From lifting cranes to elevators, direct current (DC) motors are all around you. Like all motors, **DC motors** convert electrical energy into another form of energy, typically mechanical motion such as the lifting of an elevator shaft. You can describe how much energy they produce by calculating the torque of these DC motors, a measure of rotational force.

## Torque Equation

A DC torque motor works by passing an electric current through a coil in a magnetic field. The coil is shaped in a rectangle outline between the two magnets with the rest of the coil extending out and away from the magnets. The torque is the magnetic force that causes the coil to spin and create energy.

The torque equation of DC motor designs is

for each turn of the motor with the electric current *I* in amps, magnetic field *B* in teslas, area outlined by the coil *A* in m^{2} and angle perpendicular to the coil wire "theta" *θ*. To use the calculate torque of DC motor designs, make sure you understand how the underlying physics works.

Electric current describes the flow of electric charge, and you direct it in the opposite direction of electron flow in units of amperes (or charge/time). The magnetic field describes the propensity for a magnetic object to influence a force on a moving charged particle using units of teslas just as how electric field describes the force that would affect an electric charge. Magnetic force describes this fundamental force that lets magnets exert properties like torque.

## DC Motor Design

For a DC motor, the magnetic force causes the coil of wire to move, but because the coil would otherwise move back and forth because the force direction continuously reversing upon it, DC motors use a **commutator**, a split-ring material, to reverse the current and keep the coil rotating in one direction.

The commutator uses "brushes" that remain in contact with the electric current to reverse the direction. Most present-day motors make these parts of carbon and use spring-loaded mechanisms to continuously reverse the direction.

You can also use the right-hand rule to calculate the direction of torque. The **right-hand rule** is a way of telling you the direction of a magnetic force using your right hand. If you extend your thumb, index finger and middle finger outward on your right hand, the thumb will correspond to the direction of current, the index finger shows you the direction of magnetic field and the middle finger will be magnetic force direction.

## Deriving the Torque Equation

You can derive the equation for torque from the Lorentz equation,

for electromagnetic force *F*, electric field *E*, electric charge *q*, velocity of the charged particle *v* and magnetic field *B*. In the equation, the *x* refers to a cross product, which will be explained later.

Treat the current as line of moving, charged particles that create a force from a magnetic field. That lets you rewrite *qv* (that has units of charge-distance/time) as the product of charge current and the length of wire (which would also be charge-meter/time).

Because you're only dealing with a magnetic force, you can ignore the *qE* electrical component and rewrite the equation as

*f*or current I and length of wire *L*. By the definition of a **cross product**, you can re-write the equation as

with the lines surrounding each variable denoting the absolute value. For a DC motor, you can rewrite it as *torque = IBA*sin*θ.*

To perform a motor torque calculation online, you can use an online calculator for your specific purposes. jCalc.net offers one that outputs motor torque for input motor rating in kW and motor speed in RPM.

References

- Hyperphysics: DC Motor Operation
- National High Magnetic Field Laboratory: Magnet Academy: DC Motor
- Processing Magazine: The DC Motor Advantage
- Motion Control Tips: Four laws of electromagnetism that you should know
- Top Electrical Engineering: Applications of Dc motor (dc motor applications everywhere)

Resources

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.