**7. Torque quality**

Permanent magnet servomotors are widely used in many industrial applications for their small size, higher efficiency, noise-free operation, high speed range and better control. This makes quality of their torque an important issue in wide range of applications including servo applications. For example, servomotors used in defense applications, robotics, servo systems, electric vehicles all require smooth torque operation.

One of the most important issues in PM servomotors is the pulsating torque component which is inherent in motor design. If a quality work is not completed during the design stage, this component can lead to mechanical vibrations, acoustic noise, shorter life and drive system problems. In addition, if precautions are not taken, it can lead to serious control issues especially at low speeds. Minimization of the pulsating torque components is of great importance in the design of permanent magnet servomotors.

In general, calculation of torque quality is a demanding task since the torque quality calculation does not only consider the torque density of the motor but also consider the

Brushless Permanent Magnet Servomotors 287

situations such as converter caused disturbed stator current waveform and disturbed back EMF waveform arising from the design cause non-sinusoidal current and airgap flux density waveforms which result in undesired pulsating torque components at the motor output. In other words, if the airgap flux density waveform is disturbed, pulsating torque at

Definitions of the torque components will be given before getting into the subject any deeper. First, "cogging torque" is defined as the pulsating torque component produced by the variation of the airgap permeance or reluctance of the stator teeth and slots above the magnets as the rotor rotates. In other words, there is no stator excitation involved in cogging torque production of a PM motor. Second, "ripple torque" is the pulsating torque component generated by the stator MMF and rotor MMF. Ripple torque is mainly due to the fluctuations of the field distribution and the stator MMF which depends on the motor structure and the current waveform. This component can take two forms, one of which resulting from the MMF created by the stator windings and the other from MMF created by the rotor magnets. The second form is the torque created by stator MMF and rotor magnetic reluctance variation. In surface mounted PM servomotor, since there exists no rotor reluctance variation, there is no second form of the ripple torque and the ripple torque is mainly created by the first form. The third definition is "pulsating torque" which is defined as the sum of both cogging and ripple torque components (Sebastian et al., 1986 and Ree &

In the following analysis, it is assumed that a Y connected three phase unsaturated PM motor used and it has a constant airgap length and symmetrical stator winding. It is also assumed that stator currents contain only odd harmonics and current harmonics of the order of three does not exist. Finally, armature reaction is assumed negligible. For PM motors, at instant t, the instantaneous electromagnetic torque produced by phase A can be written as the interaction of the magnetic field and the phase current circulating in *N* turns

> /2 /2 ( ) 2 ( ) ( , ) *mp a a <sup>g</sup> er r mp T t pNi t R L B t d*

where *Rg* is airgap radius of the servomotor, *Le* is effective stack length, *p* is pole pairs and *m*

*e t pN R LB t d*

*<sup>m</sup>* is the rotor angular speed. The back-EMF in phase a can also be written as

*<sup>m</sup> mean R r r mp*

/2 <sup>a</sup> /2 () 2 ( ,) *mp*

*e t pN R L B td* 

> () () ( ) *a a <sup>a</sup> m*

*et it T t* 

*mean R r m r mp*

 (7)

 (8)

(9)

 (6)

The back-EMF of the motor induced in phase A at the time instant *t* is given by

Substituting (8) into (6), the torque expression becomes

/2 <sup>a</sup> /2 () 2 ( ,) *mp*

the motor shaft becomes inevitable (Jahns & Soong, 1996).

**7.1 Electromagnetic torque in PM motors** 

Boules, 1989).

is the number of phases.

where 

Fig. 12. General procedure for any commercially available FEA software

pulsating torque component. Therefore, a mathematical approach about torque quality should include harmonic analysis of electric drive system rather than a simple sizing of the motor.

Output torque of a PM servomotor has an average torque and pulsating torque components. The pulsating torque consists of cogging torque and ripple torque components. Cogging torque occurs from the magnetic permeance variation of the stator teeth and the slots above the permanent magnets. Presence of cogging torque is a major concern in the design of PM motors simply because it enhance undesirable harmonics to the pulsating torque. Ripple torque, on the other hand, occurs as a result of variations of the field distribution and the stator MMF. At high speed operations, ripple torque is usually filtered out by the inertia of the load or system. However, at low speeds 'torque-ripple' produces noticeable effects on the motor shaft that may not be tolerable in smooth torque and constant speed applications. Servomotors can also be categorized by the shape of their back EMF waveforms which can

take different forms such as sinusoidal and trapezoidal as seen in Fig. 13. Any non-ideal

Fig. 13. Current and back EMF waveform options in PM servomotors: (a) sinusoidal back-EMF and current and (b) trapezoidal back-EMF and current

Fig. 12. General procedure for any commercially available FEA software

motor.

pulsating torque component. Therefore, a mathematical approach about torque quality should include harmonic analysis of electric drive system rather than a simple sizing of the

Output torque of a PM servomotor has an average torque and pulsating torque components. The pulsating torque consists of cogging torque and ripple torque components. Cogging torque occurs from the magnetic permeance variation of the stator teeth and the slots above the permanent magnets. Presence of cogging torque is a major concern in the design of PM motors simply because it enhance undesirable harmonics to the pulsating torque. Ripple torque, on the other hand, occurs as a result of variations of the field distribution and the stator MMF. At high speed operations, ripple torque is usually filtered out by the inertia of the load or system. However, at low speeds 'torque-ripple' produces noticeable effects on the motor shaft that may not be tolerable in smooth torque and constant speed applications. Servomotors can also be categorized by the shape of their back EMF waveforms which can take different forms such as sinusoidal and trapezoidal as seen in Fig. 13. Any non-ideal

(a) (b)

E(t) i(t)

Fig. 13. Current and back EMF waveform options in PM servomotors: (a) sinusoidal back-

EMF and current and (b) trapezoidal back-EMF and current

E(t)

i(t)

situations such as converter caused disturbed stator current waveform and disturbed back EMF waveform arising from the design cause non-sinusoidal current and airgap flux density waveforms which result in undesired pulsating torque components at the motor output. In other words, if the airgap flux density waveform is disturbed, pulsating torque at the motor shaft becomes inevitable (Jahns & Soong, 1996).

#### **7.1 Electromagnetic torque in PM motors**

Definitions of the torque components will be given before getting into the subject any deeper. First, "cogging torque" is defined as the pulsating torque component produced by the variation of the airgap permeance or reluctance of the stator teeth and slots above the magnets as the rotor rotates. In other words, there is no stator excitation involved in cogging torque production of a PM motor. Second, "ripple torque" is the pulsating torque component generated by the stator MMF and rotor MMF. Ripple torque is mainly due to the fluctuations of the field distribution and the stator MMF which depends on the motor structure and the current waveform. This component can take two forms, one of which resulting from the MMF created by the stator windings and the other from MMF created by the rotor magnets. The second form is the torque created by stator MMF and rotor magnetic reluctance variation. In surface mounted PM servomotor, since there exists no rotor reluctance variation, there is no second form of the ripple torque and the ripple torque is mainly created by the first form. The third definition is "pulsating torque" which is defined as the sum of both cogging and ripple torque components (Sebastian et al., 1986 and Ree & Boules, 1989).

In the following analysis, it is assumed that a Y connected three phase unsaturated PM motor used and it has a constant airgap length and symmetrical stator winding. It is also assumed that stator currents contain only odd harmonics and current harmonics of the order of three does not exist. Finally, armature reaction is assumed negligible. For PM motors, at instant t, the instantaneous electromagnetic torque produced by phase A can be written as the interaction of the magnetic field and the phase current circulating in *N* turns

$$T\_a \text{ (t)} = 2p \text{Ni}\_a(\text{t}) \int\_{-\pi/2mp}^{\pi/2mp} R\_\text{g} \text{ L}\_e \text{ B}(\theta\_r, \text{t}) \, d\theta\_r \tag{6}$$

where *Rg* is airgap radius of the servomotor, *Le* is effective stack length, *p* is pole pairs and *m* is the number of phases.

