Page 60 - Power Quality in Electrical Systems

Chapter
4

Harmonics and Interharmonics

In this chapter, we shall discuss harmonics (frequency
components that are integer multiples of the fundamental line
frequency) and interharmonics (high-frequency components).
For most of what we shall do in this chapter, the fundamental
frequency used will be 60 Hz.

Background
As we mentioned in previous chapters, harmonics can adversely affect
the operation of cables, capacitors, metering, and protective relays. To
summarize, a brief listing of some systems and the effects of harmon-
ics is shown in Table 4.1 [4.1].

Periodic Waveforms and Harmonics

The notion that any periodic waveform can be broken up into a series
of sine waves at the proper amplitudes and phase relationships was first
worked out by Joseph Fourier, the French mathematician and physicist
[4.2]. He showed that any periodic waveform can be expressed as a sum
of sine and/or cosine waves, with the proper amplitude, frequency, and
phase relationships between the waves. For instance, a square wave
(Figure 4.1a) can be represented by the infinite Fourier series:

4 4 4
vstd 5 a b sin svtd 1 a b sin s3vtd 1 a b sin s5vtd 1 ???
p 3p 5p
where v is the frequency in radians per second. Note that the amplitude
of the first harmonic is (4/ ), the amplitude of the third harmonic is

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