Harmonic analysis of saw function - QuickField simulation example
This example demonstrates the accuracy of the harmonic browser (included in QuickField package) How to find harmonic spectrum of sawtooth signals?
Answer Typical applications Geometry
Given
Relative permittivity of air ε = 1,
Task
Solution
Fourier analysis of this "saw" function gives the following analytical solution:*
Results
Harmonic browsers window
Harmonic browser allows calculation of the first hundred harmonics.
Harmonic
Amplitude
Error
QuickField
Theory
*Mathematical Methods for Physicists. A Comprehensive Guide. 7th Edition. George Arfken, Hans Weber, Frank Harris, p 937-938. ISBN 978-0-12-384654-9.
Rectangular wave duct consists of two electrodes. The electrodes carry electric potential in a form of saw-function. Perform Fourier analysis of electric potential distribution.
Engineering question
Set up a plane-parallel QuickField Electrostatics problem for a sawtooth signal representation and evaluate harmonic spectrum from computed field results.
sawtooth waveform signals, harmonic spectrum studies, Fourier series examples
Download
Simulation problem
Problem Type
Plane-parallel problem of electrostatics.
Boundary potentials - saw function.
Perform the Fourier analysis of the voltage distribution curve. Use the harmonic browser to calculate amplitudes of the harmonics and compare results with analytical solution.
Voltage distribution along the X axis corresponds to the "saw" function.
U(0≤x<1.507) = 2·x
U(x=1.507) = 0
U(1.507<x≤3.142) = 2·x - 2·π
f(x) = 2 · ( sin(x) - sin(2x) / 2 + sin(3x) / 3 - ... + (-1)n+1·sin(nx) / n )
Potential distribution along the bottom boundary
1
2
2
-
2
0.9999
1
-
3
0.6666
0.6666
-
4
0.4999
0.5
-
5
0.3999
0.4
-
10
0.1999
0.2
0.05%
20
0.0998
0.1
0.2%
50
0.0383
0.04
5%
100
0.0168
0.02
15%
Video
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