The back-EMF of the motor induced in phase A at the time instant *t* is given by

$$\mathcal{C}\_{\mathbf{a}}(\mathbf{t}) = 2\pi\mathcal{N} \int\_{-\pi/2\,mp}^{\pi/2\,mp} \mathcal{R}\_{m\alpha m} L\_{\mathbb{R}} \mathcal{B}(\theta\_r, \mathbf{t}) \alpha\_m d\theta\_r \tag{7}$$

where *<sup>m</sup>* is the rotor angular speed. The back-EMF in phase a can also be written as

$$\log\_a(t) = \alpha\_m \text{2} \, pN \int\_{-\pi/2\,mp}^{\pi/2\,mp} R\_{\text{mean}} L\_{\text{R}} B(\theta\_r, t) d\theta\_r \tag{8}$$

Substituting (8) into (6), the torque expression becomes

$$T\_a(t) = \frac{e\_a(t) \cdot i\_a(t)}{o o\_m} \tag{9}$$

For the Y-connected three-phase stator winding, the back-EMF in phase *A* can be written as the summation of odd harmonics including fundamental component:

$$e\_a = E\_1 \sin \alpha t + E\_3 \sin \beta \alpha t + E\_5 \sin \dots \sin \dots \tan \gamma \tan \gamma \tan \dots \tag{10}$$

Brushless Permanent Magnet Servomotors 289


0 0.2 0.4 0.6

1

0

0.5

ea ia

0

1

Fig. 14. Torque output for current and back EMF waveforms as a function of electrical cycle

T

em

<sup>1</sup> <sup>2</sup> ( ) <sup>2</sup>

Cogging torque increases due to the increased airgap flux as the magnet strength is increased. Nevertheless, the cogging torque results from the non-uniform flux density in the airgap. As the stator teeth become saturated, the flux begins to distribute evenly in the

In addition, if there is no airgap reluctance variation as in slotless motors, no cogging component occurs. For a slotted stator configuration, the airgap permeance or reluctance is non-uniform because of the shape of the stator, saturation of the lamination material, slot openings and the space between the rotor magnets. This non-uniform reluctance or magnetic flux path causes the airgap flux density to vary with rotor position. This results in

Cogging torque minimization is a significant concern during the design of brushless PM servomotors, and it is one of the main sources of torque and speed fluctuations especially at low speeds and load with low inertias. A variety of techniques are available for reducing the cogging torque of conventional PM servomotors, such as skewing the slots, shaping or skewing the magnets, displacing or shifting magnets, employing dummy slots or teeth, optimizing the magnet pole-arc, employing a fractional number of slots per pole, and imparting a sinusoidal self-shielding magnetization distribution. A summary of these

*cog r g*

*g*

(14)

0 1 2 3 4 5 6

Back EMF

current

0 1 2 3 4 5 6

0 1 2 3 4 5 6

*<sup>r</sup>* is the rotor position.

*dR*

*d*

 

*r*

Cogging torque is sensitive to varying rotor position and can be expressed by

*T*

where *g* is the airgap flux, *R* is the reluctance of the airgap and

motor airgap and the cogging torque decreases.

0 1 2 3 4 5 6

Back EMF

current

0 1 2 3 4 5 6

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>0</sup>



0.5

T

em

1

 ea i

a

**7.2.2 Minimization of cogging torque component** 

methods are displayed in Fig. 15 (Bianchi & Bolognani, 2002).

cogging torque, and generates vibration.

and likewise the current in phase *A* can be written as

$$i\_d = I\_1 \sin \alpha t + I\_5 \sin \dots \tan \dots + I\_7 \sin \gamma \cot \dots + I\_{11} \sin 11 \cot \dots + I\_{13} \sin 13 \cot \dots \dots \tag{11}$$

where *En* is the *nth* time harmonic peak value of the back EMF, which is produced by *nth*  space harmonic of the airgap magnetic flux density *Bgn* and *In* is the *nth* time harmonic peak value of current.

