Relaxation oscillation

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Relaxation Oscillation of Semiconductor Laser

Figure 6: Left: Changes in the amplitude A vs. time. Right: Changes in the number of the carriers n vs. time.

A desirable feature of a laser is the constant amplitude, shown in Figure [6]. Right after the laser turns on, the amplitude varies for a while and then gets stabilized to a constant. We call the frequency before the laser gets stabilized the relaxation oscillation frequency, which we are going to calculate in this section.

The reason we want to obtain the value of the relaxation oscillation frequency is that physically, we easily get resonance when we add a frequency similar to that of the system; in resonance we can obtain large responses.
When we add the term for the TM injection, we will get various different behaviors from the SRE system, depending on the injection strength and the detuning between the injection and the free running laser. When their detuning has a value similar to the relaxation oscillation frequency of the laser, we often observe interesting non-linear behavior.

To calculate the relaxation oscillation frequency( , hereafter), it is convenient to use the laser equation in terms of amplitude.

where . and have damped oscillations as seen in Figure [6]. The frequency changes along with time. The relaxation oscillation frequency is the frequency seen when the system relaxes close to its stable state. So we will find a steady state solution and add a small perturbation in order to calculate the relaxation oscillation frequency. First, when , we have . We can also find the steady state solutions other than this trivial one by letting and . Substituting , the equations can be written as

From Eqns. (31) and (30) , we obtain the following solution.

For ease of computation, that is, in order to transform this complicated system into a simple 2nd order ODE, we changed the variables into non-dimensional quantities.

We can also write the steady state solutions in terms of and .

where and . We differentiate and with respect to and obtain the following differential equation.

Let and be small perturbation terms and add them to and respectively.

Substituting Eqns. (34),(35) into Eqns. (32),(33), the equations are rewritten as

In the equations, since we assume p,q are small, is so small that it can be neglected. Substituting in, we obtain

Then we differentiate Eqn. (37) with respect to and substitute p from Eqn. (37) again to obtain the 2nd order differential equation of with respect to .

The equation above is dimensionless. But the value of has dimension. So We write the equation in terms of t again to obtain an ODE in terms of t.

The solution for the above ODE is

where is

To calculate a numerical value, we put and . The for our model is

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Relaxation Oscillation

• Tianshou Zhou
Reference work entry

DOI: https://doi.org/10.1007/978-1-4419-9863-7_521

How to cite

Definition

Relaxation oscillation is also a specific type of oscillation. An oscillator that exhibits relaxation oscillation is called relaxation oscillator (referring the following Fig. 1). A relaxation oscillator is an oscillator based upon the behavior of a physical system’s return to equilibrium after being disturbed. That is, a dynamical system within the oscillator continuously dissipates its internal energy. Normally, the system would return to its natural equilibrium; however, each time the system reaches some threshold sufficiently close to its equilibrium, a mechanism disturbs it with additional energy. Hence, the oscillator’s behavior is characterized by long periods of dissipation followed by short impulses. The period of the oscillations are set by the time it takes for the system to relax from each disturbed state to the threshold that triggers the next disturbance.

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Authors and Affiliations

1. 1.School of Mathematics and Computational SciencesSun Yet-Sen UniversityGuangzhouChina

How to cite

Cite this entry as:
Zhou T. (2013) Relaxation Oscillation. In: Dubitzky W., Wolkenhauer O., Cho KH., Yokota H. (eds) Encyclopedia of Systems Biology. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9863-7_521

Sours: https://www.merriam-webster.com/dictionary/relaxation%20oscillation
Limit cycles, part 4: van der Pol equation, strongly nonlinear limit - relaxation oscillations

Relaxation oscillator

The blinking turn signalon motor vehicles is generated by a simple relaxation oscillator powering a relay.

