10)

(12)

(13)

*<sup>S</sup>* (11)

 2

*lSlSN p* are the conductivity of aluminium, the longitude of

1

*<sup>l</sup> <sup>R</sup> S <sup>l</sup> <sup>N</sup> <sup>R</sup> S p*

*b*

*r*

1 ' '

'

where, 2 , , , , , , and *b br r*

**3.2. Steady state characteristics** 

' 0.737

conductivity

(Horii, 1978)

conductivity are

*b*

*b r*

*r*

the rotor bar, the cross section area of rotor bar, the longitude of the end ring, the cross section area of end ring, the number of rotor slots, and the number of poles, respectively. It is assumed that the rotor resistance is expressed by an equivalent bar with a modified

> *b b b r b l R RR*

Therefore, the modified conductivity is obtained by using the next formula (IEE Japan, 2000)

*b b*

*l S l l N S S p*

2

2 0 10 2.3 log 0.5 <sup>2</sup>

*l Nl*

*f*

Stator end leakage inductance *l* is also taken into account and given by a traditional method

where, , , and *Nl d f s* are number of stator windings, total length of coil end, and diameter of an equivalent circle whose area equals to the cross section of stator coils. If the motor has skewed slots, we should use one of the multi-slice model, the coupled method of 2D and 3D models (Yamasaki, 1996) and the full 3D model (Kometani et al., 1996). The influence that the skewing has on the radial force and vibration is not taken into account in this paper.

Electromagnetic force is calculated by the 2D non-linear finite element method coupled with voltage equations. The models are created using a triangular mesh with 13,665 elements and 6,907 nodes for the M-model see Fig. 8. One fourth of the motor is calculated because of symmetry. For the M-model these numbers are 14,498 elements and 7,333 nodes, and half of the motor is calculated, see Fig. 9. The values obtained for the aluminium relative

To corroborate the validity of the model, the measured and calculated values of the output torque and current are compared, and the results are presented in Figs. 10 and 11. The

' 0.351

for the M-model.

for the K-model, and

*b r b r*  2 2

*f*

*l*

*d*

*s*

2

$$V\_k = r \ i\_k + l \frac{d \ i\_k}{d \ t} + \frac{d \Phi\_k}{d \ t} \quad , \qquad \text{ ( $k = a, b, c$ )}\tag{7}$$

$$\text{where}\quad\Phi\_k = \frac{n}{S\_k} \iint\_{S\_k} AL \, dxdy \quad , \qquad J\_{0k} = \frac{n}{S\_k} i\_k \quad , \tag{8}$$

where, 0 *A*, , , , ,, ,, , , , *J V irl nSL* are magnetic vector potential, reluctivity, current density, conductivity, stator phase voltage, stator current, resistance of the stator winding, leakage inductance of the stator end winding, flux linkage, number of turns of stator winding, cross section area of the stator winding, and stack length, respectively. We solve equations (6) and (7) by using the time-stepping FEM. In the case where the motor is driven by the line voltage, the time step *t* is constant so that the step of rotation is about 2 / 500 at slip=0. In the case of PWM inverter, *t* is calculated from the intersection point of a sine wave and a jagged wave with a carrier frequency of 5 kHz. The transient state converged at about five cycles of the input voltage on our simulation. The radial electromagnetic force is calculated by the Maxwell's stress tensor method,

$$F\_n = \frac{1}{2\,\mu\_0} (B\_n^2 - B\_t^2) \tag{9}$$

where, 0 is the permeability of air, *<sup>n</sup> B* and *<sup>t</sup> B* are the normal and tangential component of the flux density in the air gap. In order to take into account the 3D effects, the resistance of the end ring of the rotor is considered in the 2D model by modifying the conductivity of the rotor bars. Resistances of bar and end ring can be written as

$$\begin{aligned} R\_b &= \frac{1}{\sigma} \frac{l\_b}{S\_b} \\ R\_r &= \frac{2}{\sigma} \frac{l\_r}{S\_r} \frac{N\_2}{\left(p\pi\right)} \end{aligned} \tag{10}$$

where, 2 , , , , , , and *b br r lSlSN p* are the conductivity of aluminium, the longitude of the rotor bar, the cross section area of rotor bar, the longitude of the end ring, the cross section area of end ring, the number of rotor slots, and the number of poles, respectively. It is assumed that the rotor resistance is expressed by an equivalent bar with a modified conductivity

$$R\_b \, ^\circ = \frac{1}{\sigma \, ^\circ S\_b} \frac{l\_b}{S\_b} = R\_b + R\_r \tag{11}$$