The product of back-EMF and current *ea ia* is composed of an average component and evenorder harmonics for all phases. The total instantaneous torque contributed by each motor phase is proportional to the product of back EMF and phase current. In other words, the total instantaneous torque is the sum of the torques produced by phase *a*, *b*, and *c* and given by,

$$T\_m(t) = \frac{1}{\alpha\_m} \left[ \varepsilon\_a(t) \left( i\_a(t) + \varepsilon\_b(t) \right) i\_b(t) + \varepsilon\_c(t) \left( i\_c(t) \right) \right] \tag{12}$$

Since the phase shifts between *ea ia* and *eb ib* and between *ea ia* and *ec ic* are -2/3 and 2/3, respectively, the sum ( *ea ia* + *eb ic* + *ec ic* ) will contain an average torque component and harmonics of the order of six. The other harmonics are all eliminated. Thus, the final instantaneous electromagnetic torque equation of a servomotor can be written as

$$T\_m \left( t \right) = T\_0 + \sum\_{n=1}^{m} \quad T\_{\delta n} \cos n\theta \alpha t \tag{13}$$

where *T*0 is the average torque, *T*6*<sup>n</sup>* is harmonic torque components and *n* = 1,2,3…. In the ideal case, if the back-EMF's and the armature currents are sinusoidal, then the electromagnetic torque is constant and no ripple torque exists as illustrated in Fig. 14. The same quantities are plotted for sinusoidal back EMF and square or trapezoidal stator current waveforms and the presence of the pulsating torque component is observed clearly. The resultant plots including torque pulsations for all cases are shown in Fig. 14.

#### **7.2 Cogging torque**

#### **7.2.1 Cogging torque theory**

Existence of cogging torque is always a cause of concern in the design of PM servomotors. It in fact demonstrates the quality of a servomotor. This torque component is often desired that the motor produces a smooth torque in a wide speed range. Cogging torque adds unwanted harmonic components to both torque output and the torque-angle curve, which results in torque pulsation. This produces vibration and noise, both of which may be amplified in variable speed drive when the torque frequency coincides with a mechanical resonant frequency of the stator and rotor. In addition, if rotor positioning is required at very low speeds, the elimination of cogging torque component becomes even more crucial and must be eliminated completely during the design stage (Li & Slemon, 1988).

For the Y-connected three-phase stator winding, the back-EMF in phase *A* can be written as

 *ea* = *E*<sup>1</sup> sin *t* +*E*3 sin 3*t* +*E*5 sin 5*t* +*E*7 sin 7*t* + (10)

 *ia* = *I*1 sin *t* + *I*5 sin 5*t* + *I*7 sin 7*t* + *I*11 sin 11*t* + *I*13 sin 13*t* + (11) where *En* is the *nth* time harmonic peak value of the back EMF, which is produced by *nth*  space harmonic of the airgap magnetic flux density *Bgn* and *In* is the *nth* time harmonic peak

The product of back-EMF and current *ea ia* is composed of an average component and evenorder harmonics for all phases. The total instantaneous torque contributed by each motor phase is proportional to the product of back EMF and phase current. In other words, the total instantaneous torque is the sum of the torques produced by phase *a*, *b*, and *c* and given

*T t e ti t e ti t e ti t*

Since the phase shifts between *ea ia* and *eb ib* and between *ea ia* and *ec ic* are -2/3 and 2/3, respectively, the sum ( *ea ia* + *eb ic* + *ec ic* ) will contain an average torque component and harmonics of the order of six. The other harmonics are all eliminated. Thus, the final

