Theory & Practice of Electromagnetic
Design of DC Motors &
Actuators

George P. Gogue & Joseph J. Stupak, Jr.

G2
Consulting,
Beaverton,
OR 97007

__CHAPTER
7__

__MOTOR DESIGN__

__7.1 Introduction:__

The following discussion is centered on the brushless dc motor where only some of the features are common to the brush-type dc motor and other motors. While there is a lot of similarity in the use of permanent magnets, the two types of dc motor differ in the location of the magnets between stator and rotor. The absence of the commutator in the brushless motor allows the wound stack to be in the stator, where it is electronically commutated. There is, however, a lot of similarity in the lamination design of both motors in view of the similarity of their operating theory.

__7.2 Overall dimensions:__

Barring any specific dimensional
requirement in diameter and axial length, to fit a predetermined space,
these two dimensions should be selected to reflect a reasonable ratio
between them. An initial guess can be a diameter D at the air-gap
between rotor and stator, equal to the axial length *l*
of the stator. As in the case of the brush-type motor, the output power
of the motor is proportional to and it is this relationship which
establishes the relative effects of changes in these parameters towards
meeting the performance specifications. An increase in the diameter
would result in a much bigger increase in the output power than an
equal increase in the axial length would cause. This can be readily
seen from the above relationship. There are, however, other factors to
take into consideration when making decisions about these dimensions.
The parameters to consider are the torque required and the operating
speed. For very high torque requirements, a large diameter is highly
recommended. This stems from the fact that a particular amount of force
developed in the air-gap would result in a larger amount of torque if
the distance between it and the axis is increased, e.g. by increasing
the diameter.

On the other hand, at high speed operations, it is advisable to increase the axial length rather than the diameter if more power is needed. This is intended to hold the surface speed to a reasonable value (since it is directly proportional to the diameter D) and avoid very high centrifugal forces and localized iron losses. The argument for a small diameter and long stator cannot be carried out too far since a rotor of this type may experience mechanical oscillation and possible failure as the operating speed reaches very high levels. Similarly, the argument for a large diameter and short axial length has its limits when high torque and low speed operation is required. In motors of this design, the short axial length causes high values of leakage flux at the air-gap to be experienced.

__7.3 The magnetic circuit:__

A traditional approach to motor design is to use the so-called "lumped reluctance model" of the magnetic circuit. In this approach, the circuit is divided into several sections, each pertaining to a distinct material in the circuit. If the path of flux in that material is long or is changing in width, then it is divided further into small sections.

It is intended in this exercise to determine the reluctance of each part of the circuit and from that the value of the reluctance of the complete circuit. Usually some parts of the circuit and in particular air-gaps and low permeability materials, constitute the largest percentage of the total reluctance. This can sometimes make the accuracy acceptable if the contribution of the low reluctance parts is ignored. However, in motors where the iron is allowed to be almost saturated, the reluctance of these ferromagnetic paths starts to become appreciable and must, therefore, be included in the calculation.

As the magnetic circuit is divided into more segments, the amount of calculation increases rapidly. The geometry of the little segments becomes more complicated and thus requires more calculation time. Moreover, since the reluctance of each segment of the circuit is a function of the amount of flux it is carrying, and vice versa, the calculation process becomes an iterative process with a large number of computations. Personal computers speed up this process tremendously by eliminating the mathematical errors which commonly plague repetitive and laborious calculations.

The first step in this process is to draw the equivalent circuit representing the magnetic circuit of the motor. The parts that need to be included in the circuit can be determined from the cross-sectional view of the motor. This two-dimensional view can be further divided into sectors which enclose similar details and, therefore, repeat the basic circuit. A cross-sectional view of a brushless dc motor is given in Figure 7.1 where a 90° sector of this 4-pole motor contains all the necessary information needed.