Therefore, the modified conductivity is obtained by using the next formula (IEE Japan, 2000)

$$\sigma' = \frac{\frac{l\_b}{S\_b}}{\frac{l\_b}{S\_b} + 2 \times \frac{l\_r}{S\_r} \times \frac{N\_2}{\left(p\pi\right)^2}} \sigma \tag{12}$$

Stator end leakage inductance *l* is also taken into account and given by a traditional method (Horii, 1978)

$$M = \frac{2.3}{2\pi} \mu\_0 N^2 l\_f \left( \log\_{10} \frac{l\_f}{d\_s} - 0.5 \right) \tag{13}$$

where, , , and *Nl d f s* are number of stator windings, total length of coil end, and diameter of an equivalent circle whose area equals to the cross section of stator coils. If the motor has skewed slots, we should use one of the multi-slice model, the coupled method of 2D and 3D models (Yamasaki, 1996) and the full 3D model (Kometani et al., 1996). The influence that the skewing has on the radial force and vibration is not taken into account in this paper.

### **3.2. Steady state characteristics**

232 Induction Motors – Modelling and Control

the smallest natural frequency [Hz]

600

650

700

750

where, 0 *A*, 

where, 0 

**Figure 7.** Relationship between the smallest natural frequency and space factor.

*k k*

voltage, the time step *t* is constant so that the step of rotation

the rotor bars. Resistances of bar and end ring can be written as

calculated by the Maxwell's stress tensor method,

<sup>0</sup> () () , *<sup>k</sup>*

 

0.4 0.5 0.6 0.7 <sup>550</sup>

*k k*

*di d V ri l k abc dt dt* 

where 0 , ,

conductivity, stator phase voltage, stator current, resistance of the stator winding, leakage inductance of the stator end winding, flux linkage, number of turns of stator winding, cross section area of the stator winding, and stack length, respectively. We solve equations (6) and (7) by using the time-stepping FEM. In the case where the motor is driven by the line

slip=0. In the case of PWM inverter, *t* is calculated from the intersection point of a sine wave and a jagged wave with a carrier frequency of 5 kHz. The transient state converged at about five cycles of the input voltage on our simulation. The radial electromagnetic force is

> 0 <sup>1</sup> ( ) <sup>2</sup> *n nt F BB*

of the flux density in the air gap. In order to take into account the 3D effects, the resistance of the end ring of the rotor is considered in the 2D model by modifying the conductivity of

2 2

is the permeability of air, *<sup>n</sup> B* and *<sup>t</sup> B* are the normal and tangential component

*k k k k k S n n AL dxdy J i S S*

*k*

*AA A <sup>J</sup> xx yy t*

space factor

*k*

, ( , , )

, , , ,, ,, , , , *J V irl nSL* are magnetic vector potential, reluctivity, current density,

 

(6)

(7)

(8)

(9)

 is about 2 / 500 

at

Electromagnetic force is calculated by the 2D non-linear finite element method coupled with voltage equations. The models are created using a triangular mesh with 13,665 elements and 6,907 nodes for the M-model see Fig. 8. One fourth of the motor is calculated because of symmetry. For the M-model these numbers are 14,498 elements and 7,333 nodes, and half of the motor is calculated, see Fig. 9. The values obtained for the aluminium relative conductivity are ' 0.737 for the K-model, and ' 0.351 for the M-model.

To corroborate the validity of the model, the measured and calculated values of the output torque and current are compared, and the results are presented in Figs. 10 and 11. The

graphic shows a good agreement between the measured values and the calculated values for both models. This paper does not consider the effect of skewing, then this produces some error around 1400min-1 rotating speed in the M-model.

