---
title: "DC motor working principle and torque equation"
subject: "Electrical Machines"
chapter: 4
key: "semester_4/electrical_machines:dc-motor-working-torque-equation"
source: https://www.bctengineer.com/q/semester_4/electrical_machines/dc-motor-working-torque-equation
site: BCT Engineering
---

# DC motor working principle and torque equation

Electrical Machines · Chapter 4: DC Motor

- Asked in: 2078 Chaitra, 4+4 marks; 2069 Bhadra (asked 2 times)
- 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_machines/dc-motor-working-torque-equation
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/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 DC motor works on the principle that a current-carrying conductor in a magnetic field experiences a force (F = B I l, Fleming’s left-hand rule). The commutator ensures unidirectional torque. Torque equation: T = (Z P Φ I_a) / (2π A) = K_a Φ I_a, where K_a = ZP/(2πA).

## Answer

#### Working principle

A DC motor converts electrical energy into mechanical energy. It works on the principle that **a current-carrying conductor placed in a magnetic field experiences a mechanical force**:

$$
F = BIl\ \text{newton}
$$

The direction of the force is given by Fleming's left-hand rule (forefinger = field, middle finger = current, thumb = force).

```text
       N                       S
   +--------+             +--------+
   |   ->   |   (x) F up  |        |
   |  field |  (.) F down |        |
   +--------+   armature  +--------+
              conductors
```

1. The field winding produces flux $\Phi$ in the air gap.
2. Current from the supply enters the armature conductors through the brushes and commutator.
3. Conductors under the N pole carry current in one direction and those under the S pole in the opposite direction, so forces on both sides act in the same sense and produce a **torque** that rotates the armature.
4. The commutator reverses the current in a conductor as it passes from one pole to the next, so the torque stays **unidirectional**.
5. The rotating conductors cut flux and induce a **back emf** $E_b$ opposing the supply, so $V = E_b + I_aR_a$.

#### Torque equation

Let $\Phi$ = flux per pole, $Z$ = armature conductors, $P$ = poles, $A$ = parallel paths, $I_a$ = armature current, $r$ = armature radius, $l$ = effective length.

Current per conductor $= I_a/A$. Force on one conductor $F = B\,l\,\dfrac{I_a}{A}$. Torque of one conductor $= F\times r$.

Total torque of $Z$ conductors, with average flux density $B = \dfrac{P\Phi}{2\pi r l}$:

$$
T_a = Z\cdot B\,l\,\frac{I_a}{A}\cdot r = Z\cdot\frac{P\Phi}{2\pi r l}\cdot l\cdot\frac{I_a}{A}\cdot r
$$

$$
T_a = \frac{ZP}{2\pi A}\,\Phi I_a = 0.159\,\frac{ZP}{A}\,\Phi I_a\ \text{N·m}
$$

Writing $K_a = \dfrac{ZP}{2\pi A}$:

$$
T_a = K_a\Phi I_a
$$

Equivalent form from power: $T_a\omega = E_bI_a$ with $E_b = \dfrac{\Phi ZNP}{60A}$ gives the same result. For a shunt motor ($\Phi$ constant) $T \propto I_a$; for an unsaturated series motor ($\Phi\propto I_a$) $T \propto I_a^2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A dc motor works on the principle that:
   A. A changing flux induces an emf in a stationary coil
   B. Charges repel each other
   C. A current-carrying conductor in a magnetic field experiences a force
   D. Voltage is proportional to current
   Answer: C. F = BIl. (Key point: Principle)

2. The commutator ensures:
   A. Reduced back emf
   B. Constant flux
   C. Lower armature resistance
   D. Unidirectional torque by reversing conductor current under each pole change
   Answer: D. Without reversal the torque would alternate. (Key point: Commutator role)

3. The torque developed by the armature is:
   A. Ta = (ZP / 2 pi A) Phi Ia
   B. Ta = Phi Ia / 2 pi
   C. Ta = ZNP Phi / 60A
   D. Ta = Phi Ia^2
   Answer: A. Ta = Ka Phi Ia with Ka = ZP/(2 pi A). (Key point: Torque equation)

4. For an unsaturated series motor, torque is proportional to:
   A. Ia
   B. Ia^2
   C. 1/Ia
   D. constant
   Answer: B. Flux is proportional to Ia, so T is proportional to Phi Ia = Ia^2. (Key point: Torque dependence)

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