As you tinker around with DC
motors, you'll start to run across some interesting
relationships. Namely you'll discover that
torque
and current
are linearly proportional to each other, as are
speed and voltage. Under a fixed load (torque),
voltage and current
will also be proportional to each other.
Digging into the math (and I'll spare you this),
it turns out that the current
a motor draws is ultimately determined by the
torque
the motor produces. The generated torque
is dependent upon the current
I, and factors determined by the materials and
internal geometry of the motor. Since the
construction of a finished motor will not (!)
change during operation, a constant of
proportionality between the motor current
and the materials / geometry dependent factors can
be calculated for a given motor. This constant, the
torque
constant Kt, describes the torque
generated by the motor for a specific motor
current:
Kt = T / I
Or to put it another way,
Current
through motor = torque
produced / torque
constant
I (Amps) = Torque
(ozin) / Kt (ozin/A) in imperial units
I (Amps) = Torque
(Nm) / Kt (Nm/A) in SI units
Because of the interrelationship of torque,
speed, current,
and voltage, the constant current
operation of a DC
motor produces constant output torque
regardless of speed. Given a constant load (i.e.
torque)
the speed of a motor is solely dependent on the
voltage applied to the motor. For DC
motors operated at a constant voltage, the speed
and torque
produced are inversely related (the higher the
torque,
the lower the speed of the motor).
We earlier saw that an EMF
will be developed across a motor's brushes when its
coil is rotated by an external torque
 the magnitude of this EMF
is dependent upon materials / geometry factors, and
upon the speed at which the coil is rotated. Once
again, there is a constant of proportionality which
describes the relationship between coil rotational
speed 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 speed (which in turn is generally
expressed either in RPM
or radians
/ second).
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 speed of the coil. This
version of the motor constant is commonly known as
the velocity constant, Kv. The velocity constant is
given in units of rotational speed (again, either
RPM
or radians
/ second) per volt.
Since the motor construction does not change,
regardless of what we're measuring, it turns out
that these three constants (Kt, Ke, Kv) are all
essentially the same number. The differences
between the torque
constant and the back EMF
constant are simply a matter of the units used,
while the velocity constant is simply a useful form
of the back EMF
constant.
If the torque
constant is specified in Nm / A and the back
EMF
constant in Vsec / rad,
then:
Kt = Ke = 1 / Kv
Those of us who live in the U.S., though, are
stuck with using more colorful units. Commonly used
units for small motors are ozin for torque
and RPM
for rotational speed. Using these units of measure,
torque
constants are often given in ozin / A, back
EMF
constants in mV / RPM,
and velocity constants in RPM
/ V. In imperial units, the relationships between
motor constants are then as follows:
With
Kt in units of ozin / A
Ke in units of mV / RPM
Kv in units of RPM
/ volt
Then
Kt = 1352.4 / Kv
Ke= Kt / 1.3524
Ke = 1000 / Kv
Kv = 1000 / Ke
So what good is all this? It means that given a
source of known rotational speed (an electric
drill, or drill press if you have one), you can
compute Ke for a given motor (clamp the motor shaft
in the drill's jaws, measure the resulting
opencircuit voltage, then do the math).
Ke, along with the above information will then
give you Kt (so you can compute your motor's
theoretical torque
at any given current),
and Kv (so you can compute your motor's maximum
speed at any given voltage). If you can measure
stall torque,
you can then compute motor efficiency (measured
torque
expressed as a percentage of the theoretical
torque).
Knowing all this, you can then pick the best
motor for your BEAMbot's
own specific needs.