Analysis of Natural Frequency, Radial Force and Vibration of Induction Motors Fed by PWM Inverter 235

speed[min-1]

0 500 1000 1500

I[A]

: measured : calculated

and is approximately flat in the teeth and becomes a small value at the positions where the

Torque[N.m]

Fig. 14 shows the time variation of the force at the different teeth. It is shown that the force at tooth 1 is the same as that at tooth 4 and is bigger than those at teeth 2 and 3, because the

position

0 0.5 1 1.5

θ

Figs. 15 and 16 show the radial force and its spectrum at slip=0. It is shown that the force at the teeth is bigger than that at the slots and has a fundamental frequency of 2 times the line frequency of 50Hz. Figs. 17 and 18 show the radial force and its spectrum at slip=0.05. It is

rotor slot exists.

**Figure 11.** Steady state characteristic for K-model motor

Torque [N.m], I [A]

2

4

6

**Figure 12.** Space variation of radial force.

stator winding is distributed in three slots as shown in Fig. 1.

force [N]

0

2

4

6

8

shown that the radial force at the slip of 0.05 is very different from that at 0.

**Figure 8.** Mesh partition for M-model motor

**Figure 9.** Mesh partition for K-model motor

**Figure 10.** Steady state characteristic for M-model motor

## **3.3. Radial magnetic force under line source**

The space variation of the radial electromagnetic force is presented in Fig. 12. It is shown that the radial force is big at the position where the flux density is big as shown in Fig. 13, and is approximately flat in the teeth and becomes a small value at the positions where the rotor slot exists.

**Figure 11.** Steady state characteristic for K-model motor

234 Induction Motors – Modelling and Control

**Figure 8.** Mesh partition for M-model motor

**Figure 9.** Mesh partition for K-model motor

**Figure 10.** Steady state characteristic for M-model motor

Torque [N.m], I [A]

10

20

0

**3.3. Radial magnetic force under line source** 

The space variation of the radial electromagnetic force is presented in Fig. 12. It is shown that the radial force is big at the position where the flux density is big as shown in Fig. 13,

speed [min-1]

1350 1400 1450 1500

I

Torque

: measured : calculated

error around 1400min-1 rotating speed in the M-model.

graphic shows a good agreement between the measured values and the calculated values for both models. This paper does not consider the effect of skewing, then this produces some

**Figure 12.** Space variation of radial force.

Fig. 14 shows the time variation of the force at the different teeth. It is shown that the force at tooth 1 is the same as that at tooth 4 and is bigger than those at teeth 2 and 3, because the stator winding is distributed in three slots as shown in Fig. 1.

Figs. 15 and 16 show the radial force and its spectrum at slip=0. It is shown that the force at the teeth is bigger than that at the slots and has a fundamental frequency of 2 times the line frequency of 50Hz. Figs. 17 and 18 show the radial force and its spectrum at slip=0.05. It is shown that the radial force at the slip of 0.05 is very different from that at 0.

frequency [kHz]

: slots

time [s]

0.18 0.185 0.19 0.195 0.2

0 2 4 6 8 10 12 <sup>10</sup>-3

slip=0.05 : teeth

**Figure 16.** Spectrum of radial magnetic force of M model motor at slip=0.

force [N]

0

force [N]

10-2

10-1

10<sup>0</sup>

2

4

6

10-2

10-1

10<sup>0</sup>

force [N]

**Figure 17.** Waveform of radial magnetic force of M model motor at slip=0.05.

**Figure 18.** Spectrum of radial magnetic force of M model motor at slip=0.05.

Here we discuss the frequencies of radial force. The electromagnetic flux harmonics are produced due to the relative movement between the rotor and stator. Seeing it from the

0 2 4 6 8 10 12 <sup>10</sup>-3

frequency [kHz]

**Figure 13.** Flux distribution at slip=0.

**Figure 14.** Radial magnetic force at different teeth.

**Figure 15.** Waveform of radial magnetic force of M model motor at slip=0.

**Figure 16.** Spectrum of radial magnetic force of M model motor at slip=0.

θ

: tooth 1 : tooth 2 : tooth 3 : tooth 4

time [s]

: slots

time [s]

0.18 0.185 0.19 0.195 0.2

0.18 0.185 0.19 0.195 0.2

slip=0.0 : teeth

**Figure 13.** Flux distribution at slip=0.

**Figure 14.** Radial magnetic force at different teeth.

force [N]

0

force [N]

