Torque Generation by DC Motor

The fundamental law of the motor is

$$F=Bli$$

where $F$ is the electromagnetic force, $B$ is the magnetic flux density, $l$ is the length of the conductor within the magnetic field, and $i$ is the current.

The EMF induced in the moving conductor is given by

$$\text{EMF}=Blv$$

where $v$ is the velocity of the conductor within the magnetic.

In DC motos, torque $T$ and EMF $E$ can be expressed as

$$\begin{array}{r}T=k_t{i}_a\\ E=k_e\omega \end{array}$$

where $k_t$ is the torque constant, $i_a$ is the EMF constant, $i_a$ is the armature current and $\omega$ is the angular speed.

Notably, in DC motors, the torque constant $k_t$ and the EMF constant $k_e$ are equal, implying that their units are also equivalent:

$$\left[\frac{\text{Nm}}{\text{A}}\right]=\left[\frac{\text{V}}{\text{rad/s}}\right]$$