Electric motors: brushed DC, brushless and stepper

Sensors let a robot perceive; actuators let it act. The great majority of robot actuators are electric motors: they are clean, quiet, efficient, easy to control with the electronics of ROB-200 and available in every size from a few millimeters to the hundreds of kilowatts that drive excavators. This lesson develops the simple model that describes almost every motor, uses it to read motor curves and datasheets, and compares the three main types used in robots: brushed DC, brushless DC and steppers. Lesson 9 adds gearboxes, and lesson 12 puts both together to size a real actuator.

The DC motor model

A permanent-magnet DC motor converts current into torque and speed into voltage. Two constants capture the electromechanical conversion. The torque constant \( k_t \) says how much torque each ampere produces, and the back-EMF constant \( k_e \) says how much voltage the spinning motor generates per unit speed. In SI units, with torque in N·m, current in A, voltage in V and speed in rad/s, the two constants are numerically equal; datasheets that use mixed units hide this, so convert before calculating.

The third equation is the electrical side: the applied voltage V is shared between the winding resistance R, the winding inductance L and the back-EMF e. In steady state the inductance term vanishes. Combining the equations gives the motor's steady-state torque-speed relationship at a fixed voltage, a straight line:

The no-load speed \( \omega_0 \) occurs when the motor produces no torque, and the stall torque \( \tau_s \) when it is held still and draws the stall current \( V/R \). Mechanical output power \( P = \tau\omega \) is zero at both ends and peaks at half the stall torque and half the no-load speed. Efficiency peaks at a much lighter load, typically 10 to 20 % of stall torque, which is where motors should run continuously.

Heat: continuous and peak ratings

The torque-speed line describes what a motor can do instantaneously. What it can do continuously is limited by heat. The copper loss \( i^2R \) in the windings heats them, and the insulation has a maximum temperature, often 125 to 155 °C. Because loss rises with the square of current, and current is proportional to torque, doubling the torque quadruples the heating. Datasheets therefore give a continuous (rated) torque that the motor can sustain indefinitely at a stated ambient temperature, and a peak torque several times larger that it can deliver for seconds. Thermal resistance and time constant describe how quickly it heats up. For a robot whose load varies, the relevant figure is the root-mean-square (RMS) torque over the duty cycle, which lesson 12 uses to size motors.

Brushed DC motors

In a brushed DC motor, the windings are on the rotor and the magnets on the stator. Carbon brushes rubbing on a segmented commutator switch the current between rotor coils as it turns, so the torque always pushes in the same direction. Brushed motors are cheap and simple to drive: apply a DC voltage, through an H-bridge for direction and PWM for speed (ROB-200), and the motor runs. Their weaknesses all come from the brushes: wear and limited life, sparking that creates electrical noise and rules them out in explosive atmospheres, friction, and heat generated in the rotor where it is hard to remove. They remain common in toys, small grippers, low-cost hobby robots and automotive accessories.

Brushless DC and permanent-magnet synchronous motors

A brushless motor turns the arrangement inside out: the magnets are on the rotor and the windings on the stator, where heat escapes easily to the housing. Without a mechanical commutator, the drive electronics must switch the current between three phases at the right moments, using rotor position from Hall sensors, an encoder or a sensorless estimate from back-EMF. Simple six-step (trapezoidal) commutation gives some torque ripple; field-oriented control (FOC), which ROB-200 introduced, uses an encoder or a good estimator to keep the stator field exactly at right angles to the rotor field, giving smooth torque even at very low speeds. Motors designed for sinusoidal drive are often called permanent-magnet synchronous motors (PMSMs); in robotics both are casually called brushless or BLDC.

Brushless motors offer high efficiency, long life, high power density and excellent controllability, and they dominate modern robots: drone propulsion, robot arm joints, legged robots and electric vehicles. Two shapes are common. Inrunners have the rotor inside the stator and spin fast with low inertia. Outrunners, and the flat 'pancake' or frameless motors used in robot joints, have a large diameter, which increases torque because torque scales with the rotor's radius and area. Motor torque density depends strongly on diameter, and for this reason quasi-direct-drive legged robots use large-diameter, flat motors (lesson 10).

Brushless motor datasheets often quote a speed constant \( K_V \) in rpm per volt, especially for drone motors. It is simply the reciprocal of the back-EMF constant in mixed units: \( k_t \approx 60/(2\pi K_V) \) N·m/A, or about 9.55/\( K_V \).

Stepper motors

A stepper motor moves in fixed increments, typically 200 full steps per revolution (1.8° per step), as its driver energizes the coils in sequence. Because each pulse moves the shaft by a known angle, a stepper can position open-loop, without an encoder, which makes it popular in 3-D printers, camera sliders, small linear stages and laboratory automation. Microstepping drivers divide each full step into 16, 32 or more microsteps by shaping the coil currents, giving smoother motion, though the positional accuracy of microsteps is limited.

The catch is that open-loop positioning works only while the motor's torque exceeds the load. A stepper's torque falls steeply with speed, because the coil inductance stops the current from rising fully at high step rates, and if the load or acceleration demands more torque than is available, the rotor slips and loses steps, silently. The controller then no longer knows where the axis is. Steppers also draw full current even when stationary, so they run hot and are inefficient, and they can resonate at certain speeds. Closed-loop steppers with encoders, sometimes called hybrid servos, detect and correct lost steps.

Reading a motor datasheet

Motor datasheets combine the quantities above with mechanical data. Look for nominal voltage, no-load speed and current, continuous (rated) torque and current, peak or stall torque, torque and speed constants, terminal resistance and inductance, rotor inertia, thermal resistances and time constants, and maximum winding temperature. The electrical time constant \( L/R \) sets how fast current, and therefore torque, can change; the mechanical time constant \( J R/(k_t k_e) \) sets how quickly the unloaded motor reaches speed. Both matter for the bandwidth of the control loops in ROB-300. Check the conditions for the continuous rating, such as ambient temperature, mounting on a heat-sinking flange and natural or forced cooling, because a motor mounted inside a sealed plastic robot body may manage only half of its catalog rating.

import numpy as np
V, kt, R = 24.0, 0.050, 0.60            # volts, N*m/A, ohms
tau = np.linspace(0, kt * V / R, 9)     # torque from 0 to stall
w = V / kt - R / kt**2 * tau            # rad/s (kt = ke in SI)
i = tau / kt
P_out = tau * w; P_in = V * i
eff = np.divide(P_out, P_in, out=np.zeros_like(P_out), where=P_in > 0)
print(" torque N*m   rpm   current A   P_out W   efficiency")
for t, s, c, p, e in zip(tau, w, i, P_out, eff):
    print(f"   {t:5.2f}    {s*60/(2*np.pi):6.0f}   {c:6.1f}    {p:7.1f}     {100*e:5.1f} %")

 torque N*m   rpm   current A   P_out W   efficiency
    0.00      4584      0.0        0.0       0.0 %
    0.25      4011      5.0      105.0      87.5 %
    0.50      3438     10.0      180.0      75.0 %
    0.75      2865     15.0      225.0      62.5 %
    1.00      2292     20.0      240.0      50.0 %
    1.25      1719     25.0      225.0      37.5 %
    1.50      1146     30.0      180.0      25.0 %
    1.75       573     35.0      105.0      12.5 %
    2.00         0     40.0        0.0       0.0 %

The table confirms the theory: power peaks at half the stall torque, while efficiency in this idealized model is highest at light load and falls linearly toward zero at stall. A real motor's friction and iron losses make efficiency fall again at very light loads, which produces the hump-shaped efficiency curve in the figure.