DC motors: characteristics and speed control

Why it matters

Brushed DC motors drive wipers, small robots, conveyors, toys and countless positioning stages, and their equations are the template for every other drive: brushless and servo motors are analysed with exactly the same constants. If you can predict how speed falls with load and how voltage or flux changes speed, you can size a motor and design its controller.

Key ideas

Construction and working principle

• The stator provides the field: permanent magnets (PMDC) or a field winding. The rotor (armature) carries the conductors; the commutator and brushes reverse the current in each coil as it passes the neutral axis so that torque stays in one direction.
• A current-carrying conductor in a magnetic field feels a force (F = B·I·L), giving torque proportional to flux × armature current.
• A conductor moving in the field generates an EMF. In a motor this back EMF opposes the applied voltage and is proportional to flux × speed. Back EMF is what makes the motor self-regulating: as load rises, speed falls, back EMF falls, and current rises to supply the extra torque.

Motor constants With flux φ constant, T = K_t·I_a and E_b = K_e·ω. In SI units (N·m/A and V·s/rad) K_t = K_e numerically; this is energy conservation (E_b·I_a = T·ω). Catalogues often quote K_e in V/krpm, so convert before using it.

Steady-state equations (separately excited or PMDC) V = E_b + I_a·R_a (brush drop and armature inductance ignored in steady state). Combining gives a straight-line speed–torque characteristic: ω = V/K − (R_a/K²)·T

• No-load speed ω₀ = V/K (where T = 0).
• Stall torque T_s = K·V/R_a, with stall current V/R_a (often 10–20 times rated current, which is why starting current must be limited in large machines).
• The slope R_a/K² is the speed drop per unit torque; a small R_a gives a "stiff" motor.
• Maximum mechanical power occurs at half no-load speed and half stall torque, but motors are run well below that point for efficiency and heating reasons.

Types by field connection

• Separately excited / PMDC: nearly constant speed with load; linear characteristic; used in servo and position control.
• Shunt: field across the supply; behaves like separately excited at constant supply voltage.
• Series: field carries armature current, so φ rises with load; very high starting torque (T ∝ I_a² below saturation) and speed falling steeply with load. Never run a series motor at no load: speed can rise dangerously. Used in traction and hoists.
• Compound: mix of shunt and series.

Speed control (from N ∝ (V − I_a·R_a)/φ)

• Armature voltage control: vary V with flux at rated value; speed from near zero to base speed at rated torque (constant-torque region). Efficient when the variable voltage comes from a converter.
• PWM chopper: a transistor switches the DC supply at several kHz; the average armature voltage is D·V_dc, where D is the duty cycle. The armature inductance smooths the current. An H-bridge adds reversing and regenerative braking. This is how armature voltage control is done in practice.
• Field weakening: reduce φ to go above base speed. Torque capability at rated current falls in proportion, so power stays roughly constant (constant-power region). Not possible with permanent magnets.
• Armature resistance control: add series resistance; simple but wasteful, since the extra I²R is lost, and speed regulation becomes poor. Mainly for starting.

Dynamics (link to control topics): with inductance L_a and inertia J, the motor has an electrical time constant L_a/R_a and a mechanical time constant J·R_a/K²; the voltage-to-speed transfer function is second order and is often approximated as first order because the electrical time constant is much smaller.

Formulas

T = K_t·I_a — T: torque (N·m), K_t: torque constant (N·m/A), I_a: armature current (A).

E_b = K_e·ω — E_b: back EMF (V), K_e: back-EMF constant (V·s/rad), ω: speed (rad/s). In SI, K_t = K_e = K.

V = E_b + I_a·R_a — V: armature voltage (V), R_a: armature resistance (Ω). Steady state.

ω = (V − I_a·R_a)/K and ω = V/K − (R_a/K²)·T — speed in rad/s.

N = 60·ω / (2π) — speed in rpm.

T_stall = K·V / R_a, ω₀ = V / K — stall torque (N·m) and no-load speed (rad/s).

