t is extremely common to control a system involving a DC motor, or if not

an actual brushed DC motor, a motor that can be modeled from the control perspective as

a DC motor. The difference between these cases has largely to do with the use of power

electronic drives to replace brushes, which are prone to failure.

Some equations to consider for an ideal motor are

$=$ K$1$i

v $=$ K$2$

where is the torque of electromagnetic origin, is the shaft speed, v is the ideal motor

terminal voltage, and i is the motor current. If the motor is $100\%$ efficient, what relationship

can be inferred between K$1,$ K$2?$ Write equations for mechanical and electrical power.

Often, modeling proceeds by assuming an $$ideal$$ motor and capturing the departure

from the ideal in lumped terms. For example,

vt $=$ v $+$ iR

captures the effect of an armature resistance R and introduces vt as the terminal voltage

taking this effect into account.

Similarly, a mechanical model may include inertia and damping,

J d

dt $=$ $$ $.$

Finally, the relationship between and the shaft angle is

d

dt $=$ $.$

Collect these equations to write first$-$order state$-$space equations for the DC motor