Formula sheet

Everything the site derives, in one place, with this site's conventions applied: amplitude-invariant scaling, per-phase parameters, d along the rotor flux.

The worked numbers beside them are computed from src/core when this page is built, so they are the same values the figures use rather than a copy that can drift.

Download the printable sheet PDF, two pages, 217 KB — typeset from this same source, so it carries the same numbers.

Frames

Clarke, amplitude-invariant

i_{α} = \frac{2}{3} (i_{a} - \frac{1}{2} i_{b} - \frac{1}{2} i_{c}), i_{β} = \frac{1}{3} (i_{b} - i_{c})

The 2/3 keeps the peak. With a floating neutral the third current is redundant.

Park

i_{d} = i_{α} cos θ_{e} + i_{β} sin θ_{e}, i_{q} = - i_{α} sin θ_{e} + i_{β} cos θ_{e}

A rotation, so it preserves magnitude and the dot product — and therefore power.

Power

p = \frac{3}{2} (v_{d} i_{d} + v_{q} i_{q})

The 3/2 is the amplitude-invariant convention, not a fudge.

The machine

dq voltage equations

v_{d} = R_{s} i_{d} + L_{d} \frac{d i _{d}}{d t} - ω_{e} L_{q} i_{q} v_{q} = R_{s} i_{q} + L_{q} \frac{d i _{q}}{d t} + ω_{e} (L_{d} i_{d} + λ_{m})

The cross terms are what decoupling cancels; the $λ_{m}$ term is the back-EMF.

Torque

T_{e} = \frac{3}{2} p [λ_{m} i_{q} + (L_{d} - L_{q}) i_{d} i_{q}]

Magnet term plus reluctance term. On this machine $L_{q} > L_{d}$ , so negative $i_{d}$ adds torque. Torque constant 0.051 N·m/A at $i_{d} = 0$ .

saliency $L_{q} / L_{d}$ = 1.57

Characteristic current

i_{c h} = - \frac{λ _{m}}{L _{d}}

Decides the whole field-weakening envelope. A machine whose rated current reaches it can be weakened indefinitely.

70.8 A on the reference machine

Back-EMF constant

K_{e} = \frac{3}{2} λ_{m} ω_{e, 1000}

Line-to-line RMS per 1000 rpm — the form a datasheet quotes. Three conversions stand between it and $λ_{m}$ .

The inverter

Linear voltage limit

∣ v ∣_{ma x} = \frac{V _{d c}}{3}

The inscribed circle of the hexagon. SPWM reaches only $V_{d c} /2$ — 15.5% less.

27.7 V against 24 V at 48 V

SVPWM dwell times

t_{1} = \frac{3 ∣ v ∣}{V _{d c}} sin (\frac{π}{3} - γ), t_{2} = \frac{3 ∣ v ∣}{V _{d c}} sin γ t_{0} = 1 - t_{1} - t_{2}

γ is the angle within the sector. The switching order is decided by the switch words, not by which vector is trailing.

Deadtime voltage error

Δ v = \frac{t _{d}}{T _{s}} V_{d c} sgn (i)

Sign follows the current, so it is a square wave at the electrical frequency — 6× ripple in dq.

Bridge loss

P = 3 I_{r m s}^{2} R_{d s (o n)} + 3 f_{pw m} V_{d c} \overline{∣ i ∣} t_{s w}

Square-law loss takes the RMS; linear loss takes the mean of the absolute sine, 0.6366. A lower $R_{d s (o n)}$ moves the crossover down.

Control

Current-loop gains

k_{p} = ω_{c} L, k_{i} = ω_{c} R_{s}

Pole-zero cancellation leaves a pure integrator, so the closed loop is first order at $ω_{c}$ .

$k_{p}$ = 1.319, $k_{i}$ = 2199 for 1 kHz on the d axis

Anti-windup tracking gain

k_{a w} = \frac{k _{i}}{k _{p}} = \frac{R _{s}}{L}

Back-calculation. Too low and the integrator still runs away, just slower.

Phase margin from delay

PM = 9 0^{\circ} - ω_{c} T_{d}

The whole stability story of a well-tuned current loop. A timing property, not a gain property.

75 µs of delay gives 63° at 1 kHz

Angle prediction

θ_{o u t} = θ_{m e a s u r e d} + ω_{e} T_{d}

Removes a speed-dependent rotation entirely. Does nothing for phase margin.

Cascade separation

\frac{ω _{inn er}}{ω _{o u t er}} \geq 5 to 10

10× costs nothing, 5× starts to cost, under 3× doubles overshoot.

Sensing

Low-side measurement window

t_{w in d o w} = (1 - d) T_{s}

Closes as duty approaches 1. The phase with the largest duty fails first.

Converter resolution

q = \frac{2 I _{f s}}{2 ^{N}}, σ = \frac{q}{12}

In dq the RMS is $2/3$ of the per-phase figure, and the peak bound is $2/3$ of a code — not a half.

Encoder speed step

Δ ω = \frac{2 π}{CPR \cdot T}

Independent of speed, so it grows as a fraction of the measurement as the machine slows.

Link capacitor current

I_{c a p} = I_{r m s}^{2} - I_{m e an}^{2}

Exact, not approximate — mean and ripple are orthogonal. Peaks near half modulation depth.