Notation

Every page here uses the same symbols and the same conventions. This is the list, and the parameters of the machine the figures run on.

Conventions, fixed everywhere

These hold on every page. A page that needs a different one says so in its own text.

Amplitude-invariant scaling. The Clarke transform carries the 2/3 factor, so α and β hold the phase peak. Power is then 3/2 (v_d i_d + v_q i_q). The power-invariant convention scales by √(2/3) and has no factor — neither is wrong, and mixing them is.
α lies along phase a. β leads α by 90°.
d lies along the rotor flux, q leads d by 90°.
θ is measured from α, positive counter-clockwise.
Positive i_q makes positive torque on a machine with the rotor turning positive. Negative i_d weakens the field.
Duty is 0 to 1, centre-aligned, with the pulse centred on the middle of the period — so the null vector straddles the boundary.
Currents are per phase, and so are R_s, L_d and L_q. A line-to-line measurement is two windings in series.

Quantities

Symbol	Meaning	Unit
$i_{a}, i_{b}, i_{c}$	Phase currents, as measured at the machine terminals	A
$i_{α}, i_{β}$	The same current in the stationary two-axis frame	A
$i_{d}, i_{q}$	The same current in the rotor frame — d along the magnet flux	A
$v_{d}, v_{q}$	Voltage the controller requests, in the rotor frame	V
$V_{d c}$	DC-link voltage at the bridge, not at the supply	V
$θ_{e}$	Electrical angle of the rotor flux, measured from the α axis	rad
$θ_{m}$	Mechanical rotor angle, related by theta_e = p*theta_m + offset	rad
$ω_{e}, ω_{m}$	Electrical and mechanical angular velocity	rad/s
$ω_{c}$	Closed-loop crossover, the loop bandwidth in rad/s	rad/s
$T_{e}$	Electromagnetic torque produced at the airgap	N·m
$d$	Duty cycle, 0 to 1, as written to a compare register	—

Machine parameters

Symbol	Meaning	Unit
$R_{s}$	Stator resistance, per phase — not line to line	Ω
$L_{d}, L_{q}$	Axis inductances, per phase. Equal on a surface-magnet machine	H
$λ_{m}$	Magnet flux linkage seen by one phase	Wb
$p$	Pole pairs — half the pole count	—
$J$	Rotor plus load inertia	kg·m²
$B$	Viscous friction coefficient	N·m·s

Timing and control

Symbol	Meaning	Unit
$f_{pw m}$	Carrier frequency. The control ISR runs once per period	Hz
$T_{s}$	Control period, 1/f_pwm for single-update PWM	s
$T_{d}$	Transport delay, sample instant to the centre of the resulting voltage	s
$t_{d}$	Deadtime, the interlock between one device turning off and the other on	s
$k_{p}, k_{i}$	Proportional and integral gains of a PI controller	V/A, V/A·s

The reference machine

One machine throughout the site, so numbers on different pages can be compared. A 48 V interior-magnet PMSM. Every value below is read from src/core/machine.ts at build time — the same constants the figures run on, not a copy.

Quantity	Value
$R_{s}$	0.35 Ω
$L_{d}$	210 µH
$L_{q}$	330 µH
$λ_{m}$	8.5 mWb
$p$	4
$J$	120 g·cm²
$L_{q} / L_{d}$	1.57×
$\frac{3}{2} p λ_{m}$	0.051 N·m/A
$K_{e}$	4.36 V/krpm, line-to-line RMS
$i_{c h} = - λ_{m} / L_{d}$	70.8 A

See also the formula sheet and what goes wrong.