The flux path is also shown in Figure 7.1 and is seen traveling through the magnet and air-gap into the stator and then back across the air-gap into an adjacent magnet completing the circuit via the back iron. As the number of poles increases, the length of the path is reduced. More importantly, as the number of poles increases, the amount of flux traveling between adjacent magnets is reduced. The advantageous consequence is that less back iron is required as the flux between magnets is reduced and, therefore, the weight of the rotor is reduced.

Figure 7.2 shows a representation of the magnetic circuit of Figure 7.1. It is common to use electrical symbols to represent different parts of the magnetic circuits. The following table gives the main terms of electrical circuits and the corresponding terms of the

Electrical |
Magnetic |

Voltage | MMF |

Resistance | Reluctance |

Current | Flux |

Due to the complicated nature of the magnetic circuits, various assumptions are made to simplify the equations. These assumptions either result in a very small error or limit the usefulness of the equations to a certain range with some of the parameters. The reluctances have the basic formula.

where *l* is the
length of the magnetic path in m and A is the cross sectional area in m. The
reluctance is, therefore, in Henry/m and its reciprocal is the
permeance. Mu () is the permeability of the material of the particular
section of the magnetic circuit and would be
in air as seen in Equation (3.16). The parameter is totally
dependent on the reluctance of the circuit. Perhaps Figure 7.3 is a
helpful representation of the interaction between those parameters.

The other basic equations are

where MMF is the magnetomotive force in Ampere-turns and the flux in Webers. This equation is the magnetic analog of the electric Ohm's law equation. And:

where B is the flux-density in Tesla. Refer to Chapter 3 for a more complete analysis of these parameters.

The reluctance terms of Figure 7.2 correspond to different parts of the motor, relevant to the magnetic circuit. The following is a discussion of these reluctances:

i. The tooth reluctance is perhaps the most difficult to determine accurately; the reason being the wide variation in flux-density of the different parts of each tooth. The tooth tips are the most likely parts of the tooth to reach saturation because of their narrow cross section. Unchecked flux-density in the tooth tips can lead that part into saturation making the slot opening effectively larger than the physical opening. This has the disadvantage of reducing the effective area of the steel facing the magnets. With the reduced area of steel the reluctance of the path increases, leading to a drop in the air-gap flux. To avoid this sequence of events the tooth tips are made as thick as reasonably possible.

Other sections of the tooth also experience different levels of flux-density, despite the fact that most teeth are parallel-sided. This is the result of varying amounts of leakage flux outside the tooth and along its radial surfaces. For added accuracy of the reluctance calculation, the tooth is divided into sections that have differing leakage factors. In addition to the tooth tip section, this division defines the conditions in the tooth sufficiently accurately.

Since there are usually several teeth facing each magnet, it is important to determine the minimum number of teeth including fractions of teeth that constitute the flux-carrying steel.

ii. The core is the steel part behind the teeth which completes the magnetic circuit between adjacent magnets. The reluctance of this path is and is represented by one value in Figure 7.2. In reality, the reluctance of the core depends on the flux-density (and permeability ) of the particular segment of the core. To simplify the problem, the core is divided into a number of segments depending on the number of teeth chosen in the design. For each segment the flux-density is assumed constant, allowing and the reluctance to be calculated. The reluctance of the core is then the sum of the reluctances of all the parts.

iii. The back-iron (or housing) is usually a carbon-steel cylinder around the magnets that forms the return path of flux between adjacent magnets. Flux-density levels are different at the different parts of the iron. The values are highest at interpolar zones and least along the centerline of the magnet.

