DC Motor
========

In BAM, servomotor sources are modeled using DC motors dynamics. A DC motor turns current into torque, according
to the following equation:

.. math::

    \tau = k_t i

where :math:`\tau` is the torque, :math:`k_t` is the torque constant, and :math:`i` is the current. When a motor
rotates at a velocity of :math:`\omega`, it generates a back electromotive voltage of :math:`k_t \omega`.

As a result, if you apply a voltage :math:`V` to the motor, the current is given by:

.. math::

    i = \frac{V - k_t \omega}{R}

where :math:`R` is the motor resistance. Thus, the torque generated by a DC motor is given by:

.. math::

    \tau = k_t \frac{V}{R} - k_t^2 \frac{\omega}{R}

Torque vs speed limit
---------------------

There are typically two limits in play in a DC motor:

* The maximum **voltage** you can apply, this is typically limited because of your battery that, for example,
can only provide 12V
* The maximum **current** you can apply, this is typically enforced by firmwares to avoid overheating the motor.

Let's first look at the voltage limit, using above equation, we get:

.. image:: ../_static/velocity_vs_torque.svg
    :alt: Torque vs speed limit
    :width: 400
    :align: center

The feasible green area here is depicting the limit caused by **voltage** limit. Note that the torque you can apply actually depends on the current speed of the motor. At a given velocity :math:`\omega_0`, you will not even be
able to apply positive torque (this is the no-load speed of the motor). Conversely, at zero speed, there is a
maximum torque :math:`\tau_{stall}` you can apply (this is the stall torque of the motor).

It is in general a good idea to use voltages as high as possible. Firstly because it allows to use smaller wires,
but also because it can make the voltage-related limit less restrictive. However, many servo-actuators run with
low-voltage batteries and fall into that functioning regime.

The second limit is the **current** limit, that is typically enforced by the firmware of the motor. Since the torque
is proportional to current, this limit can be seen as a maximum torque limit :math:`\tau_{max}`:

.. image:: ../_static/velocity_vs_torque_custom_tau_max.svg
    :alt: Torque vs speed limit
    :width: 400
    :align: center

Note that this is, however a *soft* limit. The firmware will *attempt* to limit the current, but it is not guaranteed that it will always be able to do so. 

For example, if the motor rotates at a very high speed, the best strategy to provide the less torque is to provide
as much volt as possible. However, your voltage supply is limited, and you might have no choice that to provide a voltage that will generate a current above the limit.