Sawtooth wave

As tilting vibration or sawtooth wave is referred to a specific form of periodic non- sinusoidal oscillations. In contrast to harmonic oscillation, which run reciprocation symmetric, follows in the wagging of a slow charging a very fast discharge, which is typical of a process in which the discharge is triggered by a time by reaching a threshold. According to the appearance of their graphical representation it is also called " sawtooth ". The curve of the deflecting oscillation is generally low to high, i.e. that the signal is continuously increases and then drops abruptly.

Mathematical Description

The ideal tilting vibration can be expressed without scaling factors as a sectionally continuous and linear function with the parameter t as

Where the term represents the floor function ( rounding function). In English, the term floor ( t) is used therefor.

Like any periodic function can be expressed by Fourier series and the sawtooth waveform in an equivalent representation to

It is worth mentioning by a scaling factor c ≠ 0, f that occur in the sawtooth even-numbered and odd-numbered multiples of the fundamental frequency in the spectrum.

In real systems, passed through the band limiting to only a finite number of summands. The sum of harmonic oscillations then results in a distorted sawtooth. This effect is illustrated graphically in the adjacent figure with only a finite number of sinusoids.

In the area of the discrete signal processing, it is sufficient to limit the number of harmonics to be determined half the sampling rate.

Applications

A peculiarity of this waveform is that it theoretically contains all integer multiples of a fundamental frequency in the harmonics, which are called harmonics. The large spectrum generally has advantages when used in electronic instruments, such as in particular of the electronic organ with the subtractive synthesis of sound. By filtering the sawtooth different overtones may be preferred from a single vibration generated as they are needed for the colors of the organ, such as trumpet or flute direction rather sound. Typical relaxation oscillations are generated, among other things also on the strings

In addition, the tilting vibration is used in oscilloscope for horizontal deflection of the electron beam. In cathode ray tube screens (such as in conventional CRT television ), the electron beam is controlled in both the horizontal, and vertical direction with a deflecting oscillation, the frequency of the horizontal deflection is significantly higher in the control.

Examples

A simple example of a deflecting oscillation is an asymmetrical container, which is rotatably mounted and is uniformly filled with water. Achieved the content of a certain limit, tipping the container and the water from running.

In the electronic relaxation oscillations can be generated using a neon lamp or by more complex circuits such as blocking oscillator, Miller Transitron, sawtooth generator or relaxation oscillator.