Electrical
The Brushed DC Motor
Commutator and brushes — mechanical current switching that made electric motion cheap
A brushed DC motor is a rotating electric machine in which a segmented copper commutator and two spring-loaded carbon brushes mechanically reverse the current in the rotor (armature) windings twice per revolution, so the torque against a fixed permanent-magnet field always turns the shaft one way. Torque is proportional to armature current through the torque constant Kt (T = Kt·Ia, in N·m/A), while back-EMF and therefore steady-state speed are proportional to voltage through the same constant (E = Ke·ω). Speed follows ω ≈ (V − Ia·Ra)/Ke, so you control it by varying terminal voltage, almost always with pulse-width modulation through an H-bridge. It needs only two wires and no electronics, which is why it powers billions of toys, fans, tools, and car actuators — but the sliding brushes wear, arc, and radiate EMI, capping efficiency near 70–85% and life at a few hundred to a few thousand hours.
- TorqueT = Kt · Ia (N·m)
- Back-EMFE = Ke · ω, Kt = Ke (SI)
- Speedω ≈ (V − Ia·Ra) / Ke
- FieldPermanent magnet or wound stator
- ControlPWM + H-bridge, 1–20 kHz
- Efficiency~70–85% peak
- Brush life~few hundred–2000 h
Interactive visualization
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Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Why the brushed DC motor matters
The brushed DC motor is the machine that made electric motion cheap and universal. Connect two wires to a battery and the shaft spins; swap the wires and it reverses. No inverter, no position sensor, no firmware — the motor commutates itself. That mechanical self-switching is why a brushed motor still costs cents and sits inside almost everything that moves on a small scale.
- Ubiquitous. Toys, cordless drills, cooling fans, vibration motors in phones, RC cars.
- Automotive body. Window lifts, mirror adjusters, seat motors, fuel pumps, wipers, blowers — a modern car has dozens.
- Two-wire simplicity. Torque set by current, speed by voltage, direction by polarity — no controller strictly required.
- High starting torque. Series-wound variants deliver enormous stall torque, historically prized for starter motors and traction.
- Trivial reversibility. An H-bridge gives bidirectional drive and regenerative braking from a single low-cost stage.
- Predictable. The linear torque-current and speed-voltage laws make it easy to model and control.
How it works, step by step
A brushed DC motor turns electrical energy into rotation through the Lorentz force on current-carrying conductors sitting in a magnetic field. The genius — and the flaw — is the commutator that keeps that force pointing the right way as the rotor spins.
- 1. Fixed field. The stator provides a steady magnetic field B, from permanent magnets (ferrite or neodymium) in small motors or from field windings in large ones.
- 2. Armature current. The rotor carries insulated copper coils. Current I in a conductor of length L in field B feels a force F = B·I·L, and the sum of these forces around the rotor is torque.
- 3. Brushes feed current. Two carbon brushes, pushed by springs, slide against the rotating commutator and connect the external supply to whichever coils are best positioned to make torque.
- 4. Commutation. As a coil rotates past the neutral plane between poles, the brushes hand off from one commutator segment to the next, reversing that coil's current. The force therefore keeps driving the rotor the same way — the commutator is a position-triggered mechanical inverter.
- 5. Back-EMF appears. The moment the rotor spins, its conductors generate a voltage opposing the supply: the back-EMF E = Ke·ω. This is why current — and heat — are highest at stall and drop as the motor speeds up.
- 6. Equilibrium. The motor settles where the supply balances back-EMF plus resistive drop, V = Ke·ω + Ia·Ra, delivering exactly the torque the load demands.
The governing equations
Two coupled linear relations describe the ideal brushed DC motor. Both spring from the same physics — force on a conductor and the voltage induced in a moving conductor — which is why the two constants are numerically identical in SI units.
