Editing RC oscillator (section)

==Low distortion oscillators==
The Barkhausen criterion mentioned above does not determine the amplitude of oscillation.  An oscillator circuit with only ''[[linear circuit|linear]]'' components is unstable with respect to amplitude.  As long as the loop gain is exactly one, the amplitude of the sine wave would be constant, but the slightest increase in gain, due to a drift in the value of components will cause the amplitude to increase exponentially without limit.  Similarly, the slightest decrease will cause the sine wave to die out exponentially to zero.  Therefore, all practical oscillators must have a nonlinear component in the feedback loop, to reduce the gain as the amplitude increases, leading to stable operation at the amplitude where the loop gain is unity.

In most ordinary oscillators, the nonlinearity is simply the saturation (clipping) of the amplifier as the amplitude of the sine wave approaches the power supply rails.  The oscillator is designed to have a small-signal loop gain greater than one. The higher gain allows an oscillator to start by exponentially amplifying some ever-present noise.<ref>{{citation |last=Strauss |first=Leonard |title=Wave Generation and Shaping |edition=second |publisher=McGraw-Hill |year=1970 |chapter=Almost Sinusoidal Oscillations — the linear approximation |pages=663–720}} at page 661, "It follows that if {{math|''A''&beta; > 1}} in the small-signal region, the amplitude will build up until the limiter stabilizes the system...."</ref>

As the peaks of the sine wave approach the supply rails, the saturation of the amplifier device flattens (clips) the peaks, reducing the gain. For example, the oscillator might have a loop gain of 3 for small signals, but that loop gain instaneously drops to zero when the output reaches one of the power supply rails.<ref>{{harvnb|Strauss|1970|p=694}}, "As the signal amplitude increases, the active device will switch from active operation to the zero-gain regions of cutoff and saturation."</ref> The net effect is the oscillator amplitude will stabilize when average gain over a cycle is one. If the average loop gain is greater than one, the output amplitude increases until the nonlinearity reduces the average gain to one; if the average loop gain is less than one, then the output amplitude decreases until the average gain is one. The nonlinearity that reduces the gain may also be more subtle than running into a power supply rail.<ref>{{harvnb|Strauss|1970|pp=703–706}}, ''Exponential limiting—bipolar transistor''.</ref><!-- Strauss uses a transcribing function. There's also a harmonic balance approach. -->

The result of this gain averaging is some [[harmonic distortion]] in the output signal.  If the small-signal gain is just a little bit more than one, then only a small amount of gain compression is needed, so there won't be much harmonic distortion.  If the small-signal gain is much more than one, then significant distortion will be present.<ref>{{harvnb|Strauss|1970|p=664}}, "If gross nonlinear operation is permitted, the limiter will distort the signal and the output will be far from sinusoidal."</ref>  However the oscillator must have gain significantly above one to start reliably.

So in oscillators that must produce a very low-distortion [[sine wave]], a system that keeps the gain roughly constant during the entire cycle is used.  A common design uses an [[incandescent lamp]] or a [[thermistor]] in the feedback circuit.<ref>{{harvnb|Strauss|1970|p=664}}, "Alternatively, an amplitude-controlled resistor or other passive nonlinear element may be included as part of the amplifier or in the frequency-determining network."</ref><ref>{{harvnb|Strauss|1970|pp=706–713}}, ''Amplitude of Oscillation—Part II, Automatic Gain Control''.</ref> These oscillators exploit the [[electrical resistance|resistance]] of a [[tungsten]] [[Electrical filament|filament]] of the lamp increases in proportion to its [[temperature]] (a [[thermistor]] works in a similar fashion). The lamp both measures the output amplitude and controls the oscillator gain at the same time. The oscillator's signal level heats the filament. If the level is too high, then the filament temperature gradually increases, the resistance increases, and the loop gain falls (thus decreasing the oscillator's output level). If the level is too low, the lamp cools down and increases the gain.<!-- Not the whole story! Amplifier nonlinearity/compression is still needed. --> The 1939 HP200A oscillator uses this technique. Modern variations may use explicit level detectors and gain-controlled amplifiers.<!-- the control equations are a significant issue (even for radio AGC circuits) -->

[[File:Wien Bridge Oscillator.png|right|thumb|225px|Wien bridge oscillator with automatic gain control. Rb is a small incandescent lamp. Usually, R1 = R2 = R and C1 = C2 = C. In normal operation, Rb self heats to the point where its resistance is Rf/2.]]

===Wien bridge oscillator===
{{main|Wien bridge oscillator}}
One of the most common gain-stabilized circuits is the [[Wien bridge oscillator]].<ref>{{Harvnb|Department of the Army|1962|pp=179–180}}</ref> In this circuit, two RC circuits are used, one with the RC components in series and one with the RC components in parallel. The Wien Bridge is often used in audio [[signal generator]]s because it can be easily tuned using a two-section [[variable capacitor]] or a two section variable potentiometer (which is more easily obtained than a variable capacitor suitable for generation at low frequencies). The archetypical [[HP200A]] audio oscillator is a Wien Bridge oscillator.

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