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# Describing the harmonics of a signal

Thursday, August 14, 2014

Q. I am studying Lesson 2405 on amplifiers. In it there is a description of the harmonics of a signal. I don’t understand this.

A. A signal that is not a perfect sine wave can be broken down into a group of sine waves. These sine waves are called the spectrum of the signal. The frequencies of the sine waves are related in a special way. The one with the lowest frequency is called the fundamental. It has the same frequency as the original signal.

Next comes a sine wave at twice the frequency of the fundamental. This is called the second harmonic. Above this there is the third harmonic, a sine wave whose frequency is three times that of the fundamental. There are harmonics at 4, 5, 6 and 7 times the fundamental frequency, and in principle this continues on up to infinity. Note that there are no fractional-number harmonics; they are all integer multiples.

The shape of the signal determines the strengths of the individual harmonics. Usually the fundamental is the strongest, and the strengths get smaller as we go up in frequency. We find certain waveforms occurring very often in electronics.

The relative strengths of the harmonics for these waveforms are well known, and are listed in electronics reference books. For example, the harmonics for a square wave are all odd multiples of the fundamental. There are no harmonics for even multiples of the fundamental. The spectra for triangle, sawtooth, variable-width pulse, and rectified sine waves are all cataloged in reference books.

The big reason we are interested in the spectrum of a signal is that all the harmonics must be present, in the right strengths, to reproduce the signal’s waveform accurately. In an audio amplifier, We want the sound coming from the amplifier to be a good copy of the input signal. This referred to as the amplifier’s fidelity.

An amplifier has “high fidelity” if it produces an accurate copy of the input signal. In a video ampliñer, if the signal is not accurately reproduced, the picture will be smeared, blurry, or lack detail. If an ampliñer does not change the relative strengths of the fundamental and harmonics in a signal, it will accurately reproduce the signal.

In practical terms, it is obviously impossible to amplify all the harmonics of a signal up to infinity. So an upper limit is chosen. The signal waveform is considered good enough if the fundamental and all the harmonics up to a given number are amplified by the same amount. In some cases it may only be necessary to go up to the 10th harmonic (ten times the fundamental frequency.) In others we may need all the harmonics up to the 100th. This would be the case for an oscilloscope’s internal amplifier handling a square wave. For example, if the scope were to be used to display the shape of a 1 MHZ square wave accurately, it should be able to handle all the frequencies up to 100 MHZ.

The spectrum of a signal can be derived mathematically by using integral calculus. There are also instruments which allow you to see the spectrum of the signal on a CRT. These are called spectrum analyzers. There are analog spectrum analyzers, which are mostly used in radio-frequency work like cable TV. There are also digital spectrum analyzers, which are often used for lower-frequency work.