How to select the right magnetic torquers to desaturate a wheel?

Question

I want to select actuators for a micro-satellite (around 40 kg) orbiting in LEO.

I proceeded in calculating the maximum disturbance torques that affect my spacecraft and selecting proper reaction wheels for three axis stabilization.

Now I want to select a second set of actuators, to activate them during the desaturation of the wheels and I choose a magnetic torquer (since it'll be in LEO and I want to minimize the mass of the fuel). The problem is that I can't find an appropriate explanation about how to select a magnetic torquer for this purpose.

I didn't quite understand if:

1. Does the magnetic torquer desaturate the wheel?
2. Is the motor that resets the speed of the wheel and the magnetic torquer counteract the unwanted rotation of the spacecraft due to the spin down of the wheel?

If the right answer is the first one, I don't understand essentially how it works and if the right answer is the second one (the one that I suppose it to be), I don't understand how to take into account the torque that generate the wheel during the desaturation to select the magnetic torquer.

I hope that my question is well asked and I want to thank in advance those who will answer me.

Answer 1

For sure, there are many possible implementations, but probably the most straightforward approach is to realize that reaction wheels are simply integrating (accumulating) over time torque necessary to keep the spacecraft in a desired attitude. Saturation means that the value of this integral over time reached a maximal allowed value.

Instead of thinking of "desaturation" as a specific event, you make sure the time-integral of torque never runs out of bounds. Your magnetorquers will add a "attitude-destabilizing" torque which reaction wheels will compensate for inside their normal attitude stabilization feedback loop as always. The trick is that extra torque exerted by MTQs will result in opposite sign to be integrated into wheel spin so they will gradually decelerate.

Roughly speaking, the time-integral of available MTQ torque over a "long enough" time period needs to be higher than the integral of external disturbances (intentional attitude control) of torque over the same time.