Harmonic Distortion & Nonlinearities

Along the way

1
Tones, not a floor. Quantization error (Part 3) spreads. Distortion is deterministic and lands at exact multiples of \(f_{\mathrm{in}}\). A single sine in → a comb out.
2
Cubic → HD3. \(\sin^3\theta=\tfrac{3}{4}\sin\theta-\tfrac{1}{4}\sin 3\theta\). The fundamental gains a little; a brand-new 3rd harmonic appears. Odd powers stay odd.
3
SFDR vs THD vs SINAD. SFDR = fundamental / worst spur. THD sums harmonic power. SINAD lumps noise in too. A part can have great SNR and terrible SFDR.
4
Even-order needs asymmetry. Drag \(a_2\) — the transfer curve stops being odd, a DC offset appears, and HD2 lights up. Real devices are slightly lopsided; that is why datasheets quote HD2.
5
Slew looks like HD3, but isn't. If odd harmonics climb with amplitude and the waveform triangulates, check slew rate before blaming the static cubic. Same trap in Part 14.

Intermodulation, memory / slew, and DNL-generated “harmonic-like” spurs. SFDR here (vs HD3 only) is a lower bound on how messy a real spectrum can look.

Next: Part 7 — DC transfer errors: offset, gain, INL, DNL. Calibratable vs not.

A sine in, a comb out

Transfer + time · residual = the new harmonic · spectrum like a analyzer (dBc, 0 at top)

Fundamental \(a_1\) Even \(a_2\) (HD2) Odd \(a_3\) (HD3)

Ideal sine Distorted output Residual (the harmonic) HD2

Transfer \(v(x)=a_1 x+a_2 x^2+a_3 x^3\)

Time · two cycles of \(f_0\)

Residual = output − fundamental · this is the new tone(s)

Spectrum · dBc · 0 at top · needles at \(k f_0\)

Fundamental—

HD2—

HD3—

SFDR—

THD—

\[ \sin^3\theta = \tfrac{3}{4}\sin\theta - \tfrac{1}{4}\sin 3\theta \]

\[ v = \big(a_1+\tfrac{3}{4}a_3\big)\sin\omega t \; - \; \tfrac{1}{4}a_3\sin 3\omega t \]

\[ \mathrm{SFDR}_{\mathrm{dBc}} = 20\log_{10}\!\dfrac{a_1+\tfrac{3}{4}a_3}{\tfrac{1}{4}a_3} \]

= —

Odd \(a_3\) → only HD3 (and a little extra fundamental). Even \(a_2\) → DC + HD2 from \(x^2=(1-\cos 2\theta)/2\). SFDR is fundamental to the worst spur, not always HD3.

Try it: “HD3 only” — residual is a clean \(3f_0\) sine (three wiggles per cycle) and only the rose needle rises. “HD2 only” — residual is \(2f_0\), amber needle. “Messy” — SFDR is whoever is taller. Nothing like Part 3’s noise floor.

Analog Intuition

A sine in, a comb out