Linear alternator phases

A row of pickup coils on a train of slugs: what phases, and what do one-phase, three-phase and rectified DC come to?

The formula

θ = 360^{°} \cdot \frac{s}{p} \mod 360^{°}, εLL = 2 \cdot ε \cdot \sin \frac{π}{n}

LaTeX

θ = 360^{\circ} \cdot \frac{s}{p} \bmod 360^{\circ}, \qquad εLL = 2 \cdot ε \cdot \sin\frac{\pi}{n}

Work it out

n Pickup coils

How many pickup coils in the row, and so how many phases at most.

s Coil spacing

Centre to centre between adjacent coils along the tube.

p Slug pitch

Centre to centre between slugs — flow speed over pulse rate. One pitch is one electrical cycle.

ε Per-coil peak EMF

What one coil gives on its own — the pickup-EMF page's figure, or the loaded-output page's if a load is meant.

or see a worked example with typical numbers

One slug pitch is one electrical cycle, so the angle between adjacent coils is 360° times the spacing over the pitch, reduced modulo 360°.
The one-phase series sum is the phasor sum of n equal EMFs stepping by θ: ε·|sin(nθ/2)/sin(θ/2)|, which is nε when θ is a multiple of 360° and can be zero — three coils at 120° in series cancel exactly.
A balanced n-phase set has its phases 360°/n apart; the peak between any two adjacent lines is 2ε sin(π/n), which is √3 ε for three phases and ε itself for one.
Full-wave rectifying an n-phase set gives m = 2n ripple pulses per cycle, and the mean of the rectified waveform is the line peak times (m/π)·sin(π/m): 0.637 of the peak for one phase, 0.955 of the line-to-line peak for three.
The set is called balanced when θ is within 5° of 360°/n (or of its complement, which is the same set in the other order); otherwise the line-to-line and rectified figures are for a set this row does not make, and the page says so.

Identical coils, identically loaded, with no mutual coupling between them — each sees only the slugs, not its neighbours' currents.
A sinusoidal approximation of the slug waveform. The true output of one coil is the bipolar pulse of the pickup-EMF page, whose fundamental at the slug rate is what these figures describe; the harmonics add to the rectified mean and do not cancel in series the way the fundamental does.
"DC" here means rectified. A steady flow of uniformly magnetised medium through a coil changes no flux and induces nothing — Faraday's law has no term for it — so a claim of DC from an unrectified coil is a claim this model gives zero for, and it is stated here so that the zero is not mistaken for a missing calculation.
Nothing here is Meyer's. The one-phase, three-phase and DC claim for the EPG has, so far, no holding in the estate material that this archive can cite for how the coils were connected; this is what the geometry of any alternator gives for a row of coils at a spacing on a pitch, and a reader can put in the spacing the photographs suggest.