I must admit that my question sounds purposely a little provocative.
But I accidentally came across the circuit described below, which exclusively works in the linear part of its operational range (see the attached circuit). The first block represents an active inverting first-order lowpass filter, where the summation can be done directly at the input node of the used opamp. In this case, the second block must be implemented as a non-inverting integrator circuit (Hint: Both polarities can also be swapped, thereby allowing the convenient MILLER integrator).
The closed feedback loop then represents a second-order lowpass function with a high pole quality factor (e.g. Qp=10). Once triggered, this high-Q lowpass would react with a decaying oscillation - unless it receives new "kick impulses" at the „right“ time slot. Therefore, at each zero crossing of the lowpass output voltage v(out1) the comparator adds a voltage Vc to the feedback signal v(out2) for half a period with: Vc negative for v(out1)>0 and vice versa. (Now the analogy to the pendulum clock becomes obvious) .
Demonstration example: Lowpass: H(s)= -10/(1+0.01*s), Integrator: H(s)=+1/0.001*s, Closed-loop function (2nd order): Ao=1, pole frequncy 160 Hz, Qp=10 Comparator output: Vc=(+-)1V.
The result is a sinusoidal output signal (frequency fo=160 Hz) with very good quality: Amplitude Vmax(out,2)=12.5 V (supply voltages 20V). It is to be mentioned that the observed frequency and amplitude are in full accordance with theoretical considerations (calculation). The frequency is set by the time constants of low pass and integrator. The oscillation amplitude can be varied over a large range (without changing the frequency) by the comparator voltage Vc, but it also depends on the lowpass gain Ao and the closed-loop quality factor Qp (which determines the decaying time constant).
Final comment: I think, the shown circuit belongs to the class of oscillators which are stabilized by „restoring initial conditions“ (see IEE papers from Filanovski) - however, this circuit is much simpler in design compared to other related oscillator configurations.
Final question: Does the output signal contain any intentional systematic non-linearities? Sorry - but I couldn`t manage to make the figure smaller.
All oscillators are nonlinear. We are suggested to think the harmonic oscillators are linear, but they are not either. The idealization of a pure imaginary pole is just a way to enable the analysis of the circuit. If the pole is on left half plane, the oscllation will cease after some time. If it is on right half plane, where it usual is, the amplitude will grow until some nonlinear mechanism acts and forces a limitation. In the case presented, the nonlinear mechanism is the action of the comparator. It forces the circuit to behave as in its transitory state repetitively, compensating the losses of the circuit.
I would love to see an implementation of the circuit and some waveforms from it.
Dear Sir,
thank you fpr your answer.
I agree to everything you wrote.
As I have mentioned - my question (linear/non-linear) was a bit provocative. It was my intention to direct the reader`s attention to this - as I think - remarkable topology.
As requested by you - I will show a circuit and the corresponding waveform soon.
Here are two pdf files showing the circuit and the oscillator output signal. The IDEAL opamp block in the short feedback loop works as a comparator (supply +- 1V). The amplitude of the output signal (12.7 volts) is in full acordance with the calculated amplitude value.
Thank you for the images, Lutz. They are very interesting.
There is classical practical example of using inertial non-linearity of heated metallic conductor (filament bulb) to stabilize amplitude in harmonic generator.
Another way which is more complicated but does not require inertial non-linearity is to introduce and combine two balanced 'friction' forces : one behaving like A^2*sin(t)^2 another one like A^2*cos(t) ^2 thus riding on the fact sin(t)^2+cos(t)^2=1 and that will stabilize amplitude without raising other harmonics. It was demonstrated experimentally but is not popular.
Thank you for your response...
If I am not wrong....I remember an article from the late Barrie Gilbert in which he proposes something similar for estabishing a unity loop gain (sin²+cos²=1).
@ Nikolay Pavlov
Please find below an excerpt from a Patent Document (Barrie Gilbert, Analog devices):
"A quadrature oscillator includes an amplitude control circuit that is based upon the trigonometric identity sin2 ωt+cos2 ωt=1. The amplitude control circuit, referred to as a Pythagorator, includes two squaring circuits. Each squaring circuit receives a respective quadrature oscillator signal and squares it. The outputs of the two squaring circuits are joined together so as to sum the outputs of the two squaring circuits to produce a sum of squares signal. This signal, a current in the preferred embodiment, is provided to damping diodes coupled to the outputs of the quadrature oscillator. "
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