Table of Contents

Overview
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A Brushless DC (BLDC) motor replaces the mechanical brushes and commutator of a traditional DC motor with electronic commutation. This eliminates the primary failure point (brush wear) while providing higher efficiency, better torque-to-weight ratio, and longer lifespan.

However, removing the mechanical commutator means the controller must actively manage which motor coils are energized at any given moment. This post covers the fundamentals of BLDC motor operation and progressively builds up to advanced precision control techniques.

Brushed DC Motor:                    BLDC Motor:
┌─────────────────┐                 ┌─────────────────┐
│                 │                 │                 │
│  Voltage ──→ Brushes ──→ Rotor   │  Voltage ──→ Controller ──→ Stator
│             (mechanical)         │             (electronic)
│                 │                 │                 │
│  Simple!        │                 │  Complex, but:  │
│  Self-commutates│                 │  - No brush wear│
│                 │                 │  - Higher power  │
│                 │                 │  - Precise ctrl  │
└─────────────────┘                 └─────────────────┘

1. BLDC Motor Construction
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1.1 Physical Structure
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A BLDC motor consists of a permanent magnet rotor and a wound stator with three phases (A, B, C):

        Stator (fixed, with coils)
    ┌──────────────────────────────┐
    │         ┌─────┐              │
    │    C ───┤     ├─── A        │
    │         │  N  │              │
    │         │  ↑  │  Rotor      │
    │         │  S  │  (rotating  │
    │    B ───┤     ├─── magnets) │
    │         └─────┘              │
    └──────────────────────────────┘

    Three-phase winding: A, B, C
    Rotor: Permanent magnets (N-S poles)

1.2 Electrical Model
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Each phase can be modeled as an inductor (\(L\)), resistor (\(R\)), and back-EMF source (\(e\)) in series:

$$ V_a = R \cdot i_a + L \frac{di_a}{dt} + e_a $$$$ V_b = R \cdot i_b + L \frac{di_b}{dt} + e_b $$$$ V_c = R \cdot i_c + L \frac{di_c}{dt} + e_c $$

The back-EMF is proportional to rotor speed:

$$ e = K_e \cdot \omega $$

where \(K_e\) is the back-EMF constant and \(\omega\) is the angular velocity.

1.3 Torque Production
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Torque is produced by the interaction between stator current and rotor magnetic field:

$$ \tau = K_t \cdot i $$

where \(K_t\) is the torque constant. In a three-phase BLDC:

$$ \tau = K_t (i_a \cdot f_a(\theta) + i_b \cdot f_b(\theta) + i_c \cdot f_c(\theta)) $$

where \(f(\theta)\) is the back-EMF waveform shape as a function of rotor angle \(\theta\).

2. The Inverter: Power Electronics
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The motor is driven by a three-phase inverter (also called a 6-switch bridge):

          DC Bus (V_DC)
    ──────────┬──────────┬──────────
              │          │          │
           ┌──┴──┐   ┌──┴──┐   ┌──┴──┐
           │ Q1  │   │ Q3  │   │ Q5  │   ← High-side MOSFETs
           └──┬──┘   └──┬──┘   └──┬──┘
              │          │          │
              ├── A      ├── B     ├── C   ← Motor phases
              │          │          │
           ┌──┴──┐   ┌──┴──┐   ┌──┴──┐
           │ Q2  │   │ Q4  │   │ Q6  │   ← Low-side MOSFETs
           └──┬──┘   └──┬──┘   └──┬──┘
              │          │          │
    ──────────┴──────────┴──────────┴──── GND

Each motor phase is connected between a high-side and low-side MOSFET (or IGBT). By turning specific switches on/off, the controller determines which phases are energized and in which direction.

Critical rule: Never turn on both MOSFETs of the same leg simultaneously — this creates a shoot-through short circuit that can destroy the inverter.

