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Tatsuya Kikuchi, *Member, IEEE*, and Takashi Kenjo, *Member, IEEE*

*Abstract -* The necessity of in-depth learning of cogging/detent torque for mainly undergraduates is discussed with particular attention to its method and tools. The experimental tools developed include a cogging-torque tester, variable-skew rotors for a DC motor and a three-phase hybrid stepping motor having a special rotor construction to eliminate certain harmonic components of cogging torque. Two pieces of software are presented for use in the class and laboratory to supplement the experiment set-ups: one is a motion simulator for a watch stepping motor to demonstrate the notion of detent torque and the other is for quick computation of the cogging torque primarily to investigate pole-and-slot number combinations for various pole-arc to pole-pitch ratios. The latter can be used to see the effects of construction asymmetry on the cogging-torque's amplitude and frequency. The usage of these tools and software is discussed from the viewpoint of engineering education. The major conclusion is that the variable skew rotor is very suited to in-depth learning of cogging torque but the supplemental use of the computation software is invaluable for gaining deeper insight into this subject.

The level of sophistication and precision attained in the field of automation technology is rising yearly, which is why it is important for today's engineers to have a good working knowledge of servomotors and stepping motors. Many of these motors employ permanent magnets. The torque generated in such motors consists of two components: the major part which is basically proportional to the supplied electric current, and that portion generated independent of the current and arising from the interaction between the permanent magnet and the teeth in the rotor/stator core. The former current-generated torque is the usual object of adjustment or control in motion-control technology. The latter is called by various terms depending on how it works or what causes it, e.g. cogging torque, detent torque, salient-pole torque, reluctance torque, etc.

Here we shall use as a general expression "no-current torque" for this latter type of torque, discussing cases in which it has a (positive) detent or (undesirable) cogging effect, mainly from the perspective of engineering education.

In the stepping motor used in a wristwatch, the no-current torque is used for 'detenting' or holding the arm at a fixed position, which is why it is traditionally called the detent torque. In most conventional and brushless DC servomotors, however, this sort of torque is undesirable since it causes speed ripples or cogging motions. This is the origin for the expression 'cogging torque'.

In the design of single-phase brushless DC motors, used for example in small ventilating fans, this sort of torque is put to positive use in achieving a stable operation [1]. Meanwhile, in the microstep operation of hybrid stepping motors, the detent torque has no positive effect and is regarded as a cogging torque which should preferably be suppressed.

The cogging/detent torque is clearly not a trivial engineering problem, and we believe that obtaining a firm understanding of the no-current torque is essential in the education of electrical/electronic as well as mechanical engineers.

Over the years, we have developed a set of teaching materials (experiment set-ups and texts) mainly through a process of trial-and-error, for use in educating undergraduates, post-graduates as well as working engineers. From this process, we have identified a few issues. In this paper, we discuss some of these issues and our latest solution, which resulted by incorporating some research results we had obtained on the dynamic behavior of brush and brushless DC motors and cogging torque computations, and by adapting Microsoft Visual Basic (VB) to develop user-friendly, visual instructional materials.

The most important subject in the study of electric motors is the theory and mechanism of torque production. University and vocational high-school textbooks give the torque *T* generated in a DC motor as:

This formula, based on design parameters, is not so useful to someone using the motor since *k* and cannot be measured from outside the motor. When dealing with recent DC servomotors, which use permanent magnets in their field system, we often cite the following similar formula:

The proportionality constant ,which is known as the "torque constant" and theoretically identical to the back e.m.f. constant in a consistent unit system, can be measured. We tell students that this parameter, which is always specified in the manufacturer's data sheets, is the most important parameter for DC servomotors.

The above formulas can be modified or extended to apply to other motor types such as synchronous, brushless DC, and stepping motors. These topics are discussed at length in many textbooks and monographs.

The torque expressed by these formulas is that created by the interaction between the field system's magnetic flux and armature currents. Yet, there is another torque component, that is, the no-current torque, which is generally ignored in university education.

The no-current torque is passed over as a subject mainly because it does not affect the net torque of a motor, another reason being that it cannot be reduced to a simple mathematical expression suitable to beginners. Nevertheless, with increasingly higher performance levels demanded in today's control-use machinery (e.g., accurate positioning, quiet operation, etc.), it is becoming increasingly important for electrical and mechanical engineers to have a basic working knowledge of the no-current torque. There are basically two considerations from the educational standpoint. The first is how to introduce the no-current torque as an interesting subject in class, and the second is imparting some in-depth and practical understanding to the student.