In a similar fashion to the procedure followed in the stator core, the back-iron is divided into sections that are individually assumed to have a constant flux-density. This enables the calculations of and the individual reluctances leading to the total reluctance of the back-iron.

iv. The air-gap referred to here includes the length of both the physical air-gap and the actual radial thickness of the magnet. The implied assumption here is that most magnet materials used in dc motors have a permeability equal to that of air. The combined reluctance of the air-gap and magnet are represented by in Figure 7.2. The magnet arc is usually smaller than the full-pitch (unless a ring magnet is used, which results in a near full-pitch arc). In addition, the axial length of the magnet is usually longer than that of the stack. The combination of the above two features results in the cross sectional area of the magnet being different from that of the air-gap. Besides this difference in area, an account must also be taken of the flux fringing at the edges of the magnet. The most traditional way is to use Carter's coefficients (Reference 22) which approximate the effect of fringing to an additional air-gap length. An alternate technique of accounting for fringing flux is discussed in Reference 23.

The reluctance of this part of the magnetic circuit represents by far the largest portion of the total reluctance of the circuit. Special care, therefore, must be given to the assumptions and approximations which can result in a large percentage error in this calculation. For example, assuming a flat 5% or 10% increase in airgap area is only a first order approximation based on the designer's experience.

__7.4 Magnet performance:__

With the total reluctance determinable from the values of the individual segments of the circuit, it is possible to determine the magnet performance at different flux-density levels. As mentioned in the previous section, the magnet performance and the flux-density level affect the reluctance of the circuit and vice versa. In this iterative process, the operating point of the magnet is determined from the permeance or the load line of the magnet. The permeance is calculated from the following relation:

__7.5 Design Features:__

In the course of designing a brushless dc motor, a few principles and tips that have been developed over the years can be followed to get further improvements on the design and solve some simple but nagging problems. Following is a discussion of some of these issues:

*1. Material choice:*

An important part of the magnetic circuit calculation is the knowledge of the material characteristics. This determines the reluctance of the flux path and the flux-density value possible at different parts of the circuit. The choice of materials also determines the amount of iron losses expected under those conditions. The tradeoffs between saturation flux-density levels and losses are made with a view on the particular application: e.g. a high speed application would require a low loss material, a high power density application would require a high saturation steel. A more complete discussion of the different materials is found in Section 2.2.

*2. Air-gap Length:*

The radial length of the air-gap between stator and rotor is usually made as small as possible in dc motors. With consideration given to the overall size of the motor, the manufacturing process and the tolerance on the assembly determine what the smallest length can be to ensure that the rotor does not rub against the stator. The advantage of a short air-gap is a lower reluctance value since the air-gap constitutes the major part of the total circuit reluctance. In very small motors, it is not unusual to have air-gaps less than 0.01 inch.

It is also important to make the air-gap uniform around the complete periphery of the rotor. Any asymmetry would result in unequal forces, radial and axial, around the rotor and cause additional losses and mechanical vibration.

*3. Magnet Arc and Ring:*

Many electromagnetic devices use a circle of magnets or a ring with various poles, magnetized radially, as shown in Figure 7.4. Brushless dc motors are made in this way, as well as some brush-type motors, rotary actuators and linear actuators of the voice-coil variety. In the past, magnet rings were built up of arc segments using three or more pieces. The magnetic materials could not be made into continuous rings because they could not be pressed in the direction of radial orientation. For several years, some manufacturers have been producing complete magnet rings capable of significant magnetization in the radial direction. These rings offer a number of advantages over the use of arc segments. They also have a few drawbacks which must be considered but for many purposes they are clearly superior. The arc of each pole would, in the case of a ring, be very close to 180° electrical. The transition in polarity between adjacent poles can be made very small and in the order of one or two air-gap lengths, resulting in a nearly full pole arc. However, magnet rings are not available for large size motors and, therefore, arcs must be used. In such cases, the arc lengths are usually smaller than 180° electrical by 20 to 30%. This reduction in pole arc from its maximum, results in a flux-density distribution in the air-gap, resembling a sinusoid. Similar wave shapes would then be seen in the back emf and torque per phase.