3. Commutation: The Fundamental Challenge
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3.1 Six-Step (Trapezoidal) Commutation
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The simplest BLDC control method. At any instant, only two of three phases are active — one sourcing current, one sinking, one floating:

Step 1: A+ B-    Step 2: A+ C-    Step 3: B+ C-
Step 4: B+ A-    Step 5: C+ A-    Step 6: C+ B-

Phase Voltages:
     A:  ┌──┐         ┌──┐
─────┘   └────────────┘   └─────────
     B:        ┌──┐         ┌──┐
─────────────┘   └────────────┘   └──
     C:  ──┐         ┌──┐         ┌─
───────┘   └────────┘   └────────┘

Each step = 60° electrical rotation
Full cycle = 6 steps = 360° electrical

3.2 Rotor Position Detection
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To commutate correctly, the controller must know the rotor position. Three main approaches:

Hall Sensors (Most Common for Six-Step):

Three Hall sensors placed 120° apart on the stator:

Hall A  Hall B  Hall C  │  Step  │  Active Phases
   1       0       1    │   1    │  A+ B-
   1       0       0    │   2    │  A+ C-
   1       1       0    │   3    │  B+ C-
   0       1       0    │   4    │  B+ A-
   0       1       1    │   5    │  C+ A-
   0       0       1    │   6    │  C+ B-

The 3 Hall sensors provide 6 unique combinations — one per commutation step. Resolution: 60° electrical.

Encoder (For precision):

  • Incremental: Provides relative position via A/B quadrature signals
  • Absolute: Provides exact position at power-on
  • Typical resolution: 1,000–10,000 CPR (counts per revolution)

Sensorless (Back-EMF Zero-Crossing):

  • Monitors the floating phase’s back-EMF
  • Detects zero-crossing to determine rotor position
  • Does not work at zero/low speed (no back-EMF generated)

4. PWM Speed Control
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Speed is controlled by varying the duty cycle of a PWM signal applied to the active switches:

100% Duty Cycle:     50% Duty Cycle:      25% Duty Cycle:
┌────────────────┐   ┌────┐    ┌────┐     ┌──┐      ┌──┐
│                │   │    │    │    │     │  │      │  │
│    V_DC        │   │    │    │    │     │  │      │  │
│                │   │    │    │    │     │  │      │  │
└────────────────┘   └────┘    └────┘     └──┘      └──┘

V_avg = V_DC         V_avg = 0.5·V_DC     V_avg = 0.25·V_DC

Speed ∝ V_avg (approximately, at steady state)

PWM frequency is typically 10–100 kHz — fast enough that the motor’s inductance smooths the current into a near-DC waveform.

5. Field-Oriented Control (FOC)
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Six-step commutation is simple but produces torque ripple because only two phases are active at a time, and the current waveform is trapezoidal. For precision applications, Field-Oriented Control (FOC) — also known as vector control — is the gold standard.

5.1 Core Idea
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FOC transforms the three-phase AC problem into a DC control problem using coordinate transformations. The goal: independently control torque-producing current and flux-producing current.

Three-Phase World (complex):        d-q Frame (simple):
┌──────────────────────┐           ┌──────────────────────┐
│  i_a(t) = sin(ωt)    │           │  i_d = constant (DC) │
│  i_b(t) = sin(ωt-120)│  ──→     │  i_q = constant (DC) │
│  i_c(t) = sin(ωt+120)│  Park    │                      │
│                      │  Transform│  Much easier to       │
│  AC signals,         │           │  control with PID!    │
│  time-varying        │           │                      │
└──────────────────────┘           └──────────────────────┘

5.2 The Transformation Chain
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FOC uses two coordinate transformations:

         Clarke              Park
  a,b,c ──────→ α,β ──────→ d,q
 (3-phase)     (2-phase     (rotating
               stationary)   frame, DC)

Step 1: Clarke Transform (3-phase → 2-phase stationary)

$$ \begin{bmatrix} i_\alpha \\\\ i_\beta \end{bmatrix} = \frac{2}{3}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\\\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i_a \\\\ i_b \\\\ i_c \end{bmatrix} $$

This maps three phase currents to a two-axis stationary reference frame.

Step 2: Park Transform (stationary → rotating)

$$ \begin{bmatrix} i_d \\\\ i_q \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \\\\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} i_\alpha \\\\ i_\beta \end{bmatrix} $$

where \(\theta\) is the electrical angle of the rotor. This rotates the reference frame to align with the rotor, converting AC signals to DC values.