In Japan and perhaps in other countries as well, students in primary or secondary school have the opportunity in their science class to construct a DC motor from parts including a permanent magnet. Having also played with motorized toy cars and other toys as youngsters, most entering university students are familiar with the Mabuchi DC motor. They must have directly felt the cogging torque with their fingers as they played with their toys, but in most cases thought no more about it.

When these students are shown in class motors as in Fig. 1, a few of them are always interested in knowing why the armature core is skewed in one of the motors. The instructor explains that the cogging felt by hand is caused by the interaction between the permanent magnets and the rotor's teeth, that this is undesirable in terms of a smooth operation, and that this cogging can be suppressed by skewing at a certain angle. Thus what the student had been vaguely aware of from his youth is presented to him or her as cogging torque, along with a practical design solution.

Fig. 1. Two types of DC motor armature (rotor). (a) Straight lamination. (b) Skewed lamination.

To bring up the notion of detent torque, we usually use the example of watches. We tell students that "Your watches contain a highly efficient electric motor which runs for more than three years on a single tiny battery. Let's take a look at this motor." Instead of actually opening the back of a watch, the instructor shows Fig. 2 and remarks, "The stator core has two peculiarly shaped grooves. What are they for?" (Answer: they ensure that the watch arms move in the clockwise direction.) This question can lead to a discussion on cogging and detenting.

Fig. 2. Schematic diagram of single-phase stepping motor used in a watch.

If some students wish to study the characteristics of the cogging torque in more depth, suitable experimental tools are needed. To this end, we designed several experimental set-ups for use in class, one in the form of a final-year project. At the same time, for research purposes we produced several pieces of software using languages BASIC and C for computing cogging torque and the dynamic behavior of motors. By incorporating VB into this software, we found we could produce useful supplemental teaching materials to be used together with the experimental tools.

The following section describes the experimental set-ups, section IV presents experiment results, and section V discusses the software.

We have designed an experiment bench called Mechatro Lab, which is discussed in Reference [2]. A number of regular experiments on electric motors and generators can be demonstrated using this Lab, but some students are always interested in pursuing some topics in further depth. To provide such advanced students with instruction on the subject of no-current torque, we have designed a cogging tester and two optional topics. One involves cogging torque measurement of a DC motor structure of various schemes, and the other entails advanced rotor design for decreasing the detent torque in a three-phase hybrid stepping motor.

A. Cogging-torque Tester

Our cogging/detent tester has the configuration shown in Fig. 3. Although the principle involved is not novel, the use of a 500-step, five-phase stepping motor geared down by a harmonic gear (by a factor of 50) may be unique. This is used for driving the test motor at the desired (slow) speed. Stepping command is provided by a PC. Between the stepping motor and the test motor is placed a torque transducer, which detects the torque from the strain in the shaft coupling the two motors; torque is converted to digital values with an A/D converter and displayed on the PC monitor. When the tester is run without a test motor only a horizontal line appears on the PC screen; thus, although the driving stepping motor has by its nature some cogging torque, it does not affect the measurement of the cogging torque of the test motor. The tester requires no position sensor because the stepping motor determines the angular position of the test motor. Although some positional error is produced by the torsional strain at the shaft, the error can be mostly corrected by incorporating an error-compensation software. The cogging torque tester is used for measuring torque in both motor types below.

Fig. 3. Cogging/detent torque tester. (a) Hardware configuration. (b) Photograph of the tester.

B. DC Motor Construction

Fig. 4 shows the stator with a two-pole scheme and two types of test rotors, both without coils so that only the no-current torque is produced. Rotor-1 can be given a skew at any desired angle or inclination. Each sheet of the lamination core can be rotated individually with respect to the core axis, then the entire core is fastened with a nut.

In rotor-2, the core lamination is divided into two blocks which are shifted against each other at the desired tooth pitch. (Although the block-rotor construction of rotor-2 makes it impossible to install windings, the effect of the shifted blocks can be achieved with a "straight" normal rotor (which will take windings) by using differently shaped magnets in the stator.)

