Relaxation oscillator

Inelectronics,arelaxation oscillatoris anonlinearelectronic oscillatorcircuit that produces anonsinusoidalrepetitive output signal, such as atriangle waveorsquare wave.The circuit consists of afeedback loopcontaining a switching device such as atransistor,comparator,relay,op amp,or anegative resistancedevice like atunnel diode,that repetitively charges acapacitororinductorthrough a resistance until it reaches a threshold level, then discharges it again.Theperiodof the oscillator depends on thetime constantof the capacitor or inductor circuit.The active device switches abruptly between charging and discharging modes, and thus produces a discontinuously changing repetitive waveform.This contrasts with the other type of electronic oscillator, the harmonic orlinear oscillator,which uses anamplifierwith feedback to exciteresonantoscillations in aresonator,producing asine wave.

Simple relaxation oscillator made byfeeding backan invertingSchmitt trigger's output voltage through aRC networkto its input.

The blinkingturn signalon some motor vehicles is generated by a simple relaxation oscillator powering arelay.

Relaxation oscillators may be used for a wide range of frequencies, but as they are one of the oscillator types suited to low frequencies, below audio, they are typically used for applications such as blinking lights (turn signals) andelectronic beepers,as well asvoltage controlled oscillators(VCOs),inverters,switching power supplies,dual-slope analog to digital converters,andfunction generators.

The termrelaxation oscillator,though often used in electronics engineering, is also applied todynamical systemsin many diverse areas of science that produce nonlinear oscillations and can be analyzed using the same mathematical model as electronic relaxation oscillators.For example, geothermalgeysers,networks of firingnerve cells,thermostatcontrolled heating systems,coupled chemical reactions,the beating human heart,earthquakes,the squeaking of chalk on a blackboard,the cyclic populations of predator and prey animals, andgene activationsystemshave been modeled as relaxation oscillators. Relaxation oscillations are characterized by two alternating processes on different time scales: a longrelaxationperiod during which the system approaches anequilibrium point,alternating with a short impulsive period in which the equilibrium point shifts.Theperiodof a relaxation oscillator is mainly determined by therelaxation timeconstant.Relaxation oscillations are a type oflimit cycleand are studied innonlinear controltheory.

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).

Originalvacuum tubeAbraham-Bloch multivibrator oscillator, from their 1919 paper

The first relaxation oscillator circuit, theastable multivibrator,was invented byHenri AbrahamandEugene Blochusingvacuum tubesduringWorld War I.Balthasar van der Polfirst distinguished relaxation oscillations from harmonic oscillations, originated the term "relaxation oscillator", and derived the first mathematical model of a relaxation oscillator, the influentialVan der Pol oscillatormodel, in 1920.Van der Pol borrowed the termrelaxationfrom mechanics; the discharge of the capacitor is analogous to the process ofstress relaxation,the gradual disappearance of deformation and return to equilibrium in aninelasticmedium. Relaxation oscillators can be divided into two classes

• 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 approximates asawtooth 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 approximates atriangle waveform,while the current through the switching device approximates a square wave.

Before the advent of microelectronics, simple relaxation oscillators often used anegative resistancedevice withhysteresissuch as athyratrontube,neon lamp,orunijunction transistor,however today they are more often built with dedicated integrated circuits such as the555 timerchip.

Applications

Relaxation oscillators are generally used to produce lowfrequencysignals for such applications as blinking lights and electronic beepers. During the vacuum tube era they were used as oscillators in electronic organs and horizontal deflection circuits and time bases forCRToscilloscopes;one of the most common was the Miller integrator circuit invented byAlan Blumlein,which used vacuum tubes as a constant current source to produce a very linear ramp.They are also used involtage controlled oscillators(VCOs),invertersandswitching power supplies,dual-slope analog to digital converters,and infunction generatorsto produce square and triangle waves. Relaxation oscillators are widely used because they are easier to design than linear oscillators, are easier to fabricate onintegrated circuitchips because they do not require inductors like LC oscillators,and can be tuned over a wide frequency range.However they have morephase noiseand poorerfrequency stabilitythan linear oscillators.

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 acapacitiveorresistive-capacitive integrating circuitdriven respectively by a constantcurrentorvoltage source,and a threshold device withhysteresis(neon lamp,thyratron,diac,reverse-biasedbipolar transistor,orunijunction 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 repeatsad infinitum.

