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For a motor that turns S radians
per step, the plot of torque versus angular position for the
rotor relative to some initial equilibrium position will
generally approximate a sinusoid. The actual shape of the
curve depends on the pole geometry of both rotor and stator,
and neither this curve nor the geometry information is given
in the motor data sheets I've seen! For permanent magnet and
hybrid motors, the actual curve usually looks sinusoidal,
but looks can be misleading. For variable reluctance motors,
the curve rarely even looks sinusoidal; trapezoidal and even
asymmetrical saw tooth curves are not uncommon.
For a three-winding variable
reluctance or permanent magnet motors with S radians per
step, the period of the torque versus position curve will be
3S; for a 5-phase permanent magnet motor, the period will be
5S. For a two-winding permanent magnet or hybrid motor, the
most common type, the period will be 4S, as illustrated in
Figure 2.1:
Figure 2.1
Again, for an ideal 2 winding
permanent magnet motor, this can be mathematically expressed
as:
T = -h sin( (( /2)
/ S)
)
Where:
T -- torque
h -- holding torque
S -- step angle, in radians
= shaft angle, in radians
But remember, subtle departures
from the ideal sinusoid described here are very common.
The single-winding holding
torque of a stepping motor is the peak value of the
torque versus position curve when the maximum allowed
current is flowing through one motor winding. If you attempt
to apply a torque greater than this to the motor rotor while
maintaining power to one winding, it will rotate freely.
It is sometimes useful to
distinguish between the electrical shaft angle and
the mechanical shaft angle. In the mechanical frame
of reference, 2
radians is defined as one full revolution. In the electrical
frame of reference, a revolution is defined as one period of
the torque versus shaft angle curve. Throughout this
tutorial,
refers to the mechanical shaft angle, and ((/2)/S)
gives the electrical angle for a motor with 4 steps per
cycle of the torque curve.
Assuming that the torque
versus angular position curve is a good approximation of a
sinusoid, as long as the torque remains below the holding
torque of the motor, the rotor will remain within 1/4 period
of the equilibrium position. For a two-winding permanent
magnet or hybrid motor, this means the rotor will remain
within one step of the equilibrium position.
With no power to any of the
motor windings, the torque does not always fall to zero! In
variable reluctance stepping motors, residual magnetization
in the magnetic circuits of the motor may lead to a small
residual torque, and in permanent magnet and hybrid stepping
motors, the combination of pole geometry and the permanently
magnetized rotor may lead to significant torque with no
applied power.
The residual torque in a
permanent magnet or hybrid stepping motor is frequently
referred to as the cogging torque or detent torque
of the motor because a naive observer will frequently guess
that there is a detent mechanism of some kind inside the
motor. The most common motor designs yield a detent torque
that varies sinusoid ally with rotor angle, with an
equilibrium position at every step and an amplitude of
roughly 10% of the rated holding torque of the motor, but a
quick survey of motors from one manufacturer (Phytron) shows
values as high as 23% for one very small motor to a low of
2.6% for one mid-sized motor.
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