I have extracted this general question from the more specific discussion about the validity of the Barkhausen criterion in the circuit of a Wien bridge oscillator:
https://www.researchgate.net/post/Is_the_Barkhausen_criterion_about_the_loop_gain_right_in_the_case_of_the_Wien_bridge_oscillator1
Here are my speculations that can be used as an initial point for this discussion:
In LC oscillators, the sinusoidal oscillations arise in an LC tank as a result of exchanging the initially stored energy alternatively between two dual storage elements. I have explained this process in the story below:
http://en.wikibooks.org/wiki/Circuit_Idea/How_do_We_Create_Sinusoidal_Oscillations%3F (user name Circuit-fantasist)
So, in LC oscillators, the very LC tank reverses the direction of the voltage change and the active electronic circuit only follows and sustains the existing oscillations by adding so much energy as it dissipates in the internal resistance.
RC oscillators contain only one storage element (a capacitor) that cannot reverse its voltage by itself like an LC tank; so, the charging source has to change alternatively its polarity. In relaxation RC oscillators the source is constant during the transition and it is reversed sharply at the peaks by threshold circuits:
http://en.wikipedia.org/wiki/Talk:Electronic_oscillator#Relaxation_versus_LC_oscillations (user name Circuit dreamer)
In sinosuidal RC oscillators, the charging source (amplifier's output) follows with advance the voltage across the storage element during the transition and slows down when approaching the peaks:
http://en.wikipedia.org/wiki/Talk:Wien_bridge_oscillator#How_do_RC_oscillators_produce_sine_wave.3F
It is obvious that this is achieved by changing dynamically the amp's gain when the voltage changes between the rails. But the main question is, "How is the voltage 'movement' reversed at the peaks?"
A quite good explanation regarding your question is given in the theory part at:
http://www.visionics.a.se/html/curriculum/Experiments/RC%20Phase%20Shift%20Oscillator/RC%20Phase%20Shift%20Oscillator1.html
I think the explanation about the noise through the resistor is the valueable hint!
Thank you, Andreas!
The material is good but still too conventional. I need more detailed "instantaneous" explanations at the possibly lowest level...
Maybe, once we started talking about phase shift circuits, we have to clarify the phase shift phenomenon of the humble RC circuit. I intend to ask a separate question, "Why is there a phase shift in RC circuits?" I have posed some speculations about it in the Wikipedia talk page about RC circuit:
http://en.wikipedia.org/wiki/Talk:RC_circuit#Why_there_is_a_phase_shift_between_the_current_and_voltage_in_a_capacitor
Regards, Cyril
To help the discussion, I would like to add more speculations about the voltage reversal at the peaks of the sinusoid (I address them particularly to Lutz).
Imagine the output voltage has reached the top (the positive rail). It cannot continue increasing since the gain is one and will decrease even more (figuratively speaking, the voltage "has touched the ceiling" and cannot "move up" anymore:) But it can freely decrease ("move down") and the positive feedback will "encourage" it to go down; it will "help" the movement in this direction. So, the point at the top is unstable only in the downward direction.
Now remember there are always some AC noises (disturbances) superimposed on the voltage. They make the voltage slightly increase and decrease alternatively. In the first case, nothing will happen since the gain will even decrease below one and there is no reinforcing (positive) feedback. But in the second case, the gain begins increasing; the reinforcing feedback comes into operation and begins "pulling down" the voltage...
Hi Cyril,
I understand your question. You are looking for some similarities between LC and RC oscillators - in particular regarding the "direction reversal" at the peak of the output signal. But - as far as I remember - I have given already my attempt to explain this process in another sequence of question and answers. As an example - there are different time constants associated with both C`s in the WIEN network causing a voltage at the positive opamp input node that goes negative if the input voltage of the WIEN bridge (opamp output) reduces (due to discharging of the grounded capacitor).
To my understanding - no surprising behaviour. This also can be explained with the step response of an RC bandpass****.
To me - a much more challenging question is NOT why some RC oscillators oscillate, but why some other RC circuits do NOT oscillate - in spite of the fact that they are able to meet Barkhausen`s condition.
This leads, of course, to the question why Barkhausen is only necessary but not sufficient.
Regards
Lutz
**** Correction: Instead of "step response" please read: Response to a single positive impuls (triangle or trapezoid), which allows charging and subsequent discharging while the driving voltage reduces. It can be observed, that the voltage at the grounded capacitor goes negative - thus explaining the discussed effect of reversal of the direction.
