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COGGING TORQUE CONTROL IN BRUSHLESS DC MOTORS A. Hartman, W. Lorimer Quantum Corp., Milpitas CA, USA
itself primarily with the magnetically generated forces Abstract- During the past five years, cogging torque in HDD spindle motors has decreased drastically as designers and manufacturers have developed a better understanding of the cogging phenomenon. This paper examines various methodologies of cogging torque reduction. Timing techniques such as dead zones and tooth
sinusoidal magnetization, and geometric techniques such as tooth crowning and slot skewing are identified. Cogging torque waveforms are examined in frequency and time domains. Recommendations for accurate cogging torque prediction and design of magnetizing fixtures for low-cogging motors are forwarded. I.
cause excessive motor vibrations if not properly controlled. Cogging occurs when the saliency of the stator steel interacts with the multiple permanent magnet (PM) poles on the rotor. Each tooth of the stator undergoes an identical cyclic torque fluctuation as the rotor turns. The only difference is that the tooth forces are displaced from each other in time. The instantaneous sum of all the individual tooth torques is the total torque, and is in general nonzero. This cyclic torque pattern is referred to as “cogging”. Each tooth of the stator is strongly attracted to each passing PM pole. The tangential component of that
known as “cogging”. These forces are large and can
initially an acoustics concern. Motor vibrations were often key sources of audible noise in hard disk drives (HDD). This was particularly problematic when drives were used in quiet environments such as offices. In time, as HDD track densities steadily increased, these vibrations also became a major problem from a data tracking perspective. The vibrations added position errors that the data-tracking system of the HDD had to correct. During the last several years, considerable effort has been put into understanding the key contributors to magnetically-induced motor vibrations and developing design techniques for reducing them. The main motor vibration sources are ball bearing roughness, air turbulence, electrical commutation, and steel-magnet reluctance forces. This paper concerns
attractive force contributes to the overall motor cogging torque. There are two positions of the rotor pole that will result in zero tangential force on a stator tooth. The first— PM pole centered on tooth— is a stable point. If the rotor is moved from this position, a restoring torque will occur on tooth and magnet attempting to reestablish this position. The second— transition between two poles centered on tooth— is an unstable point. If the rotor is moved from this position, a nonrestoring torque will develop on tooth and magnet seeking to depart from this position and attract the nearest pole to the tooth center. As shown in Figure 1, the magnitude of the tooth reluctance torque is quite large; its peak is similar in magnitude to the total motor output torque. This is surprising
completely undesirable and contributes nothing to
Using the Maxwell stress tensor and assuming
µ air << µ iron , shear stress is given by (1).
motor performance. 0.008
σ rθ =
1 Br Bθ µ air
tooth motor output
Since flux lines flow generally perpendicular to the
steel-air boundary of the tooth, we can conclude that
tooth torque is produced primarily at the tooth tips, and
is outwardly directed, i.e., magnets always attract steel. This observation is verified by finite element analysis
(FEA). As shown in Figure 2, reluctance torque is
electrical angle [°]
Figure 1. Motor output and single tooth reluctance torque − 4-Watt HDD spindle motor
concentrated over a relatively small area at the tooth tips.
Cogging torque reduction consists of manipulating the shape of the tooth torque waveforms so that the sum over all the stator teeth is as close to zero as possible. This is done by manipulating PM magnetization pattern and stator tooth geometry, alone or in combination. Several techniques of analysis, design, and testing have been applied to this end, and numerous cogging reduction schemes have been utilized. Depending on the technique, they can be best understood as TimeBased or Frequency-Based. II.
COGGING TORQUE REDUCTION VIEWPOINTS
Figure 2. Flux lines and shear force distribution− − Pole aligned with tooth. Net torque equals zero.
As a simple model, consider a series of alternatelyoriented poles with dead zones in between passing by a
single tooth. Shear forces are generally constant and act
Of the two methods, Time-Based is more physical.
in opposite directions at the tooth tips. When the dead
It involves looking in detail at the forces acting on each
zone passes over the tooth tip, however, the shear force
tooth. Recognizing that passing PM poles will
disappears ( Br = 0 ). Therefore, this tip’s shear force
unavoidably cause tooth reluctance forces, we focus on reducing the magnitude of these forces. Secondly, we focus on introducing timing relationships between the tooth waveforms so that torque cancellations can occur.
cannot cancel the other tip’s shear force, and a net tooth force results. The resultant tooth force waveform during one pole pass is illustrated in Figure 3.