2

4

6

8

**Figure 15.** Waveform of radial magnetic force of M model motor at slip=0.

0

2

4

6

**Figure 17.** Waveform of radial magnetic force of M model motor at slip=0.05.

**Figure 18.** Spectrum of radial magnetic force of M model motor at slip=0.05.

Here we discuss the frequencies of radial force. The electromagnetic flux harmonics are produced due to the relative movement between the rotor and stator. Seeing it from the stator's side where the main flux is generated, the permeance varies periodically due to the presence of the slots in the rotor. Following this reason, the frequency of the harmonics in the electromagnetic flux is obtained by the product of the fundamental stator magnetmotive force (MMF) and the rotor slot permeance. The fundamental stator MMF *F* is proportional to cos( / 2 2 ) *p ft* , where is the stator angle. The permeance *P* is proportional to <sup>2</sup> <sup>2</sup> 1 cos 1 2 *Ak kN s ft p* , where *N*2, s and k are the number of rotor

slots, slip and the order of space harmonics, respectively. Considering that the radial electromagnetic force is proportional to <sup>2</sup> ( ) *F P* , the next three frequencies are obtained,

$$2f,\ \frac{2kN\_2}{p}(1-s)f,\ \frac{2kN\_2}{p}(1-s)f \pm 2f\tag{14}$$

Analysis of Natural Frequency, Radial Force and Vibration of Induction Motors Fed by PWM Inverter 239

: mode=4 : mode=8 : mode=12

frequency [Hz]

force [N]

10-5

force [N]

10-5

10-4

10-3

10-2

10-1

10<sup>0</sup>

10-4

10-3

10-2

10-1

10<sup>0</sup>

frequency [Hz]

frequency [Hz]

0 1000 2000 3000 4000 5000 6000 <sup>10</sup>-6

0 1000 2000 3000 4000 5000 6000 <sup>10</sup>-6

0 500 1000 1500 2000 <sup>10</sup>-5

**Figure 21.** Radial force and its spectrum of K model motor at slip=0.

: slots

time [s]

0.18 0.185 0.19 0.195 0.2

slip=0.5 : teeth

force [N]

10-4

: slots

slip=0.0 : teeth

10-3

10-2

10-1

10<sup>0</sup>

10<sup>1</sup>

**Figure 22.** Radial force and its spectrum of K model motor at slip=0.5.

time [s]

0.18 0.185 0.19 0.195 0.2

**Figure 20.** Enlarged one of Fig. 19.

force [N]

0

force [N]

0

0.5

1

1.5

0.5

1

1.5

Since the rotor has 44 slots, when slip is 0, the combination of the slot permeance and the fundamental stator MMF produces the peaks at 100, {1000, 1100, 1200}, {2100, 2200, 2300}, and so on, see Fig. 16. When the slip is 0.05, the frequencies are 100, {945, 1045, 1145}, {1190, 2090, 2190}, see Fig. 18.

In the vibration problems, small space harmonics, namely, small modes are important. Then, we calculate the space and time spectrum of the radial electromagnetic force in the air gap, and show the time spectrum for several space harmonics in Figs. 19 and 20. It is shown that time harmonics of mode 4 are 100, 200, 400, and so on, and the mode of harmonics of 300, 600 and 900Hz is 12.

**Figure 19.** Frequency spectrum for different mode.

For the K-model the rotor has 34 slots, when slip is 0, the combination of the slot permeance and the fundamental stator MMF produces the peaks at 100, {750, 850, 950}, {1600,1700,1800}, {2450,2550,2650}, and so on, see Fig. 21. When the slip is 0.5, the frequencies are 100, {325,425,525}, {750,850,950}, see Fig. 22. When slip=1.0, only the first frequency remains and this is appreciated in Fig. 23.

**Figure 20.** Enlarged one of Fig. 19.

proportional to cos( / 2 2 ) *p ft*

2090, 2190}, see Fig. 18.

300, 600 and 900Hz is 12.

