DC Machines, Excitation & Drive Characteristics

A brushed DC machine uses mechanical commutation to keep armature torque aligned with the field. The same electromechanical structure can motor or generate: applied electrical power can become shaft power, or external shaft power can become electrical power. Its model is compact enough to derive, but complete enough to expose the dangerous cases that a drive must handle.

The coupled model #

For a permanent-magnet or constant-field machine,

$$v_a = R_a i_a + L_a\frac{di_a}{dt} + k_e\omega,$$

$$\tau_e = k_t i_a,$$

$$J\frac{d\omega}{dt}=\tau_e-\tau_L-b\omega.$$

$k_e\omega$ is back electromotive force (back EMF): rotation generates a voltage opposing the applied voltage in motoring operation. $k_t i_a$ is developed electromagnetic torque. In coherent SI units and under matching sign and flux conventions, $k_t$ in N·m/A and $k_e$ in V·s/rad have the same numerical value; that equality is an energy-conversion result, not a coincidence.

The equations hide the mechanism that makes torque unidirectional. The commutator is a segmented rotary switch connected to the armature coils; stationary brushes transfer current to whichever conductors are in the correct magnetic position. As a coil passes the neutral zone, the brush bridges adjacent segments and reverses that coil's current. The energy-model diagram above shows the resulting terminal behavior; the commutation section below explains the brush/segment transition and its failure modes.

Torque-speed characteristic #

At steady state with constant field and supply,

$$i_a=\frac{V-k_e\omega}{R_a},$$

so

$$\tau_e=\frac{k_t}{R_a}V-\frac{k_tk_e}{R_a}\omega.$$

The idealized torque-speed curve is a line between stall torque at $\omega=0$ and no-load speed near $V/k_e$. Mechanical loss shifts the real no-load point, and saturation, brush drop, armature reaction, commutation limits, temperature, and converter voltage drop bend the line.

Excitation changes the characteristic #

Field weakening reduces flux. Since back EMF is proportional to flux times speed, weakening allows higher speed for a given voltage. But torque per ampere also falls, current demand rises for the same torque, and the mechanical speed limit remains. It is a constant-power-region technique, not free speed.

Four-quadrant operation #

Use signs of speed $\omega$ and torque $\tau$:

The machine equations allow all four. The converter and power source may not. A one-quadrant supply cannot absorb regenerative energy; an H-bridge can reverse terminal voltage and current, but its DC bus still needs a battery, active front end, braking resistor, clamp, or other explicit energy sink.

H-bridge states and PWM #

A four-switch H-bridge applies either polarity to the armature and provides recirculation paths for inductive current. The logical states are not just forward and reverse:

PWM controls average terminal voltage, but winding current follows $R_a$, $L_a$, back EMF, switching state, and decay path. Therefore equal duty cycle does not imply equal current at different speeds. Current sensing and a fast inner current loop turn the bridge into a torque actuator; an outer loop can then regulate speed or position.

Dynamic time scales #

With the rotor initially fixed, the electrical time constant is

$$\tau_e=\frac{L_a}{R_a}.$$

With current or torque established, mechanical acceleration is set by $J$ and net torque. A reduced mechanical time constant for the linear constant-field model depends on $JR_a/(k_tk_e)$, modified by damping and load. Treating the motor as an instantaneous gain hides starting current; treating it as only an electrical $RL$ load hides inertia and back EMF.

Commutation is a current-transfer transient #

During mechanical commutation a brush temporarily shorts the coil connected between two adjacent commutator bars. That coil has inductance, so its current cannot reverse instantaneously. If current is still substantial when the trailing segment leaves the brush, the opening contact arcs; if brush position is wrong, the shorted coil also sees rotational EMF that drives additional circulating current. Brush width, neutral-plane position, armature reaction, interpoles or compensating windings, load current, and speed all influence the result.

Visible sparking is therefore a symptom, not a diagnosis. Check brush seating and pressure, commutator runout and surface, neutral setting, current ripple, overload, field strength, and the waveform during current transfer. A drive can limit average armature current while PWM edges and poor commutation still create local brush stress and electromagnetic interference.

Losses and limits #

• Copper loss is $i_a^2R_a$ and rises with winding temperature through $R_a$.
• Brush and commutator loss includes voltage drop, friction, wear, sparking, and electromagnetic interference.
• Core loss grows with magnetic reversal and speed.
• Mechanical loss includes bearings, windage, and driven-load friction.
• Peak torque is limited by current, commutation, magnetic saturation, and mechanics; continuous torque is primarily thermal.

The correct operating envelope is a speed-torque region with separate continuous, intermittent, and absolute limits. A single nominal power number cannot validate acceleration, stall, reversing, or regenerative braking.

What to identify on a real motor #

Measure terminal resistance with temperature recorded, inductive current response with the shaft fixed, back EMF versus speed while externally driven, torque constant or torque-current slope, no-load loss versus speed, inertia, and thermal rise under representative duty. Fit all tests with one sign convention and one parameter set; if the model fits only one plot, it is not yet a drive model.