Fundamental component of a Quasi-Square Wave (QSW) voltage as shown below has a peak value of (4/𝝅)Vdc*cos(α) = 1.27Vdc*cos(α)

#FourierSeries #PowerElectronics #Inverter

Source: Slides 8-9 → https://www.slideserve.com/lapis/inverters-dc-ac-converters

Teaching Transforms

We’re about two-thirds of the way into the Autumn Semester here at Maynooth and, by a miracle, I’m just about on schedule with both the modules I’m teaching. It’s always difficult to work out how long things are going to need for explanation when you’re teaching them for the first time.

One of the modules I’m doing is Differential Equations and Transform Methods for Engineering Students. I’ve been on the bit following the “and” for a couple of weeks already. The first transform method covered was the Laplace transform, which I remember doing as a physics undergraduate but have used only rarely. Now I’m doing Fourier Series, as a prelude to Fourier transforms.

As I have observed periodically, the differential equations and transform methods are not at all disconnected, but are linked via the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

In the module I’m teaching, the applications are rather different from when I taught Fourier series to Physics students. Engineering students at Maynooth primarily study electronic engineering and robotics, so there’s a much greater emphasis on using integral transforms for signal processing. The mathematics is the same, of course, but some of the terminology is different from that used by physicists.

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

http://www.youtube.com/watch?v=LznjC4Lo7lE

This reminds me of a point I’ve made a few times in popular talks about astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form of a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding. What makes for a good physical theory is, in my view, largely a matter of economy: if two theories have equal predictive power, the one that takes less chalk to write it on a blackboard is the better one!

Anyway, soon I’ll be moving onto the complex Fourier series and thence to Fourier transforms which is familiar territory, but I have to end the module with the Z-transform, which I have never studied and never used. That should be fun!

#FourierSeries #FourierTransforms #LaplaceTransforms

The Fourier Series (demo) for the ZX81

https://www.youtube.com/watch?v=TMfqFbCIK78

#TheFourierSeries #FourierSeries #ZX81 #Demo #SinclairUser

FOURIER SERIES AND TAYLOR SERIES

#FourierSeries #TaylorSeries #TaylorExpansion

The Fourier Series and the Taylor Series have something in common. Each of them arose with this train of thought: "Let's assume I can represent a given function by the sum of terms of a given type [polynomials with Taylor, sines/cosines with Fourier]. The only question is how to weight those terms properly. How, HOW can I possibly torture my function to get it to confess the appropriate weight of each term?"

In the case of Taylor, you can torture your function by taking its derivative: to get a given polynomial term's weight, you take the derivative of the function however many times, so that it's coughed up a term that is just a constant. Then that constant becomes the weight of that polynomial term. Do that for all the polynomial terms and you get all the polynomial term weights, and that lets you construct a complete representation of your original function out of polynomial terms.

In the case of Fourier, the form of torture is different. If you look at sin(x), sin(2x), sin(3x), etc, you'll notice that they all repeat every 2*pi (and in fact most of them will repeat more than once every 2*pi, but the important thing is, they all happen to line up with that 2*pi). Now, one interesting property of the integral from 0 to 2*pi is, if your integrand is the product of any pair of those sine terms, the integral evaluates to zero ... BUT if it's a term times itself, the integral is non-zero. That can be the our instrument of torture: you can, for example, take the integral of sin(3x)*f(x) to get f(x) to confess how much sin(3x) it contains, and that will tell us how to weight sin(3x).

The Fourier transform (FT), explained in one sentence: 🔗 https://blog.revolutionanalytics.com/2014/01/the-fourier-transform-explained-in-one-sentence.html
\[\boxed{\hat{f}(\xi)=\displaystyle\int_{-\infty}^\infty e^{-i2\pi\xi t}f(t)\ \mathrm{d}t}\]

Discrete Fourier transform (DFT):
\[\displaystyle x_n = \sum_{k=0}^{N-1} X_k \cdot e^{-i2\pi \tfrac{n}{N}k}\]

Inverse transform:
\[\displaystyle X_k = \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{i2\pi \tfrac{n}{N} k}\]

#FourierTransform #FourierSeries #Transform #MathematicalTransform #Signal #SignalProcessing #Frequency #Energy #FourierAnalysis #Series #Analysis

Fourier vase https://www.thingiverse.com/thing:5451891

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#dotSCAD
#FourierSeries