DC Motor Model and Torque-Speed Curve

A DC motor runs fastest with no load and slows down as you load it, until it stalls. That trade-off is a straight line, the torque-speed curve, and it tells you almost everything about whether a motor fits a robot.

Supply voltage Load torque

The motor equations

Torque is proportional to current, and the spinning motor makes a back voltage proportional to speed. With winding resistance $R$:

$$\tau = k_t\, i \qquad V = i R + k_e\,\omega \qquad \Longrightarrow \qquad \omega = \omega_0\left(1 - \frac{\tau}{\tau_s}\right)$$

The no-load speed is $\omega_0 = V / k_e$ and the stall torque is $\tau_s = k_t V / R$. Output power $P = \tau\,\omega$ peaks at half the stall torque.

What to notice

As load rises, speed falls in a straight line and current climbs. Power peaks in the middle, but efficiency peaks at light load, often around 10 to 20 percent of stall torque. Running a motor near stall makes lots of torque but almost no power, and the current turns into heat, which is how motors burn out. Raising the voltage shifts the whole speed line up but keeps its slope.

When choosing a motor, pick one whose usual operating point sits near peak efficiency, with peak demands staying well below stall.