Can a practical steady-state electronic oscillator be a linear circuit?

Question

Can a practical steady-state electronic oscillator, made of an amplifier (an OA) and a feedback loop be a linear circuit?

Apparently, this is not possible as long as a linear oscillator can never satisfy in practice the Barkhausen condition which, in consequence, can not fully explain how a real oscillator work.

(The Barkhausen condition is equivalent to balancing a bearing ball on the point of a needle. This is possible only in theory.)

Answer 1

In theory it can be a linear circuit, although the amplitude of the oscillation will not be controlled.

In practice, it cannot be a perfectly linear circuit.

Therefore there is a tradeoff between 'practical' and steady-state. If you could make the loop gain precisely 1.00..., then the amplitude would remain constant, and you would have a linear system that is oscillating. In practice, you can't make the loop gain precisely 1.00, and some non-linear circuit (such as a thermistor, lamp or other amplitude-detecting device) is used modulate the gain around 1.00 to maintain steady-state oscillation. This non-linearity can be made arbitrarily small, and in practice introduce no detectable distortion (e.g. non-linearity) in the oscillator.

You cannot measure (and therefore regulate) the amplitude of the oscillation with purely linear circuits or elements. It is possible to make the non-linearity required arbitrarily small, but there is no practical reason to do so.

Answer 2

My answer is NO.

A practical "harmonic" oscillator can continuously produce an oscillation signal only when there is a certain amount of non-linearity within the loop. Barkhausens necessary oscillation condition can be met only when the loop gain is unity (mean value over several periods). This is identical to the case where the closed-loop poles are moving (swinging) left and right from the jw-axis. (This is the rsult of the finite time constant within the control loop).

With respect to the question it is, therefore, important to realize if the non-linearity of the circuit is intentionally introduced into the loop or not (because - strictly speaking - each circuit and each part is non-linear). With other words: Do we exploit this non-linearity for amplitude control purposes (like power supply rails)?

Comment to the phase shift oscillator (see Tony Stewarts answer): This double-integrator loop is a very interesting one.

In most (if not in all) contributions where this oscillator is explained, nothing is said about amplitude control. Indeed, it is a very special effect which enables the circuit to meet Barkhausens condition. It must be stated that the circuit would NOT oscillate for IDEAL opamps.

Fortunately, opamps are always real introducing at least another pole and additional negative phase shift into the loop. As a result: The oscillator will always safely start at a frequency where the loop phase is zero and the loop gain is larger than unity.

The rising amplitude will reach the supply rail - and the amplitude limiting process will cause a positive phase shift which brings the frequency to a value where both oscillation conditions (gain and phase) will be met. Therefore, a very small clipping effect can be observed only.

Details about this effect can be found here: https://www.researchgate.net/publication/342550625_DIO_why_does_it_oscillate?_sg=sUT7Z1wWrdX5TsjWEM1sY21_TjKcs0fj6jjIjPTd9nU6H6Yh04y47XSksDNspeXXWDYkFUCeHr8LJcGgI-ECbhKpCowbVJYMERn2WuLf.er-Je7bqhaMeaw86rCaLlAxzZz-_rz5R8oWGZBn7hExQuMkApF5DlQX9QWsX5GZhDVofFB_z_v4RKZwDIPZLmQ

Answer 3

Can a practical ...oscillator, ... be a linear circuit?

Certainly! As long as you define the spec for linearity it has to meet.

Sine Linearity is sometimes defined by THD in -dB or a result of a very high Q BPF from a square wave with another THD result.

• The ideal infinite and real gain Op Amps alike all go to zero gain when saturated.
• At that edge, the gain changes from 0 to that closed-loop gain value
• utilizing these facts allows you to choose, a slightly higher than unity gain that grows from some initial condition to the saturation point. This gain error from "Unity Loop Gain" also controls the THD and the rate of change of the envelope growing to saturation.
  • or you can use a 1% gain increase and then have a soft limiter to reduce THD even greater.
  • There are many improvements to make a linear oscillator even using LED or Zeners back to back, with a shared resistor, to provide "soft" limiting.
  • The higher the Q or smaller the gain error from 1 , the longer the number of cycles it takes to reach saturation.
  • the improvement for this is to set an initial condition for Vmax rather than gnd. This step voltage allows it to start oscillating at this peak voltage and settle very fast to steady-state even if the Q is 100k as in SC cut crystals used in OCXO's. (The only latency there is the oven settling time.)

AMPLITUDE CONTROL

This is controlled by the abrupt change near limiting, but never clips. (added again for clarity)

Thus the ability to regulate the tolerance on a gain error from 1 and the preset voltage pullup for a cap, enables a simple RC 3stage phase shift oscillator to be as stable as and low THD as the total tolerance stackup you choose. Even if using a resistor hybrid with matched values to 100 ppm.

example of the R-value for the needle balancing act vs Aol for 2 OA's. Here you can reset and see how stable the output is. I chose a gain balancing R tuned to a few ppm to get unity loop gain overall, which is balanced enough to change very slowly in mV from the 10V reset initial condition on +/-15V that it will take millions of cycles to reach Vsat and then stop increasing without noticeable clipping. so THD could be -60dB better or worse depending on the amount of nonlinearity added.

You can change the cap's V+ to 15V and see the result and not have to wait.

There are dozens of other ways to make very low THD sine oscillators and that depends on specs for THD, f, power etc.