5.3 The d-q Current Components
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After the Park transform:

  • \(i_d\) (direct axis): Controls magnetic flux. For surface-mount PM motors, set \(i_d = 0\) to maximize efficiency (no need to strengthen or weaken the permanent magnet field).
  • \(i_q\) (quadrature axis): Controls torque. Torque is directly proportional to \(i_q\):
$$ \tau = \frac{3}{2} \cdot p \cdot \lambda_m \cdot i_q $$

where \(p\) is the number of pole pairs and \(\lambda_m\) is the permanent magnet flux linkage.

5.4 The Complete FOC Loop
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                    ┌─────────────────────────────────────┐
                    │           FOC Control Loop            │
                    │                                       │
  Speed Ref ──→ [Speed PI] ──→ i_q_ref ──→ [Current PI] ──┤
                                                           │
  i_d_ref = 0 ─────────────────────────→ [Current PI] ──┤
                                                           │
                    ┌──────────────────────────────────────┘
                    │
                    ▼
              Inverse Park      Inverse Clarke      PWM
  V_d, V_q ──────────→ V_α, V_β ──────────→ V_a, V_b, V_c ──→ Inverter
                    ↑
              [Park Transform]
                    ↑
              [Clarke Transform]
                    ↑
              i_a, i_b, i_c ←── Current Sensors
                    ↑
              θ (rotor angle) ←── Encoder / Sensorless Estimator

The loop runs at 10–40 kHz (current loop), with the speed loop typically running 10x slower.

5.5 Space Vector Modulation (SVM)
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Instead of simple sinusoidal PWM, FOC typically uses Space Vector Modulation to maximize DC bus utilization:

Sinusoidal PWM:                  Space Vector PWM:
Max output = V_DC / 2            Max output = V_DC / √3 ≈ 0.577 · V_DC

SVM achieves ~15% more voltage utilization than sinusoidal PWM

SVM treats the three-phase inverter as producing 8 possible voltage vectors (6 active + 2 zero) and synthesizes any desired output voltage by time-averaging adjacent vectors within each PWM period.

6. Precision Techniques
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6.1 Current Sensing
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Accurate current measurement is fundamental to precision control. Three main approaches:

Low-Side Shunt Resistors:         Phase Shunt Resistors:
┌──────────────────┐              ┌──────────────────┐
│ Q1    Q3    Q5   │              │ Q1    Q3    Q5   │
│  │     │     │   │              │  │     │     │   │
│  ├─A   ├─B   ├─C │              │  ├─A   ├─B   ├─C │
│  │     │     │   │              │  R     R     R   │
│ Q2    Q4    Q6   │              │ Q2    Q4    Q6   │
│  │     │     │   │              │  │     │     │   │
│  R     R         │              └──┴─────┴─────┴───┘
│  │     │         │
└──┴─────┴─────────┘              Pros: Full 3-phase measurement
                                  Cons: More components, routing
Pros: Cheap, simple
Cons: Only 2 of 3 phases directly
      (3rd computed: i_a+i_b+i_c=0)

For highest precision, isolated current sensors (e.g., hall-effect based ACS712, or sigma-delta modulator based) provide galvanic isolation and bandwidth up to several hundred kHz.

6.2 Dead-Time Compensation
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When switching MOSFETs, a dead time (typically 0.5–2 \(\mu\)s) is inserted to prevent shoot-through. This dead time causes voltage distortion and torque ripple:

Ideal PWM:        Actual (with dead time):
┌────────┐        ┌────────┐
│ Q1 ON  │        │ Q1 ON  │
└────────┘        └───┐    │  ← Dead time Δt
┌────────┐            └────┘
│ Q2 ON  │            ┌────────┐
└────────┘            │ Q2 ON  │
                      └────────┘

Voltage error per PWM cycle = ±V_DC · (Δt / T_PWM)

Compensation: Measure or estimate current direction, then add/subtract the dead-time voltage error from the PWM reference:

$$ V_{comp} = \text{sign}(i_{phase}) \cdot V_{DC} \cdot \frac{\Delta t}{T_{PWM}} $$