It is expensive to produce ground, accurate edges on arc segments: the segment width, therefore, is usually variable. If each arc segment is magnetized as a single pole, which is usually the case, the poles will have a somewhat variable strength. It is also difficult to accurately adjust the gaps between segmented magnets, especially if they are magnetized before assembly. If the magnets are on a rotor, considerable imbalance may result and must be corrected. Ring magnets, on the other hand, are generally very well balanced and require only fine adjustment. Often the balancing operation can be eliminated altogether. It is possible to pre-magnetize the ring magnets before assembly without adding greatly to the manufacturing difficulty. If a ring magnet is magnetized in place, there is no need to orient the part in the fixture, as must be done with segments. Because only one part is handled, assembly time is reduced. Since only one part is needed per electromagnetic device, ring magnets often cost less to manufacture and ship (see Ref. 40).

At present, FDK produces ferrite ring magnets with radial orientation. In the manufacturing process the ferrite material is first mixed with a binder and rolled out into a thin, flexible sheet (about .008 inch thick). The platelets of ferrite material which comprise the magnet are oriented in the process. The sheets are then wound on a mandrel to produce a tube which can be cut and shaped. The parts are then sintered at high temperature to produce solid ferrite rings with radial magnetic orientation.

Several companies make magnetic rings with radial orientation by injection-molding of magnetic particles in a plastic binder. Magnetic particle materials presently available include various grades of ferrite, samarium-cobalt and neodymium-iron. The resulting magnets are dimensionally accurate (typically +/-0.001 inch), mechanically strong and shock-resistant and inexpensive. They are not very powerful, however, because of the relatively low volume of magnetic material to the total volume as allowed by the process (typically 40% or less). Some companies which produce parts in this way are Gries-Dynacast, Seiko-Epson and Hitachi.

Another process used to make ring magnets is more expensive but produces parts which are magnetically more powerful. Magnetic powder, which is usually precoated with a plastic binder, is measured and poured into a mold cavity. An orienting magnetic field is applied and the parts are then pressed axially (sometimes with added heat) to form the part. Magnets made in this way are dimensionally accurate without grinding and have magnetic strengths intermediate between injection-molded parts and those made of solid, sintered material. They are often more fragile, however, than parts made by other means. Companies which produce such rings are: Xolox, Seiko-Epson, TDK and Hitachi.

*4. Magnet Overhang:*

Due to the fringing of flux at the edges of magnets, it is common to make the axial length of the magnets longer than that of the lamination stack. Since it is required to allow space for the end turns of the winding, the overhang of the magnet does not constitute any additional burden on the design. The need to economize on the magnet material limits unnecessarily long overhang but it is not unusual to have overhangs that are 2 - 3 times the air-gap radial length. A rigorous calculation of the fringing flux between the magnet overhang and the end of the stack can be done using Rotor's method. Figure 7.4 shows the locations of the above parameters on a typical motor.

*5. Skewing:*

Skewing the stator slots with respect to the rotor magnets is a practice adopted from the design of conventional brush-type motors and is now used in brushless-type motors also. Typically the slot openings are skewed from one end of the lamination stack to the other by about half a slot pitch. This practice is followed where the numbers of poles and slots are such that the rotor can find stable positions when it comes to rest. Torque is required to overcome the natural reluctance or cogging torque to break the rotor loose from one of those stable positions. Under running conditions, this torque has a mean value of zero and, therefore, does not constitute a loss of torque. The variation in the instantaneous value can, however, cause an unacceptable increase in the percentage ripple torque.

Skewing the stator lamination results in averaging out the reluctance torque between stator and rotor as the magnets face a fairly constant amount of stator steel. The skewing of the lamination stack can make the winding process more difficult since the slot openings are not parallel to the axis anymore. If this is a problem, skewing the magnets becomes a viable alternative. After all, it is not important whether the rotor or the stator is skewed as long as there is a relative amount between them.

__7.6 Motor Winding:__

The most common winding patterns of brush type dc motors are the wave and lap windings. These are explained thoroughly in textbooks such as Reference 20. These alternatives and their various derivatives provide options for achieving design parameters and for satisfying manufacturing constraints.

