Motor Constants

where A is the included angle between B and I, and L is the length of coil segments AB and CD. Assume for the sake of discussion that A is 90¡, then sin A = 1, and the sin A factor can be dropped. Further, assume that multiple conductors are used in the coil. The force on the conductor AB is then given by:

where N is the number of conductors in the coil. The torque generated in the coil as a result of this force F is the product of the force F and the distance from segment AB to the rotational axis O - O’. If this distance is called R (radius), then the resulting torque is given by:

Coil segment CD also contributes to the generated torque, however, and has a contribution equal to that of segment AB. Therefore, the total torque generated is simply twice that generated by segment AB and is given by:

Note that the generated torque is dependent upon the current I, the radius of the coil R, the number of conductors (turns) N, the magnetic field flux density B, and the length of the conductors L. With the exception of the motor current, all other factors are determined by the geometry of the motor and the materials from which it is made. Since it is generally safe to assume that construction features of a finished motor will not change, a constant of proportionality between the motor current and the materials/geometry dependent factors can be assigned to the motor. In the case at hand, there is a constant which describes the torque generated by the motor for a specific motor current, the torque constant.

In the early part of this section, it was shown that an EMF E is developed in a conductor moving through a magnetic field of flux density B at velocity V. If one uses the same assumptions adopted to develop the crude motor using a coil, brushes, and commutator, it can be shown that an EMF will be developed across the brushes when the coil is physically rotated by an external torque. Like the principles involved in the development of the torque constant, the magnitude of the EMF is dependent upon materials/geometry factors. It is also dependent upon the velocity at which the coil is rotated. Once again, there is a constant of proportionality which describes the relationship between coil rotational velocity and materials/geometry factors, commonly known as the back EMF constant (Ke). The back EMF constant is typically given in volts per unit of rotational velocity.

If one takes the reciprocal of the back EMF constant, the result is a proportionality constant which relates the voltage applied to the motor terminals to the rotational velocity of the coil. This version of the motor constant is commonly known as the velocity constant. The velocity constant is given in units of rotational velocity per volt.

There is, of course, one constant for a motor. The differences between the torque constant and the back EMF constant are simply a matter of the units used to make routine calculations while the velocity constant is simply a useful form of the back EMF constant. In fact, it can be shown that if the torque constant is specified in Nm/A and the back EMF constant in V-sec/rad, then:

A common selection of units of torque and speed in the United States for small motors is oz-in for torque and rpm for rotational velocity. Using these English units of measure, torque constants are often given in oz-in/A, back EMF constants in mV/rpm, and velocity constants in rpm/V. These units are selected on the basis of common usage in the United States. The relationships between motor constants given in these units is as follows: