So long as no part of the
magnetic circuit saturates, powering two motor windings
simultaneously will produce a torque versus position curve
that is the sum of the torque versus position curves for the
two motor windings taken in isolation. For a two-winding
permanent magnet or hybrid motor, the two curves will be S
radians out of phase, and if the currents in the two
windings are equal, the peaks and valleys of the sum will be
displaced S/2 radians from the peaks of the original curves,
as shown in Figure 2.2:
Figure 2.2
This is the basis of
half-stepping. The two-winding holding torque is
the peak of the composite torque curve when two windings are
carrying their maximum rated current. For common two-winding
permanent magnet or hybrid stepping motors, the two-winding
holding torque will be:
h2 = 20.5
h1
where:
h1 --
single-winding holding torque
h2 -- two-winding holding torque
This assumes that no part of the
magnetic circuit is saturated and that the torque versus
position curve for each winding is an ideal sinusoid.
Most permanent-magnet and
variable-reluctance stepping motor data sheets quote the
two-winding holding torque and not the single-winding
figure; in part, this is because it is larger, and in part,
it is because the most common full-step controllers always
apply power to two windings at once.
If any part of the motor's
magnetic circuits is saturated, the two torque curves will
not add linearly. As a result, the composite torque will be
less than the sum of the component torques and the
equilibrium position of the composite may not be exactly S/2
radians from the equilibria of the original.
Microstepping allows
even smaller steps by using different currents through the
two motor windings, as shown in Figure 2.3:
Figure 2.3
For a two-winding variable
reluctance or permanent magnet motor, assuming no saturating
magnetic circuits, and assuming perfectly sinusoidal torque
versus position curves for each motor winding, the following
formula gives the key characteristics of the composite
torque curve:
h = ( a2
+ b2 )0.5
x = ( S / ( /2)
) arctan( b / a )
Where:
a -- torque applied
by winding with equilibrium at 0 radians.
b -- torque applied by winding with equilibrium
at S radians.
h -- holding torque of composite.
x -- equilibrium position, in radians.
S -- step angle, in radians.
In the absence of saturation,
the torques a and b are directly proportional
to the currents through the corresponding windings. It is
quite common to work with normalized currents and torques,
so that the single-winding holding torque or the maximum
current allowed in one motor winding is 1.0. |