Brushless motor designers have, over the years, devised a number of different winding patterns to suit the variety of applications encountered. The most prominent types are the lap winding and the concentrated winding. The lap winding here is similar to that in the brush-type dc motor as represented in Figure 7.5.

In Figure 7.5, the number of poles is 4 and the number of slots is 12. Thus, for a 3-phase winding, there are 4 coils/phase. Each coil has n turns and there are two coil sides/slots. For this example, the smallest number of slots is 12 since there must be at least 1 slot/pole/phase. The coils in this winding are uniformly distributed around the lamination stack and therefore, would, as a result of their interaction with the magnets, produce symmetrical torque around the air-gap. If the motor is large enough, there could be more than 1 slot/pole/phase i.e. 2 or 3 etc. resulting in a total number of slots (for 4 poles) of 24 or 36 etc. The above parameters are related to each other by the following relationship:

The other and more recent type of winding is the concentrated winding. Here, the coils are around individual teeth and thus do not span over other teeth. This eliminates the problems associated with coils overlapping each other and resulting in a large overhang. The absence of long end turns results in a reduction of the total length of wire and therefore, of the winding resistance also. This type of winding, however, is only suitable for selected numbers of poles and slots that have severe restrictions in space.

Whatever type of winding may be used in the design of the motor, it is important to divide up the available space for the stator in the most optimal way between steel and copper. A large amount of steel allows the use of higher energy magnets and therefore more flux in the magnetic circuit. This, however, restricts the available space for the slots and, therefore, reduces the amount of copper for the winding.

The slot includes the necessary space for the copper wire plus slot lining or insulation. Unfortunately, the slot area cannot be totally utilized for the copper wire especially if it is intended to wind the wire with automatic winding machines. In such cases it is not unusual that only 30-40% of the total slot area is for the copper wire (including its own enamel insulation). However, this packing or fill factor may be allowed to go as high as 80-90% in the extreme case of hand winding of square wire. The most suitable value for a particular design is dependent, not only on the intended manufacturing method but also on the size of the wire and number of turns. For a small wire gage the insulation around the wire constitutes a higher percentage of the wire size than for a large gage. Counteracting this trend is the fact that it is easier to pack small wires together than large ones since they nest inside each other without leaving much void. In some cases a square wire is used to restrict the voids between the wires. Most likely in such a case, hand winding is expected. In many applications, the resistance of the winding is limited to a maximum value thus imposing a limit on the ratio of number of turns to wire size,

Equation (7.6) says that the limit on R also imposes a limit on the maximum number of turns and on the smallest cross-sectional area of wire. The other constraints on these 2 parameters are: the back emf required determines the number of turns, and the manufacturability of the winding determines the smallest wire size allowed. These considerations and judgments are made by the designer with a view to the particular application and do not necessarily follow any formulae.

In most designs of laminations, the slot opening is made as small as possible but still wide enough to allow an easy winding process. For large wire sizes, 2-3 times the wire diameter for the slot opening is recommended. But for small wire sizes, the slot opening should not be made any smaller than 0.025" even if the wire diameter is only a fraction of that.

__7.7 Winding Connections__:

Regardless of the winding pattern used inside a brushless dc motor, the winding terminals must be brought out and connected to the drive electronics. The switching circuit of the electronics can be one of many. The basic choice is that of the number of phases, and the following are the most common:

a) 2-phase switching

b) 3-phase switching

c) 4-phase switching

Each of the above types can be either unipolar or bipolar. This means that for the 2-phase switching either one winding or both windings are used at any one time. For the 3-phase switching it means either one winding or two windings are used at any one time. The 4-phase switching can be achieved either by switching one winding or two windings on at any one time. The selection of winding configuration is based on some or all of the following parameters:

a) The space, i.e. diameter and axial length, available for a motor capable of satisfying the output requirements

b. The required starting torque at some or all of the possible starting positions.

