# What Makes a Reaction Wheel a Reaction Wheel?

A DC motor is the core of all momentum and reaction wheels. If you apply a voltage a, current will be produced which will cause the wheel to change speed. At the same time, the back electromotive force (EMF) will build up, eventually driving the motor torque to zero.

This is evident from the dynamical equation for a DC motor.

$J\dot{\omega} = \frac{K_T}{R}\left(V - K_T\omega\right) + T_F$

$J$ is the inertia, $K_T$ is the torque constant, $V$ the voltage, $T_F$ the friction torque, $R$ the motor impedance and $\omega$ is the angular rate of the shaft.

You can turn this into a reaction wheel by adding current feedback as shown in the following block diagram.

$G$ is the forward gain. The input is the desired torque. This is divided by the torque constant to get the desired current. The difference between the motor current and the desired current is integrated. How do you pick the gain? If you work through the equations you will get this equation for the voltage, $V$

$\dot{V} + \frac{G}{R}V = G\frac{T_C}{K_T} + \frac{G}{R}K_T\omega$

$R/G$ is the time constant. The response is shown in the following plot. Even as the speed increases, the difference between the desired torque and motor torque is nearly zero.

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Michael Paluszek is President of Princeton Satellite Systems. He graduated from MIT with a degree in electrical engineering in 1976 and followed that with an Engineer's degree in Aeronautics and Astronautics from MIT in 1979. He worked at MIT for a year as a research engineer then worked at Draper Laboratory for 6 years on GN&C for human space missions. He worked at GE Astro Space from 1986 to 1992 on a variety of satellite projects including GPS IIR, Inmarsat 3 and Mars Observer. In 1992 he founded Princeton Satellite Systems.

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