# oscillator

An oscillator (from the Latin oscillare ' to rock' ) is a system that can vibrate . This means that it usually enables a temporal oscillation of its state variables . Oscillation means that there is a constant change between two states or around a central point, which usually corresponds to the rest position of the system.

If the behavior of the oscillator can be described with differential equations , then from a mathematical point of view it is a dynamic system . Such a system is called an oscillator if it has a stable limit cycle . A state in which a limit cycle is reached is called a steady state . In such a state, the oscillation of the oscillator is necessarily periodic .

Oscillators are mainly found in electrical engineering or electronics and mechanics . However, systems with periodic behavior are also known from other areas of technical time systems , in chemistry , in biology and in sociology .

Vibrations in mechanical or electrical systems are always dampened without additional measures . This means that the amplitude of the oscillation decreases over time if no external energy is actively added. An oscillator therefore always has a device for supplying energy. This can be done for example by mechanical force, as in a clockwork , or by electrical voltage.

The balance wheel of a watch which, together with the armature and steering wheel, forms an oscillator
Watch crystal , which together with an amplifier a quartz oscillator forms

## Mathematical definition

Phase space of the van der Pol oscillator with trajectories (blue lines). States that start near the limit cycle (thin lines) approach the limit cycle (bold line) for t → ∞.

Consider a system of ordinary differential equations

${\ displaystyle {\ frac {\ mathrm {d} x} {\ mathrm {d} t}} = f (x), \ quad x \ in \ mathbb {R} ^ {n}, \ quad n \ geq 2 }$

or

${\ displaystyle {\ frac {\ mathrm {d} x} {\ mathrm {d} t}} = f (x, t), \ quad x \ in \ mathbb {R} ^ {n}, \ quad n \ geq 1}$

with a smooth function . The quantity is the state of a physical system. The set of all states is called the state or phase space . The input variable can be viewed as time or, more generally, can also be selected from . A solution or trajectory is periodic if there is a constant such that holds ${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ times \ mathbb {R} \ to \ mathbb {R} ^ {n}}$${\ displaystyle x}$${\ displaystyle t \ in \ mathbb {R}}$${\ displaystyle \ mathbb {R} ^ {p}}$${\ displaystyle x_ {0} (t)}$${\ displaystyle T> 0}$

${\ displaystyle x_ {0} (t) = x_ {0} (t + T)}$.

The constant is the period , the reciprocal is the frequency of the oscillation. The set of states (or the flow ) of such a solution is a periodic orbit , also called an orbital or limit cycle. The system under consideration is called an oscillator if there is an asymptotically orbitally stable orbit. This means that a trajectory that is sufficiently close to the periodic orbit also remains sufficiently close for all or, more precisely, fulfills the following conditions: ${\ displaystyle T}$ ${\ displaystyle 1 / T}$${\ displaystyle t = 0}$${\ displaystyle t> 0}$

• For every value there exists such that for is true for all .${\ displaystyle \ varepsilon> 0}$${\ displaystyle \ delta> 0}$${\ displaystyle | x (0) -x_ {0} (0) | <\ delta}$${\ displaystyle | x (t) -x_ {0} (t) | <\ varepsilon}$${\ displaystyle t> 0}$
• There is an asymptotic phase shift , so that applies .${\ displaystyle \ phi}$${\ displaystyle \ lim _ {t \ to \ infty} \ left | x (t) -x_ {0} (t + \ phi) \ right | \ to 0}$

## Oscillators in Physics

The Morse potential (blue line) for modeling a diatomic molecule can be approximated for small energies using a harmonic oscillator (green line). The horizontal thin lines mark the energy levels ν = 0… 6 of the quantum mechanical oscillator

Asymptotic stability means attractive and Lyapunov stable . The former applies to systems whose energy approaches a limit value, the latter to systems whose energy is conserved . Conservation of energy means that no work is done when moving along a closed path . These systems are called conservative . As can be seen, for example, from the model of dynamic billiards , not all conservative systems have a stable periodic orbit and are therefore an oscillator.

Due to the conservation of energy, the force field of a conservative system can be described by a potential . An energy can thus be assigned to each periodic orbit. This model can be used for an electrically charged particle that moves in an electrical potential . In quantum mechanics , this model can be used for B. Calculate atomic orbitals . Classically, the state of the oscillator is determined by the deflection of the particle from its rest position and its speed or its momentum . ${\ displaystyle x = (q, p)}$ ${\ displaystyle q}$ ${\ displaystyle p}$

On closer inspection, almost all real oscillators are anharmonic . However, they can often be approximated using the model of a harmonic oscillator :

{\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {d} q} {\ mathrm {d} t}} & = p \\ {\ frac {\ mathrm {d} p} {\ mathrm {d } t}} & = - \ omega ^ {2} q \\\ end {aligned}}}

Here, the particle mass is de-dimensioned 1 selected so that

• ${\ displaystyle \ omega / (2 \ pi)}$ the frequency,
• ${\ displaystyle \ langle p ^ {2} \ rangle = \ omega ^ {2} \ langle q ^ {2} \ rangle}$the total energy

of an orbit. The brackets stand for the mean value over time or the quantum mechanical expected value . The total energy follows from the equipartition theorem or virial theorem for any oscillator. In quantum mechanics only are for the total energy energy levels with allowed. The constant is Planck's quantum of action . ${\ displaystyle \ langle \; \ rangle}$ ${\ displaystyle \ hbar \ omega (\ nu + {\ tfrac {1} {2}})}$${\ displaystyle \ nu = 0.1, \ dotsc}$${\ displaystyle 2 \ pi \ hbar}$

## Oscillators in electronics

An oscillator in electronics generates undamped, mostly sinusoidal electrical oscillations. It works on direct voltage and generates alternating voltage and can consist of a single self-oscillating component or of several components that are joined together to form an oscillator circuit. These components must therefore have a gain> 1 (output amplitude greater than input amplitude) and amplify the amplitude of the vibration signal until a physical limitation occurs. This ultimately leads to a stable output signal.

The requirements placed on oscillators are constancy of the output signal in terms of frequency and amplitude and low temperature dependence. Some oscillators are used to generate alternating voltage or to convert voltage with high efficiency (for example magnetron , Royer oscillator ).

An oscillator always contains frequency-determining components, a limitation of the amplitude and a negative differential resistance . This is implemented either by a feedback amplifier or by a component with a negative differential resistance such as a tunnel diode or lambda diode .

The amplitude is limited by passive or active measures. There can be an amplitude control (typically e.g. with RC oscillators ), but in most cases the property of the circuit itself will be sufficient to limit the amplitude (operating point shift, limitation to non-linear characteristics, decrease in voltage gain as the amplitude increases).

The frequency-determining components of electronic oscillators can be:

## Examples (selection)

Potassium titanyl phosphate crystals in the resonator of an optical parametric oscillator . With such an oscillator an optical signal can be amplified and the energy or frequency of the photons can be changed.