c) The starting current available from the power supply and whether this current is controlled electronically or by the motor winding resistance.

d) The maximum number of power switching devices allowed as well as their type which determines the voltage drop across them.

e) The maximum tolerable torque ripple under running condition. However, the speed-control circuit is also important in minimizing this ripple.

f) The electrical and mechanical time constants required are based on switching and acceleration requirements. A high inductance can cause switching problems and hence is specified by a low electrical time constant. A high motor moment of inertia can result in a long acceleration time if the mechanical time constant is not specified.

g) Non-repetitive runout of the hub can be affected by the type of switching circuit. The speed-control circuit is also a factor in limiting the runout to a low level.

h) The temperature rise of the motor and the switching circuit reflects directly on the temperature rise of the adjacent parts in the drive unit. Thus the limit on heat dissipation can affect the choice of winding resistance and continuous current and hence the type of winding.

A different comparison can be made between the three major types of switching circuit based on the ratios of minimum and average torque to the maximum torque. These ratios reflect on the starting torque obtainable from the motor and on the percentage torque ripple. Table 7.1 shows this comparison and also gives the number of switching devices required in each circuit.

2-ph | 2-ph | 3-phY | 3-phY | 4-ph | 4-ph | |

Parameter | Uni |
Bi |
Uni |
Bi |
Uni |
Bi |

% Winding utilization |
50 |
100 |
33 |
66 |
25 |
100 |

No.of Switching Device |
2 |
4 |
3 |
6 |
4 |
8 |

No.of Commutations per pole-pair (m) |
2 |
2 |
3 |
6 |
4 |
4 |

Tave/Tmax |
0.64 |
0.64 |
0.83 |
0.95 |
0.9 |
0.9 |

Tmin/Tmax |
0 |
0 |
0.5 |
0.87 |
0.71 |
0.71 |

(Tmax-Tmin)/Tave |
1.56 |
1.56 |
0.6 |
0.14 |
0.32 |
0.32 |

Table 7.1

The equations used in calculating the torque ratios in Table 7.1 are as follows:

In both Equations (7.7) and (7.8), m is the number of commutations per pole-pair. Changing from unipolar to bipolar switching for any of the three major types may, however, require additional devices beyond the number given in the table. For example in the 2-phase motor, a fifth device of power rating equal to the other four is required. The above equations assume that the emf waveforms of the motors are sinusoidal. This, however, is not always the case. The values for the above 2 equations, given in the table are in fact lower than would be obtained from motors with trapezoidal emf waveforms and the percentage ripple is higher. This is why trapezoidal waveforms are desired in brushless motor designs.

No account is taken in Table 7.1 of the effect of the reluctance torque on the total torque, and the ratios given are solely due to the electromagnetic torque. The reluctance torque is the bi-directional torque produced by the interaction of the magnets with the lamination stack, and the electromagnetic torque results from the interaction of the current-carrying winding with the magnetic field. For further reading on the design and performance of dc motors, References 24 to 31 are recommended.

__7.8 Motor characteristics:__

After determining the magnetic circuit parameters, flux and flux densities in all parts of the circuit, the next step is to calculate the motor performance. Two methods will be discussed here. First, either the back emf constant or the number of turns can be assumed while the other is calculated from Equation (7.9). The value of flux is the parameter that links the two together.

where p is the total number of poles, m the number of coils/phase, n the number of turns/coil, is flux/pole in Weber, and is the back emf constant in V/krpm.

Next, , the torque constant, can be determined from using one of the two following equations:

The developed torque can be determined approximately from the following Equation.

where I is the winding current. This approach assumes that is a constant value not affected by current since it is based on the no-load value of . Naturally, this is not absolutely true since the load current has a continuously changing effect on and . Alternatively, the torque can be calculated from the stored energy in the magnetic circuit.

where is the magnetic energy and is the angle of rotation. The energy itself can be determined from the parameters already calculated.

