CR · Non-sinusoidal

Harmonic Synthesis and Fourier

The signals entering a circuit are not always clean sinusoids. Yet there is a reason it is enough to handle only sinusoids: any periodic waveform can be split into a sum of several sinusoids. Learn the principle of that synthesis.

Add a few sines to build a square wave

Add harmonics to the fundamental one at a time. It starts as a round sine, but as you add odd harmonics the sum approaches the flat square wave (faint line). Turn on more harmonics to make it hug the square.

Tap to switch the number of harmonics.

Harmonics included

n = 1  (≤ 1)

Still rounded

Only integer-multiple frequencies are allowed

To build a waveform of period T from sinusoids, the only sines you may use are those that fit neatly into the same period T. That means only frequencies at integer multiples of the fundamental — twice, three times, four times… The wave at the fundamental frequency is the fundamental, and its integer multiples are the harmonics. Choose each harmonic’s amplitude rightly and their sum traces the shape you want.

A square wave uses only odd harmonics

A waveform with up-down symmetry like a square wave has all its even harmonics equal to zero, leaving only the odd ones. And the amplitude of the n-th harmonic falls off as 1/n. So a square wave is sin(ωt) + (1/3)sin(3ωt) + (1/5)sin(5ωt) + …. The higher harmonics add in smaller amounts, but they are exactly what sharpen the corners.

Why sinusoidal analysis covers every signal

Here impedance and frequency response meet. Once you know how a circuit responds to a sinusoid at each frequency (its gain and phase), you split any input into harmonics, pass each through, and add them back to get the output. If a filter trims the highs, the square wave’s high harmonics shrink and its corners round off. So mastering a single sinusoid means, in effect, mastering every periodic signal.

Observef = Σ a_n sin(nωt)

A periodic wave is a sum of sinusoids.

Choosesquare: n = ?

A square wave has only odd harmonics.

Fill ina_n ∝ ?

The n-th amplitude falls as 1/n.

On your ownf_n = n · ?

Harmonic frequencies are integer multiples.

Back to the first screen

With just the fundamental it was a round sine, but as you added odd harmonics one by one, the sum gained a flat top and steep corners and approached a square wave. Each harmonic, though shrinking as 1/n, did its part in shaping the form. The fact that any complicated waveform is, in the end, a sum of sinusoids extends every tool for solving a circuit with one sinusoid to an arbitrary signal.

The Fourier series states that any periodic waveform can be written as a sum of a fundamental and integer-multiple harmonics (sines and cosines). A square wave is a sum of odd harmonics only, with the n-th amplitude falling as 1/n: f = (4/π)(sin ωt + ⅓ sin 3ωt + ⅕ sin 5ωt + …). Adding terms converges to the square. This extends sinusoidal analysis to every periodic signal.