# Tilting oscillation

A special form of periodic, non-sinusoidal oscillations is referred to as tilting oscillation or sawtooth oscillation . In contrast to the harmonic oscillation , in which the reciprocating movement takes place symmetrically, in the tilting oscillation a slow charge is followed by a very fast discharge, which is typical for a process in which the discharge is triggered all at once by reaching a threshold value. According to the appearance of its graphic representation, it is also called “sawtooth oscillation”. The curve of the relaxation oscillation is generally ascending; That is, the signal rises continuously and then falls abruptly.

Schematic graph of a tilting oscillation

## Mathematical description

Sawtooth oscillation as a superposition of different numbers of harmonics. ( Fourier synthesis )
(even and odd) harmonics of a sawtooth oscillation of 1000 Hz

The ideal breakover oscillation can be expressed as a section-wise continuous and linear function with the parameter t without scaling factors as

${\ displaystyle x (t) = t- \ lfloor t \ rfloor \,}$

where the expression represents the Gaussian bracket (rounding function). In English, the term floor ( t ) is also used for this. ${\ displaystyle \ lfloor t \ rfloor}$

Like any periodic function , this sawtooth curve can be expressed in an equivalent representation using a Fourier series :

${\ displaystyle x (t) = - {\ frac {1} {\ pi}} \ cdot \ sum _ {k = 1} ^ {\ infty} {\ frac {\ sin (2 \ pi kt)} {k }} + {\ frac {1} {2}}}$ or more generally and centered around the zero point:
${\ displaystyle x (t) = - {\ frac {2c} {\ pi}} \ cdot \ sum _ {k = 1} ^ {\ infty} {\ frac {\ sin (2 \ pi kft)} {k }}}$

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

In real systems, due to the band limitation, only a finite number of summands occurs. The sum of the harmonic oscillations then results in a distorted sawtooth curve. This effect is graphically illustrated in the figure opposite with only a finite number of sinusoidal oscillations.

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

## Applications

Audio sample for a sawtooth wave

A special feature of this form of oscillation is that it theoretically contains all integer multiples of a fundamental frequency in the harmonics , which are called harmonics . The large spectrum has general advantages when used in electronic musical instruments such as, above all, the electronic organ with subtractive sound synthesis . By filtering the sawtooth vibration, different overtones can be preferred from a single generated vibration, as required for the timbres of the organ, e.g. B. more towards trumpet or flute sound. Typical tilting vibrations are also generated in string instruments .

In addition, the tilting oscillation is used in oscilloscopes to deflect the electron beam horizontally. In cathode ray tube screens (for example in conventional CRT televisions ) the electron beam is driven in both the horizontal and vertical directions with a tilting oscillation, the frequency of the horizontal deflection usually being significantly higher.

## Realization examples

A simple example of a tilting vibration is a conical container (bucket) that is rotatably suspended above its center of gravity when empty and that is gradually filled with water. If the content reaches the height at which the center of gravity is above the pivot point , the container tips over and the water runs out.

Another application is the ram , a simple water pump.

In electronics , tilting vibrations can be generated with the help of a glow lamp or by more complex circuits such as blocking oscillators , Miller-Transitron , sawtooth generators or tilting oscillators .