In electronics a relaxation oscillator is a nonlinearelectronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave.[1][2][3][4] The circuit consists of a feedback loop containing a switching device such as a transistor, comparator, relay,[5]op amp, or a negative resistance device like a tunnel diode, that repetitively charges a capacitor or inductor through a resistance until it reaches a threshold level, then discharges it again.[4][6] The period of the oscillator depends on the time constant of the capacitor or inductor circuit.[2] The active device switches abruptly between charging and discharging modes, and thus produces a discontinuously changing repetitive waveform.[2][4] This contrasts with the other type of electronic oscillator, the harmonic or linear oscillator, which uses an amplifier with feedback to excite resonant oscillations in a resonator, producing a sine wave.[7] Relaxation oscillators are used to produce low frequency signals for applications such as blinking lights (turn signals) and electronic beepers and in voltage controlled oscillators (VCOs), inverters and switching power supplies, dual-slope analog to digital converters, and function generators.

The term relaxation oscillator is also applied to dynamical systems in many diverse areas of science that produce nonlinear oscillations and can be analyzed using the same mathematical model as electronic relaxation oscillators.[8][9][10][11] For example, geothermal geysers,[12][13] networks of firing nerve cells,[11]thermostat controlled heating systems,[14] coupled chemical reactions,[9] the beating human heart,[11][14] earthquakes,[12] the squeaking of chalk on a blackboard,[14] the cyclic populations of predator and prey animals, and gene activation systems[9] have been modeled as relaxation oscillators. Relaxation oscillations are characterized by two alternating processes on different time scales: a long relaxation period during which the system approaches an equilibrium point, alternating with a short impulsive period in which the equilibrium point shifts.[11][12][13][15] The period of a relaxation oscillator is mainly determined by the relaxation time constant.[11] Relaxation oscillations are a type of limit cycle and are studied in nonlinear control theory.[16]

Electronic relaxation oscillators

A vacuum tube Abraham-Bloch multivibrator relaxation oscillator, France, 1920 (small box, left). Its harmonics are being used to calibrate a wavemeter (center).
Original vacuum tubeAbraham-Bloch multivibrator oscillator, from their 1919 paper

The first relaxation oscillator circuit, the astable multivibrator, was invented by Henri Abraham and Eugene Bloch using vacuum tubes during World War I.[17][18]Balthasar van der Pol first distinguished relaxation oscillations from harmonic oscillations, originated the term "relaxation oscillator", and derived the first mathematical model of a relaxation oscillator, the influential Van der Pol oscillator model, in 1920.[18][19][20] Van der Pol borrowed the term relaxation from mechanics; the discharge of the capacitor is analogous to the process of stress relaxation, the gradual disappearance of deformation and return to equilibrium in an inelastic medium.[21] Relaxation oscillators can be divided into two classes[13]

• Sawtooth, sweep, or flyback oscillator: In this type the energy storage capacitor is charged slowly but discharged rapidly, essentially instantly, by a short circuit through the switching device. Thus there is only one "ramp" in the output waveform which takes up virtually the entire period. The voltage across the capacitor is a sawtooth wave, while the current through the switching device is a sequence of short pulses.
• Astable multivibrator: In this type the capacitor is both charged and discharged slowly through a resistor, so the output waveform consists of two parts, an increasing ramp and a decreasing ramp. The voltage across the capacitor is a triangle waveform, while the current through the switching device is a square wave.

Applications

Relaxation oscillators are generally used to produce low frequency signals for such applications as blinking lights, and electronic beepers. and clock signals in some digital circuits. During the vacuum tube era they were used as oscillators in electronic organs and horizontal deflection circuits and time bases for CRT oscilloscopes; one of the most common was the Miller integrator circuit invented by Alan Blumlein, which used vacuum tubes as a constant current source to produce a very linear ramp.[22] They are also used in voltage controlled oscillators (VCOs),[23]inverters and switching power supplies, dual-slope analog to digital converters, and in function generators to produce square and triangle waves. Relaxation oscillators are widely used because they are easier to design than linear oscillators, are easier to fabricate on integrated circuit chips because they do not require inductors like LC oscillators,[23][24] and can be tuned over a wide frequency range.[24] However they have more phase noise[23] and poorer frequency stability than linear oscillators.[2][23] Before the advent of microelectronics, simple relaxation oscillators often used a negative resistance device with hysteresis such as a thyratron tube,[22]neon lamp,[22] or unijunction transistor, however today they are more often built with dedicated integrated circuits such as the 555 timer chip.

