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Magnetotorquers are often used for momentum dumping of reaction wheels in smallsats, and they can use them as well for attitude control directly, per answers to Are there any modern cubesats or smallsats that have relied only on magnetotorquers for attitude control?
But has magnetic torque ever been used to spin-up a satellite, as a sort of electric motor? If so, how might that system work?
The cases you've mentioned are applied to three-axis attitude control systems.
In the past (and oddly even today), magnetic torquers were used to replenish the angular momentum of spin-stabilized spacecraft.
The simplest example that comes to mind is SCD 1 and 2:
The ACS is responsible to stabilize and control the spacecraft orientation with respect to the Sun. To perform these tasks, the ACS is provided with two digital one-axis sun sensors, one analog magnetometer, one spin axis air core magnetic coil, and a passive nutation damper. The stabilization is achieved by a rotation around its major principal axis imparted to the spacecraft by the launcher's last stage. Attitude determination and control are both performed on-ground, by using the telemetered sensor signals and commanding the appropriate coil polarity
So every now and then, people on ground compute the attitude profile of the spacecraft, figure out how much torque it needs to keep spinning as it should and command a given torque to be applied by the magnetics torquers.
The nutation damper is meant to alleviate angular rates outside the spin axis, which could cause loss of attitude control. This also amplifies the natural damping of the angular momentum the spacecraft would normally have.
