Helix Harmonic Analyzer

The Helix is a simple mechanical device for measuring the amplitude and phase of the harmonics of a periodic signal. Like the harmonic analyzers of Lord Kelvin, Henrici, and Michelson and Stratton, it finds the Fourier coefficients of the signal, but here gravity does the summing.

It is an axis with a row of perpendicular rods through it, spiralling one full turn along the axis' length. Each rod passes through the axis at its middle, so the bare helix is balanced in any position. The rods carry sliders: beads that slide along a rod, red ones and blue ones (black in the photos).

Measuring the first harmonic

1. Prepare — Draw one period of the signal as wide as the helix is long, with a horizontal line at the signal's mean (equal area above and below). Remove all sliders and hold the helix with its axis directly above that line.
2. Record — Turn the helix one full turn in one direction while working from left to right, so that the rods lie flat one after another, each pointing away from you. Looking straight down, put a red slider on each flat rod where it crosses the curve. Call the half of a rod that pointed away from you its positive half.

   

3. Phase — Hold the helix by the ends of its axis and let it turn freely. Gravity turns its heavy side down. The rod whose positive half now hangs straight down marks the peak of the first harmonic A sin(2πt + φ): if that rod sits at t along the period (0 ≤ t < 1), then φ = 90° − 360°·t. For the square wave it is the rod a quarter of the way along, so φ = 0: its first harmonic is a pure sine.
4. Amplitude — Keep the helix in its rest position and look along the axis. Put a blue slider on each rod so that the blue sliders form a circle above the axis that passes through the axis. Resize the circle until the helix is balanced and stays put at any angle. The diameter of the circle is the amplitude A.

   

5. Synthesise — Place the helix above the paper again and lay each rod flat as in step 2, but half a turn further, with its positive half pointing towards you: the blue sliders sit opposite the red weight. Mark the paper below each blue slider and join the marks with a smooth curve. This is the first harmonic.

Why it works

Rod k of N sits at t_k = (k − ½)/N and points at angle φ_k = 360°·t_k. Its red slider lies at distance f(t_k) along it, so the centre of mass of the red sliders is

(1/N) Σ_k f(t_k) (cos φ_k, sin φ_k) ≈ ½ (a₁, b₁),

half the first Fourier coefficients: a point at distance A/2 from the axis, in the direction of the harmonic's peak. Blue sliders on a circle of diameter D through the axis have their centre of mass at the circle's centre, D/2 from the axis on the other side, so they balance the red sliders exactly when D = A. A constant offset in the signal adds equal distances on rods spread evenly around the axis, which cancel; drawing the line at the mean only keeps the sliders close to the axis.

Higher harmonics

For harmonic n, the sample at t has to sit at angle n·360°·t. There are several ways to do this:

• Wider drawing — Draw one period n times as wide as the helix and record it in n passes, so each rod gets n red sliders. With one blue slider per rod, the circle balances at diameter n·A.
• More turns — Use a helix whose rods make n full turns over one period, for example two helices joined end to end for n = 2, with one period drawn over their full length.
• Variable helix — Let the rods turn freely on the axis. Record once with all rods flat in one plane, like an abacus, without turning anything. Then spread the rods evenly over n full turns and measure as above. One recording serves every harmonic; left flat (n = 0), the helix's imbalance gives the constant term.

With N rods the helix samples the signal at N points, so more rods give more accurate results. Harmonics N ± n of the signal turn up as harmonic n (aliasing), so n should stay well below N/2.