Torque: T = Kt · Ia
Back-EMF: E = Ke · ω
Voltage balance: V = Ke · ω + Ia · Ra, giving ω = (V − Ia · Ra) / Ke
Where the symbols are:
- T — output torque, newton-metres (N·m)
- Kt — torque constant, N·m per ampere (N·m/A)
- Ia — armature current, amperes (A)
- E — back-EMF (counter-EMF), volts (V)
- Ke — back-EMF (speed) constant, volt-seconds per radian (V·s/rad). Numerically Ke = Kt in SI units.
- ω — angular speed, radians per second (rad/s); rpm = ω · 60 / 2π
- V — terminal (applied) voltage, volts (V)
- Ra — armature (winding + brush) resistance, ohms (Ω)
Two limiting cases anchor the whole datasheet. At stall, ω = 0 so E = 0, and current is Istall = V / Ra — often 5–20× the rated current — producing peak torque and peak heating. At no load, torque is nearly zero, so current is tiny and speed reaches its maximum ω0 ≈ V / Ke. Between them the torque-speed curve is a straight line, which makes the brushed DC motor one of the easiest machines to model.
Worked example: a 12 V gearmotor
Take a small permanent-magnet brushed motor rated 12 V with armature resistance Ra = 1.0 Ω and torque constant Kt = Ke = 0.025 N·m/A (equivalently 0.025 V·s/rad). Suppose the load draws Ia = 2.0 A.
- Torque: T = Kt · Ia = 0.025 × 2.0 = 0.050 N·m (50 mN·m).
- Back-EMF: E = V − Ia·Ra = 12 − (2.0 × 1.0) = 10.0 V.
- Speed: ω = E / Ke = 10.0 / 0.025 = 400 rad/s ≈ 3820 rpm.
- Mechanical power out: P = T · ω = 0.050 × 400 = 20 W.
- Copper loss: I²·Ra = 2.0² × 1.0 = 4 W; electrical input = V·Ia = 24 W, so efficiency ≈ 20/24 = 83% at this point.
- Stall current: V / Ra = 12 / 1.0 = 12 A — six times the running current, and 144 W of pure heat, which is why you never hold it stalled.
Now halve the average voltage with PWM to 6 V: no-load speed drops to about ω₀ = 6/0.025 = 240 rad/s (2290 rpm), while torque per amp is unchanged. That is speed control in one line — vary V, and the whole torque-speed line slides down without changing its slope.
Brushed vs brushless: the trade table
| Property | Brushed DC | Brushless DC (BLDC) |
|---|---|---|
| Commutation | Mechanical (commutator + brushes) | Electronic (transistors + rotor sensing) |
| Windings | On rotor (armature) | On stator |
| Controller needed | None (two wires) or simple H-bridge | Multi-phase inverter + control |
| Peak efficiency | ~70–85% | ~85–95% |
| Typical life | Hundreds to ~2000 h (brush-limited) | Tens of thousands of h |
| EMI / arcing | High (brush sparking) | Low (no sliding contacts) |
| Max speed | Limited by commutation (~tens of krpm) | Very high (>100 krpm feasible) |
| Heat removal | Poor (windings spin inside) | Good (windings on outer stator) |
| Cost | Lowest | Higher (electronics) |
| Best for | Cheap, low-power, intermittent duty | Efficient, quiet, long-life, high-speed |
Common misconceptions and failure modes
- "More voltage means more torque." No — torque tracks current, not voltage. More voltage raises speed; torque rises only if the load lets current rise.
- "The commutator makes AC into DC." Backwards. It turns the external DC into the alternating current each rotor coil actually needs — a mechanical inverter, not a rectifier.
- "Stalling is fine, it just stops." At stall there is no back-EMF, so current spikes to V/Ra and the windings cook. Seconds can be fatal without current limiting.
- "Brush sparking is just cosmetic." The arc erodes copper, radiates broadband EMI that upsets nearby electronics, and in oxygen-rich or dusty environments is an ignition risk. Suppression capacitors and proper brush grade matter.
- "Kt and Ke are unrelated numbers." They are the same physical constant; in SI, N·m/A equals V·s/rad. If a datasheet lists them differently, check the units (rpm/V vs rad/s).
- Failure modes. Worn brushes (increasing resistance, then open circuit), a glazed or grooved commutator, shorted or open armature coils, demagnetised magnets from overheating, and bearing wear. Brush and commutator wear is the dominant life limit.
Frequently asked questions
What is a brushed DC motor?
A brushed DC motor is a rotating machine with windings on the rotor (the armature) and a fixed magnetic field, usually from permanent magnets. A segmented copper commutator on the shaft, contacted by two spring-loaded carbon brushes, reverses the current in each coil as it rotates past the poles. This mechanical switching keeps the Lorentz force, and therefore the torque, pointing the same way around the shaft. Feed it a DC voltage and it spins; reverse the voltage and it spins the other way.
How do the commutator and brushes work?
The commutator is a ring of insulated copper segments, each wired to a rotor coil. Two carbon brushes press against it and connect the coil to the external DC supply. As the shaft turns, a given coil passes the boundary between the north and south field poles; at that instant the brushes hand contact from one segment to the next, reversing the current in that coil. Because the current flips exactly as the coil crosses the neutral plane, the force on every conductor always drives the rotor the same direction. The commutator is a mechanical, position-triggered DC-to-AC inverter running inside the motor.
Why is torque proportional to current and speed proportional to voltage?
Each armature conductor carries force F = B·I·L, so total torque T = Kt·Ia is set by armature current Ia through the torque constant Kt. The spinning armature also generates a back-EMF E = Ke·ω opposing the supply. At steady state the applied voltage balances that back-EMF plus the resistive drop: V = Ke·ω + Ia·Ra. Solving gives ω = (V − Ia·Ra)/Ke, so raising V raises speed almost linearly, while the small Ia·Ra term makes speed sag a little under load. Kt and Ke are the same number in SI units (N·m/A equals V·s/rad).
How do you control the speed of a brushed DC motor?
Speed tracks average terminal voltage, so you vary that voltage. A series rheostat wastes power as heat and is obsolete for anything but toys. The standard method is pulse-width modulation: an H-bridge of MOSFETs switches the full supply on and off at 1 to 20 kHz, and the duty cycle sets the average voltage from zero to full. The H-bridge also reverses polarity for direction and can short the armature for braking. A current-sense resistor closes an inner torque loop, and a tachometer or back-EMF estimate closes an outer speed loop.
Why do brushes wear out and spark?
Brushes slide continuously against the spinning commutator, so friction grinds the carbon away and copper dust and heat erode the segments. Sparking, called commutation arcing, happens because each coil is inductive: at the moment the brush breaks contact, the collapsing coil current tries to keep flowing and jumps the widening gap as an arc. That arc burns the segments, radiates electromagnetic interference, and limits maximum speed and life. Brush life is typically a few hundred to a couple thousand operating hours, after which brushes and often the commutator need replacing.
Brushed vs brushless DC motor — what's the difference?
A brushed motor puts the windings on the rotor and switches them mechanically with a commutator and brushes; a brushless (BLDC) motor puts the windings on the stator and switches them electronically with transistors driven by rotor-position sensors or back-EMF sensing. Brushless has no sliding contacts, so it runs quieter, lasts tens of thousands of hours, tolerates higher speeds, sheds heat better from the stationary windings, and reaches 85–95% efficiency. Brushed wins on simplicity and cost: it needs only two wires and no controller. Brushless needs a multi-phase inverter, which is why brushed still dominates cheap and low-power devices.
What is stall current and why does it matter?
At standstill the armature is not spinning, so there is no back-EMF, and the only thing limiting current is the small armature resistance Ra. Stall current is therefore V/Ra, often five to twenty times the rated running current. It produces peak stall torque but also peak heating, so holding a brushed motor stalled for more than a second or two can burn the windings or brushes. Motor controllers add current limiting so that stalling a jammed mechanism does not destroy the motor or the driver.