Oscillator's poles (must be positive real part to begin the oscillation?)

[julian403]

oscillator's poles ( must be positive real part to begin the oscilation? )

Hello.

To have an oscillator, there must be the system's pole on the imaginary axis? or ? to start the oscilation the poles must to be on the positive semiplane (whit positive real part)?

My question it's if the poles are in the imaginary axis, with σ=0, 0 ± j ω. The oscillator will start?

If I have the funtion system transfer

\[h(j \omega=) = \frac{A(j \omega)}{1- A(j \omega) B (j \omega)}\]

So \[|A(j \omega) B (j \omega)}| = 1\] and \[<(A(j \omega) B (j \omega)) = 0 rad\]

So \[A(j \omega) B (j \omega) = 1 + j 0\]

Then, If the img part is 0 (the poles are not in the imaginary axis)

 

[LvW]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

Theoretically, it would be sufficient to have the poles directly on the axis (zero real part) - and to start oscillations externally.
However, in reality this is impossible because this would require zero tolerance of all parts.
Therefore, taking tolerances into account and to ensure a safe start of oscillations the oscillator design always starts with a "small" positive real part of the poles.
"Small" means: As small as allowed and as large as necessary to cover all uncertainties and tolerances.

Because poles in the right half of the s-plane are identical with continuous rising amplitudes - and to avoid hard limiting (clipping) of the amplitudes - we use a slight non-linearity within the circuit (diodes, thermistor, light bulb, FET as resistance) to allow soft limiting before the amplitudes reach the supply rails.

This leads to the interesting description of an oscillator: In order to function as linear as possible each harmonic oscillator must contain a small non-linearity. Sounds nice, does it not?

 

[schmitt trigger]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

More than nice. Very interesting!

Just like Bill Hewlett's master Thesis at Stanford University in California, where he used a simple light bulb as the non-linear element of a Wien-bridge oscillator, which later became the HP200C.

 

[LvW]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

Just like Bill Hewlett's master Thesis at Stanford University in California, where he used a simple light bulb as the non-linear element of a Wien-bridge oscillator, which later became the HP200C.

It is worth mentioning that there are some oscillator principles (with two or even three opamps) which can provide a pretty good sinusoidal signal without the need for an additional non-linear element.
In this case, a small "clipping" at the output of one opamp is accepted, but the signal of another opamp is used (bandpass filter oscillator or double-integrating oscillator).

 

[CataM]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

Then, If the img part is 0 (the poles are not in the imaginary axis)

You are a bit confused here.

1-A*β is the characteristic equation of the transfer function. That equation must have its poles on the imaginary axis in order for the system to oscillate (theoretically).

If 1-A*β=0 has 2 imaginary complex solutions, then, that equation is equal to zero when s=jω.. which is the same as saying A*β = 1+j0.

 

[zorro]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

Let's say that these are two different principles that govern the amplitude of an oscillator:

a) Nonlinearity (clipping or compression) at every cycle. In this principle are based many types of oscillator including usual tuned (LC) oscillators, phase-shift RC oscillators, and Wien bridges without AGC, among others.

b) Automatic gain control (AGC). With this principle works, for example, the Wien bridge with a light bulb.
The thermal time constant of the bulb is much longer than a cycle of oscillation. The current across the filament "senses" the amplitude of the oscillation at long term (many cycles) in such a way that a higher amplitude increases the resistance of the filament and the gain is reduced. Once stabilized, the amplitude reaches the level necessary to make that the gain loop has exactly the value needed for linear oscillation (i.e. poles in the imaginary axis).
In a well designed oscillator, this happens without clipping in the actve device. This makes that this type of oscillator has so low harmonic distortion. A price to be paid is that a change in frequency (or other parameter) takes some time to stabilize.

Regards

 

[LvW]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

..............
Automatic gain control (AGC). With this principle works, for example, the Wien bridge with a light bulb.
The thermal time constant of the bulb is much longer than a cycle of oscillation. The current across the filament "senses" the amplitude of the oscillation at long term (many cycles) in such a way that a higher amplitude increases the resistance of the filament and the gain is reduced. Once stabilized, the amplitude reaches the level necessary to make that the gain loop has exactly the value needed for linear oscillation (i.e. poles in the imaginary axis).

Your remark concerning the time constant of the non-linear part (light bulb or FET stabilization) is correct - however, to be exact: It is not correct that "the gain loop has exactly the value needed".
Instead, the compex pole pair is moving (swinging) in the close vicinity of the imag. axis between the left and the right half of the s-plane. And the frequency of this movement (a kind of unwanted amplitude modulation) is the inverse of the large time constant. This is a typical result of any control loop with a finite time constant.