A second more subtle method is to adjust the
trailing edge force
location of the tooth tips (slot width) and PM edges (dead zone width) so that the tooth cogging cycles will cancel each other. This strategy was first suggested by Ackermann . Finite element simulations of this 0
technique yield a multiplicity of zero-cogging solutions,
leading edge force
as shown in Figure 5.
9 Slot 12 Pole Motor - Cogging Torque 20
-0 0 0.001
] g e d[ ht di w t ol s
12 0.005 0.003 0.002
Figure 3. Sequence of shear force production at tooth tips
8 10 12 14 PM dead zone [deg]
9 Slot 12 Pole Motor - Kt per turn 20 0.00028 18
This immediately suggests a class of cogging
reduction strategies. The first is to introduce additional “tooth tips” in the form of notches in other tooth heads so that an oppositely-directed torque curve will be created and used to cancel the original tooth’s torque. The location of cancellation notch edges is illustrated in
14 0.00022 ] g e d[ ht di w t ol s
8 6 4 0.00018 2
8 10 12 14 PM dead zone [deg]
Figure 5. Cogging Torque and Kt for 9s12p Motor− − radial magnetization with variable dead zone and slot width
Although Figure 5 results are for ideal radially-
oriented PMs, similar results are observed with more realistic magnet orientations. Figure 6 shows simulated N5 N6
zero-cogging solutions for magnetization patterns produced by a family of 2-wire magnetizing fixtures.
Figure 4. Cogging reduction by introduction of notches
9 Slot 12 Pole Motor - Cogging Torque
0.001 0.0015 0
8 0.002 0.00150.001 6
] g e d[ ht di w t ol s
rotor position [° mech]
6 8 10 12 14 magnetizing wire spacing [deg]
Figure 7. Phase-flipped cogging torque waveforms
9 Slot 12 Pole Motor - Kt per turn 20
A characteristic of the Time-Based method is that it
requires extensive computer modeling to gauge the
14 ] g e d[ ht di w t ol s
differences. FEA and to a lesser extent lumped-
employed. Ever-faster computers and judicious use of
symmetry relations  make use of even highly
detailed numerical models quite practical.
6 8 10 12 14 magnetizing wire spacing [deg]
Figure 6. Cogging Torque and Kt for 9s12p Motor− − 2-wire fixture magnetization with variable slot width
This is fortunate, since cogging prediction is a difficult numerical problem. Cogging torque is the sum of numerous large forces that mostly cancel each other.
An important conclusion of the Time-Based method
Since the forces act over small areas at the tooth tips,
is that although tooth torque waveforms are always
high spatial detail is required. The cogging cycle is
present and relatively large, by manipulating the timing
faster than the mutual torque cycle, so simulations at
of these waveforms, e.g., by adjusting slot width or PM
small angular increments are needed. In addition,
dead zone, it is possible to change the cogging
accurate material saturation properties and magnet
waveform. Furthermore, the waveform can be changed
orientation pattern must be known in order to calculate
from a positive “sinusoidal” shape to a negative
cogging. Prediction of the PM orientation pattern
“sinusoidal” shape, as shown in Figure 7. This implies
generally involves simulation of the PM magnetizer and
that if we choose an intermediate geometry, we can
magnetizing process. Although precise answers are
cancel the cogging waveform completely.
difficult, close proximity to a low cogging solution can still be obtained. Typically, fine-tuning some easy-tochange parameter will be enough to reduce cogging in actual practice.
Similarly, the periodic PM transition zone function,
The Frequency-Based method is more analytical
shown in Figure 9, has frequency spectra values at the
than Time-Based. The approach involves viewing the
N poles sampling frequency. It too can have zeros at
motor cogging torque waveform as the interaction of
sampling frequencies where the basic PM transition
the PM MMF waveform and the stator slot permeance
zone spectrum also has zeros.
manipulating waveform harmonics in both the PM and stator. There is a rich set of frequency domain tools for identifying which frequencies are most important to introduce or eliminate. The tooth torque sequence of Figure 3 suggests modeling
convolution of the PM transition zone function, P, and the tooth slot (or tooth head) torque function, S . This is Figure 8. Slot Torque Function S() - space & frequency domains.
given by (2). T (λ ) =
θ = 360
P(λ - θ ) ◊ S(θ )dθ
A similar approach has been used in lumped parameter methods to model complex slot geometries interacting with PM transition zones . The PM transition zone function repeats itself N poles times per revolution. It is zero everywhere except at the transition zone where it goes to one. Whenever the PM transition zone sweeps over the slot, slot (and tooth) torque is
Figure 9. PM Transition Zone Function P() - space & frequency domains.
produced. We will now apply Fourier transform arguments to further study their frequency spectra. Consider a 9 slot / 12 pole motor. As shown in Figure 8, the periodic slot function spectrum is a sampled version of its basic shape (magnitude scaled by N slots ). The resulting frequency spectrum has discrete
values at multiples of the slot frequency. This spectrum can have some zeros if the slot width is carefully chosen such that one or more zeros of the basic spectrum align with the slot frequency harmonics.
