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The achievement of low and repeatable values of **cogging torque** probably is the major problem encountered in designing and manufacturing permanent magnet motors. Although finite element methods show that cogging cancellation is theoretically always possible, if the dimensions of the magnets are carefully determined, experimental results are very often disappointing, and sometimes so far from the objective that some motors have to be rejected. This paper tries to bring some understanding about that situation and to quantify the influence of some geometrical defects.

Obviously, the cogging torque results from the combined influence of the interaction of each individual magnet with each individual tooth. To analyse that, we shall make the assumption that the **magnetic circuit is linear** (*unsaturated*), which is a reasonable assumption at no load, to allow the use of the superposition principle.

We shall then look first at the following situation:

¤ *all stator teeth are perfectly identical, and evenly spaced.*

* ¤ only one magnet has been placed on the rotor.*

In that particular condition, it is clear that the magnet on the rotor will have a stable position in front of each stator tooth. Let Ns be the number of stator teeth; the torque T can then be written as:

where ø represents the position of the magnet. All harmonics, either odd or even, may be present, the first one being by far the largest one.

To deal with all the magnets, as the magnetic circuit has been assumed linear, it is possible to add all the corresponding contributions. Let Np be the number of poles; then, if the magnets are evenly distributed and spaced by , the cogging torque is given by:

This general expression, valid if the teeth and the magnets all are in their right position, will be useful to analyse the frequency components of the torque. This will be illustrated by a few examples hereafter. However, for a better understanding, it is easier to consider the vector approach, as is usual in electrical engineering. Each sine component is considered as the projection on a fixed axis of a revolving vector, angular speed of which being related to its harmonic order.

Replacing Ns by 18 and Np by 6 leads to the following:

¤ for any harmonic order T_{i} , the vectors representing the contribution of each magnet are shifted by a multiple of , and are then in phase concordance, leading to a zero sequence situation.

¤ all these vectors will then directly add, the total cogging being the sum of all the contributions of all the magnets. It is obviously the worst case that can be encountered, and one can conclude that for this kind of machine, the cogging is intrinsically very high, making skew absolutely necessary.

Here the situation is quite different, because 21 and 8 have no common divider. Then, for most of the harmonic orders, the corresponding space vectors form a balanced multiphase system ( here with 8 phases), either positive or negative sequence, the sum of which is zero. So, the total contribution of those harmonic orders disappears.

However, some harmonic orders will lead to a zero sequence situation, where all the vectors will directly add. The lowest frequency order corresponding to that will occur when:

which obviously occurs when i = 8. Then, the fist frequency in the cogging waveform will be 21 x 8 = 168 times the mechanical speed, and the amplitude of that contribution will be given by 8 x T_{8} .

This amplitude is likely to be very low, resulting from a fairly high harmonic order.

The situation here is, in some way, similar to the preceding one, with the difference that 12 and 8 have 4 as a common divider. Consequently, the first frequency in the cogging waveform will occur when:

which occurs when i = 2. The first frequency in the cogging waveform is then 12 x 2 = 24 times the mechanical speed, and the amplitude is now 8 x T_{2} . If the magnets are not carefully shaped, this amplitude is likely to be quite high, but if T_{2} can be minimised by shaping, the theoretical value of cogging at that frequency can be made very low.

Remember that this analysis is valid when the geometry of the motor is mathematically perfect, which of course cannot be true in practice. We are, however, now able to deal with geometrical defects, as for instance bad position of one magnet or one stator tooth..

We can very easily follow the same procedure, assuming now that the magnets are not evenly spaced, that is to say not shifted by exactly one from the other.

To illustrate that, let us take again the example of a 12 teeth 8 poles machine, and consider the influence of the first harmonic order T_{1} .

If only one magnet is not in its right position, by an arbitrary mechanical angle, the space vector corresponding to the wrong magnet will be shifted by degrees from its right position. Six of these vectors will fully compensate each other, because they are alternatively in phase opposition, and it will remain the sum of 2 vectors, of same amplitude T_{1} , but with a misalignment of degrees. The resultant of that is given by:

which is the amplitude of the cogging, at the frequency of Ns times the mechanical speed, that is to say the teeth frequency. This value can be very large, since T_{1} is the largest component in the single magnet torque waveform.

If a is 1°, the cogging will be as high as 0.21 T_{1} , which can be very large.

Note that for a rotor diameter of 25 mm for instance, 1° corresponds to 0.22 mm at the periphery of the rotor, which is not a very large geometrical defect.

Resulting cogging will be at tooth pass frequency.

Improper location of teeth may also be another important issue, especially for the segmented construction where some uncertainties can occur during the assembly process.

To look at that, one can follow a similar procedure to that developed in the first paragraph, with the difference that we now will assume that all the magnets are present on the rotor and evenly spaced, and the stator has only one tooth.

The torque due to that single tooth can be written as:

in which Np now appears instead of Ns, and where T_{i} values are different from the previous ones.

The total torque, when all the teeth are present, is now:

when teeth are evenly distributed.

A similar analysis can be conducted. If one tooth is shifted from its right position, a residual torque will appear, at the pole pass frequency.

