Torque-Speed Curve

We can model permanent magnet (PM) DC motors as:

$$\begin{array}{rl}V& =E+iR\\ & =k_e\omega +iR\\ & =k_e\omega +\frac{T}{k_t}R\end{array}$$

where $V$ is the supply voltage, $E$ is the back EMF, $i$ is the current through the armature, $R$ is the armature resistance, $T$ is the generated torque, and $\omega$ is the angular velocity.

The torque-speed of the motor can be derived as:

$$\begin{array}{rl}T& =-\frac{k_t{k}_e}{R}\omega +\frac{k_t}{R}V\\ & =\frac{k_t}{R}(V-k_e\omega )\end{array}$$

This equation shows that torque decreases linearly with increasing speed for a constant supply $V$. The negative slope $-\frac{k_t{k}_e}{R}$ indicates that torque decreases as speed increases.

Example