Figure 10. Motor cogging torque T() from slot & PM functions - space & frequency domains.
As stated earlier, motor cogging torque is the
also tried adding regularly spaced notches of the same
convolution of the slot and PM functions. The resulting
approximate shape as the tooth slots . Both
spectrum of motor cogging is found by multiplying the
techniques change the apparent number of slots and can
slot spectrum by the PM spectrum (see Figure 10).
beneficially raise the cogging frequency. Although the
Non-zero values occur only at multiples of the least-
addition of equally-spaced notches is physically similar
common multiple (LCM) of N poles and N slots .
to addition of cancellation notches suggested by the
Frequency-Based analysis yields several useful
Time-Based approach, the spacing and width of the two
insights. It shows that cogging occurs at multiples of
notch patterns are different. As shown in Figure 11,
LCM ( N poles , N slots ). These are the only places where
however, both are effective in reducing cogging torque.
the PM function and slot function spectra are both
cancellation notches equally-spaced notches no notches
simultaneously non-zero. It implies that placing a zero
at the cogging frequencies for either the slot function or torque [N-m]
the PM function is all that is necessary to eliminate
cogging since their product will be zero. Slot and PM transition functions typically have periodic zeros in
their spectra that can be judiciously placed by adjusting their widths. This observation also suggests the
rotor mech. position [°]
Figure 11. Cogging reduction by addition of notches.
existence of multiple special slot widths and transition zone widths that produce zero cogging and are
The Frequency-Based method demonstrates which
independent of each other as shown in Figures 5 and 6
harmonics in the tooth torque waveform contribute to
motor cogging. What is seen graphically in Figure 10, Frequency arguments explain why higher cogging
Hanselman showed analytically . Tooth torque
frequencies generally have lower amplitudes, and how
harmonics that contribute to cogging are given by
smoothing techniques work. Basic slot and PM spectra
values of n that solve (4).
have decreasing energy at higher frequencies, thus higher cogging harmonics naturally have lower
GCF(nN poles , N slots ) = N slots
amplitudes. Smoothing the slot shapes and PM
This equation shows the attraction of a perfectly
transition zones further reduces energy at the higher
sinusoidal tooth torque waveform ( n = 1). It will never
produce any cogging (no energy at cogging frequency),
With this in mind, preferences should be given to
hence its popularity in the literature. However, steel saturation and the desire for high Kt motors usually
pole/slot ratios with higher cogging frequencies (3).
make perfectly sinusoidal waveforms impractical. fcogging = LCM ( N poles , N slots )
A complication of the Frequency Method is that
Designers have exploited this idea by adding extra
HDD motors tend to be very saturated. Therefore, the
unwound teeth in the stator slots , . They have
actual system is nonlinear and one cannot rely on linear
superposition techniques. The usefulness of the method
resulting acoustic noise was usually worse than the
lies primarily in its ability to suggest cogging reduction
original cogging noise. A symmetric chevron magnet
strategies rather than to propose actual implementation
pattern was suggested by Jang . This pattern
details. In reality, the slot torque functions are not
produces no net vertical force, but is difficult to
independent of the PM transition zone functions and
magnetize as a single piece.
our convolution model is not completely correct. Again,
A better way to smooth the cogging is by crowning
detailed FEA to get to a good design point and further
tooth heads (also known as variable airgap). Crowning
laboratory tweaking is the best approach.
has much the same effect as skewing, but its zsymmetry eliminates any vertical forces.
