DC Motor Torque Constant

I am very new to DC motors and to stackexchange. Please correct me if anything I said does not make sense.

For DC motors, the equation looks like this:

$P = \tau\dot{\theta}$

where $P$ is power, $\tau$ is torque and $\dot{\theta}$ is angular velocity. The power can be expressed as $P=VI$ (voltage times current) and $\tau=K_\tau I$ ($K_\tau$ is the torque constant).

Thus,

$VI = K_\tau I \dot{\theta}$

It seems from the above equation that the angular velocity is directly proportional to the voltage applied to the motor and the current flowing through the motor is irrelevant.

For a PWM controlled motor, the effective voltage is $V_{in} \times \textrm{duty}\%$. For different loads applied to the motor, the angular velocity should be the same if the PWM duty is fixed. However, in reality, when the load increases, the RPM goes down for a given PWM duty.

So, the question is:

• Is $K_\tau$ constant across different loads for a motor (in this case, a DC shunt motor)?
• What is the 'effective voltage' that should be used in the equation?

If $K_\tau$ is constant with different loads for a given motor, then the 'effective voltage' must not be a simple $V_{in} \times \textrm{duty}%$. When the load increases with a fixed PWM duty, there must be a decrease in the 'effective voltage' since the RPM goes down.

If the 'effective voltage' is the voltage difference between Motor+ and Motor-, I did measure it and the voltage difference drops when the load increases for a given PWM duty. Is it that the back EMF causing the reduction in the voltage difference?

Thank you in advance.

• Hi - I re-did the math formatting in your post. Feel free to tweak it if it's not quite to your liking. This is a starting point to learning the LaTeX markup used on SE: physics.stackexchange.com/faq#notation – Kyle Oman Feb 9 '13 at 5:42