Reduce torque ripple

[dzid26]

Due to cheap stepper construction, torque ripple (under load) is quite significant. Under load, the generated torque seems to be higher at the "half-step" than at the "full step". This torque ripple can be represented by 3rd harmonics.[1][2] https://www.desmos.com/calculator/xxyvgnyl7m

[1] Matsui, N., Nakamura, M., & Kosaka, T. (1996). Instantaneous torque analysis of hybrid stepping motor. IEEE Transactions on Industry Applications, 32(5), 1176–1182. doi:10.1109/28.536880 [2] CERN-THESIS-2014-380 Modelling and Control of Stepper Motors for High Accuracy Positioning Systems Used in Radioactive Environments https://core.ac.uk/download/pdf/148670797.pdf

[dzid26]

The idea is then to subtract the electromotive force where torque ripple peaks occur by modifying the commutation sine wave.

The wave shall still be between -1..1 and its first derivative should have the same sign as the normal sine wave (aka, it should not turn around or overshoot). Electrical angle circle:

Benefits of smooth torque:

• less vibration at low speed and noise at higher speed
• reduce stress on the motor gearing
• easier and more linear change of direction under load - especially with gearing Beneficial side effects:
• total heat dissipation when holding in between steps is lower than at full step (it follows the combined sine_mod, cos_mod curve above) - resulting in lower chip/motor average temperatures

Disadvantage:

• lower peak torque (~ -20%) reduces maximum average power (~ -10%) so the motor slows down faster under heavy load But it is possible to gradually introduce torque ripples back at higher speeds to bring back the power or even boost above (~ +30%) what is normally possible - at the expense of higher noise.

For completeness - reverse example of torque ripple/power boost:

Plot script