DC motor: torque equation, characteristics and starting

Why it matters

DC motors (and their modern cousins, permanent-magnet and brushless DC motors) drive cranes, traction, rolling mills, actuators, valve positioners and servo systems, because torque and speed are simple to control. The torque, back-EMF and speed equations are the basis of every DC drive and of the transfer function of a DC servo motor used in control-system questions.

Key ideas

Motor action and back EMF. A current-carrying armature conductor in the field feels a force B·I·l; all such forces form the torque. As the armature turns, its conductors also cut flux and generate a back EMF Eb = Ka·Φ·ω that opposes the supply. Back EMF makes the motor self-regulating: if load torque increases, speed falls, Eb falls, armature current rises and torque rises until it balances the load.

Voltage and power equations. V = Eb + Ia·Ra (plus brush drop if given). Multiplying by Ia: V·Ia = Eb·Ia + Ia²·Ra. The term Eb·Ia is the gross mechanical power developed; shaft output is this minus rotational (friction, windage, iron) losses.

Torque equation. From Eb·Ia = T·ω and Eb = Ka·Φ·ω: T = Ka·Φ·Ia with Ka = P·Z/(2π·A). In rpm form, T = 0.159·P·Φ·Z·Ia/A N·m. This is the gross (electromagnetic) torque; shaft torque is less by the rotational loss torque.

Speed equation. N ∝ Eb/Φ = (V − Ia·Ra)/Φ. Speed rises if flux is reduced — the basis of field weakening, and the reason an open field is dangerous.

Characteristics (T–Ia, N–Ia, N–T)

• Shunt (and separately excited): Φ nearly constant, so T ∝ Ia (a straight line) and speed drops only 5–10 % from no load to full load — "constant speed" motor. Uses: lathes, fans, pumps, conveyors, machine tools.
• Series: Φ ∝ Ia below saturation, so T ∝ Ia² (high starting torque) and N ∝ 1/Ia — speed is very high at light load. A series motor must never be started or run without load (or on a belt that can slip): it can overspeed and burst. Uses: traction, cranes, hoists.
• Compound (cumulative): between the two — good starting torque with a definite no-load speed. Uses: presses, shears, rolling mills with flywheels. Differential compound is unstable and rarely used.

Starting. At standstill Eb = 0, so the starting current V/Ra would be 10–20 times rated (Ra is small). This damages the commutator, causes heavy sparking and a supply dip. A starter inserts external resistance in the armature circuit, cut out in steps as speed and back EMF build up. The three-point starter has a no-volt release coil in series with the shunt field (it also trips on field failure) and an overload release; the four-point starter connects the no-volt coil directly across the supply so that field-rheostat changes do not trip it (but then it does not protect against field failure). Small motors and modern drives use a soft-starting power-electronic converter that ramps the armature voltage.

Losses and efficiency. Copper losses (armature, field), iron loss, mechanical loss and brush loss. Maximum efficiency occurs when variable (armature copper) loss equals constant loss.

Formulas

• V = Eb + Ia·Ra — motor voltage equation (V); add brush drop if given.
• Eb = P·Φ·Z·N / (60·A) = Ka·Φ·ω — back EMF (V); ω = 2π·N/60 (rad/s).
• T = Ka·Φ·Ia ; Ka = P·Z / (2π·A) — electromagnetic torque (N·m); Φ in Wb, Ia in A.
• T = Eb·Ia / ω = 9.55·Eb·Ia / N — same torque from power (N in rpm).
• N ∝ (V − Ia·Ra) / Φ — speed equation; N2/N1 = (Eb2/Eb1)·(Φ1/Φ2).
• Shunt: Ia = IL − Ish ; Ish = V/Rsh.
• Series (unsaturated): Φ ∝ Ia ⇒ T ∝ Ia² , N ∝ Eb / Ia.
• R_start = V / I_start − Ra — external starting resistance (Ω) to limit starting current to I_start.

Worked examples

Example 1 (standard): shunt motor torque and starter. Given: 220 V shunt motor, Ra = 0.2 Ω, Rsh = 110 Ω, line current 52 A at 1000 rpm. Neglect brush drop.

1. Field current: Ish = 220/110 = 2 A; armature current Ia = 52 − 2 = 50 A.
2. Back EMF: Eb = V − Ia·Ra = 220 − 50 × 0.2 = 210 V.
3. Mechanical power developed: Eb·Ia = 210 × 50 = 10.5 kW.
4. ω = 2π × 1000/60 = 104.7 rad/s. Torque T = Eb·Ia/ω = 10500/104.7 = 100.3 N·m.
5. Direct-on-line starting current would be 220/0.2 = 1100 A (22 times rated). To limit it to 1.5 × 50 = 75 A: R_start = 220/75 − 0.2 = 2.73 Ω.

Example 2 (GATE level): series motor at reduced load. Given: 230 V series motor, total resistance Ra + Rse = 0.5 Ω, takes 40 A at 1000 rpm. The load is reduced until the current is 20 A. Assume an unsaturated magnetic circuit (Φ ∝ I).

1. Eb1 = 230 − 40 × 0.5 = 210 V; Eb2 = 230 − 20 × 0.5 = 220 V.
2. Speed ratio: N2/N1 = (Eb2/Eb1)·(Φ1/Φ2) = (220/210) × (40/20) = 2.095.
3. New speed N2 = 2095 rpm — halving the load current roughly doubles the speed.
4. Torque ratio: T2/T1 = (I2/I1)² = (20/40)² = 0.25, so the torque falls to a quarter.

Common mistakes

• Using the line current in place of the armature current for a shunt motor.
• Writing Eb = V + Ia·Ra (that is the generator equation).
• Treating flux as constant in a series motor; it follows the armature current.
• Computing torque from output power with ω in rpm instead of rad/s (factor 9.55 error).
• Confusing gross (electromagnetic) torque with shaft torque.
• Thinking a starter "raises the voltage gradually" — a resistance starter limits current by series resistance.
• Starting a series motor on no load.

For GATE IN

Expect speed and torque calculations using Eb = V − Ia·Ra and N ∝ Eb/Φ for shunt and series motors, starting-resistance calculations, and MCQs matching motor type to characteristic and application. DC servo motor transfer functions (Kt = Ka·Φ, Kb = Ka·Φ; numerically equal in SI units) are a common link to the Control Systems section.

Quick check

1. A shunt motor takes Ia = 40 A from 250 V with Ra = 0.25 Ω. What is Eb?
2. Why is a series motor never run on no load?
3. Torque of a series motor at 30 A is 60 N·m. What is it at 15 A (unsaturated)?
4. What does the no-volt release coil of a starter do?

Answers: 1. 240 V. 2. Flux becomes very small, so speed rises dangerously. 3. 15 N·m. 4. Holds the starter arm in the run position; if supply (or, in a three-point starter, field current) fails, it releases the arm back to the off position so the motor cannot restart without resistance.