*n* 1

where *T*0 is the average torque, *T*6*<sup>n</sup>* is harmonic torque components and *n* = 1,2,3…. In the ideal case, if the back-EMF's and the armature currents are sinusoidal, then the electromagnetic torque is constant and no ripple torque exists as illustrated in Fig. 14. The same quantities are plotted for sinusoidal back EMF and square or trapezoidal stator current waveforms and the presence of the pulsating torque component is observed clearly. The

Existence of cogging torque is always a cause of concern in the design of PM servomotors. It in fact demonstrates the quality of a servomotor. This torque component is often desired that the motor produces a smooth torque in a wide speed range. Cogging torque adds unwanted harmonic components to both torque output and the torque-angle curve, which results in torque pulsation. This produces vibration and noise, both of which may be amplified in variable speed drive when the torque frequency coincides with a mechanical resonant frequency of the stator and rotor. In addition, if rotor positioning is required at very low speeds, the elimination of cogging torque component becomes even more crucial and must be eliminated completely during the design stage (Li & Slemon,

instantaneous electromagnetic torque equation of a servomotor can be written as

resultant plots including torque pulsations for all cases are shown in Fig. 14.

 <sup>1</sup> *m aa bb cc* () () () () () () ()

(12)

*<sup>T</sup>*6*<sup>n</sup>* cos *n*6*t* (13)

the summation of odd harmonics including fundamental component:

*m*

 *Tm* (*t*) = *T*0 +

and likewise the current in phase *A* can be written as

value of current.

**7.2 Cogging torque** 

1988).

**7.2.1 Cogging torque theory** 

by,

Fig. 14. Torque output for current and back EMF waveforms as a function of electrical cycle Cogging torque is sensitive to varying rotor position and can be expressed by

$$T\_{\rm cog} \left( \theta\_r \right) = -\frac{1}{2} \phi\_\mathcal{g}^2 \frac{dR\_\mathcal{g}}{d\theta\_r} \tag{14}$$

where *g* is the airgap flux, *R* is the reluctance of the airgap and *<sup>r</sup>* is the rotor position. Cogging torque increases due to the increased airgap flux as the magnet strength is increased. Nevertheless, the cogging torque results from the non-uniform flux density in the airgap. As the stator teeth become saturated, the flux begins to distribute evenly in the motor airgap and the cogging torque decreases.

In addition, if there is no airgap reluctance variation as in slotless motors, no cogging component occurs. For a slotted stator configuration, the airgap permeance or reluctance is non-uniform because of the shape of the stator, saturation of the lamination material, slot openings and the space between the rotor magnets. This non-uniform reluctance or magnetic flux path causes the airgap flux density to vary with rotor position. This results in cogging torque, and generates vibration.

#### **7.2.2 Minimization of cogging torque component**

Cogging torque minimization is a significant concern during the design of brushless PM servomotors, and it is one of the main sources of torque and speed fluctuations especially at low speeds and load with low inertias. A variety of techniques are available for reducing the cogging torque of conventional PM servomotors, such as skewing the slots, shaping or skewing the magnets, displacing or shifting magnets, employing dummy slots or teeth, optimizing the magnet pole-arc, employing a fractional number of slots per pole, and imparting a sinusoidal self-shielding magnetization distribution. A summary of these methods are displayed in Fig. 15 (Bianchi & Bolognani, 2002).

Brushless Permanent Magnet Servomotors 291

be extracted from the finite element analysis. Even cogging torque component can precisely

Flux 2D software package by Cedrat Co., which is one of the frequently used FEA software in academia and industry, is used in the analyses of the PM servomotor given in Fig. 16 - Fig. 18 (Flux 2D and 3D Tutorial 2002). Cogging torque is obtained using no-load simulations. Rotor structure is rotated for one slot pitch and torque values are calculated using the Flux 2D. Fig. 16 shows both no-load flux density distribution of a 24 slot-8 pole PM servomotor as well as its cogging torque variation over one slot-pitch. Fig. 17 displays the rotor a disc type PM servomotor, its FEA predicted and experimentally verified cogging torque variation. The results show that FEA work well for the cogging torque predictions.

