CIM no-load current vs applied voltage

[Ether]

Piggybacking on Eric's last post, consider the following:

1) tau = F(NLS)  

2) emf = Ke*NLS

3) V = NLC*R + emf

4) tau = Kt*NLC

where:

tau: motor internal torque due to bearing and brush friction, windage, core losses, etc. Note: This is not the load torque (external torque)

NLS: no-load speed

Ke: constant (assumed). It converts NLS into back emf.

Kt: constant (assumed). It converts current into torque

NLC: no-load current through the motor

R: motor coil resistance

V: voltage applied across the motor terminals

F(): a to-be-determined function which converts NLS into motor internal torque

emf: the back emf generated by the motor's NLS

Taking equation (4) and substituting from equation (1),

we can get NLC as a function of NLS:

(5)  NLC = tau/Kt = F(NLS)/Kt

Taking equation (3) and substituting from equations (2) and (5),

we can get V as a function of NLS:

(6)  V = NLC*R + emf = (F(NLS)/Kt)*R + Ke*NLS

Equations (5) and (6) give the relation between motor no-load speed and no-load current and applied voltage.

The question is, what is F()? Is it a proportional function? A linear function? Or a non-linear function?

Remember that F() is the motor's internal torque (due to bearing and brush friction, windage, and core losses). It certainly isn't going to be proportional (since friction doesn't disappear at low NLS), and it will not be perfectly linear (since windage for example is not linear).

For the motors that are used in FRC, the no-load current is small enough, and the manufacturing tolerances which affect the nominal value of Kt (and Ke, which is numerically equal to Kt if SI units are used) are large enough, that the non-linear behavior of F() can safely be ignored for most purposes.


Globe has published an interesting primer on DC motors which addresses this topic:

The no-load-torque value shown in this catalog for each
motor series includes all no load losses and can be considered
a nominal value over the speed ranges where it is anticipated
that the unit will be used. While brush and bearing friction are
relatively independent of speed, other factors such as grease
viscosity, windage, hysteresis and electrical losses will change
as exponential functions of speed. The most noticeable variation
from unit-to-unit or test-to-test will be caused by temperature
effects on grease viscosity. When more exact calculations are
required, you may assume that one-half of the no load losses
occurs at zero rpm and that these losses will follow a linear
curve from this point to the listed catalog value

(emphasis mine)