# Does electrical sawtooth wave actually produce sinusoidal oscillations at harmonic frequencies?

According to Fourier's theorem, we know that a sawtooth wave can be represented as a sum of sine waves. These sine waves we know as harmonics (in the context of sound). My understanding is that it is the same for electrical current.

Let us take a band-limited sawtooth wave in an electrical circuit. Say its frequency is 440hz. We know that its next harmonic after fundamental is $$880$$ Hz.

Do we actually have something oscillating in this circuit at frequency $$880$$ Hz in a sinusoidal waveform? What is it then? Or this is just a mathematical concept?

I am thinking from a perception point of view: when we produce a sawtooth wave by gradually increasing the voltage from $$-1$$V to $$1$$V and then dropping it from $$1$$V to $$-1$$V almost instantly. And that is what we see in the oscilloscope: spikes of $$2V$$ at $$440$$ Hz. But we don’t see its harmonics there. Do they actually happen or this is just an abstraction?

Please, help me with some guidance.

• Re, "a sawtooth wave can be represented as a sum of sine waves." Yes, because any periodic function actually is mathematically equal to its Fourier decomposition. Nov 16, 2020 at 14:51
• You should check out D-class amplifiers. They convert audio into square wave pulses and reconvert them to audio by filtering the high frequency harmonics. Square waves are more power efficient to produce. en.wikipedia.org/wiki/Class-D_amplifier Nov 16, 2020 at 17:06

## 1 Answer

Let us take a band-limited sawtooth wave in an electrical circuit. Say its frequency is 440hz. We know that its next harmonic after fundamental is 1320 Hz. Do we actually have something oscillating in this circuit at frequency 1320 Hz in a sinusoidal waveform?

No. The period of the waveform is $$1/440\text{s}$$.

What is it then?...just a mathematical concept?

Well, it is a mathematical concept, but it's not just a mathematical concept.

If you feed that signal to a circuit that resonates at 1320 Hz, then it will excite the resonator. In fact, that's pretty much how your ears respond to complex sounds: You have thousands of cells in your ears that each have a mechanical structure (a hair) that resonates at a different frequency. When you hear a tone whose periodic function can be decomposed into a few discrete frequencies, it stimulates precisely those nerve endings that respond to those specific frequencies.

In effect, your ear is a mechanism that performs the mathematical decomposition of the periodic function into pure-sinusoidal components.