---
title: "BLDC motor torque equation and characteristics"
subject: "Electric Machines II"
chapter: 3
key: "semester_5/electric_machines_ii:bldc-torque-equation"
source: https://www.bctengineer.com/q/semester_5/electric_machines_ii/bldc-torque-equation
site: BCT Engineering
---

# BLDC motor torque equation and characteristics

Electric Machines II · Chapter 3: Fractional Kilowatt Motors

- Asked in: 2083 Baisakh (new course), 3 marks
- Video: [BLDC Torque Equation derivation](https://www.youtube.com/watch?v=-JM0aNx4Vfc) — Knowledge Share to delete ignorance
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machines_ii/bldc-torque-equation
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machines_ii/chapter-3

## Question

Derive the torque equation of permanent magnet BLDC motor. And also explain torque-speed characteristics of the same motor.

## Short answer

With two phases conducting in series, line emf is E = 4NBlr ω = ke ω. Power P = EI = ke ω I, so electromagnetic torque T = P/ω = ke I = kt I. The torque-speed curve is a straight drooping line from no-load speed ω0=V/ke to stall torque Tstall=ktV/R, with continuous and intermittent operating zones.

## Answer

A permanent-magnet **BLDC motor** has magnets on the rotor and a 3-phase stator winding switched electronically; with trapezoidal (120° conduction) drive, **two phases conduct at a time** in series.

#### Torque equation
Each conductor of length $l$ at radius $r$ in a gap flux density $B$ has emf $e = Blv = Blr\omega$. With $N$ turns ($2N$ conductors) per phase:

$$
\begin{aligned}
E_{ph} &= 2NBlr\,\omega \\
E_{line} &= 2E_{ph} = 4NBlr\,\omega = k_e\,\omega \quad (\text{two phases in series}) \\
P &= E_{line} I = 4NBlr\,\omega I \\
T &= \frac{P}{\omega} = 4NBlr\,I = k_t I
\end{aligned}
$$

Torque is proportional to current, and $k_t = k_e$ in SI units, as in a PM DC motor.

#### Torque–speed characteristic
From $V = E + IR = k_e\omega + \dfrac{R\,T}{k_t}$:

$$
\omega = \frac{V}{k_e} - \frac{R}{k_e k_t}T
$$

```text
 speed
  ^ w0 = V/ke (no load)
  |\
  |  \   continuous | intermittent
  |    \   zone     |    zone
  |      \          |
  +--------\--------+----> T
                 stall T = kt V / R
```

- Straight drooping line: speed falls slightly as torque rises (like a shunt DC motor).
- Rated operation lies in the **continuous zone**; the intermittent zone (up to about 2× rated torque) is limited by heating.

## Check yourself

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

1. In the derivation, why is the line emf taken as 2 x E_ph?
   A. Because the winding is delta connected
   B. Because two phases conduct at a time and are in series
   C. Because the flux density doubles
   D. Because there are two magnets
   Answer: B. With 120° conduction, two phases are in series at any instant, so E_line = 2E_ph = 4NBlr w. (Key point: Line emf with two phases conducting)

2. From P = E_line x I and T = P/w, the torque is:
   A. T = 4NBlr w I
   B. T = 4NBlr I = kt I
   C. T = 2NBlr I^2
   D. T = kt w
   Answer: B. Dividing P = 4NBlr w I by w removes the speed, so torque is proportional to current only: T = kt I. (Key point: Torque expression)

3. What does the torque-speed characteristic of a BLDC motor look like?
   A. A rising curve; speed increases with torque
   B. A straight drooping line from no-load speed V/ke
   C. A constant-speed vertical line
   D. A hyperbola like a series motor
   Answer: B. From w = V/ke - (R/(ke kt))T, speed falls linearly with torque, like a shunt DC motor. (Key point: Torque-speed characteristic)

4. What limits the intermittent operating zone to about 2x rated torque?
   A. Magnet saturation
   B. Heating
   C. Brush wear
   D. Slip
   Answer: B. Rated operation is the continuous zone; beyond it, the motor can work only for short periods because of heating. (Key point: Operating zones)

## Related in Electric Machines II

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- [Capacitor start capacitor run motor characteristics](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Acscr-speed-torque) — Electric Machines II, chapter 3, asked 2×
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- [Shaded pole motor principle and characteristics](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Ashaded-pole-characteristics) — Electric Machines II, chapter 3
- [Single phase IM principle, torque-slip and starting](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asingle-phase-principle-torque-slip) — Electric Machines II, chapter 3

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