# Prelab 8

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## 1) Fourier series of a square wave

For this week's lab we'll be using the phenomenon of resonance to increase the voltage created by the Teensy to a sufficient extent that we can operate a string of LED lights.

In order to generate that large voltage, we'll be using the Teensy to create a square wave of different frequencies. In class we're learning about sinusoidal steady-state, and while it seems like square waves and sinusoids are different, in fact one can show via Fourier analysis that one can create a square wave (and almost any other periodic signal) as a sum of sinusoids.

Here is a figure from here showing that if one adds up the right amplitudes and frequencies of sinusoids, then you can get a square wave:

What the figure shows is how adding a sin(\omega t) with a sin(3\omega t) and so on can start to approximate a square wave. Even four sinusoids, when added up to create the black curve, create a pretty decent square wave. You also see that the sinusoid with the largest amplitude is the one at the same frequency as the square wave.

The point of telling you this isn't so that you can do Fourier analysis, though it is admittedly an extremely useful tool. It's so that when we apply a square wave in that lab, you'll realize that we are primarily applying a sinusoid at the same frequency as the square wave.

## 2) The resonant circuit

We've learned about RLC circuits and impedance in lecture and recitation. In lab we'll be using an RLC circuit like the one shown below:

In lab, we'll eventually replace R with a string of LED lights, where we'll use v_R to light up the LED light string.

This circuit has many of the same properties as the ones we've seen in terms of having \omega_0, Q, etc. In lab we'll analyze this circuit in order to extract out all these parameters, though feel free to analyze the circuit before lab if you prefer. We'll want the differential equation, so get that if you can.