# 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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