 

[FvM]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

In my view, "unwanted amplitude modulation" refers to the case where the fundamental magnitude is actually varying, in other words you have a kind of subharmonic oscillation. With suitable setup of the magnitude control loop, the oscillator achieves steady state without subharmonic oscillations. Instead you get a certain amount of harmonic distortion, e.g. third order for the incandescent lamp.

 

[zorro]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

Very interesting, friends!

Let me explain my point of view about the Wien bridge with light bulb (or equivalent device) with the aid of the following sketch.
The left gaph plots the resistance of the filament (or equivalent) as a function of the amplitude of the oscillation. The right one plots the total gain of the loop as a function of the filament resistance. This gain is the modulus of the transfer function at the frequency at which the phase is 180 degrees, i.e. the frequency at which the system can oscillate if the gain condition is met.
These are static characteristics.



At start-up, the bulb has an initial resistante Rini that depends of the ambient temperature. With R=Rini the loop has a gain (Gini) greater than needed to start oscillation with increasing amplitude.
Long after a transient that depends mainly of the thermal time constant of the bulb and the shape of the curves, the system reaches a stable point (marked with red circles) where the gain Gosc is exactly the needed for maintain oscillation (loop gain = 1). In this condition the filament has resistance Rosc and the system holds linear oscillation with amplitude Aosc.
The fact that the curves have slopes like shown in the plots makes that the circuit reaches (asymptotically) a point of stable equilibrium.

We are assuming that one period of oscillation is much (let's say many orders of magnitude) shorter than the thermal time constant of the bulb.
Then, once the system attained steady state we can neglect the variations in the filament resistance and its consequences, as they are well below the noise level. Otherwise (if R has significant changes in a period of oscillation) this model is not appropriate and the waveform will be distorted. (By the way, this is what happens in Wien bridge oscillators when generating very low frequencies.)

Imagine that at this moment we replace the bulb by an ideal resistor that has exactly the value Rosc. What happens? The system holds its linear oscillation in the same conditions. But the mechanism assuring the existence of a stable equilibrium (the AGC) is lost: a small change in temperture, gain, etc., can provoke that the amplitude starts to decrease and oscillation dies, or it starts to grow up to a point where clipping takes control.

My conclusion about this type of oscillator (with AGC) operating according to said assumption:

= There is an internal fast loop that has a pair of poles in the imaginary axis responsible of linear oscillation
= There is an external, very slow loop that controls the gain of the internal loop. It senses the amplitude of the oscillation and controls the gain of the internal loop acting upon a voltage controlled resistor.
= The two loops can be analyzed with linear or quasi-linear models but can not be mixed or combined in a single linear model: They operate with different variables.

I hope I'm clear.

 

[LvW]

Re: oscillator's poles ( must be positive real part to begin the oscilation? )

In my view, "unwanted amplitude modulation" refers to the case where the fundamental magnitude is actually varying, in other words you have a kind of subharmonic oscillation. With suitable setup of the magnitude control loop, the oscillator achieves steady state without subharmonic oscillations. Instead you get a certain amount of harmonic distortion, e.g. third order for the incandescent lamp.

If "...suitable setup" means that the time constant of the regulating mechanism is zero, I agree.
But my remark was related to non-linear parts (or control loops) with remarkable time constants only.

- - - Updated - - -

We are assuming that one period of oscillation is much (let's say many orders of magnitude) shorter than the thermal time constant of the bulb.
Then, once the system attained steady state we can neglect the variations in the filament resistance and its consequences, as they are well below the noise level. Otherwise (if R has significant changes in a period of oscillation) this model is not appropriate and the waveform will be distorted. (By the way, this is what happens in Wien bridge oscillators when generating very low frequencies.)

Zorro, I can agree to everything you wrote - with the following restriction:

Only if the variations are below the noise level we can neglect this "amplitude modulation".
But, certainly, this is not always the case. It depends on the degree of non-linearity necessary to cope for the uncertainties, which only make this non-linearity necessary.
These "uncertainties" lead to the necessity to make at t=0 the loop gain larger than 0 dB.
But how much larger?
Answer: Larger as the worst case sum of all uncertainties: Deviation from standard R and C values, tolerances of R and C, simplifications/neglections made during design/calculation (opamp non-idealities), parasitics,..

As you can see, in some cases. the design value for the loop gain must be perhaps 1.2 or so. And after the startup phase, this value must be reduced to app. unity (within a finite time constant).
Then, the amplitude variations will certainly be larger than the noise level.
More than that, the slope of the input-output characteristics of the non-linear part (or control loop) at the desired operating point play a major role.