(a) (b) (c)

Fig. 16. 2D-FE Model of 24 slots with 8 poles servomotor (a) mesh structure, (b) flux density


Cogging Torque [Nm]

0 5 10 15

24s\_8p flux

Mechanical Angle [Degrees]

(a) (b)

Fig. 17. Rotor structure of a disc type PM servomotor (a), prediction of cogging torque with

Torque ripple is another important undesired torque element in PM servomotors. It occurs as a result of fluctuations of the field distribution and the stator MMF. In other words, torque ripple depends on the MMF distribution and its harmonics as well as the magnet flux distribution. At high speeds, torque ripple is usually filtered out by the system inertia. However, at low speeds torque-ripple may produces noticeable effects on motor shaft that

Fig. 18 shows an interior permanent magnet (IPM) servomotor geometry, flux lines and flux density distribution at no load operation. If the motor is supplied by a harmonic free

may not be tolerable in smooth torque and constant speed servo applications.

be calculated using modern FEA software.

distribution and (c) cogging torque variation

FEA and experimental data (b)

**7.3 Torque ripple** 

Fig. 15. Summary of cogging torque minimization techniques for PM servomotors

Minimization of cogging torque in PM servomotors can be accomplished by modifications either from stator side or from rotor side. Choosing the appropriate "ratio of stator slot number to rotor pole number" combination is one of the common ways to minimize cogging torque component. This is a design based choice and is the most common method to minimize the unwanted cogging torque components in PM servomotors. Utilizing dummy slots in stator teeth increases the frequency of cogging and reduces its amplitude. Similarly, "displaced slots and slot openings" is a different method to minimize cogging component. In integral slot servomotors (q=1slots/pole/phase), each rotor magnet has the same position relative to the stator slots resulting in cogging torque components which are all in phase, leading to a high resultant cogging torque. Nevertheless, in fractional slot servomotors, where q≠1slots/pole/phase rotor magnets have different positions relative to the stator slots generating cogging torque components which are out of phase with each other. The resultant cogging torque is, thus, reduced since some of the cogging components are partially cancelled out. Even uncommon combinations such as 33, 39 or 45 slots are employed for certain applications to obtain small cogging torque components even though it generates an unbalanced servomotor.

Rotor side cogging torque minimization techniques are more cost effective compared to stator side methods and classified into three different categories: variable or constant magnet pole-arc to pole-pitch ratio, pole displacement and magnet skew. Techniques applied to rotor structure are simpler and less costly than stator side techniques. One of the most effective techniques used in servomotors is to employ an appropriate magnet pole-arc to pole-pitch ratio. Reducing the magnet pole-arc to pole-pitch ratio reduces the magnet leakage flux, but it also reduces the magnet flux, and, consequently, the average torque. Another method of reducing the cogging torque is to employ variable magnet pole-arcs for adjacent magnets such that the phase difference between the associated cogging torques results in a smaller net cogging.

#### **7.2.3 Predicting cogging torque using FEA**

Finite element analysis (FEA) can correctly examine the PM servomotors. The motor designers do not need to go through cumbersome circuit type analytical methods because important parameters such as flux, inductance, force and torque can simply and accurately

Fig. 15. Summary of cogging torque minimization techniques for PM servomotors

it generates an unbalanced servomotor.

results in a smaller net cogging.