**Figure 19.** Frequency spectrum for different mode.

force [N]

10-3

10-2

10-1

10<sup>0</sup>

this is appreciated in Fig. 23.

proportional to <sup>2</sup>

 

<sup>2</sup> 1 cos 1 2 *Ak kN s ft*

stator's side where the main flux is generated, the permeance varies periodically due to the presence of the slots in the rotor. Following this reason, the frequency of the harmonics in the electromagnetic flux is obtained by the product of the fundamental stator magnetmotive force (MMF) and the rotor slot permeance. The fundamental stator MMF *F* is

slots, slip and the order of space harmonics, respectively. Considering that the radial electromagnetic force is proportional to <sup>2</sup> ( ) *F P* , the next three frequencies are obtained,

Since the rotor has 44 slots, when slip is 0, the combination of the slot permeance and the fundamental stator MMF produces the peaks at 100, {1000, 1100, 1200}, {2100, 2200, 2300}, and so on, see Fig. 16. When the slip is 0.05, the frequencies are 100, {945, 1045, 1145}, {1190,

In the vibration problems, small space harmonics, namely, small modes are important. Then, we calculate the space and time spectrum of the radial electromagnetic force in the air gap, and show the time spectrum for several space harmonics in Figs. 19 and 20. It is shown that time harmonics of mode 4 are 100, 200, 400, and so on, and the mode of harmonics of

For the K-model the rotor has 34 slots, when slip is 0, the combination of the slot permeance and the fundamental stator MMF produces the peaks at 100, {750, 850, 950}, {1600,1700,1800}, {2450,2550,2650}, and so on, see Fig. 21. When the slip is 0.5, the frequencies are 100, {325,425,525}, {750,850,950}, see Fig. 22. When slip=1.0, only the first frequency remains and

0 2 4 6 8 10 12 <sup>10</sup>-4

frequency [kHz]

mode=4

*<sup>p</sup>* , <sup>2</sup> <sup>2</sup> (1 ) 2 *kN sf f p*

  is the stator angle. The permeance *P* is

, where *N*2, s and k are the number of rotor

(14)

, where

*p* 

<sup>2</sup> *<sup>f</sup>* , <sup>2</sup> <sup>2</sup> (1 ) *kN s f*

**Figure 21.** Radial force and its spectrum of K model motor at slip=0.

**Figure 22.** Radial force and its spectrum of K model motor at slip=0.5.

time [s]

force [N]

10-5

10-4

10-3

10-2

10-1

10<sup>0</sup>

0.18 0.185 0.19 0.195 0.2

torque

: PWM : sinusoidal

**Figure 25.** Torque and stator current waveforms of K model driven by PWM inverter, slip=0.5

: slots

ia[A], torque[N.m.]


slip=0.5 : teeh

time [s]

0.18 0.185 0.19 0.195 0.2

ia

 **Figure 26.** Radial force and frequency spectrum of K-model driven by PWM inverter at slip = 0.5.

**3.6. Radial magnetic force under randomized PWM inverter source** 

Fig. 27 shows the vibration velocity measured at the centre of stator surface, when the motor is running at no-load. The vibration of 600 through 650 Hz is mainly emitted from the natural frequency, and 100, 200, 400, 500, 700, 1000 and 1200 Hz are corresponding to the frequency of the radial force with mode 4. We think that the vibration of 25Hz is emitted by the eccentricity of the rotor. Fig. 28 shows the vibration velocity emitted from the inverterfed induction motor. We can see the vibration at around *<sup>c</sup> nf* , where *n* is an integer and *<sup>c</sup>*

It is well known that a random PWM method reduces the acoustic noise emitted from an inverter drive motor. Then, we analyze the radial force of the motor fed by two types of

*f* is

frequency [Hz]

0 1000 2000 3000 4000 5000 6000 <sup>10</sup>-6

**3.5. Vibration velocity** 

force [N]

0

0.5

1

1.5

the carrier frequency.

**Figure 23.** Radial force and its spectrum of K model motor at slip=1.00.

### **3.4. Radial magnetic force under PWM inverter source**

Next, to clarify the difference between the line source and the PWM inverter, Figs. 24 and 25 show the waveforms of torque and stator current at slip=0 and 0.5 for the K-model. The PWM inverter has a currier frequency of 5kHz and the fundamental amplitude is equal to the line source. It is shown that the current and torque contain the component of the carrier frequency.

