Cooking recipes and the swarm of harmonicas

What is a harmonica? When we hear the word 'harmonica', the first thing that comes to mind is probably the musical instrument, immortalized by Billy Joel in his 'Pianoman'. But few know that harmonics are also a crucial phenomenon in electrical systems, and it is no coincidence that the musical instrument and this electrical phenomenon have the same name.

Let us travel back in time to Napoleonic France, where Jean-Baptiste Joseph Fourier, a French mathematician, physicist, essayist and politician, after accompanying Napoleon to Egypt, set about deciphering the equations of heat by means of trigonometric series. He discovered that any periodic waveform can be decomposed into an infinite series of sine waves, and vice versa. These decompositions were called Fourier series, and his formula, although it may seem frightening at first glance, is fundamental in many fields of science and engineering.

The Fourier Series formula is similar to a cooking recipe, and here we will break it down this way. No matter where you live, surely your mother or maybe your grandmother cooks a dish that is your favorite. Sometimes you just go to her house to try it; it's always delicious, with the same seasoning and no one makes it like her. It can be a cake, a mole, a rice dish, a cabrito or even in your bar, a cocktail. The trick is in precise proportions, added in a particular order.

Let's take vegetable juice as an example. Imagine that your grandmother's recipe includes three carrots, a beet, five tomatoes, a celery leaf, a parsley leaf, half a cucumber, a lettuce leaf and a broccoli leaf, and that we put them all in the blender at the same time or one by one, separated by 15 seconds of blending.

Similarly, the Fourier Series is the 'recipe' for decomposing a periodic irregular waveform into its basic components, which are the harmonics. These harmonics are the waves that repeat at regular intervals and whose sum gives us the original complex wave. For example, the second harmonic has a frequency of 120 Hz; it crosses the axis twice in the same period as the 60 Hz fundamental wave. The third harmonic has a frequency of 180 Hz, and so on.

But, just like in a cooking recipe, we need to know how much of each ingredient (in this case, harmonica) we need. This is called amplitude. For example, we can say "there is 20% harmonic fifth" or "15% harmonic seventh". This tells us how prominent each harmonic is in the final complex waveform.

In addition to frequency and amplitude, the moment each component is added to the mixture is also important, which in wave terms refers to the phase of each harmonic. This is similar to the moment we decide to add each ingredient to our recipe while cooking.

As we mentioned in a previous article, non-linear loads in electrical systems generate harmonic distortion, which can be harmful to the system. But, in order to correct this phenomenon, it is first necessary to measure it. That is where Fourier Series become essential; they allow us to decompose the complex waves of the electrical system into their harmonics, to evaluate the magnitude of this distortion and to take corrective measures.

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We can compose a square waveform by adding together many sine waves of different frequency, amplitude and phase, just as we mix ingredients in a blender to make juice. And, just as we can separate a recipe into its basic ingredients to understand what makes it up, we can decompose a complex waveform into its harmonics using Fourier analysis.

When you drink that juice, you are enjoying the combination of all its ingredients, blended to perfection. Similarly, complex waveforms circulate in our electrical grids, created by various electrical charges. Thanks to Fourier, we can 'crumble' these complex waves and understand their harmonic 'ingredients'.

So, the next time you think of Fourier, remember that his legacy goes beyond mathematics: he offers us the recipe for understanding periodic waves in our world. He is a bridge that connects abstract mathematics with very concrete and essential applications in our daily lives, from electrical circuit design to image and sound processing.

Did you like this trip between grandma's kitchen and electric waves? There's a lot more where that came from! Click the link below to explore the electrical universe we're cooking up at CIITAP (Center for Power Applied Research and Technological Innovation). From smart power electronics technology to innovations in IoT smart metering and control and energy storage, here at CIITAP we're reinventing the way we understand and use electricity.

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