# phase-shift oscillators

I'm starting to learn about phase-shift oscillators and I'm a little confused at the outset. Suppose you have an amplifier with a feedback circuit with transfer functions A and β respectively. My understanding is that the Barkhausen Criterion is a necessary condition for oscillation. In fact a simple algebraic calculation indicates that in order for there to be oscillation, you must have Aβ = 1 (A and β are complex). But this condition isn't sufficient is it?

So in the case where Aβ = 1, my questions are:

1. What additional conditions would guarantee oscillation?
2. Why is the oscillation sinusoidal?

I'm assuming there must be some 2nd order linear ODE describing the system when Aβ =1, but I don't know how to produce it.

I've been searching on the internet and all the things I've found so far are either naive explanations, which seem to suppose Barkhausen implies oscillation, which it sounds like it doesn't, or research articles in engineering journals which deal with very advanced aspects of topic.

Can someone point me to a reference that might offer some sort of "middle of the road" answer?

Thanks.

My understanding is that the Barkhausen Criterion is a necessary condition for oscillation.

You are correct. It is a necessary condition for sustained, stable oscillations, that neither decay away, nor grow without bound. But,

But this condition isn't sufficient is it?

No it isn't, and you are correct here too. If the closed loop gain is exactly one, at a given frequency, then a small oscillation would stay small, a large oscillation would stay large, and non-oscillation would stay zero!

What additional conditions would guarantee oscillation?

A way to guarantee oscillation is to have a variable gain. The gain should be above 1 when the oscillation amplitude is small, less than 1 when the oscillation amplitude is above a certain level, and exactly 1 when the oscillation is at the target amplitude. There are a number of ways to achieve this, such as taking advantage of non-linearities in certain components, or changes in component values based upon their heating.

Why is the oscillation sinusoidal?

There are a number of points to make here. Because of the necessity of adjusting the gain so that it is below 1 for "too large" oscillations, and above 1 for "too small oscillations", oscillators generally do not produce perfect sine waves. Often there is a trade off to be made. To guarantee quick start up, one wants the gain to be quite large at low amplitudes. This can mean distortion. Similarly, when the amplitude is too big, and one relies upon non-linear components to cut the gain, the drop in gain often comes about with non-zero distortion. So, the trade off is between the purity of the sine wave, and the stability of the oscillator.

The second point to be made is that the waveform is nearly sinusoidal because sinusoids are the solutions to the differential equations that correspond to the phase shift networks. Or, looked at another way, there is (hopefully) a unique frequency at which the closed loop phase shift is 0. Thus, all frequencies other than that frequency will decay and die out, leaving only a sinusoid.

Can someone point me to a reference that might offer some sort of "middle of the road" answer?

Hopefully someone else see this and knows of a good intermediate level reference.

In short:

• It is correct that the Barkhausen condition is a necessary condition only. However, there is an additional requirement which makes this condition sufficient: At the desired oscillation frtequency the phase shift of the loop gain function must have a negative slope. (This finding is relatively new and is not yet included in the textbooks).

• The produced signal is sinusoidal because the oscillation condition is fulfilled at one single sinusoidal waveform (frequency) only .

• However, it is another question how we can exactly meet the oscillation condition (unity loop gain) and - at the same time - allow a safe start of oscillation (loop gain>1). Therefore, we need some non-linear (amplitude-dependent) parts within the feedback loop. In some cases, this task is performed by supply voltage limits (amplitude clipping). Now - because the produced waveform is not sinusoidal anymore - the oscillation condition must be applied to the fundamental frequency only (first harmonic analysis).

• EDIT (Addendum): Upon request, here is a link concerning the extension of Barkhausens oscillation condition:

• Nice answer. So you are claiming that $L(j\omega_0) = 1$ and $\frac{d\phi_L(\omega_0)}{d\omega}\lt 0$ is a necessary and sufficient condition for oscillation at $\omega_0$, right? Could you provide the reference for that, please? Apr 2, 2021 at 14:20
• The mentioned requirement for the phase slope must be considered as an EXTENSION (additional condition) of Barkhausens condition for oscillation (unity loop gain).
– LvW
Apr 2, 2021 at 14:55
• I see, so it's still a necessary condition but not a sufficient one. Apr 2, 2021 at 15:01
• Why is THIS you conclusion? Read agian what I wrote: "However, there is an additional requirement which makes this condition sufficient".
– LvW
Apr 2, 2021 at 16:24