Fourier Series

Used for periodic non-sinusoidal waveforms. Suppose the periodic time is . The waveform can be represented as a sum of sine wave. The sine waves have integer multiple of the original frequency .

Useful trigonometric integrals

Unknown coefficients

.

is the DC offset.

Symmetry

For any waveform, subtracting the DC offset results in a symmetrical waveform.

Even Symmetry

When a wave is symmetric about the vertical axis.

The fourier series of an even waveform contains only cosine terms.

Odd Symmetry

When a wave is symmetric about the origin.

The fourier series of an even waveform contains only sine terms.

Half-Wave Symmetry

When a wave repeats itself with a reversal of sign after half a period.

The coefficients can be found by:

Frequency spectrum

Plot of harmonic number vs frequency.

Harmonic number

th harmonic number denotes the amplitude or strength of the th harmonic.

RMS

Total Harmonic Distortion

A measurement of the distortion present in the waveform compared to the original waveform.

Here is the rms of th harmonic. DC offset is ignored. Usually given as a percentage.

Complex form

Here can be found by: