I know that capstan equation relates $T_{in}$ and $T_{out}$ as $T_{out} = T_{in}e^{u \cdot F_c}$, where $u$ is friction coefficient and $F_c$ is wrap (slip) angle. In my situation, the capstan is driven by a motor. Input and output ends of the wrapped material are free (under tension but not supported). Both motor torque and the angular velocity of the wrapped material are positive (clockwise). The wrapped material is elongating and moving at the same time.
What I don't understand is the role of motor torque in the scenario. The equation does not take motor torque into consideration and I couldn't grasp how it has no effect at all.