Pearson–Anson oscillator

Circuit diagramof a capacitive relaxation oscillator with a neon lamp threshold device

Main article: Pearson-Anson effect

This example can be implemented with a capacitive or resistive-capacitive integrating circuit driven respectively by a constant current or voltage source, and a threshold device with hysteresis (neon lamp, thyratron, diac, reverse-biased bipolar transistor,[25] or unijunction transistor) connected in parallel to the capacitor. The capacitor is charged by the input source causing the voltage across the capacitor to rise. The threshold device does not conduct at all until the capacitor voltage reaches its threshold (trigger) voltage. It then increases heavily its conductance in an avalanche-like manner because of the inherent positive feedback, which quickly discharges the capacitor. When the voltage across the capacitor drops to some lower threshold voltage, the device stops conducting and the capacitor begins charging again, and the cycle repeats ad infinitum.

If the threshold element is a neon lamp,[nb 1][nb 2] the circuit also provides a flash of light with each discharge of the capacitor. This lamp example is depicted below in the typical circuit used to describe the Pearson–Anson effect. The discharging duration can be extended by connecting an additional resistor in series to the threshold element. The two resistors form a voltage divider; so, the additional resistor has to have low enough resistance to reach the low threshold.

Alternative implementation with 555 timer

A similar relaxation oscillator can be built with a 555 timer IC (acting in astable mode) that takes the place of the neon bulb above. That is, when a chosen capacitor is charged to a design value, (e.g., 2/3 of the power supply voltage) comparators within the 555 timer flip a transistor switch that gradually discharges that capacitor through a chosen resistor (RC Time Constant) to ground. At the instant the capacitor falls to a sufficiently low value (e.g., 1/3 of the power supply voltage), the switch flips to let the capacitor charge up again. The popular 555's comparator design permits accurate operation with any supply from 5 to 15 volts or even wider.

Other, non-comparator oscillators may have unwanted timing changes if the supply voltage changes.

Inductive oscillator

Basis of solid-state Blocking oscillator

A blocking oscillator using the inductive properties of a pulse transformer to generate square waves by driving the transformer into saturation, which then cuts the transformer supply current until the transformer unloads and desaturates, which then triggers another pulse of supply current, generally using a single transistor as the switching element.

Comparator–based relaxation oscillator

Alternatively, when the capacitor reaches each threshold, the charging source can be switched from the positive power supply to the negative power supply or vice versa. This case is shown in the comparator-based implementation here.

A comparator-based hysteretic oscillator.

This relaxation oscillator is a hysteretic oscillator, named this way because of the hysteresis created by the positive feedback loop implemented with the comparator (similar to an operational amplifier). A circuit that implements this form of hysteretic switching is known as a Schmitt trigger. Alone, the trigger is a bistable multivibrator. However, the slow negative feedback added to the trigger by the RC circuit causes the circuit to oscillate automatically. That is, the addition of the RC circuit turns the hysteretic bistable multivibrator into an astable multivibrator.

General concept

The system is in unstable equilibrium if both the inputs and outputs of the comparator are at zero volts. The moment any sort of noise, be it thermal or electromagneticnoise brings the output of the comparator above zero (the case of the comparator output going below zero is also possible, and a similar argument to what follows applies), the positive feedback in the comparator results in the output of the comparator saturating at the positive rail.