Two slip-rings can be mounted on rotor-1, which together with a search coil, are used for measuring flux distribution as shown in Fig. 5(a). The flux distribution with zero skew angle, shown in Fig. 5(b), is measured mainly to confirm the flux level or flux density, which peaks at about 0.32T in our test motor. We should note that cogging torque cannot be calculated directly from this flux distribution (this is discussed in length in section V).

Fig. 4. Experiment set-up for DC-motor construction.(a) Field system using two ferrite magnets. (b) Rotor-1: skew type with 15 teeth/slots. (c) Rotor-2: block construction.

Fig. 5. (a) Search coil installed for flux distribution measurement. (b) Flux distribution measurement.The peak flux density was about 0.32T, from which the peak magnetization was estimated to be about 0.34T or 3400 Gauss.

C. Hybrid Stepping Motor Construction

The rotor set shown in Fig. 6 is used to compare, for the three-phase hybrid stepping motor, cogging torque characteristics of the special rotor construction of (b) with the normal rotor of (a). Instead of silicon-steel laminations, which is commonly used in the toothed portions of rotors, normal carbon steel rings are used for the core in the test rotor. A stack of two rings, with each ring set at a certain tooth pitch angle against each other, is installed at each rotor end. The stator is shown in Fig. 6(c).

The cogging torque can be thought to be comprised of harmonic components. As a result of symmetrical three-phase construction, shown in Fig. 6(d), the fundamental and odd-number order harmonic components of the cogging torque are all canceled. The second or fourth order harmonic components are also canceled because of three-phase construction. If, in addition, the two rings of each end stack are shifted against one another by 1/12 tooth pitch (Fig. 7(a)), the remaining major component, which is the 6th component, is eliminated (see Fig. 7(b)). This is the rationale behind the two-ring arrangement described above.

Fig. 6. Test rotors for three-phase hybrid stepping motor. (a) Normal rotor: A disk ferrite magnet is sandwiched by two lamination cores. (b) Special rotor. (c) Stator of the three-phase hybrid stepping motor, and (d) Cross-sectional diagram.

Fig. 7. Special rotor construction. (a) Cross-sectional diagram. (b) The 6th component of the detent torque is enhanced at the S and N sides but others are canceled between them.

The torque tester was quite successful, but a few problems were found with the DC motor constructions and stepping motor set-up.

A. Problems with the DC Motor's Rotors

The rotors shown in Fig. 1 have 16 teeth and fit stators designed for general purposes (i.e., stepping motor or brushless DC motor in addition to DC motor). The test rotors of Fig. 4 are 15 toothed and were designed to be installed into the permanent magnet stator shown in the same figure.

Ideally, a rotor with 15 slots in a two-pole configuration will produce 30 ripples per rotation when the no-current torque is measured. As seen in Figs. 8 and 9, however, there are 15, not 30, major ripples, showing that the actual measurements are somewhat different from the ideal. When the skew angle is zero for rotor-1, a 30-ripple cogging exists superposed with a 15-ripple waveform (see Fig. 8). Meanwhile, the two-block arrangement of rotor-2 produces a 15-ripple non-sinusoidal waveform, and the 30-ripple waves are not apparent (see Fig. 9). We expected that shifting by half tooth-pitch, as in rotor-2, would drastically reduce the cogging, which would have been the case for a perfectly symmetrical machine construction and sinusoidal waveform. However, there still exists considerable irregular cogging.

This sort of irregular cogging torque is often observed in real DC and brushless DC motors, which we assumed was caused by asymmetry or imbalances (in the magnet positions/magnetization, eccentricity, etc.) that inevitably occur during manufacturing. To examine this problem, we produced simulation software for computing cogging torque, which is discussed in section V.

We were able to demonstrate satisfactorily that skewing the core decreases the cogging torque, as shown in Fig. 10. The amplitude of cogging is minimized with a one tooth-pitch skew.

Fig. 8. Measured cogging torque curves for three different skew angles using rotor-1: no skew, 2/3-tooth and one-tooth pitch angle (one-tooth pitch corresponds to a 24-degree difference in the two lamination sheets at the extreme ends).

Fig. 9. Comparison of cogging torque for two cases of the DC motor using rotor-2: the above is for the no-shift case, and the lower is for a half-tooth pitch shift angle.