Its amplitude will be given by:

Again, this can lead to quite unacceptable values.

As a conclusion, motor designers of course need to carefully optimise the machine geometry, including number of teeth, number of poles, magnet shaping, and also possibly some mitigation means such as helical skew, or step skew for instance.

But this is not sufficient. Designers must also investigate on sensitivity of any possible geometrical defect, or magnet strength dispersion, etc.…

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The behaviour of permanent magnet motors can be represented by the single phase vector diagram hereunder. For generality purpose, armature reaction is supposed to be anisotropic, according to the 2 axis theory.

The main control parameter is the angle a between stator current and EMF. For simplicity, one generally can neglect the stator resistance and leakage reactance, so that the power per phase and the line voltage can be expressed as:

Introducing the effective number of turns, the stack length, the magnet flux and the respective permeance of axes, EMF and reactances can be expressed as:

Consequently, power and voltage can be alternatively written as follows, where *AT* are the amp-turns:

Obviously enough, amp-turns are the most relevant parameter for the following discussion.

This can for instance be the case for surface mounted magnets, but also for radial magnets mounted in flux concentration, like in HPM motors.

In this situation, the 2 permeance are equal to, and writings can be simplified:

For given machine size, the power will be constant if is constant. Let *a *be this quantity. Then:

Of course, copper losses of the motor will increase, due to more turns and in spite of lower current, but in the same time motor core losses decrease (less flux) and inverter losses decrease as well (less current).

In other words, both inverter size and total efficiency can be optimized.

So, the traditional “zero angle” strategy for isotropic machines is not necessarily optimal, if the motor and the drive are mutually optimized, taking into consideration the motor winding definition.

This is the case, for instance, of permanent magnet assisted synchronous reluctance machines.

Here, the saliency is intentionally made high, in order to get strong reluctance torque, in addition to magnet torque.

This topology allows either to get very high specific torque with rare earth magnets, or high specific torque with low cost magnets (ferrite for instance, or plastic bonded magnets).

Typical torque and voltage capability curves are shown hereunder.

The same kind of procedure as previously can of course be conducted, but equations are a little bit more complicated.

For this reason, they are not shown here.

We just show how losses (hereunder individually in per-unit) can be impacted with optimized number of turns, together with optimized current angle (around 27° initially with non-optimized winding).

This motor topology is even more subject to optimization than traditional motors, due to increased degrees of freedom brought by saliency.

Again, a very nice global optimization of the power drive system can be achieved.

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Most magnetic noise studies assume that currents have pure sinusoidal shape; in other words that no time harmonics are present in the calculation of MMF waves. Of course, PWM introduces time harmonics that we will now introduce.

Here you can see the voltage waveform and spectrum of the PWM voltage, using space vector modulation, for a motor driven at 150 Hz with a chopping frequency at 3 kHz.

As can be seen, the voltage between phases (in blue) is a 3 level stepwise function, whereas the phase to neutral waveform is a five level stepwise function.

The next figure shows the frequency spectrum of the phase to phase voltage;

As can be seen, one finds a bunch of frequencies following the two relationships:

In the above, i, j, m and n are integers. Obviously from the spectrum, not all of these frequencies are present. It is then important to determine which of those are really present, and also to be able to predict if they lead to forward or backward revolving waves. In order to do that, it is necessary to go more in depth in the study of PWM algorithm.

It turns out that:

- For frequencies around an odd multiple of chopping frequencies, the sequence is forward, backward, zero and so on for the negative sign in front of motor frequency multiples, and backward, forward, zero and so on for positive sign
- For frequencies around an even multiple of chopping frequencies, the sequence is forward, zero, backward and so on for the positive sign in front of motor frequency multiples, and backward, zero, forward and so on for negative sign

This is shown for some of those on the following spectrum;

Of course, these voltage harmonics will produce current harmonics, according to the equivalent machine impedance, and these latter will in turn produce revolving MMF forces, with the same pole number as the machine principal winding, forward or backward revolving.

Hence, in addition to terms in the MMF linked to the fundamental frequency, one will also find terms like:

Some of these terms are of course not present, as explained above.

The MMF expression now has much more terms, and it is still worse for the MMF squared, which allows to determine the travelling waves of magnetic pressure acting on the stator bore.

And we will see hereafter that these contributions can lead to harmful noise waves, as they will combine high frequencies with low vibration orders (or number of poles), which are known to be by far the most redoubtable.

Let us for instance consider the combination of the main revolving field with one of the PWM waves;

These terms will result in the following waves:

Both of these are undesirable:

- The last one is purely radial, and can excite the zero order resonance mode of the stator;
- The first one can excite the 2 p order, which is the most likely to be excited in a 2 p motor.

Finally, all of the PWM induced waves are in the same situation, as can easily be imagined. Consequently, the risk is rather high, because these frequencies are speed dependent and are very numerous, to excite, at one speed or another, one of these harmful resonances. As the matter of fact, PWM induced waves can be encountered around any multiple of the chopping frequency, with a spacing corresponding to the motor frequency. This is true for any kind of machine. For this reason, the choice of the proper chopping frequency is very important for any electric machine in order to avoid unaffordable noise.

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