Two performance curves are critical to defining the motor formants: the speed versus torque curve, and the current versus torque curve. Developing the equation for motor speed versus torque curve starts with a motors back EMF constant, Ke. The simplest equation for motor speed (N):

That is, when all losses are ignored, the motor speed is independent of torque and purely a function of the applied voltage. While this equation is too simple to be useful, it highlights an important concept, that the motor speed is controlled by the supply voltage. Performance curves define the motor’s overall operating envelope. The controller varies the voltage to obtain the desired motor speed.

When winding losses, the largest source of motor losses, are added to the torque to the simplest equation we get:

We see that as the motor current increases with increasing load, the motor speed decreases.

Next lets add some additional sources of losses, losses due to motor friction tf and losses due to windage, lamination hysteresis, lamination eddy current losses, and fields circulating in the motor housing, which are all lumped together as Kd, which has units on in-oz/kRPM.

Doing some equation manipulation we get for N:

And for I:

All these equations are plotted below for a motor (no drive losses are considered) operating from a 140 VDC supply with a Kt of 17.5 in-oz/A and therefore a Ke of 12.94V/kRPM.

The black traces are the simplest equations, the red traces are the most complex form. We see the effects of winding losses have significant impact while that of tf and Kd are minor. This is important because adding Kd to the equation complicates things, and we do not want things to be overly complicated. We will lump Kd effects into tf by adding a torque of Kd*V/Ke, removing the N term in the equation for current. By doing so, we simplify the equations, allowing for reasonable closed form solutions for N and I when we add the effects of motor inductance, which have a major effect on the curves but is often neglected.

Next: The effects of motor inductance