How a brushed DC motor works: commutator, brushes and the torque–speed line

Armature, field and the force on a conductor

A brushed DC motor has two magnetic parts. The field is stationary: either permanent magnets or wound pole coils fixed to the housing. The armature is the rotor, an iron core with copper coils in slots, mounted on the shaft.

When a conductor carrying current I sits in a magnetic field of flux density B, it feels a force F = B × I × L, where L is the length of the conductor in the field. The armature coils sit on opposite sides of the shaft, so the force on one side pushes up and on the other pushes down. That pair of forces makes a torque about the shaft.

If the current direction in each coil never changed, the torque would reverse every half turn and the rotor would simply rock back and forth. The commutator is what prevents that.

Commutator and brushes

The commutator is a ring of copper segments on the shaft, insulated from each other. Each armature coil is connected to a pair of segments. Two stationary carbon brushes press against the commutator and bring the DC supply in.

As the rotor turns, a coil moves past the neutral zone between the field poles. At that moment the brushes slide from one segment to the next, which reverses the current in that coil. A coil that was under a north pole carrying current one way is now under a south pole carrying current the other way, so the force on it keeps the same direction.

The result is that the armature's own magnetic axis stays fixed in space, roughly 90° to the field, while the rotor itself spins. Maximum torque per ampere comes from exactly this perpendicular arrangement. A brushed motor is, in effect, a mechanical inverter: the commutator converts DC into the alternating current each coil needs.

Back-EMF and the equation V = IR + E

A conductor moving through a magnetic field also generates a voltage. The armature coils are moving through the field, so they generate a back-EMF that opposes the supply:

E = Ke × ω

where ω is the shaft speed in rad/s and Ke is the voltage constant in V·s/rad. The armature current then follows from the circuit equation

V = I × R + E

where R is the armature resistance. Rearranged, I = (V − E) / R.

The torque equation mirrors it:

T = Kt × I

In SI units Kt (N·m/A) and Ke (V·s/rad) are numerically equal. This is the key to the motor's self-regulation. At standstill E = 0, so the current is only limited by R, and the starting current V / R is very large. As the rotor accelerates, E grows, the current falls and the torque falls with it, until the motor reaches the speed where torque equals load torque plus friction.

The torque–speed line and a worked example

Combine the three equations and eliminate I:

ω = V / Ke − (R / (Ke × Kt)) × T

For a permanent-magnet motor this is a straight line. It touches the speed axis at the no-load speed V / Ke and the torque axis at the stall torque Kt × V / R. Its slope depends only on the winding resistance, so a low-resistance motor has a stiff characteristic: speed drops little when load increases.

Raising the supply voltage shifts the whole line outward without changing the slope. This is why a lower voltage gives a lower speed at the same torque, and why a brushed motor is so easy to control.

Efficiency, however, is best at moderate loads and falls toward stall, because copper loss I² × R grows with the square of current.

Worked example. Take a small motor with R = 0.5 Ω and Ke = Kt = 0.05 in SI units, run from 12 V.

1. No-load speed (ignoring friction): ω = 12 / 0.05 = 240 rad/s, which is 240 × 60 / (2π) ≈ 2,292 rpm. 2. Stall current: V / R = 12 / 0.5 = 24 A. Stall torque: 0.05 × 24 = 1.2 N·m. 3. Now load it with 0.2 N·m. The current is I = T / Kt = 0.2 / 0.05 = 4 A. 4. Back-EMF: E = V − I × R = 12 − 4 × 0.5 = 10 V. 5. Speed: ω = E / Ke = 10 / 0.05 = 200 rad/s ≈ 1,910 rpm. 6. Electrical input is 12 × 4 = 48 W; mechanical output is 0.2 × 200 = 40 W; copper loss is 4² × 0.5 = 8 W. Efficiency is 40 / 48 ≈ 83%, before friction and brush losses.

Halve the supply to 6 V at the same 0.2 N·m: the current is still 4 A, E = 6 − 2 = 4 V, and speed falls to 80 rad/s (about 764 rpm). Same torque, lower speed, just from lower voltage.

Speed control, reversing and brush wear

Speed control. Because speed follows E = V − I × R, changing the armature voltage changes speed almost proportionally. In practice the voltage is varied by PWM: a transistor switches the full supply on and off thousands of times per second, and the motor's inductance averages it. A higher duty cycle means a higher average voltage and a higher speed. Weakening the field of a wound-field motor raises speed above base speed, at the cost of torque per ampere.

Reversing. The torque direction depends on the product of field direction and armature current direction. Reverse the supply polarity of a permanent-magnet motor and it spins the other way, which an H-bridge does electronically. In a wound-field motor, reversing the whole supply flips both field and armature, so the direction does not change; you must swap the armature leads or the field leads alone.

Wound-field variants. Shunt motors have the field in parallel with the armature and run at nearly constant speed. Series motors have the field in series with the armature, giving very high starting torque but a dangerous runaway speed at no load. Compound motors combine both.

Brush wear. The brushes rub on the commutator, so they wear down, create carbon dust, make electrical noise and spark at the segment edges. That wear, and the limit it places on speed and maintenance-free life, is the main reason many designs have moved to brushless motors.

How the simulator models a brushed DC motor

Curvo Studio's brushed DC motor simulator uses the same compact model described above. The default Field excitation is Magnet (a permanent magnet), with Kt and Ke treated as numerically equal in SI units. From the nameplate values and the conditions you set, it solves the armature current, back-EMF, torque and rotor speed, and flags the motor as stalled if the current needed for the load would reach the stall current V / R at the present supply voltage.

The simulator also offers Shunt, Series and Compound field connections, and a Field rheostat for the shunt and compound types. A chart shows the speed–torque curve for the selected connection with the other three faded behind it for comparison. Start-up is shown as a simple exponential approach to the steady-state speed and current, starting from the high initial current V / R. It is an explanatory model, not a design or protection calculation.

The 3D model shows the field poles or magnets, the armature core and windings, the segmented commutator and the stationary carbon brushes as separate parts. The commutator turns with the armature while the brushes stay still.

Try it in the brushed DC motor simulator

Open the brushed DC motor simulator and try these experiments.

• Drag the Supply voltage slider down and up and watch the rotor speed change almost in proportion, while the Back EMF readout follows it.
• Raise the Mechanical load slider. The Armature current and Torque readouts rise, and the speed sags slightly because of the voltage drop across the armature resistance.
• Push the load above what the motor can deliver at the present voltage (for example a few volts of supply at 120% load) and the motor stalls.
• Switch Polarity between Positive and Reversed. A permanent-magnet motor reverses direction; select Shunt, Series or Compound under Field excitation and you see the hint that a wound-field motor does not.
• Compare the four excitation types on the Speed–torque by field connection chart and note how a series motor behaves at light load.
• Use the Cutaway or Exploded view of the 3D model to find the commutator segments and the brushes.

Try it in the simulator