--- title: "DC motor principle and torque equation" subject: "Electrical Machine" chapter: 4 key: "semester_4/electrical_machine:dc-motor-principle-torque-equation" source: https://www.bctengineer.com/q/semester_4/electrical_machine/dc-motor-principle-torque-equation site: BCT Engineering --- # DC motor principle and torque equation Electrical Machine · Chapter 4: DC Motor - Asked in: 2069 Bhadra - Video: [Torque equation of DC Motor](https://www.youtube.com/watch?v=NbQr0Xf2T_Y) — Chandan Bobby - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machine/dc-motor-principle-torque-equation - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-4 ## Question Explain the working principle of a d.c. motor and derive the equation of torque developed by the armature of the d.c. motor. ## Short answer A current-carrying conductor in a magnetic field experiences force F=BIl, producing torque on the armature; commutation keeps rotation unidirectional. Torque developed is T_a = (PZ/(2πA))Φ I_a = 0.159 (PZ/A) Φ I_a N·m, derived from force per conductor and total conductors. ## Answer #### Working principle A DC motor converts electrical energy into mechanical energy. Its action rests on this principle: **a current-carrying conductor placed in a magnetic field experiences a force**: F = B I l (newton), whose direction is found by Fleming's left-hand rule. 1. The field poles produce flux Φ in the air-gap. 2. The armature conductors, connected to the DC supply through brushes and commutator, carry current. 3. Each conductor experiences a force. The conductors under the N pole are pushed one way and those under the S pole the other way, which produces a torque on the armature and it rotates. 4. The commutator reverses the current in a conductor as it passes from one pole to the next, so the torque always acts in the same direction. 5. As the armature rotates, a back emf E_b is induced; the supply voltage must balance it: V = E_b + I_aR_a. ```text N pole S pole ( x ) force up (.) force down conductor current in conductor current out -> torque turns armature ``` #### Torque equation Let r = armature radius (m), l = active length of a conductor (m), Z = total conductors, A = parallel paths, I_a = armature current, P = poles, Φ = flux per pole. - Current in each conductor: I_a / A - Force on each conductor: F = B l (I_a/A); B = average flux density = PΦ / (2πrl) - Torque by one conductor: F × r - Total torque (Z conductors): T_a = Z × B l (I_a/A) × r = Z (PΦ/2πrl) l (I_a/A) r **T_a = (PZ / 2πA) Φ I_a newton-metre** = 0.159 (PZ/A) Φ I_a N·m So T_a = KΦI_a, with K = PZ/(2πA). Torque is proportional to flux and armature current. Check using power: E_b I_a = T_a ω, with E_b = PΦZN/(60A) and ω = 2πN/60, giving T_a = PΦZI_a/(2πA), which matches. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. On which principle does a DC motor work? A. A current-carrying conductor in a magnetic field experiences a force B. Emf induced when flux changes C. Eddy currents in the armature D. Residual magnetism Answer: A. Force F = BIl, direction by Fleming's left-hand rule. (Key point: Principle) 2. What does the commutator do in a motor? A. Reduces the supply voltage B. Reverses current in a conductor as it passes between poles, so torque stays in one direction C. Produces flux D. Stops the armature Answer: B. Without reversal, the torque would alternate. (Key point: Commutator) 3. What is the torque equation? A. T_a = (PZ/60A) Φ N B. T_a = PΦ/(2πZ) C. T_a = (PZ/2πA) Φ I_a D. T_a = ΦI_a/PZ Answer: C. Torque is proportional to flux and armature current. 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