Electric motors and DC motor types

The idea

Weeks 1–3 got electrical power from the battery or the grid to the motor’s terminals. Week 4 is the motor itself: how it turns current into torque, the main types a robot might use, and where the energy is lost on the way.

The motors used in robotics

Motor	How it works	Strengths	Weaknesses
Brushed DC	A commutator and brushes switch the coil current as the rotor turns. Speed set by voltage (PWM)	Cheap, simple drive, easy speed control, good low-speed torque with a gearbox	Brushes wear and spark (electrical noise); needs maintenance
Stepper	The driver energises the stator phases in sequence; the rotor moves one fixed step per pulse	Positions without a sensor; strong holding torque	Can miss steps; torque falls with speed; draws current even at rest
Brushless DC (BLDC)	Magnets on the rotor; an inverter switches the stator coils in sequence (six-step)	Efficient, long-lived, high power density and speed	Needs an inverter and position sensing; harder at low speed
PMSM	Like BLDC but with smooth sinusoidal currents (field-oriented control)	Very smooth, precise torque	Complex controller, needs accurate position feedback
Servo	A system: motor + encoder + drive in a closed loop	Corrects its own position, speed or torque errors	Cost, tuning and set-up

Steppers get their own page: Stepper motors.

How a brushed DC motor makes torque

• The stator (the stationary part) makes a magnetic field B, from permanent magnets or field windings.
• The rotor or armature carries coils. A conductor of length L carrying current I across a field B feels a force $F=BIL$, at right angles to both the wire and the field. Force × radius = torque.
• After half a turn, the same current would push the coil backwards. The commutator and brushes reverse the coil current every half-turn, so the torque always acts the same way. (A brushless motor does this switching electronically, with an inverter: see Motor drives.)

The generator effect happens at the same time. A conductor moving through a field has a voltage induced in it, $e=BLv$. In a motor this back-EMF opposes the supply, and it’s the key to how a DC motor regulates itself.

Worked example: a single loop

A single rectangular coil of active length 0.06 m and diameter 0.05 m spins at 2000 rpm in a 0.4 T field, carrying 3 A. Find the peak EMF and the torque.

Answer

$\omega =2000\times 2\pi /60=209.4$ rad/s · $r=0.025$ m → $v=\omega r=5.24$ m/s Each side: $e=BLv=0.4\times 0.06\times 5.24=0.126$ V. Two sides in series: $E_{max}=0.251$ V After commutation the average is $\frac{2}{\pi }{E}_{max}=0.637\times 0.251=0.160$ V (the same factor as a rectified sine; see AC values) Force per side: $F=BIL=0.4\times 3\times 0.06=0.072$ N Two sides at radius r: $T=2Fr=2\times 0.072\times 0.025=0.0036$ N·m Real motors have many coils and commutator segments, so the torque and EMF are much larger and smoother.

The two equations that run everything

$\boxed{V=I_a{R}_a+K_e\omega }\boxed{T=K_t{I}_a}$

• The electrical side: the supply voltage is shared between the drop across the winding resistance and the back-EMF, which grows with speed.
• The mechanical side: torque is proportional to current.
• In SI units, $K_t=K_e$ (N·m/A = V·s/rad). See DC motor performance and selection.

What they mean in practice:

• Speed is set by voltage. As the motor speeds up, the back-EMF rises until it nearly cancels the supply. That’s the no-load speed, ${\omega }_0\approx V/K_e$. This is why PWM (average voltage) controls speed.
• Torque is set by current, and the motor adjusts itself. Load it → it slows → back-EMF falls → more current flows ($I_a=(V-K_e\omega )/R_a$) → more torque. It settles where the motor’s torque matches the load.
• Stall is dangerous. At $\omega =0$ there’s no back-EMF, so $I_{stall}=V/R_a$, often many times the rated current. Drivers, MOSFETs, wiring and fuses must be sized for the stall current, or the driver must limit current.

Curves for a 24 V motor with $R_a=1.2$ Ω and $K_t=K_e=0.06$: no-load speed about 3800 rpm, stall current 20 A. Speed falls in a straight line as torque rises. Output power peaks at half the stall torque, but efficiency peaks much earlier, at light load. That’s why motors are chosen to run well below stall, not at peak power.

