George P. Gogue
Joseph J. Stupak, Jr.

G2 Consulting,
Beaverton, OR


The rotors of permanent magnet dc electric motors have been constructed by a number of means in the past. Perhaps the most popular has been to bond a set of magnet arc segments onto a steel backing. Now, however, a few magnet manufacturers have begun to produce complete rings of radially-oriented magnet materials. Some are isotropic, others anisotropic. These rings have definite advantages (and a few disadvantages) compared with arc segments and other construction methods. The effect of the magnet shape on the performance of the motors is discussed. Magnetic and physical properties of presently available ring magnets are also briefly discussed.


Many electromagnetic devices require a permanent magnet field which is cylindrical and radially oriented as shown in Figure 1. These devices include permanent-magnet motors, linear actuators, limited-angle torquers and magnetically-coupled clutches. In the past, such fields were generated by a variety of means, some of which are shown in Figure 2. It was found that some magnet materials became much stronger if processed in such a way as to result in a preferred direction of magnetization. In contrast, these magnets could only be magnetized weakly, if at all, along one of the other two orthogonal directions, as shown in Figure 3. The flux lines in such materials were essentially parallel to each other. In a radial field, however, the flux lines are required to point at or from the center, like a fan. Oriented magnets are called anisotropic and non-oriented magnets are called isotropic.

FIGURE 1 Radial magnetic orientation

FIGURE 2 Examples of magnetic devices

FIGURE 3 Nonisotropic magnets with uniaxial and radial orientation

As magnet materials were being developed, it was found possible to cause a radially-oriented preferred direction of magnetization in that material as it was formed in the press. Segments with areas as large as 120° were and are produced with radial magnetic orientation and then assembled into rings. Rings of this type normally have sizable gaps between segments, however. Magnets are produced from a slurry and then are sintered at high temperature to form a solid magnet. In the course of sintering, the magnet undergoes a high degree of shrinkage, which may be different in different directions and which is somewhat variable. In order to produce accurate parts, the outer and inner surfaces of the arc segments are ground to shape with close tolerances. The ends of the arcs are usually not ground because this operation would be very expensive. As a result, the included angle of the segments usually has a considerable variation (perhaps several degrees) and space must then be left between the arcs to accommodate this variation. Because of this, rings made of magnet arcs have some variation in magnet pole strength. Rotating parts may also have considerable imbalance, which must be corrected by measurement and the addition or removal of mass. If the segments are magnetized before assembly, they may be difficult to assemble evenly due to the magnetic attraction between parts. If the magnets are bonded in place and then magnetized, the assembly must be oriented in some way in the fixture so that the pole transitions will coincide with the gaps between arcs.


Permanent magnet materials are manufactured in a variety of processes. These include injection-molding, bonding, quenching and sintering.

Rings of injection-molded and bonded magnet materials have been available for some time now, and are effective in avoiding many of the problems of arc segments. Materials which are injection-molded consist of magnetic particles dispersed in a plastic such as nylon, and are injected into molds at high temperature and pressure. Bonded materials, on the other hand, consist of magnetic particles which are pre-coated with a thin layer of bonding agent. The coated powder is measured into a mold and pressed into shape, possibly with added heat. The magnet particles may be oriented to a preferred field direction with either method, or may remain unoriented. The ratio of magnetic particles to surrounding plastic is less with injection-molded parts than with bonded material, and so the injection-molded parts are usually less powerful magnetically than bonded parts of the same magnetic base material. Injection-molded parts are often made to excellent tolerances, with smooth, non-porous surfaces, and are relatively tough and flexible. Bonded parts, on the other hand, tend to be magnetically stronger, but are rather grainy, may be porous and are more easily broken.

Within the last few years, processes have been discovered which allow the production of radially-oriented rings of solid, sintered magnet material. These new rings combine the high magnetic strength of the best solid materials with the advantages of continuous rings. One ring usually costs less initially than the set of segments it replaces, and the cost advantage increases with larger numbers of smaller arcs. Installation of a ring is easier and less costly than for arc segments. The rings are so uniform that only minimal balancing may be required, or none at all.

