FOC on a BLDC: Feeding Sine Currents to a Trapezoidal Machine
July 12, 2026 · Mihna Neerulpan · views
A BLDC and a PMSM are the same class of machine - both are permanent-magnet synchronous motors. The only engineering difference is the shape of the back-EMF their windings generate - trapezoidal-ish or sinusoidal-ish. And the only question that matters when you pick a control strategy is not what the machine is called, but whether the current waveform you feed it matches the EMF waveform it generates. This post is about what happens when they don’t match - specifically, running field-oriented control (sine currents) on a trapezoidal machine - and why that mismatch is usually still the right choice.
Torque is an inner product
Everything follows from one equation. Instantaneous torque is airgap power over speed:
T = (e a·i a + e b·i b + e c·i c) / ω m
Torque is the inner product of the EMF waveform and the current waveform. Constant torque requires the three products to sum to a constant - a matching condition between machine and drive. Two pairings satisfy it exactly, on paper:
Where the 6× comes from: beat frequencies
A trapezoid is a sine plus odd harmonics - the largest by far is the 3rd (≈22% of the fundamental, versus ≈4% for the 5th), but the 3rd and its fellow triplens are about to die, leaving the 5th and 7th as the biggest survivors. Torque is a product of waveforms, and multiplying sinusoids creates sum and difference frequencies: sin(5θ)·sin(θ) contains components at 4θ and 6θ; sin(7θ)·sin(θ) at 6θ and 8θ. Then the three-phase sum acts as a filter: the 4θ and 8θ pieces arrive 120° apart across the phases and cancel; the 6θ pieces arrive in phase and reinforce. The 3rd, 9th, 15th EMF harmonics never even reach the torque: shifted by 3×120° = 360°, they are identical in all three phases - zero-sequence - and a floating neutral gives them no current path.
The rotating-frame view makes the bookkeeping effortless. Harmonic order sets rotation direction: the 7th rotates forward at 7ω, the 5th backward at 5ω. Seen from the rotor (which turns at ω), the fundamental current is DC, the 5th appears at −6ω, the 7th at +6ω: a DC vector multiplied by vectors spinning at ±6ω produces torque wobbling at 6ω. The 11th/13th pair lands at 12ω, and in general 6n±1 → 6n. Note they land at the same frequency but do not cancel each other - they are independent phasors at 6ω that generally add. At 200 Hz electrical, this is a 1.2 kHz torque wobble.
Why case C’s ‘perfect’ torque isn’t
Block commutation’s flat torque assumes phase currents that step between 0 and full value every 60°. But windings are inductors: current cannot step (v = L·di/dt would need to be infinite). Every commutation event is really a fight - one phase’s current ramping down while the next ramps up at whatever rate the available voltage allows - and during that fight the torque dips or spikes. Six commutation transients per electrical cycle, growing worse with speed (less voltage headroom to slew current against the rising back-EMF). So the practical contest is not “zero ripple vs. 15% ripple” - it is smooth, gentle 6th-harmonic ripple (FOC on the trapezoidal machine) versus theoretically-zero-plus-six-commutation-spikes (blocks on the same machine). On noise, vibration, and high-speed behavior, FOC frequently wins even on the “wrong” machine: its currents never slew violently, its ripple is a polite wobble rather than a click, and it brings along the whole FOC toolchain - field weakening, dq current limits, torque linearity.
Measure, don’t trust the datasheet
Most machines sold as “BLDC” are not very trapezoidal. True flat-top EMF requires concentrated, full-pitch windings and particular magnet arcs; plenty of production motors have EMFs much closer to sine than the name suggests. The EMF shape is a measurement, not a label: spin the machine open-terminal on a bench, scope the line-line voltage, and look. If the waveform is near-sinusoidal, FOC is simply the right control and this whole post’s tension evaporates. The drive should match the machine you measured, not the acronym on the box.
If the residual ripple actually matters: shape the current to the EMF
The matching condition never said “sine” - it said the current should be proportional to the EMF shape (which also minimizes copper loss for a given torque). A controller that knows the measured EMF harmonics can ride on top of ordinary FOC and inject deliberate 5th/7th current harmonics, phased so their 6ω torque contribution is equal-and-opposite to the EMF’s. Torque at 6ω is just a phasor sum; harmonic injection adds a third phasor that zeroes it. That is the standard technique wherever torque smoothness is a specification rather than a nicety - steering assist, control-surface actuation, gimbals - and it is the constructive reading of the 6n±1 rule: the machine’s own arithmetic funnels every imperfection to one known frequency per order, which is exactly where you aim the cure.
Takeaways
Torque is the inner product of EMF and current, so control quality is a matching question, not a naming question. FOC on a trapezoidal machine leaves a 6th-harmonic ripple of a few percent, born from the 5th/7th EMF harmonics beating against the fundamental current. Block commutation’s ideal flat torque is destroyed in practice by inductance-limited commutation transients, which is why FOC often wins anyway. Measure the EMF before believing the label; and if the residual 6× wobble violates a spec, shape the current to the measured EMF - the ripple lives at one frequency per harmonic pair, and phasor cancellation there is a solved problem.