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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 is set by the time it takes for the system to relax from each disturbed state to the threshold that triggers the next disturbance.
Many electronic relaxation oscillators store energy in a capacitor and then dissipate that energy repeatedly to set up the oscillations. For example, the capacitor can be charged toward a positive power supply until it reaches a threshold voltage sufficiently close to the supply. At that instant, the capacitor can be quickly discharged (e.g., shorted). 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. In all such capacitor-based relaxation oscillators, the period of the oscillations is set by the dissipation rate(s) of the capacitor. Implementations of these two types of relaxation oscillators are shown here, but relaxation oscillators need not be electronic in general. Any oscillator whose oscillations are driven by a system that almost always is dissipating energy may be called a relaxation oscillator.
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, 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.
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 the capacitor reaches a certain value, comparators within the 555 timer cause the latch to activate a transistor switch that discharges the capacitor through a resistor to ground. At the instant the capacitor falls to a sufficiently low value, the switch deactivates to let the capacitor charge again.
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.
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, but different from, an op-amp). 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.
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 electromagnetic noise 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 series RC 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.
is set by across a resistive voltage divider:
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 .
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,
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 .
This type of circuit was used as the time base in early oscilloscopes and television receivers. Variants of this circuit find use in stroboscopes used in machine shops and nightclubs. Electronic camera flashes are a monostable version of this circuit, generating one cycle of the sawtooth. The rising edge develops as the flash capacitor is charged, and the rapid falling edge as the capacitor is discharged. The flash is produced upon receiving the firing signal from the shutter button. Use as a timebase in oscilloscopes was discontinued when the much more linear Miller Integrator timebase circuit (invented by A.D. Blumlein), using "hard" valves (vacuum tubes) as a constant current source, was developed. 
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