N₂/N₁ = (E_b2/E_b1)·(φ₁/φ₂) — speed ratio for the same machine when voltage, current or flux change.

V_avg = D·V_dc — average armature voltage from a PWM chopper; D: duty cycle (0 to 1).

η = T·ω / (V·I_a) — efficiency, ignoring field-circuit losses.

Worked examples

Example 1 (standard): PMDC motor operating point Given: PMDC motor, V = 24 V, R_a = 1.2 Ω, K = 0.05 N·m/A (= 0.05 V·s/rad), load torque T = 0.3 N·m. Neglect friction.

1. Current: I_a = T/K = 0.3/0.05 = 6 A.
2. Back EMF: E_b = V − I_a·R_a = 24 − 6 × 1.2 = 16.8 V.
3. Speed: ω = E_b/K = 16.8/0.05 = 336 rad/s; N = 60 × 336/(2π) = 3209 rpm.
4. No-load speed: ω₀ = V/K = 480 rad/s = 4584 rpm. Stall torque: T_stall = K·V/R_a = 0.05 × 24/1.2 = 1.0 N·m.
5. Efficiency: output = 0.3 × 336 = 100.8 W; input = 24 × 6 = 144 W; η = 70 %. Answer: N ≈ 3209 rpm, I_a = 6 A, η = 70 %.

Example 2 (GATE level): voltage control and field weakening Given: separately excited DC motor, R_a = 0.5 Ω, runs at 1500 rpm on 220 V drawing 20 A. The load torque is constant. (a) Armature voltage reduced to 180 V, field unchanged.

1. E_b1 = 220 − 20 × 0.5 = 210 V.
2. Constant torque and constant flux mean I_a stays 20 A, so E_b2 = 180 − 20 × 0.5 = 170 V.
3. N₂ = N₁·E_b2/E_b1 = 1500 × 170/210 = 1214 rpm. Answer: about 1214 rpm. (b) Instead, at 220 V the flux is reduced to 80 % of its original value.
4. Torque ∝ φ·I_a is constant, so I_a2 = 20/0.8 = 25 A.
5. E_b2 = 220 − 25 × 0.5 = 207.5 V.
6. N₂ = N₁·(E_b2/E_b1)·(φ₁/φ₂) = 1500 × (207.5/210) × (1/0.8) = 1853 rpm. Answer: about 1853 rpm, with armature current up to 25 A. Weakening the field raises speed but also raises current for the same torque; this is why the constant-power region limits torque.

Common mistakes

• Using a speed in rpm with K in V·s/rad (or the reverse). Convert with ω = 2πN/60.
• Writing N = V/K and forgetting the I_a·R_a drop.
• Assuming current stays the same when flux changes; for constant torque it must rise as φ falls.
• Thinking field weakening reduces speed; reducing flux increases speed.
• Believing armature resistance control is efficient; the power in the added resistor is wasted.
• Running a series motor without load, or opening the field of a shunt motor while it runs: flux collapses to residual and speed and current shoot up.
• Treating K_t and K_e as different quantities in SI units.

For GATE ME

Expect numericals on back EMF, current, speed and torque from V = E_b + I_a·R_a, speed changes under voltage or flux control with constant load torque, PWM average voltage, and efficiency. Conceptual MCQs test the speed–torque line, series versus shunt behaviour, and which control method gives constant torque or constant power. Practise ratio methods (N ∝ E_b/φ) so you do not need K explicitly.

Quick check

1. A motor has K = 0.1 V·s/rad. What is its torque at 8 A?
2. Which speed-control method raises speed above base speed?
3. A 48 V chopper runs at 25 % duty cycle. What is the average armature voltage?
4. Why is the starting current of a DC motor so high?
5. Why should a series motor never be started without load?

Answers: 1. 0.8 N·m. 2. Field weakening. 3. 12 V. 4. At standstill there is no back EMF, so only R_a limits the current. 5. Flux falls as current falls, so speed rises to dangerous values.