The electrical and mechanical sides of a DC machine are linked by two central proportional relationships.

Generated EMF

The rotational EMF is approximately

$$E=k_e\Phi\omega,$$

where $\Phi$ is the magnetic flux, $\omega$ the mechanical angular speed and $k_e$ a machine constant.

Electromagnetic torque

The electromagnetic torque is approximately

$$\tau=k_t\Phi I_a,$$

where $I_a$ is the armature current.

For consistent SI definitions, the constants are closely related so that electromagnetic power is conserved in the ideal conversion:

$$EI_a=\tau\omega.$$

Armature circuit

The armature winding has resistance $R_a$. In motor operation a simple steady relation is

$$V=E+I_aR_a,$$

while in generator operation the generated EMF must supply both the terminal voltage and internal resistance drop:

$$E=V+I_aR_a$$

for corresponding current directions.

Coupled behavior

These equations cannot be interpreted independently. Speed changes EMF, EMF changes armature current through the electrical equation, and armature current changes torque.

This feedback between the electrical and mechanical variables explains much of the characteristic behavior of DC motors and generators.