Motors

Physics Review

Flux and Lorentz

Electric motors turn electrical power into torque via Lorentz forces. Core relationships:

\[V_{\text{applied}} = k_e \omega + I R_w\]
\[T = k_t I\]
\[P_{\text{mech}} = T \omega\]
\[P_{\text{elec}} = V I\]

Variables:

  • V_applied supply voltage (V); k_e back-EMF constant (V·s/rad); ω speed (rad/s)

  • I current (A); R_w winding resistance (Ω)

  • T torque (N·m); k_t torque constant (N·m/A)

  • P_mech shaft power, P_elec electrical power (W)

Linear Motor Model Derivation

Inuitively, if we increase the voltage to a motor, it will spin faster; rather, the speed is proportional:

\[V = \alpha \omega\]

Similarly, if we grab the axle and increase the torque, we see the current rise; again this relationship is largely linear:

\[I = \frac{1}{\alpha} \tau\]

We can recall that electrical power (in Watts) is just voltage x current, then we can follow that:

\[P_{\text{elec}} = V I = (\alpha \omega)(\frac{1}{\alpha} \tau) = \omega \tau = P_{\text{mech}}\]
\[P_{\text{mech}} \approx P_{\text{elec}}\]

We can see that in order for power to be conserved, the linear relationship between the voltage and speed must match the linear relationship between torque and current. This value is the K_v of the motor. You can find it in units of rpm/V, but it could just as easily be measured in N*m/A.

Motor Types

Motor

Figure

Typical use / notes

Brushed DC

Brushed DC Motors Brushed DC Motors gif

  • Two-wire input

  • High starting torque

  • Cheap and simple drive electronics

  • Brushes wear causing electrical noise + slow degredation

  • Easy to drive/control

Brushless DC (BLDC)

drone_motors Brushless dc run Brushless dc motor vectors

  • Higher efficiency and power density than brushed DC

  • Requires an ESC (electronic speed controller)

  • Generally more customizable for weight-constrained systems

  • Add Hall sensors or encoders for precise low-speed control

  • Used in most production-grade high-performance systems

Stepper

Stepper Motors Stepper Motor gif

  • Moves in discrete steps

  • Built-in step count serves as odometer

  • Can skip steps and throw off odometer count

  • Add feedback if loads are uncertain

  • NEMA Steppers used in most 3D printers

  • Microstepping provides half-notch at the cost of reducing holding torque

Encoders

Encoders can be classified as either linear or rotary, and as either incremental or absolute

  • Linear Encoders - measure lateral distance traveled along a slider

  • Rotary Encoders - measure an anglular position about an axis

  • Incremental - measure distance traveled along a path back and forth

  • Absolute - measure with respect to a global origin without drift

Incremental

Absolute

Linear

cap_calipers

Capacitive strip encoder (digital calipers, DROs)

ele-enc

Optical / magnetic scale (CNC, elevators)

Rotary

optical_incremental

Optical quadrature (motors, velocity FB)

pot_abs rot_opt Potentiometer Optical absolute encoder

Servos

Servos solve a special problem between cheap, high torque, position-accurate motors. They serve as a single unit combining a DC brushed motor, absolute encoder, and (typically planetary) high torque gearbox to provide built-in position control.

150kg_servo 9g_servo