In other words, because the output of the comparator is now positive, the non-inverting input to the comparator is also positive, and continues to increase as the output increases, due to the voltage divider. After a short time, the output of the comparator is the positive voltage rail, .

The inverting input and the output of the comparator are linked by a seriesRC circuit. Because of this, the inverting input of the comparator asymptotically approaches the comparator output voltage with a time constant RC. At the point where voltage at the inverting input is greater than the non-inverting input, the output of the comparator falls quickly due to positive feedback.

This is because the non-inverting input is less than the inverting input, and as the output continues to decrease, the difference between the inputs gets more and more negative. Again, the inverting input approaches the comparator's output voltage asymptotically, and the cycle repeats itself once the non-inverting input is greater than the inverting input, hence the system oscillates.

Example: Differential equation analysis of a comparator-based relaxation oscillator

Transient analysis of a comparator-based relaxation oscillator.

is set by across a resistive voltage divider:

is obtained using Ohm's law and the capacitordifferential equation:

Rearranging the differential equation into standard form results in the following:

Notice there are two solutions to the differential equation, the driven or particular solution and the homogeneous solution. Solving for the driven solution, observe that for this particular form, the solution is a constant. In other words, where A is a constant and .

Using the Laplace transform to solve the homogeneous equation results in

is the sum of the particular and homogeneous solution.

Solving for B requires evaluation of the initial conditions. At time 0, and . Substituting into our previous equation,

Frequency of oscillation

First let's assume that for ease of calculation. Ignoring the initial charge up of the capacitor, which is irrelevant for calculations of the frequency, note that charges and discharges oscillate between and . For the circuit above, Vss must be less than 0. Half of the period (T) is the same as time that switches from Vdd. This occurs when V charges up from to .

When Vss is not the inverse of Vdd we need to worry about asymmetric charge up and discharge times. Taking this into account we end up with a formula of the form:

Which reduces to the above result in the case that .

Notes

1. ^When a (neon) cathode glow lamp or thyratron are used as the trigger devices a second resistor with a value of a few tens to hundreds ohms is often placed in series with the gas trigger device to limit the current from the discharging capacitor and prevent the electrodes of the lamp rapidly sputtering away or the cathode coating of the thyratron being damaged by the repeated pulses of heavy current.
2. ^Trigger devices with a third control connection, such as the thyratron or unijunction transistor allow the timing of the discharge of the capacitor to be synchronized with a control pulse. Thus the sawtooth output can be synchronized to signals produced by other circuit elements as it is often used as a scan waveform for a display, such as a cathode ray tube.