**7.2.3 Predicting cogging torque using FEA** 

Minimization of cogging torque in PM servomotors can be accomplished by modifications either from stator side or from rotor side. Choosing the appropriate "ratio of stator slot number to rotor pole number" combination is one of the common ways to minimize cogging torque component. This is a design based choice and is the most common method to minimize the unwanted cogging torque components in PM servomotors. Utilizing dummy slots in stator teeth increases the frequency of cogging and reduces its amplitude. Similarly, "displaced slots and slot openings" is a different method to minimize cogging component. In integral slot servomotors (q=1slots/pole/phase), each rotor magnet has the same position relative to the stator slots resulting in cogging torque components which are all in phase, leading to a high resultant cogging torque. Nevertheless, in fractional slot servomotors, where q≠1slots/pole/phase rotor magnets have different positions relative to the stator slots generating cogging torque components which are out of phase with each other. The resultant cogging torque is, thus, reduced since some of the cogging components are partially cancelled out. Even uncommon combinations such as 33, 39 or 45 slots are employed for certain applications to obtain small cogging torque components even though

Rotor side cogging torque minimization techniques are more cost effective compared to stator side methods and classified into three different categories: variable or constant magnet pole-arc to pole-pitch ratio, pole displacement and magnet skew. Techniques applied to rotor structure are simpler and less costly than stator side techniques. One of the most effective techniques used in servomotors is to employ an appropriate magnet pole-arc to pole-pitch ratio. Reducing the magnet pole-arc to pole-pitch ratio reduces the magnet leakage flux, but it also reduces the magnet flux, and, consequently, the average torque. Another method of reducing the cogging torque is to employ variable magnet pole-arcs for adjacent magnets such that the phase difference between the associated cogging torques

Finite element analysis (FEA) can correctly examine the PM servomotors. The motor designers do not need to go through cumbersome circuit type analytical methods because important parameters such as flux, inductance, force and torque can simply and accurately be extracted from the finite element analysis. Even cogging torque component can precisely be calculated using modern FEA software.

Flux 2D software package by Cedrat Co., which is one of the frequently used FEA software in academia and industry, is used in the analyses of the PM servomotor given in Fig. 16 - Fig. 18 (Flux 2D and 3D Tutorial 2002). Cogging torque is obtained using no-load simulations. Rotor structure is rotated for one slot pitch and torque values are calculated using the Flux 2D. Fig. 16 shows both no-load flux density distribution of a 24 slot-8 pole PM servomotor as well as its cogging torque variation over one slot-pitch. Fig. 17 displays the rotor a disc type PM servomotor, its FEA predicted and experimentally verified cogging torque variation. The results show that FEA work well for the cogging torque predictions.

Fig. 16. 2D-FE Model of 24 slots with 8 poles servomotor (a) mesh structure, (b) flux density distribution and (c) cogging torque variation

Fig. 17. Rotor structure of a disc type PM servomotor (a), prediction of cogging torque with FEA and experimental data (b)

### **7.3 Torque ripple**

Torque ripple is another important undesired torque element in PM servomotors. It occurs as a result of fluctuations of the field distribution and the stator MMF. In other words, torque ripple depends on the MMF distribution and its harmonics as well as the magnet flux distribution. At high speeds, torque ripple is usually filtered out by the system inertia. However, at low speeds torque-ripple may produces noticeable effects on motor shaft that may not be tolerable in smooth torque and constant speed servo applications.

Fig. 18 shows an interior permanent magnet (IPM) servomotor geometry, flux lines and flux density distribution at no load operation. If the motor is supplied by a harmonic free

Brushless Permanent Magnet Servomotors 293

to know which winding will be energized. This requires precise information of the rotor position using hall sensors or resolvers. When the rotor magnet poles pass, the position information of the sensor is provided to the controller and inverter drives the motor

Brushless PM servomotors can have both trapezoidal and sinusoidal back EMF waveforms and are excited with either rectangular or sinusoidal currents. A current regulated voltage source inverter is used to drive the servomotors. Power stage of the converter is combined by a rectifier, DC link and an inverter. Current sensors are used in each phase and fed back to the DSP controller. Position information is frequently obtained either by a resolver or an encoder although hall sensors are preferred for trapezoidal brushless servomotors. The