Fig. 10. Cogging amplitude as a function of skew angle using rotor-1. The skew angle of 48 degrees corresponds to a two-tooth pitch inclination. The amplitude is minimum at one-tooth pitch (24 degrees).

B. Hybrid Stepping Motor

The three-phase hybrid stepping motor set-up of Fig. 6 was designed as the final-year project of an undergraduate student. Since the three-phase scheme was fundamental for the AC and brushless DC motors of Mechatro Lab, the same scheme was adopted for the stepping motor. The stator also was built for Mechatro Lab. The project produced some valuable findings, but as seen in Fig. 11, there still exists an irregular torque pattern with the shifted-tooth rotor.

Fig. 11. Comparison of detent torque for two cases of the three-phase stepping motor: the above is for the no-shift case (the two rings are aligned), and the lower is for 1/12-tooth pitch shift, where the amplitude of the short-period ripples is minimized but the long-period ripples has become larger.

We describe below two pieces of software we developed as part of the instructional material.

A. Watch Stepping Motor

Instead of showing the actual tiny motor, its movement can be simulated by numerical computation of electrical and dynamic differential equations. Sample software no.1, shown in Fig. 12, can demonstrate to the student that simulation principles are much simpler than what he or she expects. Here we use the Euler implicit method to numerically solve differential equations (see program coding) instead of the well-known Runge-Kutta method for the following reasons:

1) Computations can be done directly from the equations. As long as the student has an understanding of Newton's principles of force, *LR* circuits and electromotive force, this method follows naturally.

2) Cases of zero inductance or zero inertia (though unrealistic) can be dealt with.

We compared results with the Runge-Kutta and other methods, and found that this method is quite sufficient for this level of simulation.

The equations for this model are:

The voltage applied to the coil consists of alternating pulses, as shown in Fig. 12. In our simulation program, about 10-20% of each half cycle is used for excitation. In an actual wristwatch, the motor is excited for only a few milliseconds each second, while the tiny rotor is held at the detent poles the rest of the time, which is why the battery lasts for a long time.

Since some students still found this level of modeling difficult to understand, as a preliminary assignment we prepared a problem which involves linear motion of a coil placed in a magnetic field and connected to a spring (see Fig. 13). Only a simple stepwave voltage is applied to the coil to investigate the motional behavior. The equations and voltage waveforms are much simpler than in the wristwatch case.

The wristwatch simulation program shown in Fig. 12 is placed in the sample1 directory folder in the CD-ROM.

Fig. 12. Display for the wristwatch stepping motor simulation.

Fig. 13. Model of a moving coil placed in a homogeneous magnetic field, as a preparatory assignment to illustrate the principles of dynamic computation. (The coil is pulled by a spring.)

B. Cogging Computation

Sample software no.2, shown in Fig. 14, enables quick computation of the cogging torque of a DC motor construction with two- and four-pole configurations. This software is for examining the interactive effects of the number of teeth, number of poles, and pole-arc to pole-pitch ratio.

This algorithm for computing torque *T* is based on the following formula:

which is applied to the field's magnet region, and can be approximated as:

where *L* is the magnet length, *w* the magnet width, and *R* the average magnet radius as shown in Fig. 15. indicates the integration over one rotation with respect to .

Here *M* is the magnetization vector in the field's permanent magnet, and the flux density *B* is related to it as follows:

where is the permeability of free space.

For sake of simplicity, only the radial components of *M* and *H* are considered here. For *H*, we assume the distribution pattern shown in Fig. 16. The flux density distribution of Fig. 5(b) can be used as reference data, but the *M*-pattern near the magnet edges must be different from such a flux pattern taken by a coil wound on a tooth. Although the use of a Hall-element mounted on a non-slotted cylindrical core should provide a more accurate magnetization pattern, this was not employed since we thought that it was too complex for educational purposes.

The field intensity *H* is calculated using a classical method, as shown in Fig. 16; it is determined by the intercepts of eqn. (12) and the permeance line.

where *g* is the effective airgap length. The effective gap length is modeled from the toothed construction as illustrated in Fig. 17.

While this method may not be as accurate as the finite-element method, it has certain features:

1) It is convenient for quickly examining tendencies caused by small changes in the magnet size, magnetization, position deviations, etc.

