ABSTRACT
At the end of this chapter you should be able to: • define a complex wave • recognize periodic functions • recognize the general equation of a complex waveform
• use harmonic synthesis to build up a complex wave • recognize characteristics of waveforms containing odd, even or odd and even harmonics, with or without phase change
• determine Fourier series for simple functions
• calculate r.m.s. and mean values, and form factor of a complex wave
• calculate power associated with complex waves • perform calculations on single-phase circuits containing harmonics
• define and perform calculations on harmonic resonance
• list and explain some sources of harmonics
36.1 Introduction
In preceding chapters a.c. supplies have been assumed to be sinusoidal, this being a form of alternating quantity commonly encountered in electrical engineering. However, many supply waveforms are not sinusoidal. For example, sawtooth generators produce ramp waveforms, and rectangular waveforms may be produced by multivibrators. A waveform that is not sinusoidal is called a complex wave. Such a waveform may be shown to be composed of the sum of a series of sinusoidal waves having various interrelated periodic times. A function f (t) is said to be periodic if f (t +T )= f (t)
for all values of t, where T is the interval between two successive repetitions and is called the period of the function f (t). A sine wave having a period of 2π/ω is a familiar example of a periodic function. A typical complex periodic-voltage waveform, shown
inFigure 36.1, has periodT seconds and frequency f hertz. A complex wave such as this can be resolved into the sum of a number of sinusoidal waveforms, and each of the sine waves can have a different frequency, amplitude and phase.