The ferrite ring magnets made by FDK are produced by an ingenious method. The particles of ferrite are tiny flat plates, much thinner in the preferred direction than in others. The ferrite powder is mixed with a binder and then rolled into a very thin sheet (a few thousands of an inch thick). As the material is rolled, the platelets tend to align themselves in the direction of stretch. The sheets are then coiled around a mandrel to form cylinders, cut to size and fired. In firing, the binder is burned out and layers fuse to form a solid, radially-oriented ring.

One manufacturer, TDK, is now producing ring magnets with special magnetic orientation which they call "pole-to-pole". The magnetic slurry in the mold is exposed to a magnetic field with the same pole configuration as will be used in the magnetized part, orienting the particles accordingly. The result is a magnet which they claim to be somewhat stronger (on the order of 15-20%) than other radially-oriented rings. For proper magnetization these parts must be aligned in the magnetizing fixture, with the pre-oriented regions in line with the magnetizing poles. The alignment is done from a molded reference mark. For effective use of the magnet material, the width of the pole face cannot be large compared to the magnet thickness, and so the present applications tend to have relatively large numbers of poles (10 to 24, or even more), and relatively thick magnets. This new type of ring magnet has been available in Japan, but has not yet been publicly announced for sale by TDK in the US.



Table 1 lists the most commonly used magnetic characteristics of available ring magnets. They are listed in an increasing order of energy product. The maximum and minimum possible diameter is determined by the process and the vendor. The range of allowable dimensions is constantly increasing as technical advances are made. One column lists the field necessary to fully saturate the magnet material. This value should be achieved when charging the magnet with a proper magnetizing fixture.

Table 2 gives the average values of the physical parameters of these ring magnets. The amount of eddy currents induced in the magnet under operating conditions varies inversely with the electric resistivity of the material. The coefficient of thermal expansion determines how much space to allow between the ring and the surrounding parts. The maximum safe operating temperature is usually much lower than the Curie temperature. The reason is the presence of bonding agents or plastics for some, and the possible chemical reactions for others.

The dimensional tolerances of these rings are generally good but highly dependent on the process. Secondary machining, if possible, results in parts with very small tolerances.



If there are no enclosed electric currents, the integral of the coercivity H, i.e. The magneto motive force, around a closed loop must be zero:

FIGURE 4 Geometry of radially oriented ring and gap for sample operations

A magnetic circuit with radial field induced by a permanent magnet is shown in Figure 4, with both center pole and outer sleeve of a high-permeability material. The flux is assumed to be linked between them by a high-permeability path and the mmf drop in the poles is assumed to be negligibly small. If this should not be the case in an actual situation, the mmf drop can be calculated and subtracted from the magnet mmf. In the example of Figure 4, the complete integral consists of two parts, the path through the magnet and the path across the air gap:

A. Calculation by direct means

FIGURE 5 B-H curve of typical moment

The B-H curve for the magnet, shown in Figure 5, is represented by a straight, sloping line over the region of interest (above the knee of the curve). If the line were to be extended down until it intersected the H axis, it would cross the axis at a point marked as . The value of H, over the fully-magnetized part of the curve, for any value of flux-density B, may then be calculated from:


: coercivity at any point j

: flux-density at any point j

: remnant flux-density

Since the flux lines are converging toward the center, and are continuous, the flux crossing each concentric circle must be the same:


between any two radially-displaced points 1 and 2, at radii & .

The mmf difference across the gap is:


a is at inside surface of air-gap

b is at outside surface of air-gap

The mmf difference across the magnet is:

solving these equations for at radius ,

Where has a negative value.