References

1. ^Graf, Rudolf F. (1999). Modern Dictionary of Electronics. Newnes. p. 638. ISBN .
2. ^ abcdEdson, William A. (1953). Vacuum Tube Oscillators(PDF). New York: John Wiley and Sons. p. 3. on Peter Millet's Tubebooks website
3. ^Morris, Christopher G. Morris (1992). Academic Press Dictionary of Science and Technology. Gulf Professional Publishing. p. 1829. ISBN .
4. ^ abcDu, Ke-Lin; M. N. S. Swamy (2010). Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies. Cambridge Univ. Press. p. 443. ISBN .
5. ^Varigonda, Subbarao; Tryphon T. Georgiou (January 2001). "Dynamics of Relay Relaxation Oscillators"(PDF). IEEE Transactions on Automatic Control. Inst. of Electrical and Electronic Engineers. 46 (1): 65. doi:10.1109/9.898696. Retrieved February 22, 2014.
6. ^Nave, Carl R. (2014). "Relaxation Oscillator Concept". HyperPhysics. Dept. of Physics and Astronomy, Georgia State Univ. Retrieved February 22, 2014.
7. ^Oliveira, Luis B.; et al. (2008). Analysis and Design of Quadrature Oscillators. Springer. p. 24. ISBN .
8. ^DeLiang, Wang (1999). "Relaxation oscillators and networks"(PDF). Wiley Encyclopedia of Electrical and Electronics Engineering, Vol. 18. Wiley & Sons. pp. 396–405. Retrieved February 2, 2014.
9. ^ abcSauro, Herbert M. (2009). "Oscillatory Circuits"(PDF). Class notes on oscillators: Systems and Synthetic Biology. Sauro Lab, Center for Synthetic Biology, University of Washington. Retrieved November 12, 2019.,
10. ^Letellier, Christopher (2013). Chaos in Nature. World Scientific. pp. 132–133. ISBN .
11. ^ abcdeGinoux, Jean-Marc; Letellier, Christophe (June 2012). "Van der Pol and the history of relaxation oscillations: toward the emergence of a concept". Chaos. American Institute of Physics. 22 (2): 023120. arXiv:1408.4890. Bibcode:2012Chaos..22b3120G. doi:10.1063/1.3670008. PMID 22757527. Retrieved December 24, 2014.
12. ^ abcEnns, Richard H.; George C. McGuire (2001). Nonlinear Physics with Mathematica for Scientists and Engineers. Springer. p. 277. ISBN .
13. ^ abcPippard, A. B. (2007). The Physics of Vibration. Cambridge Univ. Press. pp. 359–361. ISBN .
14. ^ abcPippard, The Physics of Vibration, p. 41-42
15. ^Kinoshita, Shuichi (2013). "Introduction to Nonequilibrium Phenomena". Pattern Formations and Oscillatory Phenomena. Newnes. p. 17. ISBN . Retrieved February 24, 2014.
16. ^see Ch. 9, "Limit cycles and relaxation oscillations" in Leigh, James R. (1983). Essentials of Nonlinear Control Theory. Institute of Electrical Engineers. pp. 66–70. ISBN .
17. ^Abraham, H.; E. Bloch (1919). "Mesure en valeur absolue des périodes des oscillations électriques de haute fréquence (Measurement of the periods of high frequency electrical oscillations)". Annales de Physique. Paris: Société Française de Physique. 9 (1): 237–302. doi:10.1051/jphystap:019190090021100.
18. ^ abGinoux, Jean-Marc (2012). "Van der Pol and the history of relaxation oscillations: Toward the emergence of a concepts". Chaos 22 (2012) 023120. arXiv:1408.4890. Bibcode:2012Chaos..22b3120G. doi:10.1063/1.3670008.
19. ^van der Pol, B. (1920). "A theory of the amplitude of free and forced triode vibrations". Radio Review. 1: 701–710, 754–762.
20. ^van der Pol, Balthasar (1926). "On Relaxation-Oscillations". The London, Edinburgh, and Dublin Philosophical Magazine 2. 2: 978–992. doi:10.1080/14786442608564127.
21. ^Shukla, Jai Karan N. (1965). "Discontinuous Theory of Relaxation Oscillators". Master of Science thesis. Dept. of Electrical Engineering, Kansas State Univ. Retrieved February 23, 2014.
22. ^ abcPuckle, O. S. (1951). Time Bases (Scanning Generators), 2nd Ed. London: Chapman and Hall, Ltd. pp. 15–27.
23. ^ abcdAbidi, Assad A.; Robert J. Meyer (1996). "Noise in Relaxation Oscillators". Monolithic Phase-Locked Loops and Clock Recovery Circuits: Theory and Design. John Wiley and Sons. p. 182. Retrieved 2015-09-22.
24. ^ abvan der Tang, J.; Kasperkovitz, Dieter; van Roermund, Arthur H.M. (2006). High-Frequency Oscillator Design for Integrated Transceivers. Springer. p. 12. ISBN .
25. ^http://members.shaw.ca/roma/twenty-three.html
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Relaxation Oscillations

Simulation of Relaxation Oscillations

Relaxation oscillations can be numerically simulated with the software RP Fiber Power, which can not only calculate steady-state solutions, but also perform dynamical simulations.Arbitrary time dependencies e.g. of the pump power are possible.One could also simulate the effect of electronic feedback systems for suppressing those oscillations.

