# Optimizing dc-brushless servomotors

June 1, 2000
The plot shown is for a stator pole count of 4.
 The plot shown is for a stator pole count of 4. Similar plots can be generated for greater pole counts. Each curve on the plot represents a different ratio of air-gap magnetic flux density to flux density in the iron stator. Higher-quality iron materials give proportionately smaller ratios and higher peaks. Optimal designs fall on the peak of one of these curves. The curves also permit quick performance-cost trade-off studies to be done. For example, an off-the-shelf lamination could give adequate yet less-than-peak performance than a custom lamination without the tooling costs.

Although newly designed electric motors typically resemble earlier models, occasionally a requirement falls outside the familiar envelope. To find an optimal motor solution then, requires a return to basic principles. Consider dc-brushless servomotors, for example.

Optimized dc-brushless servomotors respond faster, deliver higher torque at lower speed, and use less current than nonoptimized designs. Such motors also require a minimum reduction-gear ratio for inertia matching to a load.

Optimizing, by this definition, means to select a rotor-to-stator diameter ratio that gives the motor these characteristics while using a minimum quantity of iron stator (core), copper (windings), and magnetic (rotor) materials. This seemingly complex design problem can be solved by maximizing only one equation containing all the relevant terms: the motor constant, KM,

The output torque, R is specified in oz-in., and the motor copper loss power, LC in watts. The torque term is independent of the motor voltage, current, power, and speed. It includes stator and rotor geometry, magnetic material properties, motor damping, rotor inertia, and winding resistance. Also of interest is the mechanical time constant. Motors with a relatively shorter mechanical time constant have greater dynamic response and wider bandwidth. The mechanical time constant, Tm is related to the motor constant by,

TM = JM/KM 2

THE MOTOR CONSTANT, KM

DO = Stator lamination OD, in.
DI
= Stator lamination ID, in.
L = Stator length, in.
BG
= Air-gap magnetic flux density, lines/in.
CMO
= ACU 1/ 2 , in.
= DI/DO ACU = Sum of all areas of lamination slots, (theoretical copper windings area), in. 2

KCU = Ratio of actual copper windings area to lamination slot area
KW
= Winding factor, product of motor winding pitch and distribution factors
K = Ratio of windings resistance at operating temperature to that at room temperature

Ratio of motor coil length plus end turns to active coil length which is twice the stator stack length :

p = Stator pole count

where JM is rotor inertia. Because both inertia of a cylindrical rotor and the square of KM are proportional to the fourth power of diameter, TM of an optimized brushless-dc motor, based on geometry only, is fixed. However, the amount of copper windings filling the stator slots changes the mechanical time constant. More windings, as would be used in higher quality motors, increase the motor constant independent of geometry and therefore reduce the mechanical time constant.

Information for this article was provided by Sidney Davis, Island Components Group Inc., Bohemia, N.Y.; www.islandcomponents.com

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