**Figure 24.** Torque and stator current waveforms of K model driven by PWM inverter, slip=0.

Fig. 26 shows the radial force and its spectrum at slip=0.5 for the K-model. The waveform of radial force driven by the PWM inverter is approximately the same as that driven by the line source. We can find a small noise in the waveform, and find that the amplitude around 5 kHz, that is, carrier frequency is bigger than that of the line source in the spectrum.

**Figure 25.** Torque and stator current waveforms of K model driven by PWM inverter, slip=0.5

**Figure 26.** Radial force and frequency spectrum of K-model driven by PWM inverter at slip = 0.5.

### **3.5. Vibration velocity**

240 Induction Motors – Modelling and Control

frequency.

force [N]

0

0.5

1

1.5

**Figure 24.** Torque and stator current waveforms of K model driven by PWM inverter, slip=0.

ia

kHz, that is, carrier frequency is bigger than that of the line source in the spectrum.

Fig. 26 shows the radial force and its spectrum at slip=0.5 for the K-model. The waveform of radial force driven by the PWM inverter is approximately the same as that driven by the line source. We can find a small noise in the waveform, and find that the amplitude around 5

0.18 0.185 0.19 0.195 0.2

time [s]

: PWM : sinusoidal

torque

Next, to clarify the difference between the line source and the PWM inverter, Figs. 24 and 25 show the waveforms of torque and stator current at slip=0 and 0.5 for the K-model. The PWM inverter has a currier frequency of 5kHz and the fundamental amplitude is equal to the line source. It is shown that the current and torque contain the component of the carrier

force [N]

10-5

10-4

10-3

10-2

10-1

10<sup>0</sup>

frequency [Hz]

0 1000 2000 3000 4000 5000 6000 <sup>10</sup>-6

**Figure 23.** Radial force and its spectrum of K model motor at slip=1.00.

time [s]

0.18 0.185 0.19 0.195 0.2

slip=1.0 : teeth

: slots

**3.4. Radial magnetic force under PWM inverter source** 

ia[A], torque[N.m.]


Fig. 27 shows the vibration velocity measured at the centre of stator surface, when the motor is running at no-load. The vibration of 600 through 650 Hz is mainly emitted from the natural frequency, and 100, 200, 400, 500, 700, 1000 and 1200 Hz are corresponding to the frequency of the radial force with mode 4. We think that the vibration of 25Hz is emitted by the eccentricity of the rotor. Fig. 28 shows the vibration velocity emitted from the inverterfed induction motor. We can see the vibration at around *<sup>c</sup> nf* , where *n* is an integer and *<sup>c</sup> f* is the carrier frequency.

### **3.6. Radial magnetic force under randomized PWM inverter source**

It is well known that a random PWM method reduces the acoustic noise emitted from an inverter drive motor. Then, we analyze the radial force of the motor fed by two types of random PWM method, namely, a randomized pulse position PWM and a randomized switching frequency PWM. The randomized pulse position PWM changes the pulse width as

$$duty = \frac{\upsilon + V\_{\text{max}}}{2V\_{\text{max}}} + (\upsilon - 0.5) \, ^\ast k \tag{15}$$

Analysis of Natural Frequency, Radial Force and Vibration of Induction Motors Fed by PWM Inverter 243

force [N]

10-4

10-3

10-2

10-1

10<sup>0</sup>

frequecy [kHz]

mode=4

frequecy [kHz]

0 2 4 6 8 10 12 <sup>10</sup>-5

0 2 4 6 8 10 12 <sup>10</sup>-5

mode=4

 **Figure 30.** Spectrum of radial force spectrum for the randomized switching frequency PWM.

force [N]

10-4

10-3

10-2

10-1

10<sup>0</sup>

Frequency 100Hz 400Hz *f*c 2 *f*<sup>c</sup> Line 0.897 0.0437 -- -- PWM 0.896 0.0423 0.00366 0.00260 P PWM 0.904 0.0418 0.00375 0.00185 F PWM 0.899 0.0423 0.00321 0.00125

P PWM: Randomized pulse position PWM, F PWM: Randomized switching frequency PWM **Table 5.** Comparison of radial force at slip=0.0.

force [N]

force [N]

10-4

10-3

10-2

10-1

10<sup>0</sup>

10-4

10-3

10-2

10-1

10<sup>0</sup>

**Figure 29.** Spectrum of radial force for the randomized pulse position PWM.

mode=4

mode=4

frequecy [kHz]

frequecy [kHz]

