Brushless DC Motors

i Learning Objectives

By the end of this chapter you should be able to:

Explain how a brushless DC (BLDC) motor replaces the mechanical commutator with electronic switching.
Describe its construction, Hall-sensor position detection and the six-step (trapezoidal) commutation sequence.
Use the torque and EMF constants to predict speed and torque.
Compare the BLDC motor with the brushed DC motor and the PMSM.

Section 89-1

An Inside-Out DC Motor

In a brushed DC motor the field is stationary and the commutator switches the current in the rotating armature so that armature MMF always stays 90° ahead of the field. The BLDC motor turns this inside out:

Rotor

Permanent magnets (ferrite or NdFeB), surface-mounted, producing a trapezoidal back-EMF.

Stator

A three-phase winding (usually concentrated, star-connected) fed from a three-phase inverter that acts as an electronic commutator.

No brushes, no commutator. All the copper loss is in the stator, where it is easily cooled; there is no brush wear, no sparking and no speed limit set by commutation. The price is an inverter and a rotor-position sensor.

Section 89-2

Position Sensing and Six-Step Commutation

Three Hall-effect sensors, spaced 120° (electrical) apart, detect the magnet poles and give a 3-bit code that changes every 60° electrical. At any instant two phases conduct — one connected to the positive rail, one to the negative — and the third floats.

Table 89.1 — A typical six-step commutation sequence (one electrical revolution).
Electrical angle	Hall code (H_AH_BH_C)	+ rail	− rail	Floating
0°–60°	101	A	B	C
60°–120°	100	A	C	B
120°–180°	110	B	C	A
180°–240°	010	B	A	C
240°–300°	011	C	A	B
300°–360°	001	C	B	A

Each phase conducts for 120° in each half-cycle — the flat top of its trapezoidal EMF — so the product of EMF and current, and hence the torque, is nearly constant. The exact Hall-code mapping depends on sensor placement; manufacturers publish it with each motor.

Sensorless control eliminates the Hall sensors by detecting the zero crossing of the back-EMF in the floating phase. It cannot work at standstill, where there is no EMF, so the motor is started open-loop.

Section 89-3

Torque and Speed Equations

With two phases in series, the line-to-line EMF and the torque are both proportional to the same machine constant. In SI units \(K_t\) (N·m/A) equals \(K_e\) (V·s/rad):

≈

BLDC behaves like a separately excited DC motor

Linear speed–torque characteristic

\[E = K_e\omega_m, \qquad T = K_tI, \qquad V = E + IR_{LL}\]

\[\omega_m = \frac{V}{K_e} - \frac{R_{LL}}{K_eK_t}\,T\]

\(R_{LL}\) is the line-to-line resistance (two phases in series). Speed is controlled by varying the average voltage with PWM of the inverter switches.

The electrical frequency depends on the number of pole pairs \(p\): \(f_e = p\,N/60\).

1 Worked Example 89.1 — Speed and Torque of a 24 V BLDC Motor

Problem. A 24 V BLDC motor has \(K_e = K_t = 0.05\) (SI units) and \(R_{LL} = 0.6~\Omega\). Find (a) the ideal no-load speed and (b) the torque and speed when the motor draws 3 A. (c) If the motor has 8 poles, what is the commutation frequency at 3000 rev/min?

\[\text{(a)}\quad \omega_0 = \frac{24}{0.05} = 480~\text{rad/s} = 4584~\text{rev/min}\]

\[\text{(b)}\quad T = 0.05(3) = 0.15~\text{N·m}, \qquad \omega = \frac{24 - 3(0.6)}{0.05} = 444~\text{rad/s} = 4240~\text{rev/min}\]

\[\text{(c)}\quad f_e = \frac{4 \times 3000}{60} = 200~\text{Hz}, \qquad 6 \times 200 = 1200~\text{commutations/s}\]

Section 89-4

BLDC, Brushed DC and PMSM Compared

Table 89.2 — Comparison of permanent-magnet motors.
Feature	Brushed DC	BLDC	PMSM (Chapter 90)
Commutation	Mechanical	Electronic, six-step	Electronic, sinusoidal
Back-EMF shape	DC	Trapezoidal	Sinusoidal
Phase current	—	Quasi-square, 120°	Sinusoidal
Position sensing	Not needed	3 Hall sensors (coarse)	Encoder/resolver (fine)
Torque ripple	Low–medium	Commutation ripple	Very low
Maintenance	Brush wear	None	None

Applications

Ceiling fans (energy-efficient "BLDC fans"), computer cooling fans and drives, drones, e-bikes and e-scooters, pumps, HVAC blowers, power tools.

Limitations

Cost of the inverter and magnets; torque ripple at each commutation; risk of magnet demagnetisation at high temperature or fault current.

Section 89-5

Summary and Common Mistakes

Key Points

BLDC = PM rotor + three-phase stator + inverter acting as commutator.
Hall sensors give six states per electrical cycle; two phases conduct at a time.
\(T = K_tI\), \(E = K_e\omega\), and \(K_t = K_e\) in SI units — DC-motor-like behaviour.

Common Mistakes

Using phase resistance instead of line-to-line resistance in the speed equation.
Mixing up mechanical and electrical degrees: Hall states change every 60° electrical.
Quoting \(K_e\) in V/krpm and \(K_t\) in N·m/A and expecting them to be numerically equal.

Section 89-6

Chapter Review

Practice Problems

P89.1 A 48 V BLDC motor has \(K_e = K_t = 0.1\) SI and \(R_{LL} = 0.4~\Omega\). Find its stall torque and no-load speed.

Show answer
Stall current \(48/0.4 = 120\) A; \(T_\text{stall} = 12\) N·m. \(\omega_0 = 480\) rad/s = 4584 rev/min.
P89.2 A motor's EMF constant is quoted as 10 V per 1000 rev/min. Convert it to SI units and state \(K_t\).

Show answer
1000 rev/min = 104.7 rad/s; \(K_e = 10/104.7 = 0.0955\) V·s/rad, so \(K_t = 0.0955\) N·m/A.

Multiple-Choice Questions

MCQ 1. In six-step commutation, the number of phases conducting at any instant is:
(a) 1 (b) 2 (c) 3 (d) 6

Show answer
(b)
MCQ 2. The back-EMF of a BLDC motor is ideally:
(a) sinusoidal (b) trapezoidal (c) square (d) triangular

Show answer
(b)

Conceptual Questions

Why does sensorless BLDC control fail at standstill?
Why is it easier to cool a BLDC motor than a brushed DC motor?
What causes torque ripple at each commutation instant?

Looking Ahead

Shape the magnets for a sinusoidal EMF, drive the phases with sinusoidal currents and measure rotor position finely, and torque ripple almost disappears. Chapter 90 develops this machine — the permanent-magnet synchronous motor — with the d–q model used for its control.