For the dimensions of Figure 4, with = 0.53 in, = 0.375 in, = 0.275 in, = -3,800 oersted, = 4,200 gauss, and solving for at the outer radius of 0.53 in.,

B = 1,783 gauss

B. Solution by iteration

A solution by iteration is longer, but can be used for any B-H curve, even if it is not representable by a straight line. It may include other factors as well, such as the mmf drop of partially saturated pole material, effect of windings, etc. To solve the problem, the space between inner and outer poles (containing both the magnet and the gap) is broken up into a number of intervals, the number being large enough to permit the required accuracy of solution. The intervals do not need to be equal to each other.

A value of B is assumed at a convenient radius (in the example, the outer surface of the magnet is chosen), and from this assumption the value of B at the center of each interval is calculated. The corresponding value of H, assumed to be an average value for the interval, is found from the B-H curve in the magnet, and from the permeability of free space (1 gauss per oersted) in the gap. The values of H are then multiplied by the width of the interval, and the mmf changes are added up for the gap and magnet. If the sum of these two are not zero, a new estimate of the value of B at the chosen point is made, based on the direction and size of the error, and the calculation is repeated to drive the error toward zero.

First iteration:

B(0) assumed = 2,500 gauss

Note: smaller intervals were taken at smaller radii to improve accuracy.

Values of H versus B were found by reading them from the B-H curve, and are less accurate than would be found by calculation. This accuracy is acceptable, however, because of the limits of precision of the data represented.

Second iteration:

Evidently, the estimate of 2,500 gauss was too high. A new estimate of 2,000 gauss will be tried.

The estimate for B at the outer diameter is still too large. A decrease of 500 gauss caused a change of (231.2 - 83.2) = 148 Oe-in., and so a new estimate is made:

After several more iterations, the solution converges (sum of mmfs approaches zero) for B(0) = 1,740 gauss. The error from the exact solution is about 2.4%. The error could have been reduced by using a larger number of smaller intervals but this accuracy is probably sufficient in light of the magnetic tolerance.

5. Leakage flux

As a natural consequence of using ring magnets with multiple poles, a certain amount of leakage flux occurs between adjacent poles at the inter polar zone. The leakage subtracts from the total flux produced by the magnets, resulting in a reduced amount of flux reaching the opposite side of the air-gap. The amount of leakage flux is highly dependent on the sharpness of the polarity transition in the inter polar zone.

There are two contradicting requirements in this situation. Since it is always desirable to maximize the amount of flux per pole, a wide pole arc is required. This however, results in a very sharp transition giving rise to a large amount of leakage flux. A balance, therefore, needs to be attained between those requirements to best suit the particular application. An additional point to consider in this decision is the effect of the sharpness of the transition on the cogging torque. This will be discussed further in a subsequent paper.

To demonstrate the interdependency between leakage flux and the inter polar zone, finite element analyses of a computer model are performed. The model is that of an 18-pole machine with varying inter polar zones and varying air-gaps between magnets and core. Figure 6 shows the flux plot at 0° transition in the inter polar zone and Figure 7 shows the flux plot at 10° transition.

Figure 7, therefore, is the condition where the inter polar zone is as wide as the pole arc itself. It is also equivalent to twice the radial thickness of the magnet, which is as wide as is normally acceptable. The two figures show a slice of the motor around the area being studied. Also shown is the condition of the same air-gap (0.025 inch) between the magnets and the core.

The analysis is done at 4 values of transition in the inter polar zone and at 5 values of air-gaps between the magnets and the core. For each configuration, the flux values calculated were the total flux-pole, the leakage flux across the inter polar zone, and the useful flux crossing the air-gap. Figure 8 shows the leakage flux as a percentage of the total flux/pole vs. the air-gap width. The percentage leakage increases as the air-gap width increases because the further the core is from the magnets, the easier it is for flux to cross the inter polar zone between adjacent poles. The percentage leakage is reduced, however, as the inter polar zone is made wider.

Looking at the leakage flux as a per unit of the maximum flux/pole, as in Figure 9, shows the same trend as above. But these two figures still do not present a complete picture. Figure 10 shows how the useful flux varies with air-gap width.