6.3 Flux Weakening
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To operate above the rated speed, the back-EMF exceeds the available bus voltage. Flux weakening injects negative \(i_d\) current to counteract the permanent magnet flux:

$$ i_d = -\frac{\lambda_m - \sqrt{V_{max}^2 / \omega^2 - (L_q i_q)^2}}{L_d} $$
Torque vs Speed:
τ │
  │  ┌──────────┐
  │  │ Constant  │  ┌──────────────────┐
  │  │ Torque    │  │ Flux Weakening   │
  │  │ Region    │  │ (constant power) │
  │  │           │  │                  │
  └──┴───────────┴──┴──────────────────┴──→ ω
     0         ω_base                ω_max

  Below ω_base: i_d = 0, full torque available
  Above ω_base: i_d < 0, torque decreases as 1/ω

6.4 Observer-Based Sensorless Control
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For applications where mechanical sensors are impractical (cost, size, harsh environment), sensorless FOC uses state observers to estimate rotor position from voltage and current measurements.

Back-EMF Observer:

$$ \hat{e}_\alpha = V_\alpha - R \cdot i_\alpha - L \frac{di_\alpha}{dt} $$$$ \hat{e}_\beta = V_\beta - R \cdot i_\beta - L \frac{di_\beta}{dt} $$$$ \hat{\theta} = \arctan\left(\frac{-\hat{e}_\alpha}{\hat{e}_\beta}\right) $$

Sliding Mode Observer (SMO): More robust to parameter variations:

$$ \frac{d\hat{i}_\alpha}{dt} = \frac{1}{L}(V_\alpha - R\hat{i}_\alpha - \hat{e}_\alpha) + k \cdot \text{sign}(i_\alpha - \hat{i}_\alpha) $$

The sliding mode term \(k \cdot \text{sign}(\cdot)\) forces the estimated current to converge to the actual current, and the back-EMF estimate can be extracted from the switching function.

Limitations: Sensorless methods struggle at zero and very low speeds where back-EMF is negligible. High-frequency injection (HFI) techniques can extend the operating range to standstill.

6.5 Anti-Cogging Compensation
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Permanent magnet motors exhibit cogging torque — a position-dependent reluctance torque caused by the interaction between rotor magnets and stator teeth. This causes vibration and position errors at low speeds.

Cogging Torque Profile:
τ_cog │    ╭╮    ╭╮    ╭╮    ╭╮
      │   ╱  ╲  ╱  ╲  ╱  ╲  ╱  ╲
──────┼──╱────╲╱────╲╱────╲╱────╲──→ θ
      │ ╱      ╲      ╲      ╲
      │╱        ╰╮    ╰╮    ╰╮
      │          ╰╯    ╰╯    ╰╯

Period = 360° / LCM(poles, slots)

Compensation technique: Pre-calibrate the cogging torque profile by slowly rotating the motor and recording the torque at each position. Store as a lookup table and inject compensating current during operation:

$$ i_{q,comp}(\theta) = -\frac{\tau_{cog}(\theta)}{K_t} $$

6.6 Vibration Suppression via Notch Filters
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Mechanical resonances in the drivetrain can cause instability at specific frequencies. Notch filters in the control loop attenuate these resonances:

$$ H_{notch}(s) = \frac{s^2 + 2\zeta_z \omega_n s + \omega_n^2}{s^2 + 2\zeta_p \omega_n s + \omega_n^2} $$

where \(\omega_n\) is the resonant frequency, \(\zeta_z < \zeta_p\) (the zero damping is less than the pole damping, creating a notch).

Magnitude Response:
|H| │
    │──────────────────────────────
    │              ╲    ╱
    │               ╲  ╱
    │                ╲╱   ← Notch at resonant frequency
    │
    └──────────────────────────────→ f
                     f_n

7. Control Loop Tuning
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7.1 Cascaded Loop Structure
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A precision BLDC controller typically uses three cascaded loops:

Position Reference ──→ [Position PID] ──→ Speed Reference
                                              │
                         Speed Feedback ←─────┤
                                              ▼
                       ──→ [Speed PI] ──→ i_q Reference
                                              │
                         i_q Feedback ←───────┤
                                              ▼
                       ──→ [Current PI] ──→ PWM Duty
                                              │
                         i_d,q Feedback ←─────┤
                                              ▼
                                          [Inverter + Motor]