DC motor types: how the field is made

Type	Field	Behaviour	Watch out
Permanent magnet (PMDC)	Magnets	Straight-line speed–torque curve, compact, efficient (no field winding loss), simple PWM control	The field can’t be adjusted; magnets can demagnetise with over-current or heat
Shunt-wound	Winding in parallel with the armature	Speed stays fairly steady as the load changes	If the field circuit fails, the motor overspeeds
Series-wound	Winding in series, so it carries the armature current	Very high starting torque, roughly ∝ I². Used for traction and lifting	Speed varies hugely with load and it runs away with no load
Compound	Both windings	A mix of starting torque and steady speed	Some connections can be unstable

Why a series motor runs away: with no load, the current falls, so the field weakens. A weaker field needs more speed to make enough back-EMF to balance the supply, so the motor keeps speeding up. Lose the load on a winch or a robot and it can destroy itself.

PMDC is the robotics default: predictable speed (a fixed field), simple to drive with PWM and an H-bridge, efficient, and compact. Small mobile robots, AGVs and educational platforms nearly all use geared PMDC motors.

Construction

• Iron-core: windings on a laminated iron core. Robust and cheap, from about 1 W to kW, 50–85% efficient. More inertia and some “cogging” (lumpy torque at low speed). Mobile robots and geared actuators.
• Coreless (ironless): a self-supporting hollow winding. Very low inertia for fast, smooth response, 60–90% efficient, but expensive and limited by heat. Grippers, medical and precision robots.

Losses and temperature

Not all the input power becomes shaft power:

$P_{loss}=I^2{R}_a+V_{brush}I+P_{iron+mech}$

The three parts are copper loss in the windings, brush loss from the voltage drop at the brushes, and iron and mechanical loss (magnetising the core, bearing friction and air drag).

$\eta =\frac{P_{out}}{P_{in}}=\frac{T\omega }{VI}\Delta T=P_{loss}{R}_{\theta }{T}_{motor}=T_{amb}+\Delta T$

It’s the same thermal idea as a heatsink (see Thermal design and heatsinks), with the motor’s own thermal resistance $R_{\theta }$.

$T_{motor}$	Meaning
40–80 °C	Normal
80–100 °C	Hot: check the manufacturer’s limits
100–130 °C	Concerning: check the insulation class
> 130 °C	Beyond typical insulation ratings

Efficiency isn’t one fixed number: it changes with load and speed (see the curves above).

Worked example: losses and temperature

A motor on 24 V draws 2 A and delivers 0.12 N·m at 3000 rpm. $R_a=1.5$ Ω, the brush drop is 0.8 V, $R_{\theta }=6$ °C/W and the room is at 25 °C. Find the efficiency, the losses and the motor temperature. What’s the stall current?

Answer

$P_{in}=24\times 2=48$ W · $\omega =3000\times 2\pi /60=314.2$ rad/s · $P_{out}=0.12\times 314.2=37.7$ W $\eta =37.7/48=78.5\%$ · total loss $=48-37.7=10.3$ W Copper $=2^2\times 1.5=6.0$ W · brushes $=0.8\times 2=1.6$ W · iron and mechanical (the rest) $=2.7$ W $\Delta T=10.3\times 6=61.8$ °C → $T_{motor}=86.8$ °C: hot, so check the rating $I_{stall}=24/1.5=16$ A, eight times the running current. At stall all 384 W ($16^2\times 1.5$) becomes heat.

Common mistakes

• Thinking torque rises with speed. It’s the other way round: more load means less speed and more current.
• Sizing the driver for running current. Size it for stall, or limit the current.
• Mixing up shunt and series faults. Shunt overspeeds if its field fails; series runs away if it loses its load.
• Forgetting rpm → rad/s before $P=T\omega$.

Check yourself

1. Why does a DC motor draw more current when it's loaded?

The load slows it, so the back-EMF ($K_e\omega$) falls. More of the supply voltage is left across $R_a$, so more current flows, which gives the extra torque ($T=K_{t}I$).

2. Why is a PMDC motor preferred over a series motor for a mobile robot?

Its fixed field gives a predictable, straight-line speed–torque relationship that’s easy to control, and it won’t run away if the wheels lose contact or the load drops. A series motor’s huge starting torque isn’t worth that risk.

3. A 12 V motor has $R_a=0.5$ Ω. What's its stall current, and why does it matter?

$12/0.5=$ 24 A. The driver, wiring and fuse must survive it, or the driver must limit current; otherwise a jammed wheel burns out the motor or the MOSFETs.

Next: using these equations to analyse and choose motors, in DC motor performance and selection.