2) It can be easily extended to account for the armature-current effect as presented in Reference [3], where Hall-element measurements were taken to determine the magnetization pattern .(The torque ripple calculation for a six-tooth four-pole brushless DC motor was in fairly good agreement with measurements.)

3) Since the computation speed is fast, it can be incorporated in dynamic calculations. (We used this to simulate the dynamic behavior of the three-phase brushless DC motor.)

This software was initially designed to demonstrate the effects that various combinations of the number of teeth and number of magnetic poles with typical pole-arc to pole-pitch ratios have on magnetization patterns.

With such test motors as in our experimental set-up, repeated disassembly and assembly often results in a loose fit between the end-caps and the stator core, and the armature core tends to be biased to one side of the magnetic poles. To examine the effects of asymmetry, therefore, we extended the program so one can also examine the effects of deviation of the magnet position from symmetry and eccentricity of the rotor shaft for the 2-pole scheme (see Fig. 18). Fig. 19 shows the computed cogging torque for three cases: 1) perfectly symmetric configuration; 2) one magnet is displaced by one-degree; and 3) there is a 0.3-mm eccentricity for an average 0.6-mm airgap length.

The cogging torque computation program shown in Fig. 14 is placed in the sample2 directory folder in the CD-ROM.

Fig. 14. Display sample (with 40% eccentricity) for the computation software for DC motor cogging torque.

Fig. 15. Model for field magnets. For software-2 the following values are set: *L*=40 mm, *w*=8.5 mm, and *R *(magnet's midway radius)=31 mm.

Fig. 16. Determining magnetic operating points as the intersects of the magnetic characteristic curve and permeance lines.

Fig. 17. The effective gap length is modeled from the toothed construction.

(a)

(b)

Fig. 18. Asymmetry in (a) magnet position's deviation, (b) airgap eccentricity.

Fig. 19. Simulated cogging torque for three cases: perfectly symmetric configuration, one magnet is displaced by one-degree and a 0.3 mm-eccentricity for an average 0.6 mm airgap length.

We shall discuss the usage of the above-mentioned experiment tools and software from various angles.

A. Cogging Torque as a Classroom Subject

The class instructor should thoroughly understand the nature of the no-current torque, particularly when compared to the current-generated torque given by eqns. (1) or (3). In the former equation, the torque is proportional to both the field fluxand armature current *I*, and the proportionality constant *k* is determined only from the winding parameters ** z**, and

The no-current torque waveforms are in turn determined by the slot- and pole-number combination as well as construction symmetry/asymmetry; the former is a design issue, and although the latter depends on fabrication accuracy, it critically affects the motor's performance regarding the no-current torque. We believe that such issues should be discussed in undergraduate classes, especially for the benefit of students who display interest in such specific matters. The main goal at the undergraduate level should be to provide a basic understanding of these matters, rather than carrying out detailed computations of the cogging torque for a particular construction.

B. Cogging Torque in Industry

Some instructors may think that cogging torque is not a very suitable subject for undergraduates. Few would probably disagree, however, that in post-university education targeting working engineers, this kind of fine-structure problem concerning electrical machines is highly important. Yet, in our experience, even working engineers often lack knowledge of the no-current torque, sometimes leading to irrational practices in an entire industry.

Howe and Zhu have proposed a simple factor *C*=*2pQ/N* as the "goodness" of the slot and pole number combination, where 2*p* is the number of magnetic poles, *Q* the number of slots and *N* the smallest common multiple of *Q* and *2p* [4]. They state that the larger the factor *C* the larger the cogging torque. The four-pole (2*p*=4) motor with sixteen-slots (*Q*=16) is used extensively in automobiles, e.g. in fans. This construction, with *C*=4, generates a high cogging torque. The cogging torque can be reduced considerably by changing the slot number to 14 or 18, for both of which *C*=2. The sixteen-tooth core has persisted mainly because of the conservatism of engineers and the automotive industry. By demonstrating that the cogging can be reduced, our software (no.2) can prove useful for introducing suitable design changes into the automotive and other industries.