7.2 Bandwidth Hierarchy
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Each outer loop must be slower than its inner loop (typically 5–10x) to maintain stability:

LoopBandwidthUpdate RateTuning Priority
Current1–5 kHz10–40 kHzFirst (innermost)
Speed100–500 Hz1–5 kHzSecond
Position10–50 Hz100–500 HzThird (outermost)

7.3 Current Loop Tuning
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The current loop plant model:

$$ G_{plant}(s) = \frac{1}{Ls + R} $$

With a PI controller \(G_{PI}(s) = K_p + \frac{K_i}{s}\), pole-zero cancellation gives:

$$ K_p = L \cdot \omega_{bw}, \quad K_i = R \cdot \omega_{bw} $$

where \(\omega_{bw}\) is the desired bandwidth in rad/s.

8. Practical Implementation Architecture
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A typical precision BLDC control system:

┌──────────────────────────────────────────────────────┐
│                  MCU (e.g., STM32G4)                  │
│                                                      │
│  ┌────────────┐   ┌────────────┐   ┌──────────────┐ │
│  │ ADC        │   │ Timer/PWM  │   │ Encoder      │ │
│  │ (12-16 bit)│   │ (center-   │   │ Interface    │ │
│  │ i_a, i_b   │   │  aligned)  │   │ (QEP/SPI)   │ │
│  └─────┬──────┘   └─────┬──────┘   └──────┬───────┘ │
│        │                │                  │         │
│  ┌─────┴────────────────┴──────────────────┴───────┐ │
│  │              FOC Algorithm                       │ │
│  │                                                  │ │
│  │  1. Read currents (ADC)                         │ │
│  │  2. Read position (Encoder)                     │ │
│  │  3. Clarke transform (abc → αβ)                 │ │
│  │  4. Park transform (αβ → dq)                    │ │
│  │  5. PI current controllers (d, q)               │ │
│  │  6. Inverse Park (dq → αβ)                      │ │
│  │  7. SVM (αβ → PWM duties)                       │ │
│  │  8. Update PWM registers                        │ │
│  │                                                  │ │
│  │  All within one PWM cycle (~25-100 μs)          │ │
│  └──────────────────────────────────────────────────┘ │
│                                                      │
│  Communication: CAN / EtherCAT / UART               │
└──────────────────────────────────────────────────────┘
         │              │              │
    ┌────┴────┐   ┌────┴────┐   ┌────┴────┐
    │ Gate    │   │ Current │   │ Encoder │
    │ Driver  │   │ Sensors │   │         │
    └────┬────┘   └─────────┘   └─────────┘
         │
    ┌────┴────┐
    │ 3-Phase │
    │ Inverter│
    └────┬────┘
         │
    ┌────┴────┐
    │  BLDC   │
    │  Motor  │
    └─────────┘

Timing Requirement
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The entire FOC computation must complete within one PWM period. At 20 kHz PWM, that is 50 microseconds for:

  • 2–3 ADC conversions
  • Clarke + Park transforms
  • 2 PI controllers
  • Inverse transforms
  • SVM calculation
  • PWM register update

This is why FOC is typically implemented on dedicated motor control MCUs (STM32G4, TI C2000, Infineon XMC) with hardware-accelerated math peripherals.

9. Summary
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BLDC Control Techniques — Complexity vs Performance:

Performance │
            │                          FOC + Anti-cogging
            │                       ●  + Dead-time comp
            │                    ●     + Flux weakening
            │                 ●
            │              FOC (Sinusoidal)
            │           ●
            │        ●  Six-Step + Current Control
            │     ●
            │  Six-Step (Trapezoidal)
            │●
            └──────────────────────────────────→ Complexity
TechniqueTorque RippleSpeed RangePosition ControlComplexity
Six-StepHigh (~15%)LimitedNoLow
Six-Step + CurrentMedium (~10%)LimitedNoMedium
FOC (basic)Low (~2%)Full rangeYesHigh
FOC + advancedVery low (<1%)Extended (flux weakening)Sub-degreeVery High

For robotics applications like joint actuators in humanoid robots or precision manipulators, FOC with encoder feedback, anti-cogging compensation, and cascaded position/speed/current control is the standard approach — providing the smooth, precise, and responsive motion that modern robotic systems demand.