﻿VK5AJL - Geometry of waves - multiplication

WAVE GEOMETRY - FREQUENCY MULTIPLICATION
PLAYING WITH FREQUENCY - CIRCUIT TECHNIQUES AND GEOMETRY

GEOMETRY

RULE:- Any repeating wave form can be expressed as a combination of sin waves.

The importance of this rule can't be expressed enough. It is stated in several places on this site and is the basis of many techniques in electronics. This rule can be used to easily multiply frequencies.

There are two very important words to mention before starting. These are "harmonic" and "overtone". (Wikipedia definitions) They are essentially the same thing but are counted differently. Throughout this text, the word "harmonic" will be used because it is less confusing to say the third harmonic is three times the base frequency called the "fundamental".

ODD NUMBERS

If we start with some fundamental frequency and add 1/3 of the 3rd harmonic, we get the following result:-

+=
Figure 1 Adding 1/3 of 3rd harmonic to fundamental

If we now add 1/5th of the 5th harmonic and add we get:-

+=
Figure 2 Adding 1/5 of 5th harmonic to result achieved above

Continuing the process with the 7th harmonic:-

+=
Figure 3 Adding 1/7 of 7th harmonic to result achieved above

If we now continue this process, without the intervening similar diagrams, until the 29th harmonic we get the waveform:-

If this process is continued to infinity, we will end up with a perfect square wave.

ELECTRONICS

Frequency multiplication is done by amplifying a sin wave signal using an amplifier with massive gain to produce a square wave then filtering out the desired harmonic. Note, the square wave was produced by using odd multiples so this technique can only be used for multiplying by odd values. The first step is to find a suitable amplifier. Any sort of amplifier can be used but probably the best is a class AB push/pull because it provides the richest mix of odd harmonics. For ease of circuitry a simple common emitter amplifier has been shown. Whether it is class A, B or C will depend on the bias set by the reisitor values. If the resistors are adjusted so only a small current flows with no signal ie. the output remains close to the supply voltage, it will be class B:-

During the low part of the input cycle, the transistor is turned off and the output goes high. During the other half a cycle, the transistor turns on pulling the output low. All that has to be done now is to filter the output for the desired harmonic and amplify the result using an amplifier with more fidelity. (See filters later.)

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