Incidentally, all of the small motors mass-produced today have even-number slots with no skew. This is not without reason. For instance, even-number toothed motors can be manufactured faster since two flyers can be used simultaneously in the coil winding process. We saw earlier that odd-numbered cores require skewing to control the cogging torque (see Fig. 9). Yet, the benefits of skewing are somewhat dubious (for instance, there is the issue of commutation ripples; see below). Even with even-toothed rotors, there is a possibility of decreasing the cogging level by controlling the magnetization although this sacrifices some of the current-generated torque. For example, our simulation program shows that sinusoidal magnetization results in no cogging.

C. Cogging Torque and Commutation Rippling Torque as Causes of Speed Ripples and Noise

It is often forgotten that in a DC motor in which the number of commutator segments and teeth are equal, torque ripples caused by commutation are produced with the same frequency as the cogging torque. 'Commutation' here means the mechanism by which DC current is supplied to the armature coils from the mechanical contact between the commutator and brushes. As seen from a certain coil, the current alternates between positive and negative. While it is desirable that this shift takes place without causing torque fluctuations, this is usually not the case. This torque ripple cannot be suppressed by skewing, and is often comparable to or higher than the no-current cogging torque.

We should note that:

1) While a moving-coil type DC motor has no cogging torque, commutation torque ripples (and speed fluctuations and noise caused by this) can be considerable in certain applications.

2) It is often thought that cogging is the major cause of mechanical vibrations and acoustic noise coming from a motor or its load. Noise, however, is a very complex problem and commutation may in fact be a larger source of noise. Other factors such as resonance can cause noise as well.

Thus it is necessary to explain to students that skewing is not always the best or the only solution for reducing torque ripples and noise.

D. Comparison of the Three-phase Stepping Motor and DC Motor Construction

The three-phase hybrid stepping motor set-up described in section III was originally invented by M. Sakamoto and A. Tozune [5]. As stated earlier, we used this as the final-year project for an undergraduate student to investigate the validity of its design concept. The other purpose was to make a comparison with the DC motor. If cogging were always sinusoidal in a DC motor, a 180-degree shift between the two blocks in rotor-2 (Fig. 4(c)) should eliminate cogging. However, this is not the case because of higher/lower harmonic components and asymmetry. In three-phase construction for the hybrid stepping motor, except for the 6th component, most of the major higher harmonics can be canceled by symmetrical 120-degree arrangement of the stator poles. The two-stack rotor tooth construction, shown in Fig. 6(b), was designed to eliminate the 6th harmonic.

While the 1/12 tooth-pitch shift will not impair the stepping function of a stepping motor, a half-pitch shift, although this would in theory entirely eliminate cogging, is not an option for stepping motors since the steps would in effect disappear. Instead of shifting by 1/12 tooth pitch, a skewed yoke construction with a skew angle corresponding to 1/12 tooth pitch should perform better in terms of controlling cogging torque (because cogging is reduced even with asymmetric magnetization; this was shown in the DC motor construction using the cogging simulation program). This has not been tried in industrial practice because of the obvious expenses this would require.

To sum up, the DC motor and three-phase stepping motor are similar in that shifting or skewing can effectively suppress cogging; the difference is in the optimum shift/skew angle: one tooth pitch for DC motors, and a small fraction of a pitch for polyphase stepping motors.

E. Machining Accuracy and Asymmetry Problem

It is necessary to address the issue of the machining quality of experimental set-ups. Neither precise symmetry nor accurate machining can be reasonably expected even in industrial precision motors. By repeated opening and closing, the fit between the end-cap and stator housing can be loose in an experiment-use machine such as ours, and this seemed to produce pronounced effects in the measured cogging torque. For example, large 15-ripple components are evident superposed with the 30-ripple components in the DC motor measurement (see Figs. 8 and 9).

A stepping motor requires a very high level of machining accuracy. With perfect construction in both symmetry and 120-degree shifting, the third-harmonic components should cancel out one another. Tooth machining must also be more precise than for a DC motor. Yet our educational-purpose motor was not machined at this level because of the expenses it would have required (small-lot production of precision motors is very costly). One may thus argue that a hybrid stepping motor is too complicated in its construction to be used in undergraduate education. For the working engineer, however, this set-up could prove useful in gaining some insight of practical significance. Sakamoto's invention has not yet seen commercial production, perhaps because of the costs and precision level required. There may be another problem: the hybrid motor has a double-salient structure, which tends to generate acoustic noise, and this may be a drawback for applications requiring quiet operations. Our example illustrates the need to make a careful evaluation when choosing research-level topics as materials for general education.

