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The Phase Shift Oscillator 

Consider the circuit shown to the right. This circuit uses three RC lowpass filter sections to develop a total phase shift of 180° The classical assumption is that each RC section contributes 60° of phase shift, for a total of 180°. For the techie types and those who like to see the math, we can write the corresponding feedback equation as:
Aβ  =  A ×  1 


(RCs + 1)³ 
In this sort of expression, s = jω, where ω = 2πf and j = .
If we assign values of R = 10K and C = 0.01µf, the frequency of oscillation should be about 2.76 kHz (ω = 1.732/RC, where 1.732 = tan 60°). Also, since the output voltage of the low pass RC filter segment is 1/2 at a 60° phase angle, the gain of the amplifier should be 8 to balance the loss through three RC segments. However, if we actually build this circuit as shown and test it, we find that the operating frequency is more like 3.76 kHz, and the required gain is around 26 to 30. Why is this the case? What is wrong with this circuit?
The real problem was the assumption that the three RC phase shift segments remain independent of each other. In reality, they don't. They can't. The first segment is OK, as it is driven directly from the op amp. However, the second segment can't help but load the first, and the third segment loads the second. This means a different phase shift response and increased attenuation.
One way to keep this from happening is to add buffer amplifiers between the individual RC segments, as shown to the left. As shown here, these are unitygain amplifiers. Their only purpose is to isolate the RC segments from each other. When this is done, the required gain for the original amplifier is reduced to very close to 8, and the operating frequency drops to appoximately the calculated value. The main error now comes from component tolerences.
A possible variation is to use inverting amplifiers at each step, and to set the gain of each amplifier to 2, with one amplifier being adjustable to accurately set the gain around the entire loop. If this is done correctly, each amplifier will output a sine wave of the same amplitude as the others, and they will be phaseshifted from each other by 120°. The circuit can then serve as a threephase signal source.
It is not necessary to limit the circuit to three RC sections. Three is simply the minimum number of sections required to permit oscillation to occur. However, if we use four sections producing 45° of phase shift each, the overall circuit gain is reduced to 4. This is generally known as a Bubba Oscillator, and is commonly built with a single quad opamp package.
The Bubba oscillator allows you to obtain quadrature outputs, so you can get simultaneous sine and cosine signals. If the gain is distributed among all op amps, each one needs to be set to a gain of = 1.414.


 
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