The above two methods have differing accuracy levels that may still not match the values determined by tests. In these tests, torque and current are measured at various values of load. The ratio of torque to current is then the at that particular amount of load.

Various shapes of speed torque characteristics can be achieved with different motor designs. Figure 7.6 shows the 2 main types of motor and their characteristics.

The series motors are a family of motors in which the magnetic field is produced by the same current flowing into the motor winding. These motors can produce very high values of torque at start and low speeds. They are also capable of very high speeds when they are lightly loaded as shown in Figure 7.6(a).

The shunt motors have an independent source of power for the magnetic field. The field can be controlled to produce a constant amount of torque over the speed range, if it is so desired. The variation over speed is generally linear and is, therefore, predictable. In the ac family, the induction motor is a good example and in the dc motor family, the brushless dc motor is a good example.

The negative slope of the curve in Figure 7.6(b) can be modified by introducing changes to the motor winding. The two parameters affecting the torque are the winding emf and input current. The emf can be changed by making changes to the winding number of turns. The maximum input current for a fixed supply voltage is determined by the resistance of the winding.

Figure 7.6(b) shows the speed/torque characteristics of a typical permanent magnet dc motor. As in most cases, the relationship between speed and torque is linear. The no-load speed is very close to the point of intersection of the line with the speed axis. The starting torque is at the point of intersection of the line with the torque axis. Due to the linearity of the relationship it is quite common to measure these 2 points only and simply join them in a straight line.

A sample of the other motor characteristics is given in Figure 7.7(a) as a function of speed and in Figure 7.7(b) as a function of torque (Reference 24). The point of maximum efficiency is dependent on the shape of the output power curve. The latter is itself dependent on the loss characteristics consisting of iron and mechanical losses.

The above characteristics are common to all permanent magnet dc motors. However, there are differences between brush type and brushless dc motors, the most obvious being in type of construction. In the brushless motor, the commutator and brushes are eliminated, thus simplifying the motor design and reducing the weight of the rotor. Switching is controlled electronically with power devices connected to the winding terminals which are in the stator. The rotor, which carries the permanent magnets, as shown in Figure 7.1, also switches the sensors on and off to provide the pulses necessary for commutation. Electronic switching of the current into the winding makes the brushless motor more controllable than the brush type motor but it also causes some problems related to the smoothness of operation.

__7.9 Loss calculation:__

The losses most relevant to this discussion are the electrical and magnetic losses. The electrical losses are simply the IR losses in the copper wire. Once the winding selection of turns and wire gage are made, the winding resistance can easily be determined. The calculation of the electrical losses is then fairly straightforward even with temperature effect on the resistance taken into account.

The situation, however, is markedly different with the calculation of the magnetic losses. These are commonly called iron or core losses. The reason for this difficulty is similar to that encountered previously in the calculation of reluctance. Both are highly dependent on the flux-density level that the material is working at. Since the flux-density levels are different at different parts of the magnetic circuit and also at different times during one revolution of the rotor, an approach similar to that in the reluctance calculation is taken here.

The magnetic circuit is divided into many sections and each section is small enough to allow the following assumption to hold true. The flux-density in any particular section is assumed to be constant throughout that section. The accuracy of the results, therefore, improves as the magnetic circuit is divided into smaller and smaller sections. However, too many small sections lengthen the solution time.

The iron losses include hysteresis and eddy current losses. The variables affecting these losses are the flux-density and the frequency of flux reversals. This can be seen from the following two equations, as well as from Section 2.2:

Manufacturers of the steel materials usually publish the above losses combined. The units are Watts per unit weight and the curves are plotted against the flux-density level of the material. The curves are sometimes made available at various values of frequency. This is the exception however. In a similar fashion to the B/H curves discussed earlier, the loss curves can also be represented mathematically. Thus, the iron losses must be determined for particular values of the flux-density. In the absence of the loss data at the required frequency, the effect of the higher frequency (since the curves are given at 60 Hz) on the loss is taken into consideration on the basis of the above formulae.