Detailed introduction to brushless permanent magnet servomotors used in both industrial and servo applications is provided in this chapter. Motor classification and types, advantages and disadvantages of different PM servomotors and comparison, materials used in motor components are reviewed. Servomotor design process including electromagnetic, structural and thermal steps; softwares used in the analysis, design and optimization of such motors are also enlightened in detail. Torque quality, mathematical output torque equations, cogging and ripple torque components are investigated thoroughly since the torque quality

The authors are indebted to CEDRAT Co. for providing the Flux 2D FEA Package and MDS Motor Ltd. for providing some motor pictures and its facilities in preparing this document.

simple system set-up with the main blocks of the system is illustrated in Fig. 20.

windings in the correct sequence.

Fig. 20. Permanent magnet servomotor drive

confirms the quality of the servomotor.

Flux 2D and 3D Tutorial, Cedrat Co. 2002.

**10. Acknowledgment** 

**11. References** 

**9. Conclusion** 

excitation, almost no ripple exists at the motor output (Fig. 19). However, if inverter or motor driven harmonics, such as integer slot motors or single segmented rotors with *q*=1 with no skew, exist in the current excitation, significant torque ripple appears at the motor output and precautions must be taken to lower this component as much as possible. One of common techniques to reduce the torque ripple component and obtain smooth torque output is to use segmented rotor. In order to use this approach, rotor is divided into segments and each piece is rotated with respect to each other to obtain ripple free output. As displayed in Fig. 19, if no segment is used, more than 130% of torque ripple is observed at the motor output. When the rotor is divided into 4 segments, torque ripple is reduced to less than 6% of the average torque which is a reasonable number for most applications.

Fig. 18. Spoke type IPM servomotor (a) and flux line and flux distribution (b)

Fig. 19. Torque output variation of the IPM servomotor with low and high torque ripple

### **8. Control of PM servomotors**

Commutation of a brushless pm motor is achieved electronically. Stator winding is energized using an inverter in a sequence. It is crucial to know the position of the rotor so as

excitation, almost no ripple exists at the motor output (Fig. 19). However, if inverter or motor driven harmonics, such as integer slot motors or single segmented rotors with *q*=1 with no skew, exist in the current excitation, significant torque ripple appears at the motor output and precautions must be taken to lower this component as much as possible. One of common techniques to reduce the torque ripple component and obtain smooth torque output is to use segmented rotor. In order to use this approach, rotor is divided into segments and each piece is rotated with respect to each other to obtain ripple free output. As displayed in Fig. 19, if no segment is used, more than 130% of torque ripple is observed at the motor output. When the rotor is divided into 4 segments, torque ripple is reduced to less

than 6% of the average torque which is a reasonable number for most applications.

magnets

rotor

stator

(a) (b)

Fig. 18. Spoke type IPM servomotor (a) and flux line and flux distribution (b)

Fig. 19. Torque output variation of the IPM servomotor with low and high torque ripple

Commutation of a brushless pm motor is achieved electronically. Stator winding is energized using an inverter in a sequence. It is crucial to know the position of the rotor so as

0 10 20 30 40 50 60

1 segment 2 segments 3 segments 4 segments

Mechanical Angle [Degree]

**8. Control of PM servomotors** 

Torque output [Nm]

shaft

to know which winding will be energized. This requires precise information of the rotor position using hall sensors or resolvers. When the rotor magnet poles pass, the position information of the sensor is provided to the controller and inverter drives the motor windings in the correct sequence.

Brushless PM servomotors can have both trapezoidal and sinusoidal back EMF waveforms and are excited with either rectangular or sinusoidal currents. A current regulated voltage source inverter is used to drive the servomotors. Power stage of the converter is combined by a rectifier, DC link and an inverter. Current sensors are used in each phase and fed back to the DSP controller. Position information is frequently obtained either by a resolver or an encoder although hall sensors are preferred for trapezoidal brushless servomotors. The simple system set-up with the main blocks of the system is illustrated in Fig. 20.

Fig. 20. Permanent magnet servomotor drive
