Relaxation oscillator: Definition from Answers.com

A relaxation oscillator is an oscillator in which a capacitor is charged gradually and then discharged rapidly. It is usually implemented with a resistor or current source, a capacitor, and a "threshold" device such as a neon lamp, thyratron, diac, unijunction transistor, or Gunn diode. For simplification below, a single "threshold" device will be replaced by a set of comparators and a SR Latch.

The capacitor is charged through the resistor, causing the voltage across the capacitor to approach the charging voltage on an exponential curve. In parallel with the capacitor is the threshold device. Such devices don't conduct at all until the voltage across them reaches some threshold (trigger) voltage. They then conduct heavily, quickly discharging the capacitor. When the voltage across the capacitor drops to some lower threshold voltage, the device stops conducting and the capacitor can begin charging again, repeating the cycle. If the threshold element is a neon lamp, the circuit also provides a flash of light with each discharge of the capacitor.

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.

The electrical output of a relaxation oscillator is always a sawtooth wave. If only a small portion of the exponential ramp is used (that is, if the triggering voltage of the threshold device is much lower than the charging voltage source), the ramp will approximate a linear ramp but if a truly linear sawtooth is required, then the charging resistor should be replaced by some sort of constant current source.

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.

1 Op Amp–Based Relaxation Oscillator
- 1.1 General Concept
- 1.2 Example: Differential Equation Analysis of Op-Amp based Relaxation Oscillator
  - 1.2.1 Frequency of Oscillation
2 Practical examples of the use of the relaxation oscillator
3 See also

Op Amp–Based Relaxation Oscillator

A comparator-based hysteretic oscillator.

One example of a relaxation oscillator is a hysteretic oscillator, named this way because of the hysteresis created by the positive feedback loop implemented with the comparator (or 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.

General Concept

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

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

Series RC Circuit

The inverting input and the output of the op amp are linked by a series RC circuit. Because of this, the inverting input of the op amp asymptotically approaches the op amp 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 op amp 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 op amp'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 Op-Amp based Relaxation Oscillator

Transient analysis of an opamp based relaxation oscillator.

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

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

$\, \! V_-$ is obtained using Ohm's law and the capacitor differential equation:

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

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

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

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 $\frac{dV_-}{dt}=0$ .

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

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

Using the Laplace transform to solve the homogeneous 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.

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

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

Solving for B requires evaluation of the initial conditions. At time 0, $V o u t = V d d$ and $\, \! V_-=0$ . Substituting into our previous equation,

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

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

Frequency of Oscillation

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

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

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

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

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

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

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

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

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

Which reduces to the above result in the case that $V d d = - V s s$ .

Practical examples of the use of the relaxation oscillator

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 as the flash capacitor is charged and the rapid falling edge as the capacitor is discharged and 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 using "hard" valves, (vacuum tubes) as a constant current source, was developed.

	neon oscillator (electronics)
	glow-tube oscillator (electronics)
	relaxation inverter (electronics)

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Relaxation oscillator

Contents

Op Amp–Based Relaxation Oscillator

General Concept

Example: Differential Equation Analysis of Op-Amp based Relaxation Oscillator

Frequency of Oscillation

Practical examples of the use of the relaxation oscillator

See also