If the threshold element is aneon lamp,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 thePearson–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 a555 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)comparatorswithin the 555 timer flip a transistor switch that gradually discharges that capacitor through a chosen resistor (which determine the 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

Ablocking oscillatorusing the inductive properties of a pulsetransformerto 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. The earlier invertingSchmitt triggeranimated example operates on the same principle (since the Schmitt trigger internally performs comparison). This section will analyze a similar implementation using acomparatoras a discrete component.

A comparator-based hysteretic oscillator.

This relaxation oscillator is a hysteretic oscillator, named this way because of thehysteresiscreated by thepositive feedbackloop implemented with thecomparator(similar to anoperational amplifier). A circuit that implements this form of hysteretic switching is known as aSchmitt trigger.Alone, the trigger is abistable multivibrator.However, the slownegative feedbackadded to the trigger by the RC circuit causes the circuit to oscillate automatically. That is, the addition of the RC circuit turns the hysteretic bistablemultivibratorinto anastable 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 orelectromagneticnoisebrings 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 thevoltage divider.After a short time, the output of the comparator is the positive voltage rail,$V_{DD}$.

Series RC Circuit

The inverting input and the output of the comparator are linked by aseriesRC circuit.Because of this, the inverting input of the comparator asymptotically approaches the comparator output voltage with atime constantRC. 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.

$\,\!V_{+}$is set by$\,\!V_{\rm {out}}$across a resistivevoltage divider:

$V_{+}={\frac {V_{\rm {out}}}{2}}$

$\,\!V_{-}$is obtained usingOhm's lawand thecapacitordifferential equation:

${\frac {V_{\rm {out}}-V_{-}}{R}}=C{\frac {dV_{-}}{dt}}$

Rearranging the$\,\!V_{-}$differential equation into standard form results in the following:

${\frac {dV_{-}}{dt}}+{\frac {V_{-}}{RC}}={\frac {V_{\rm {out}}}{RC}}$

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,$\,\!V_{-}=A$where A is a constant and${\frac {dV_{-}}{dt}}=0$.

${\frac {A}{RC}}={\frac {V_{\rm {out}}}{RC}}$

$\,\!A=V_{\rm {out}}$

Using theLaplace transformto solve thehomogeneous equation${\frac {dV_{-}}{dt}}+{\frac {V_{-}}{RC}}=0$results in

$V_{-}=Be^{{\frac {-1}{RC}}t}$

$\,\!V_{-}$is the sum of the particular and homogeneous solution.

$V_{-}=A+Be^{{\frac {-1}{RC}}t}$

$V_{-}=V_{\rm {out}}+Be^{{\frac {-1}{RC}}t}$

Solving for B requires evaluation of the initial conditions. At time 0,$V_{\rm {out}}=V_{dd}$and$\,\!V_{-}=0$.Substituting into our previous equation,

$\,\!0=V_{dd}+B$

$\,\!B=-V_{dd}$

Frequency of oscillation

First let's assume that$V_{dd}=-V_{ss}$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${\frac {V_{dd}}{2}}$and${\frac {V_{ss}}{2}}$.For the circuit above, V_ssmust be less than 0. Half of the period (T) is the same as time that$V_{\rm {out}}$switches from V_dd.This occurs when V₋charges up from$-{\frac {V_{dd}}{2}}$to${\frac {V_{dd}}{2}}$.

$V_{-}=A+Be^{{\frac {-1}{RC}}t}$

${\frac {V_{dd}}{2}}=V_{dd}\left(1-{\frac {3}{2}}e^{{\frac {-1}{RC}}{\frac {T}{2}}}\right)$

${\frac {1}{3}}=e^{{\frac {-1}{RC}}{\frac {T}{2}}}$

$\ln \left({\frac {1}{3}}\right)={\frac {-1}{RC}}{\frac {T}{2}}$

$\,\!T=2\ln(3)RC$

$\,\!f={\frac {1}{2\ln(3)RC}}$

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

$T=(RC)\left[\ln \left({\frac {2V_{ss}-V_{dd}}{V_{ss}}}\right)+\ln \left({\frac {2V_{dd}-V_{ss}}{V_{dd}}}\right)\right]$

Which reduces to the above result in the case that$V_{dd}=-V_{ss}$.

See also

• Multivibrator
• FitzHugh–Nagumo model– A hysteretic model of, for example, a neuron.
• Schmitt trigger– The circuit on which the comparator-based relaxation oscillator is based.
• Unijunction transistor– A transistor capable of relaxation oscillations.
• Robert Kearns– Used relaxation oscillator in intermittent wiper patent dispute.
• Limit cycle– Mathematical model used to analyze relaxation oscillations