0 2 4 6 8 10 12 <sup>10</sup>-5

0 2 4 6 8 10 12 <sup>10</sup>-5

where, max *vV x k* , , and are the voltage reference, the amplitude of the jagged wave with 5 kHz carrier frequency, a random number and the maximum variation of pulse position, respectively. This means that the interval of switching signals is changed by *DT*, where 0.5 / 5000 0.5 / 5000 [sec] *k DT k* . The randomized switching frequency PWM changes the switching frequency as

$$f\_c = 5000 + (x - 0.5)^\*(a - 1)^\* \, 50 \tag{16}$$

Fig. 29 shows the time spectrum of the radial force of the motor fed by the randomized pulse position PWM, where the end of the interval of switching signals is changed by *DT*, 0.2 / 5000 0.2 / 5000 [sec] *DT* . Fig. 30 shows the time spectrum by the randomized switching frequency PWM, where the switching frequency is change by *DF DF* , 500 500 [Hz] . 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 around *<sup>c</sup> nf* . The reduction of radial forces is summarized in Tables 5 and 6.

**Figure 27.** Vibration velocity emitted from M model driven by the line source (measured)

**Figure 28.** Vibration velocity emitted from M model driven by the PWM inverter source (measured)

**Figure 29.** Spectrum of radial force for the randomized pulse position PWM.

**Figure 30.** Spectrum of radial force spectrum for the randomized switching frequency PWM.


P PWM: Randomized pulse position PWM,

242 Induction Motors – Modelling and Control

PWM changes the switching frequency as

The reduction of radial forces is summarized in Tables 5 and 6.

**Figure 27.** Vibration velocity emitted from M model driven by the line source (measured)

**Figure 28.** Vibration velocity emitted from M model driven by the PWM inverter source (measured)

random PWM method, namely, a randomized pulse position PWM and a randomized switching frequency PWM. The randomized pulse position PWM changes the pulse width as

> max max

where, max *vV x k* , , and are the voltage reference, the amplitude of the jagged wave with 5 kHz carrier frequency, a random number and the maximum variation of pulse position, respectively. This means that the interval of switching signals is changed by *DT*, where 0.5 / 5000 0.5 / 5000 [sec] *k DT k* . The randomized switching frequency

> 5000 ( 0.5) \* ( 1) \* 50 *<sup>c</sup> f x*

Fig. 29 shows the time spectrum of the radial force of the motor fed by the randomized pulse position PWM, where the end of the interval of switching signals is changed by *DT*, 0.2 / 5000 0.2 / 5000 [sec] *DT* . Fig. 30 shows the time spectrum by the randomized switching frequency PWM, where the switching frequency is change by *DF DF* , 500 500 [Hz] . 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 around *<sup>c</sup> nf* .

*V* 

( 0.5) \* <sup>2</sup> *v V duty x k*

(15)

(16)

F PWM: Randomized switching frequency PWM

**Table 5.** Comparison of radial force at slip=0.0.


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

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

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

F.Ishibashi, K. Kamimoto, S. Noda, and K. Itomi, Small induction motor noise calculation,

K. Shiohata, K. Nemoto, Y. Nagawa, S. Sakamoto, T. Kobayashi, M. Itou, and H. Koharagi, A method for analyzing electromagnetic-force-induced vibration and noise analysis, (in

D. M. Munoz, J. C. S. Lai, Acoustic noise prediction in a vector controlled induction machine, 2003 IEEE international Electric Machines and Drives Conference, pp.104-110,

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D. Mori and T. Ishikawa, Force and Vibration Analysis of Induction Motors, IEEE

D. Mori and T. Ishikawa, Force and Vibration Analysis of a PWM Inverter-Fed Induction Motor, The 2005 International Power Electronics Conference,pp.644-650, Niigata, 2005

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T. Horii, Electrical Machines Outline, Corona, publishing Co., Ltd., 1978

for the stator core with winding. They agree well with each other.

different at each tooth because of the distributed stator winding.

**4. Conclusion** 

frequency PWM.

**Author details** 

*Gunma University, Japan* 

Takeo Ishikawa

**5. References** 

2003.

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