To determine the iron loss of each section of the magnetic circuit, the weight of each section must be determined. This is an exercise in geometric calculation of the cross sectional areas and lengths of these sections. This calculation leads to the volume and weight since the material density is known.

In summary, the calculation of the iron losses requires a large number of formulae because of the complicated geometry and large number of variables. The following list summarizes the steps taken in those calculations.

1) Divide the magnetic circuit into a reasonable number of segments.

2) Determine the weight of each segment.

3) Determine flux-density in each segment.

4) Determine the iron loss in each segment of the soft iron at the appropriate B level.

5) Determine the effect of the operating frequency on the iron loss (eddy current and hysteresis).

6) Determine the total losses by multiplying items 2 and 5 for each segment and adding the results for all the segments.

__7.10 Armature reaction &
demagnetization__**:**

The high armature reaction and potential for magnet demagnetization are major parameters in the design of the optimum dc motor. Since most applications require the motor to run in both directions, the approach of shifting the commutation axis cannot be employed. In permanent-magnet dc machines, the effect of armature reaction is minimized by the presence of the magnets which are effectively large air-gaps.

Figure 7.8: Armature reaction

Armature reaction is illustrated in Figure 7.8. This is the interaction of the magnet field axis with the winding axis. It unbalances the magnet and winding flux patterns. It also demagnetizes the magnet in the region where the two fields oppose each other. If soft iron magnetic saturation levels were of no concern, the net impact of armature reaction would be minimal. However, that is certainly not the case in high power density motors.

Figure 7.9

Figure 7.9 displays the second quadrant B-H curve for a Nd-Fe-B magnet. The high intrinsic coercive force of modern magnet materials will eliminate the permanent demagnetization of the magnet. However, partial demagnetization of the magnet can still occur and can significantly lower the magnet flux.

Figure 7.9 also displays the effect of partial permanent-magnet demagnetization. It can be computed using Equation (7.17).

What is interesting about this equation is that it shows that a higher pole count and a thicker magnet will result in reducing the demagnetization value. The term is the traditional NI or ampere turn parameter which must be maximized for the high power performance needed in this motor.

__7.11 Acceleration:__

In some applications it is required to predetermine the acceleration time for a particular load inertia. The following calculation takes into account the limits imposed on the motor by the supply voltage and current. If the mechanical and iron losses of the motor during acceleration are ignored, then:

The mechanical time constant is determined in the following way:

where R is the motor resistance which determines the maximum current allowed, J is the combined inertia of the motor and load in (oz.in.s),and KT is the torque constant in oz.in/A.

From the above 2 equations:

To find the minimum acceleration time, the differential dt/d is equated to zero and the equation is solved to yield the following:

__7.12 Designing with computers:__

Personal computers have proved to be a very useful and convenient tool in the design process of dc motors. The number of parameters seen in this chapter to significantly affect the final design of the motor is large. It is, consequently, difficult to keep track of how they are interrelated while doing a hand calculation. The magnetic circuit can be broken down into small components, for which simple equations can be written to describe their conditions. These formulas are then entered in a computer program, e.g. a spreadsheet, where repeated calculations are made without error.

The attached Table 7.2 shows a typical example of a spreadsheet calculation for a brushless motor design. It uses the lumped reluctance method described at the beginning of this chapter and contains all the other relevant formulas to calculate the motor parameters and performance. The entered values generally refer to the required dimensions, selected materials (magnet and steel) and performance specifications. The calculated values are parameters of the magnetic circuits, winding particulars and performance characteristics.

The power loss calculations follow the method described in Section 7.9 and the results of that calculation are used in determining the efficiency of the motor.

By using a computer program as the one suggested, repeated calculations of the design are done to study the effect of one parameter on the performance. Optimization of various parameters can thus be made without the need to build and test a large number of prototype motors.