IntermediateTopic 70 of 104Engineering Mathematics
Fourier series
Represents periodic signals as sums of sines and cosines.
In depth
A periodic function f(t) of period T has f=a₀/2+Σ(aₙcos nω₀t+bₙsin nω₀t). The coefficients are integrals over one period. Square waves contain only odd harmonics with amplitude 1/n. The approach is the basis of spectral analysis.
Key points
- Gibbs phenomenon ~9% overshoot at jumps
- Even functions have only cosines
- Parseval relates power to coefficients
ExampleA 50 Hz square wave contains 150 Hz, 250 Hz, … components.
Fourier series show that any repeating signal is built from harmonics, which is the basis for filters, power quality and vibration analysis.
What is Fourier series?
Represents periodic signals as sums of sines and cosines.
Why is it important to understand fourier series?
Fourier series show that any repeating signal is built from harmonics, which is the basis for filters, power quality and vibration analysis.
What is a harmonic?
A sinusoid whose frequency is an integer multiple of the fundamental. Harmonics explain why square waves have sharp edges and why power supplies distort the grid.
Why does a square wave need infinitely many terms?
Its sharp corners require high frequencies. Truncating the series causes the Gibbs overshoot near each jump.
How is Fourier series used in practice?
Typical applications include describing the harmonic distortion in a rectifier output, predicting the response of a beam to a periodic load and designing a filter to remove unwanted harmonics from a motor drive.
What should I learn before Fourier series?
Study "Systems of ODEs" first (topic 69 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Fourier series?
Gibbs phenomenon ~9% overshoot at jumps. Even functions have only cosines. Parseval relates power to coefficients.
Can you give an example of Fourier series?
A 50 Hz square wave contains 150 Hz, 250 Hz, … components.
What comes after Fourier series?
The next topic is "Fourier transform", which builds directly on this one.
How difficult is Fourier series?
It is classed as Intermediate in Engineering Mathematics (topic 70 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.