Definition: small mutually coupled oscillations of the laser power and laser gain around their steady-state values

German: Relaxationsoszillationen

Category: laser devices and laser physics

How to cite the article; suggest additional literature

Author: Dr. Rüdiger Paschotta

URL: https://www.rp-photonics.com/relaxation_oscillations.html

When a laser is disturbed during operation, e.g. by fluctuations of the pump power, its output power does not immediately return to its steady state. Many lasers (e.g. solid-state lasers and most laser diodes) operate in the so-called class B regime, with the upper-state lifetime often being much longer than the laser resonator's damping time. In that regime, the laser dynamics are such that changes in pump power lead to so-called relaxation oscillations. These are usually damped, eventually leading back to the steady state. Particularly pronounced oscillatory behavior with relatively low oscillation frequencies (often in the kilohertz regime) occurs in doped insulatorsolid-state lasers, whereas semiconductor lasers normally exhibit strongly damped relaxation oscillations with very high frequencies in the gigahertz region. Other lasers, e.g. many gas lasers, operating in the class A regime with an upper-state lifetime below the cavity damping time, do not exhibit relaxation oscillations, but only an exponential relaxation to the steady state.

As Figure 1 shows, class B lasers can exhibit strong spiking e.g. when the pump power is suddenly turned on. After the emission of a few spikes (pulses), the laser power exhibits damped relaxation oscillations. The oscillation frequency is similar to the inverse period of the spikes.

Calculations of relaxation phenomena can be based on the dynamic equations as presented in the article on laser dynamics, which can (for small fluctuations, not for spiking) be linearized around the steady state. In the following, the main results of such an analysis for class-B lasers are given. The frequency of the relaxation oscillations is determined by the intracavity power Pint, the resonator losses ρ, the round-trip time TR of the resonator, and the saturation energyEsat and the upper-state lifetimeτg of the laser gain medium:

The cavity damping time corresponds to TR / ρ, and the first term in the radicand is larger than the second one in the mentioned class B regime. For solid-state lasers (with τg >> TR), the second term of the radicand is negligible (except for operation close to the laser threshold), so that the equation simplifies to

The equations are valid for both four-level and three-level laser gain media. However, only for four-level gain media can the former equation be transformed into

where r is the so-called pump parameter, which is the ratio of pump power to threshold pump power.

The damping time of the oscillations can be calculated from

For operation just above the laser threshold, the relaxation oscillations are slow, and their damping time is about twice the upper-state lifetime of the gain medium. For higher powers, the oscillations can faster, and the damping time gets shorter. For four-level lasers, the damping time is inversely proportional to the pump parameter r.

Note that a saturable absorber in the laser resonator, which may be used for passive mode locking, can strongly reduce the damping [2]; the oscillations can even become undamped, so that the steady state becomes unstable. This leads to the phenomenon of Q-switching instabilities and Q-switched mode locking.

The characterization of the laser dynamics can deliver useful information on the laser parameters such as the resonator losses or the gain saturation energy, thus also the transition cross sections of the laser gain medium.

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Bibliography

 [1] K. J. Weingarten et al., “In situ small-signal gain of solid-state lasers determined from relaxation oscillation frequency measurements”, Opt. Lett. 19 (15), 1140 (1994), doi:10.1364/OL.19.001140 [2] A. Schlatter et al., “Pulse-energy dynamics of passively mode-locked solid-state lasers above the Q-switching threshold”, J. Opt. Soc. Am. B 21 (8), 1469 (2004), doi:10.1364/JOSAB.21.001469 [3] A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) [4] O. Svelto, Principles of Lasers, Plenum Press, New York (1998)

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Suppression of Relaxation Oscillation Dynamics in Semiconductor Lasers with External Feedback

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