Space vectors

The Clarke and Park page opens by asserting that three phase currents describe a two-dimensional vector. That is true, and it is not obvious. It is a consequence of how the windings are arranged, and the derivation is worth doing — because it also produces the $3/2$ and $2/3$ factors that the transforms and the torque equation have been carrying ever since.

It answers a more basic question too: why three phases at all?

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Three fixed directions with time-varying amplitudes. Nothing here rotates on its own — the rotation is what the sum does, and the 1.5 is what the addition produces rather than something inserted by hand.

The physical quantity is MMF

What matters inside the machine is not current as such but magnetomotive force — current times turns — because that is what drives flux across the airgap.

A practical stator winding is distributed in slots so that the MMF it produces varies sinusoidally with position around the airgap, peaking on the winding’s own axis:

F_{a} (x) = N i_{a} cos x

with $x$ measured around the bore from phase $a$ ‘s axis. Phases $b$ and $c$ are built identically, rotated by $\pm 120°$ .

On the bench The sinusoidal distribution is a design goal, not a law. Real windings approximate it with a finite number of slots, and the leftover space harmonics are a genuine source of torque ripple. Everything downstream on this site assumes the ideal — which is standard, and worth knowing you have assumed.

One winding cannot turn a rotor

Energise phase $a$ alone with $i_{a} = I cos ω t$ and the MMF is

F (x, t) = N I cos ω t cos x

The peak is always at $x = 0$ or $x = 180°$ . It never moves — it only grows, collapses and reverses. This is a standing wave, and a standing wave exerts no consistent direction of torque.

The trigonometric identity says exactly why:

cos ω t cos x = \frac{1}{2} cos (x - ω t) + \frac{1}{2} cos (x + ω t)

A pulsating field is two fields of half amplitude rotating in opposite directions. They pull the rotor equally hard each way, and the net is nothing.

That is why a single-phase motor cannot start itself, and why every one you have met has some trick — a start winding, a capacitor, a shaded pole — to weaken one of those two components long enough to pick a direction.

Three windings, and the backward half cancels

Now energise all three with a balanced set. Adding the three distributions and collecting terms gives something remarkable:

F (x) = \frac{3}{2} [i_{α} cos x + i_{β} sin x]

where $i_{α}$ and $i_{β}$ are precisely the amplitude-invariant Clarke components.

Three sinusoidal distributions sum to another sinusoidal distribution — no harmonics, nothing left over. Its amplitude and its angular position are the only two things left to say about it.

That is the space vector. It is not a mathematical convenience applied to the currents; it is the shape of the field in the machine. The site’s test suite checks this directly: the peak of the summed MMF lands exactly where the Clarke vector points, to 1e-12, for balanced and unbalanced currents alike.

Single phase only Show counter-rotating halves Drag the airgap to move the angle by hand.

The airgap end-on, with the MMF drawn as a deviation from the bore. Right: each winding's contribution and their sum, above the currents that produce it. Tick Single phase only and watch the resultant stop turning.

Let it run. The resultant sweeps round at constant amplitude while the windings themselves do not move. Three stationary coils, fed with currents 120° apart in time, produce a field that rotates in space. That is the whole trick of polyphase machines, and it predates field-oriented control by a century.

Now tick Single phase only. The vector stops rotating and merely breathes along phase $a$ ‘s axis. Tick Show counter-rotating halves to see the two components that make it up, chasing each other in opposite directions.

Where 3/2 and 2/3 come from

The resultant of three windings each peaking at $I$ is $\frac{3}{2} I$ — not $3 I$ . The phases are 120° apart in space, so they partly oppose one another, and the sum of three unit vectors at 120° carrying $cos$ -weighted currents lands at exactly 1.5.

That single factor propagates through the whole subject:

Torque carries a $\frac{3}{2}$ : $T = \frac{3}{2} p [λ_{m} i_{q} + \dots]$
Power carries a $\frac{3}{2}$ : $P = \frac{3}{2} (v_{d} i_{d} + v_{q} i_{q})$
The Clarke transform carries a $\frac{2}{3}$ — because dividing the $\frac{3}{2}$ back out is exactly what makes $∣ i_{α β} ∣$ equal the peak phase current.

The amplitude- versus power-invariant choice that page described is a choice about where to put this factor. It has to appear somewhere; the conventions differ only in whether it lands in the transform or in the power equation.

Why the machine cannot see common mode

Add the same current to all three phases and the MMF does not change at all, because the three spatial cosines sum to zero:

cos x + cos (x - 120°) + cos (x + 120°) = 0

A zero-sequence current produces no field. Two consequences the site has already leaned on:

The SVPWM injection is free. Adding a common-mode term to all three duty cycles buys 15.5% more linear range and the machine cannot detect it, because there is nothing to detect.
A common-mode sensing offset is invisible to a two-shunt drive, and the reconstruction forces the currents to sum to zero regardless.

Why two phases would not do

Two windings at 90° would also produce a rotating field, and the maths would be simpler — αβ directly, with no Clarke transform at all.

Three wins for a practical reason. With three phases and an isolated neutral, the currents must sum to zero, so two wires can carry three phases and only three connections leave the machine. A two-phase machine needs four. Three phases is the smallest number that produces a rotating field and lets the return path be shared.

The transform back to two axes is then free — the machine was two-dimensional all along, and the third wire was only ever bookkeeping.

What to take away

The machine responds to MMF, and a distributed winding makes MMF that varies sinusoidally around the airgap.
One winding gives a standing wave, which is two counter-rotating fields of half amplitude. No net torque, hence no self-starting.
Three windings 120° apart, fed 120° apart in time, sum to a single sinusoidal distribution that rotates at constant amplitude.
Because that sum is exactly sinusoidal, two numbers describe it completely. Those two numbers are the space vector, and they are physical.
The resultant is $\frac{3}{2}$ of one winding’s peak. That factor is the ancestor of every $3/2$ and $2/3$ elsewhere on this site.
Zero-sequence current produces no field at all, which is what makes common-mode injection free and common-mode offsets invisible.

Next: Clarke and Park — turning the vector this page just derived into two numbers a PI controller can regulate.