Let us now return to the student's question on skewing when shown the two rotors of Fig. 1. As evident from the discussions above, we found that the answers are far more complex than we had anticipated when we first set out to develop the instructional materials on no-current torque.

As long as the armature core construction and field system are perfectly symmetrical, odd-number-toothed cores (with a certain pole-arc to pole-pitch ratio and a certain magnetization pattern) have a very low cogging torque compared to even-number-toothed ones in theory, and skewing is not needed. However, the experimental results display high cogging levels perhaps due to asymmetry effects. The purpose of skewing in practice is thus to counter the adversary effects caused by asymmetry rather than minimizing the remaining higher harmonics of cogging in an ideal construction.

Our study demonstrated that the variable-skew rotor shown in Fig. 4(b), which is simple and low-cost, is a highly useful experiment tool for studying cogging torque. If a 16-slot rotor is added to the set, its instructional value would be upgraded considerably. One must, however, be careful when introducing further refinements into the hardware, for example, improving the machining fit, symmetry, concentricity, control of magnetization, etc., since this would entail considerable expenses. Rather, the effectiveness of our instructional material set is heightened by the simulation/computation software such as the one that employs a classical algorithm for cogging computation.

While our cogging-torque set-up is based on the Mechatro Lab, designed by the authors and used at many vocational schools and colleges in Japan and other countries, it can be used independently. The torque tester (Fig. 3), in particular, is not very complicated and proves to be highly useful for measuring not only cogging torque but also torque ripples from the current-generated torque.

The authors are grateful to K. Egawa, who kindly helped them design manufacture the test motors. They would also like to thank R. Takeguchi, who kindly brushed up their English.

[1] T. Kenjo and S. Nagamori, *Permanent-magnet and brushless DC motors*, U.K.: Oxford Univ. Press,1985.

[2] T. Kikuchi and T. Kenjo, "A Unique Desk-top Electrical Machinery Laboratory for the Mechatronics Age," *IEEE Trans. Educ*., vol.40, no.4, CD-ROM09, Nov. 1997.

[3] T. Kenjo, H. Takahashi, H. Takeuchi and K. Marushima, "Effects of winding current on cogging torque in a brushless motor (In Japanese)," *Proceedings of JIEE Rotary Machinery Symposium*, RM-97-19, Jun. 1997.

[4] D. Howe and Z. Q. Zhu, "Cogging torque in permanent magnet machines," https://www.shef.ac.uk/uni/academic/D-H/eee/md/cogging.html.

[5] M. Sakamoto and A. Tozune, "Hybrid type stepping motor," US Patent 5410200, Apr. 1995.

Tatsuya Kikuchi

Department of Electrical and Electronics Engineering

Tokyo Polytechnic College

2-32-1, Ogawanishi, Kodaira, Tokyo 187-0035 JAPAN

Phone: 81-42-346-7139

Fax: 81-42-344-5609

E-mail: tkikuchi@ieee.org

Takashi Kenjo

Department of Electrical Engineering and Power Electronics

Polytechnic University of Japan

4-1-1, Hashimotodai, Sagamihara, Kanagawa 229-1196 JAPAN

Phone: 81-427-63-9140

Fax: 81-427-63-9150

E-mail: kenjo@uitec.ac.jp

**Tatsuya Kikuchi** (M'95) received the B.S. degree in Electronic Engineering from the Polytechnic University, Kanagawa, Japan, in 1984. From 1985 to 1992, he worked as a design engineer of servomotor controls. From 1992 to 1998, he was an instructor in the Department of Electrical Engineering and Electronics at the Polytechnic Centers in Aichi, then Kanagawa prefectures. He received the M.S. degree in Electronic Engineering from the Polytechnic Univ. in 1997. Since 1998, he has been working at the Tokyo Polytechnic College. His interests include mechatronics and multimedia computing.

**Takashi Kenjo** (M'97) was born in Japan on February 2, 1940. He received the Masters Degree in 1964 and the Doctor-of-Engineering Degree in 1971 from Tohoku University, Sendai, Japan. His area of interest is in small precision motors and their controls, and he has written several monographs published by Oxford University Press. He has been with the Polytechnic University of Japan since 1965 and is currently a Professor in the Department of Electrical Engineering and Power Electronics.