COGGING CONTROL PRACTICES
Tooth notching and the addition of unwound teeth
Since real motors have saturation and nonsinusoidal magnetization, other design strategies must be used. Some current and former cogging reduction practices
combination of techniques is employed. Modern motor design practices have largely eliminated the cogging torque problem.
high inherent cogging frequency. Additionally, the design should use symmetric tooth shapes and PM patterns. Finally, careful assembly should be done to ensure that the stator and rotor are concentric and aligned at the proper z-height. A rich spectrum of magnetic forcing frequencies can appear if these are
to simpler more effective methods. PM magnetization patterns and tooth shaping are the primary methods used today. They employ the concepts illustrated in Figure 6 and also rely on frequency arguments. The design of the tooth geometry and magnetizing fixture geometry is chosen to be near a
The first step is to pick a good pole/slot ratio with
were both used for a time but have generally given way
improvements are obtained by applying additional cogging reduction techniques. One example of an older technique now avoided is skewed magnetization . The design idea was to reduce the magnitude of cogging by smoothing it out. In this design, either the stator slots are slanted from the bottom of the stack to the top, or the north-south transition zone is slanted from the bottom magnet edge to the top magnet edge. This did indeed reduce the cogging torque, but it created a vertical force imbalance that would be excited by electrical commutation. The
zero-cogging location. As already discussed, this still admits a large selection of possible choices. The design space can be further narrowed by choosing the highest torque constant designs. From Figures 5 and 6, it is clear that low cogging does not necessarily imply low motor torque. This is a common misconception. As an actual example, Figure 12 shows the measured driven torque waveform of a motor made with three slightly different magnetization conditions. For visual clarity, one mutual torque cycle is repeated 3 times.
Cogging Torque in a Class of Brushless DC Motors,” IEEE Proceedings-B, v. 139, n. 4 (July 1992), pp. 315-20.  Hartman, A. and Lorimer, W., “Symmetry Relationships in Brushless DC Motors” Proceedings of 27th Annual IMCSD Symposium (1998), pp. 209-215.  Jufer, M.,”Brushless DC Motors – Gap Permeance and PM-MMF Distrubution Analysis”, Incremental Motion Control Systems and Devices 16th Annual Symposium Proceedings, June 1987, pp. 21-25.  Kenjo, T. and Nagamori, S., Permanent Magnet and Brushless DC Motors, Sogo Electronics Publishing Co., Tokyo, 1984, pp. 96-9. Figure 12. Line-line motor torque w/ cogging− − Three different magnetizing conditions
The arrows show clearly that the phase of the cogging torque (which is superimposed on the driven torque) has flipped 180 degrees in the top and bottom waveforms. At an intermediate condition (middle waveform), the cogging mostly disappears. This phaseflipping behavior can be observed in magnetically balanced motors ( GCF( N poles , N slots ) > 1) as well as in motors with unbalanced magnetic pull (UMP) . IV.
The cogging torque problem in HDD spindle motors has been effectively eliminated. This paper showed how considering the
Frequency-Based viewpoints leads to different insights and strategies for cogging torque reduction. The techniques discussed are very general, and work for virtually any pole / slot ratio. Some popular cogging reduction strategies were examined, and an actual example of their effectiveness was given. V.
 Ackermann, B., Janssen, J.H., Sottek, R., van Steen, R.I., “New Technique for Reducing
 Cros, J. and Viarouge, P., “Synthesis of High Performance PM Motors with Concentrated Windings,” IEEE International Electric Machines and Drives Conference Proceedings, Seattle, WA, May 1999.  Hwang, S.M. and Lieu, D.K., “Design Techniques for Reduction of Reluctance Torque in Brushless Permanent Magnet Motors,” IEEE Transactions on Magnetics, v. 30, n. 6 (Nov. 1994), pp. 4290-2.  Hanselman, D., “Fourier Decomposition of Radial and Tangential Forces in Brushless DC Motors,” Proceedings of 25th Annual IMCSD Symposium (1996), pp. 233-41.  Lorimer, W. and Hartman, A., “Magnetically Induced Vibrations in Brushless DC Motors,” Incremental Motion Control Systems and Devices 29th Annual Symposium Proceedings, July 1999, pp. 229-308.  Hanselman, D., Brushless Permanent-Magnet Motor Design, McGraw-Hill, New York, 1994, pp. 118-120.  Jang, G.H., Yoon, J.W., Ro, K.C., Park, K.C., Jang, S.M., “Performance of a Brushless DC Motor due to the Axial Geometry of the Permanent Magnet,” IEEE Transactions on Magnetics, v. 33, n. 5 (Sept. 1997), pp. 4101-3.  Chen, S., “Designing Spindle Motors for High Performance Drives”, Data Storage, v. 3, n. 11, Nov. 1996, pp. 39-44.