The general drop with air-gap width is attributed to the rise in the amount of leakage flux. What is important to note however, is the significant drop in useful flux at the transition as the inter polar zone is made wider. Hence, the desire to avoid a large drop in useful flux might make the high percentage leakage flux tolerable. These compromises are best made by the designer to suit the particular case at hand.

An alternative explanation of the above results is by thinking of inter polar zones as means for focusing flux from the magnets to the core. The presence of magnets in the inter polar zones focuses flux into the core achieving higher amounts of useful flux despite the higher leakage experienced.

6. Cogging considerations

Permanent-magnet motors are susceptible to high cogging torques, which is the tendency for the rotor to be unstable in some positions while resisting motion away from other locations. Cogging rotors tend to 'jump' from one position to another, with resulting vibration, noise and torque variation.

One effective method to reduce cogging is to 'skew' either the permanent magnet poles, or the stator teeth, or both, relative to each other (Figure 11). The unbalanced torques are spread out and tend to cancel each other. Skewing the stator lamination assembly requires that each piece undergo special handling. Winding a skewed rotor is more difficult than winding a straight lamination stack. Skewing the magnet poles, on the other hand, can be done by constructing the magnetizing fixture with the proper angle. If a different angle is found later to be more advantageous, no part change is required. A new fixture must then be made. This can only be done with ring magnets, however; skewing the poles of arc segment magnets is impractical.

Considerable work, both theoretical and experimental, is now underway to reduce noise and vibration in permanent-magnet motors, both in the disk-drive industry and elsewhere. It is much easier to do this with continuous, uniform, radially-oriented ring magnets than with discrete arc segments, and it is reasonable to expect that the percentage of new designs using these ring magnets will continue to increase.

7. Lines in the magnetic transitions

To examine magnetic patterns, a type of thin plastic sheet (typically about 0.007 in. thick) is often used. When placed over a magnet, the plastic sheet shows dark regions over magnetic poles and lighter regions where either the field is zero or where the field is parallel to the sheet (in the plane of the plastic sheet). The plastic shows some darkening at only about ten gauss and is fully dark at about one hundred gauss. When used to examine a pole transition, the actual transition zone (from several thousand gauss positive to several thousand gauss negative) may be very much wider than it appears under the sheet.

When this sheet is used to examine transitions in magnet rings which are made of neodymium-iron, it sometimes happens that two very fine dark lines are seen in the transition region (the lines may appear to merge into one line, but in fact there are two). This puzzling phenomenon has sometimes been taken as an indication of incorrect magnetization, which is not necessarily the case. When the region is examined by accurate scanning with a hall-type flux probe, if the probe is relatively thick compared to the transition width, the transition appears to be normal. If an extremely thin probe is used, however, reversals of the flux direction may be detected, a few thousands of an inch above the magnet surface but not further out. When the same magnet is measured in a gap between two permeable surfaces (which is of course the way they are normally used), the flux reversal disappears.

Neodymium-iron is not very permeable, compared to ferromagnetic materials like iron, 'soft' ferrites etc., but it is still somewhat more permeable than air. The permeability may be different in different directions, and in the transition region it may be considerably different than the published figure for the material. It also requires a very strong field to magnetize, and hence the transition zone between magnet poles may be relatively wide compared to the magnet thickness (because of the space needed for the windings of the magnetizing fixture), as shown in Figure 12. Under these conditions, when the magnet is in air, lines of flux near the transition may bend down and reenter the magnet, travel through the magnet material in the circumferential direction within the transition region, and then reemerge to link the opposite pole on the other side. This only happens very close to the surface but can be detected by the plastic sheet because its active region is only a few thousands of an inch thick. The dark lines are the regions where flux is entering and leaving the transition region (which is why two bands exist rather than one). When permeable walls are brought near the transition, the flux lines change direction, no flux reenters the transition region and the apparent reversals disappear.

FIGURE 12 Flux plot of interpolar region