Design Equations (Electric Motors)

Design Approach. In the design equations that follow, the approach is to start with basic motor geometrical constraints and a magnetic circuit describing magnet flux flow. From this circuit, the magnet operating point is found, as are the important motor dimensions and current required to generate a specific motor output power at some rated speed. Given the desired counter-emf at rated speed, the number of turns per phase are found. From the winding information, phase inductances and resistances are computed.
Radial Flux Motor Design. The radial flux topology considered here is shown in
Fig. 5.43.
Fixed Parameters. Many unknown parameters are involved in the design of a brushless PM motor. As a result, it is necessary to fix some of them and then determine the remaining ones as part of the design. Which parameters to fix is up to the designer. Usually, one has some idea about the overall motor volume allowed, the desired output power at some rated speed, and the voltage and current available to drive the motor. Based on these assumptions, Table 5.10 shows the fixed parameters assumed here.
The parameters given in Table 5.10 are grouped according to function. The required power or torque at rated speed, the peak counter-emf, and the maximum
Radial-flux motor topology showing geometrical definitions.
FIGURE 5.43 Radial-flux motor topology showing geometrical definitions.
conductor current density are measures of the motor’s input and output.Topological constraints include the number of phases, magnet poles, and slots per phase. The air gap length, magnet length, outside stator radius, outside rotor radius, motor axial length, core loss, lamination stacking factor, back-iron mass density, conductor resistivity and associated temperature coefficient, conductor-packing factor, and magnet fraction are physical parameters. Magnet remanence, magnet recoil permeability, and maximum steel flux density are magnetic parameters. Shoe parameters include the slot-opening width and shoe depth fraction. Finally, the winding approach must be specified.
Of the parameters in the table, it is interesting to note that the stator outside radius, motor axial length, and rotor outside radius are considered fixed. The stator outside radius and axial length are fixed because they specify the overall motor size. The rotor outside radius is fixed because one often wishes to either specify the rotor inertia, which increases as iC or to maximize Rro, since torque increases as R2o. Clearly, as Rro increases for a fixed Rso, the area available for conductors decreases, forcing one to accept a higher conductor current density to achieve the desired torque. Secondarily, by specifying the rotor outside radius, the design equations follow in a straightforward fashion, and no iteration is required to find an overall solution.
Geometric Parameters. From the parameters given in Table 5.10 and the dimensional description shown in Figs. 5.43 and 5.44, it is possible to identify important geometric parameters. The various radii are associated by

TABLE 5.10 Fixed Parameters for the Radial Flux Topology

Fixed Parameters for the Radial Flux Topologytmp3A-35_thumbSlot geometry for the radial-flux motor topology.
FIGURE 5.44 Slot geometry for the radial-flux motor topology.
As shown in Fig. 5.44, the stator teeth have parallel sides and the slots do not. However, the situation where the slots have parallel sides and the teeth do not is equally valid. A trapezoidal-shaped slot area maximizes the winding area available and is commonly implemented when the windings are wound randomly, when they are wound turn by turn without any predetermined orientation in a slot (Hender-shot, 1990). On the other hand, a parallel-sided slot with no shoes is more commonly used when the windings are fully formed prior to insertion into a slot.
The unknowns in the preceding equations are the back-iron widths of the rotor and stator wbi. Given these two dimensions, all other dimensions can be found. In particular, the total slot depth is given by
which must be greater than zero. In addition, the inner rotor radius Rri must be greater than zero. If either of these constraints is violated, then Rro or Rso must be changed.
Magnetic Parameters. The unknown geometric parameters wtb are determined by the solution of the magnetic circuit. The air gap flux and flux density are given by Eqs. (5.34) and (5.35), respectively, and can be evaluated using the fixed and known geometric parameters given previously.
Electrical Parameters. The electrical parameters of the motor include resistance, inductance, counter-emf, and current. All of these parameters are a function of how the motor is wound. It is assumed that no matter what winding approach is used, all coils making up a phase winding are connected in series. This assumption maximizes the counter-emf and minimizes the current required per phase to produce the required rated torque.
Before any parameters can be found, it is necessary to convert the rated motor speed to radians per second. Then, if the motor output is specified in terms of horsepower, it must be converted into an equivalent torque. Since there are 746 W/hp, the equivalent torque is
where com, rad/s, is the rated mechanical speed.
Torque. To find the electrical parameters, it is necessary to specify the relationship between torque and the other motor parameters. The torque developed by a single phase when VVspp = 1 is
where once again int () returns the integer part of its argument because the number of turns must be an integer. Due to the truncation involved in Eq. (5.42), the peak counter-emf may be slightly less than £max. The actual peak counter-emf achieved can be found by substituting the value computed in Eq. (5.42) back into Eq. (5.41).
Current. Given the desired torque, the required current can be specified in a number of ways. Conductor current, slot current, phase current, or their associated current densities can be found. In addition, these can be specified when any number of phases are conducting simultaneously. Moreover, the peak or RMS value can be specified. And finally, the shape of the current is a function of the counter-emf waveform as well as of the implemented motor drive scheme. As a result, the peak slot current and peak slot-current density under the assumption that only one phase is producing the desired torque will be computed. These values represent a worst-case condition, since more than one phase is usually contributing to the motor torque at one time. In addition, the phase current is computed under the assumption that all phases are contributing equally and simultaneously to the motor torque.
This current value is useful for estimating the ohmic or I2R losses of the motor when producing the rated output. In an actual motor, the RMS phase current is greater than Eq. (5.44), since the counter-emf is never an exact square wave.Therefore, computations using Eq. (5.44) are optimistic.
The slot current given by Eq. (5.43) is distributed among ns conductors occupying the slot cross-sectional area given by Eq. (5.30). Part of this area is occupied by conductor insulation, inevitable gaps between slot conductors, and additional insulation pieces placed around the slot periphery, called slot liners. As a result, only some fraction of the total cross-sectional area is occupied by slot conductors themselves. This fraction is taken into account by specifying a conductor-packing factor:
Typically, kcp is less than 50 percent, but it can be higher under special circumstances. The exact value of this parameter is known only through experience.
Using Eqs. (5.30) and (5.43) and the conductor-packing factor given in Table 5.10, the slot and conductor current density is
Resistance. The phase resistance and inductance of the motor windings are func-
Inductance. The phase inductance has three components due to the slots and end turns. Writing the air gap inductance on a per slot basis gives
As earlier with the phase resistance, given V sp slots per phase and one end turn per slot, the total phase inductance is
Performance. To compute the efficiency it is necessary to compute the ohmic winding loss and the core loss. Of these, the core loss is the most difficult to compute accurately. The magnets and rotor back iron experience little variation in flux and therefore do not generate significant core loss. On the other hand, the stator teeth
in driving the motor can be included in Eq. (5.55), giving a more realistic total system efficiency.
Finally, summing the ohmic and core losses and dividing by the stator peripheral area gives an estimate of the maximum heat density to be removed from the motor:

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