Magnetics Design of Brush and Brushless DC Motors

Through Lumped Reluctance

and FEM Simulation Programs

George P. Gogue
Dan Jones


When designing a permanent magnet dc motor, the magnetics design engineer usually faces a myriad of design conditions that are of an interactive nature. He must determine the flux values from the main field (magnets) as well as from the winding. Knowledge of the resultant magnetic field and the armature reaction is also required. He must also keep a continuous check on the iron losses in the different sections of the magnetic circuit. These losses must be established in concert with the copper winding configuration and fill. The outcome of these optimization activities is a magnetic design that integrates both electrical and mechanical performance. This design would define the geometry of all the parts of the magnetic circuit and also the materials involved.

A number of simple lumped reluctance simulation programs that can run on a personal computer have been developed. These programs require a very short computer time and eliminate many of the typical variations on configuration. They can achieve sufficient accuracy for the majority of motor design applications. They can be used in place of the slower but more accurate finite element mesh (FEM) simulation programs.

This paper will simulate a set of brush and brushless dc motor designs to demonstrate the development of a "high tech" (rare earth magnet) motor. Some basic formulae used in these programs will also be presented. The intent is to demonstrate that PM motors can be designed with simpler program techniques, as long as the user understands that the resulting boundary conditions will impact simulation accuracies.

The reluctance model:

There are several papers explaining the principles of the reluctance model with particular emphasis on dc motors (Refs. 1, 2, 3). An electrical equivalent of the magnetic circuit is given, enabling the development of the performance equations.

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 computer program used in this work is based on the reluctance model of Figure 1. The reluctances have the basic formula.

Figure 1

Where l is the length of the magnetic path in m and A is the cross sectional area in . 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. But is a parameter totally dependent on the reluctance of the circuit. Perhaps Figure 2 is a helpful representation of the interaction between those parameters.

Figure 2

The other basic equations are

where MMF is the magneto-motive force in Ampere-turns and the flux in Weber. This equation is the magnetic analog of the electric Ohm's law equation.

where B is the flux-density in Tesla.

The program sums up the reluctances of all the different parts of the circuit and calculates the flux corresponding to a particular value of total reluctance. But because of the interactive nature of flux and reluctance, this calculation is repeated many times in an iterative fashion. The limit on this calculation is a residue of percentage error that is pre-selected by the designer.

The reluctance terms of Figure 1 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, therefore, 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 R(core) and is represented by one value in Figure 1. 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 computer program divides up the core 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 the calculation of and the reluctance. 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 R(gap) in Figure 1. 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 2 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 which approximate the effect of fringing to an additional air-gap length. An alternate technique of accounting for fringing flux is discussed in Reference 6.

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 air-gap area is only a first order approximation based on the designer's experience.

Magnetic characteristics:

With the reluctance values of all the different parts of the motor represented as equations, the calculations now center around the amount of flux flowing in the circuit, as shown in Figure 1. The mmf referred to in Equation 2 is the sum of Ampere-turns from the main magnet and from the winding current.

The magnet mmf is a function of the flux-density level at which the magnet is operating. The B-H demagnetizing curves of the particular magnet material give the value of the magnetizing force H from which the magnet mmf can be calculated.

where is the radial thickness of the magnet (in m if H is AT/m). k is a constant, dependent on the units selected.

With regard to the winding ampere-turns, the value is determined from NI where N is the number of turns carrying current I in amperes. However, care must be taken to include only the coils within the magnetic circuit being analyzed and not the complete winding. The magnetic circuit consists of a pair of magnets, a north and a south. In addition, with some of the recently developed winding configurations, some coils could be totally subtended within one magnet, e.g. when a coil is wound around one tooth or when the number of teeth is odd and there are fractional coils facing each magnet.

Figure 3:

As mentioned in the previous section, the reluctance calculation for steel requires the value of permeability which in turn requires the B-H curve of the material. Figure 3 shows B-H curves for typical steel materials. There are many mathematical ways of representing these non-linear curves. The simplest and least accurate being the approximation of the curve to 2 straight lines joined at the knee of the typical curve. Much more accurate results can be obtained from such mathematical simulation as in Ref. 4

where kl, k2 and k3 are constants solved from the equations written by inspecting the B-H curve.

There is also the issue of orientation of steel and its effect on the calculation. In most magnetic circuit designs for dc motors, the steel used is non-oriented silicon steel. Therefore, although flux is simultaneously flowing in many directions inside the steel, the same B-H characteristics can be used in all parts of the steel.

Armature reaction:

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.

Armature reaction is illustrated in Figure 4. 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 4 Armature reaction

Figure 5 displays the second quadrant B-H curve for Neodymium iron. The high intrinsic coercive force 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 5 Demagnetization Curves

Figure 5 also displays the effect of partial permanent magnet Demagnetization. It can be computed using Equation 6.

where Z = no. of winding conductors

= motor peak current (amps)

= magnet radial thickness (cm)

p = no. of magnet poles

n = no. of winding parallel paths

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.

Motor performance:

After determining the magnetic circuit parameters, flux and flux-densities in all its parts, 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. The value of flux is the parameter that links the two together.

Where p is the total number of poles, N the number of turns/coil, m the number of coils/phase 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 10

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.

Loss calculation:

The losses most relevant to this discussion are the electrical

and magnetic losses. The electrical losses are simply the 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 enlarge the program and lengthen the solution time, and this is not intended here.

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.


= hysteresis loss (W/kG)

= hysteresis constant

f = frequency (Hz)

B = flux-density (T)

= eddy-current loss (W/kG)

= eddy-current constant

t = material thickness (mm)

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 computer program determines the iron losses for the continuously changing 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 leading to the volume and weight by knowing the material density.

Figure 6a

Figure 6b

Figure 6c

In summary, the calculation of the iron losses requires a large number of formulae because of the complicated geometry and large numbers 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.

A recent presentation (Reference 5) was written around the results obtained from using this computer program. In that work the losses were calculated for different types of steel. The material selection was demonstrated to have a significant effect on the total losses especially since the investigation was aimed at high speed motor performance.


1. C. K. Taft, H. D. Chai, "A Parametric Model For Brushless

DC Motors", 1987 IMCSD Proc.

2. A. Cassat, "Brushless DC Motors - Torque and Inductance

Determination", 1987 IMCSD Proc.

3. D. Jones, "Motor Magnetics Design Changes in Higher Energy

Rare Earth PM DC Motors", 1985 IMCSD Proc.

4. J. R. Brauer, "Simple Equations for the Magnetization and

Reluctivity Curves of Steel", 1975 IEEE Magnetics.

5. D. Jones, G. Gogue, "The Impact of High Speed Iron Losses in BLDC Motor operation at Speeds Above 10,000 RPM", 1988 IMCSD Proc.

6. J. Ireland, "Ceramic permanent magnet motors", McGraw-Hill

Book Company.