Waveform harmonics

•         Consider that an instantaneous voltage v be represented by  the formula

v = Vm sin 2πft volts.

This is a waveform which varies sinusoidally with time t, has a frequency f, and a maximum value represented by Vm. It is normally assumed that alternating voltages have wave shapes which are sinusoidal where only 1 frequency is present. If the waveform is not happened to be sinusoidal it is called a complex wave, & whatever its shape is, itmay be split up mathematically into components called the fundamental and a number of harmonics.This process is referred to as harmonic analysis.The fundamental, which is the first harmonic, is sinusoidal and has the supply frequency, f ; the other harmonics are also sine waves having frequencies which are integer multiples of f . Thus, if supply frequency is fifty hertz, then the third harmonic frequency is 150Hz, the fifth 250Hz, & so on.

·        A multifaceted waveform comprising the sum of the fundamental & a third harmonic of about half the amplitude of the fundamental is shown in diagram (a) below, both waveforms being at the start in phase with each other. If more odd harmonic waveforms of the appropriate amplitudes are added, a good approximation to a square wave results. In diagram (b), the third harmonic is shown having an initial phase displacement from the fundamental. The +ve & -ve half cycles of each of multifaceted wave-forms shown in diagram (a) and (b) are indistinguishable in shape, and this is a aspect of wave-forms holding the fundamental & only odd harmonics.

·        A complex waveform consisting of the sum of the fundamental and a 2nd harmonic of about half the amplitude of the fundamental is publicized in diagram (c), each waveform being originally in phase with each other. If further even harmonics of proper amplitudes are added a good approximation to a tri-angular wave results.

DIA

In diagram (c) the negative cycle appears just like a mirror image of the +ve cycle about point-A. In diagram-(d) the 2nd harmonic is represented with an initial phase displacement fromthe fundamental and the positive and negative half-cycles are not alike.

·        A complex waveform consisting of the sum of the fundamental, a 2nd harmonic and a 3rd harmonicis shown in Figure (e), each wave-form being at first ‘in-phase’. The negative half cycle appears as a mirror image of the positive cycle about point B. In Figure(f), a complex wavef-orm comprising the sum of the fundamental,a second harmonic and a third harmonic are shown with initial phase displacement. The positive and negative half cycles are seen to be not identical.The features mentioned relative to diagram(a) to (f ) enable to recognize the harmonics that are present in a complex wave-form displayed on an oscilloscope.