BLDC motor torque equation and characteristics

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 $V = E + IR = k_e\omega + \dfrac{R\,T}{k_t}$:

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

 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.