Dynamic Analysis of Stepper Motor Mechanism
Figure 2.6
The first thing to note about the process of taking one step is that the maximum available torque is at a minimum when the rotor is halfway from one step to the next. This minimum determines the running torque, the maximum torque the motor can drive as it steps slowly forward. For common two-winding permanent magnet motors with ideal sinusoidal torque versus position curves and holding torque h, this will be h/(20.5). If the motor is stepped by powering two windings at a time, the running torque of an ideal two-winding permanent magnet motor will be the same as the single-winding holding torque.
It shoud be noted that at higher stepping speeds, the running torque is sometimes defined as the pull-out torque. That is, it is the maximum frictional torque the motor can overcome on a rotating load before the load is pulled out of step by the friction. Some motor data sheets define a second torque figure, the pull-in torque. This is the maximum frictional torque that the motor can overcome to accelerate a stopped load to synchronous speed. The pull-in torques documented on stepping motor data sheets are of questionable value because the pull-in torque depends on the moment of inertia of the load used when they were measured, and few motor data sheets document this!
In practice, there is always some friction, so after the equilibrium position moves one step, the rotor is likely to oscillate briefly about the new equilibrium position. The resulting trajectory may resemble the one shown in Figure 2.7:
Figure 2.7
Here, the trajectory of the equilibrium position is shown as a dotted line, while the solid curve shows the trajectory of the motor rotor.
Resonance
The resonant frequency of the motor rotor depends on the amplitude of the oscillation; but as the amplitude decreases, the resonant frequency rises to a well-defined small-amplitude frequency. This frequency depends on the step angle and on the ratio of the holding torque to the moment of inertia of the rotor. Either a higher torque or a lower moment will increase the frequency!
Formally, the small-amplitude resonance can be computed as follows: First, recall Newton’s law for angular acceleration:
T = µ A
Where:
T — torque applied to rotor
µ — moment of inertia of rotor and load
A — angular acceleration, in radians per second per second
We assume that, for small amplitudes, the torque on the rotor can be approximated as a linear function of the displacement from the equilibrium position. Therefore, Hooke’s law applies:
T = -k
where:
k — the “spring constant” of the system, in torque units per radian
— angular position of rotor, in radians
We can equate the two formulas for the torque to get:
µ A = -k
Note that acceleration is the second derivitive of position with respect to time:
A = d2/dt2
so we can rewrite this the above in differential equation form:
d2/dt2 = -(k/µ)
To solve this, recall that, for:
f( t ) = a sin bt
The derivitives are:
df( t )/dt = ab cos bt
d2f( t )/dt2 = -ab2 sin bt = -b2 f(t)
Note that, throughout this discussion, we assumed that the rotor is resonating. Therefore, it has an equation of motion something like:
= a sin (2 f t)
a = angular amplitude of resonance
f = resonant frequency
This is an admissable solution to the above differential equation if we agree that:
b = 2 f
b2 = k/µ
Solving for the resonant frequency f as a function of k and µ, we get:
f = ( k/µ )0.5 / 2
It is crucial to note that it is the moment of inertia of the rotor plus any coupled load that matters. The moment of the rotor, in isolation, is irrelevant! Some motor data sheets include information on resonance, but if any load is coupled to the rotor, the resonant frequency will change!
In practice, this oscillation can cause significant problems when the stepping rate is anywhere near a resonant frequency of the system; the result frequently appears as random and uncontrollable motion.
Resonance and the Ideal Motor
Up to this point, we have dealt only with the small-angle spring constant k for the system. This can be measured experimentally, but if the motor’s torque versus position curve is sinusoidal, it is also a simple function of the motor’s holding torque. Recall that:
T = -h sin( ((/2)/S) )
The small angle spring constant k is the negative derivitive of T at the origin.
k = -dT / d = – (- h ((/2)/S) cos( 0 ) ) = (/2)(h / S)
Substituting this into the formula for frequency, we