# Relation between number of RC networks and phase shift in RC phase shift oscillator

The generalized expression for the frequency of oscillations produced by a RC phase-shift oscillator is given by the formula below

$$f=\frac1{2\pi RC\sqrt{2N}}.$$

Here $$\N\$$ is the number of RC stages.

The link for the website I have referred is RC Phase Shift Oscillator. I do not understand how we arrive at this equation. I would like to know the proof for this equation.

• A start might be found here?
– jonk
Commented May 1, 2022 at 9:22

There are many ways to determine the transfer function of a cascaded $$\CR\$$ networks such as the one you've shown. I am using the fast analytical circuits techniques or FACTs as described in my book on the subject. The principle is quite appealing since it describes a way to determine transfer functions with the least possible algebra. In some examples, you can even obtain the transfer function by inspection meaning you read the circuit and infer what its time constants are. This is the idea here, determine the time constants $$\\tau=RC\$$ (or $$\\tau=\frac{L}{R}\$$ if you had inductors) of the circuit in various conditions and assemble them to form a well-ordered polynomial expression.

Applying the method to a given circuit means splitting the network in a myriad of small individual sketches, each representing a certain combination. What is cool is that when you've identified a mistake somewhere, you can fix the guilty sketch alone and keep the rest intact. You could not do it with the classic brute-force approach. Let's how these sketches look like:

Once the sketches are done, just look at the resistance offered by each energy-storing element - capacitors in our case - and infer the time constants. You need to determine a certain set of combinations but this is easy to remember:

Then you check the transfer function you have determined with the FACTs against that of the brute-force approach:

To determine the oscillation frequency, cancel the imaginary part and find the formula you want for a 3-stage $$\RC\$$ network.

Finally, let's check if it works with a quick SPICE simulation where the gain of the amplifier corresponds to the insertion loss brought by the network at the calculated frequency. It is 1/29 in this example. If I amplify by that amount, I have exactly a 0-dB loop gain and a phase lag of 0° at the determined frequency: oscillations can be theoretically sustained according to the Barkhausen criterion. In this circuit, a .IC statement gives me the kick I need to crank oscillations:

• You should spend a moment explaining your first tour-de-force diagram. I can see how the taus are developed and how you name them. But others may not so readily see/understand your convention. For example, what does your terminology for tau13 mean, exactly. Is it "tau, with C1 shorted and looking at tau from C3's perspective?" It can be inferred. With work. But some words would help there. Other than that? Nice!
– jonk
Commented May 1, 2022 at 17:57
• Hello jonk, merci, oui, the convention tau13 is read as "when element 1 is placed in its high-frequency state (a short for a cap or an open circuit for an inductor), determine the resistance seen from element 3's connecting terminals". Then, for tau123, same but elements 1 and 2 are set in their hi-frequency state this time. My seminar from 2016 is a good starting point for the technique. Commented May 1, 2022 at 18:43
• Understood. I was thinking that it may help to add an explanation each time you write, as many readers won't view seminars in order to read your writing here. Since communication is the important goal, sometimes writing an additional sentence or two helps a great deal. Just a suggestion from an admirer. ;)
– jonk
Commented May 1, 2022 at 18:49