My idea is, in this discussion, relying on the most ordinary human reasoning, to realize in a general form the truth about the creation of RC sinusoidal oscillations... to do it freely and in a funny and human friendly manner seeing the evolution of this powerful idea in nature, engineering and electronics as I did in the Wikibooks story about LC oscillations:
http://en.wikibooks.org/wiki/Circuit_Idea/How_do_We_Create_Sinusoidal_Oscillations%3F
Here I just want to penetrate deeply into the essence of the RC oscillation phenomenon... to understand (at the possibly lowest level) how the sinusoid conceives in an RC oscillator. For this purpose, I suggest first to consider various analogical situations from our routine, then to mount an RC oscillator and finally, to investigate it following mentally the "trajectory" of the instant output voltage "point" when "moving" and “reversing” between the power supply rails...
If you do not like this “vague”, “indistinct” and maybe “confused” approach, and you think you can “explain” this phenomenon better by solving (nonlinear) differential equations, please do it... but do not deny my intuitive approach... do not forget that people are human beings, not computers... Cyril
Maybe we should first answer the questions, "Why do we need to create oscillations?" and generally, “What is an oscillation?”... and only after that discuss the issue with the creation... Of course, there are many possible answers to these “philosophical” questions and they are probably all true but I will use a slightly unusual perspective that is best suited for the next my speculations. From this viewpoint, we can think of an oscillation as of a kind of “perpetual movement”. Why?
EXAMPLES. There are not unlimited quantities in this world; so, if we try to increase endlessly and only in one direction a quantity, we will finally get to some high limit (saturation), and the change will stop. To continue the change, we should reverse the direction of “movement” and keep decreasing until we reach the low limit... and so on... Let’s consider some examples from our routine where an object moves in such an oscillating manner.
Car: Imagine that for a purpose, a car has to move endlessly (well, until it depletes its fuel) ... but we do not have enough space to do it in one direction only... we have only a limited space. So, when the car nears the one end of the space, it decelerates, stops and then goes to reverse... when it nears the other end, it decelerates, stops and then goes to reverse again... and so forth...
Swimmer: Or imagine you want to swim “endlessly” in a pool ... but the length of the pool is limited. So when you get the one end, you slow the speed, stop, reverse the direction and continue swimming back; when you reach the other end, you slow again the speed, stop, reverse the direction and continue swimming back... and so on and so forth...
PHASES. So, we can distinguish ten phases (moments) in one period of oscillation: INCREASING, SLOWING, STOPPING, REVERSING, ACCELERATING and the mirror DECREASING, SLOWING, STOPPING, REVERSING, ACCELERATING... and to produce a period of a sinusoid, we have to shape, one after another in such a sequence, these ten parts of the curve. Let’s implement it in the next fictional experiment that is slightly different from the experiments above (here we have two objects – one passive and one active).
“Forced” boat: Imagine that we want to make a light (not inert) boat do such sine movements (oscillations) between two opposite sides of a lake. So we hung it with an elastic rope for another but motor boat and begin to draw it. When we (the motor-boat) approach the opposite shore, we begin slowing the speed and loosing the rope... then stop for a while... reverse the direction... begin begin to drawing the rope again and accelerating... As a result, the boat smoothly changes its direction of motion and begin moving to the other shore; there we repeat all these actions again and the boat smoothly changes its direction again, etc. (for comparison, if the motor-boat abruptly changes its position on the opposite bank, the towed boat also will turn sharply - this is an example of a relaxation oscillation... but this is other story)...
“Self-pulling” boat: The final step of our boat analogy is to make the speeds of both the boats (the drawn and the drawing), depend on each other. Thus the drawn boat will control the drawing motor-boat thus acting as a “self-pulling” boat and the system of the two boats will operate automatically as a sine oscillator. We have only to hung the drawn boat with an additional weak elastic rope to an anchor stick in the middle of the lake. When the motor-boat approaches the opposite shore, it begins slowing the speed and loosing the rope... and then stops for a while... The drawn boat does the same and even goes back under the impact of the weak rope... the motor-boat follows it... and the whole system reverses its direction of movement...
More “self-pulling” analogies: We can see this idea in many other impressive analogical situations from our routine (most impacting analogies are the life situations in which we have ever come across). For example, imagine a politician, an employee, a businessman, an officer, a scientist:) or just a person who gradually began to grow in his career. Seeing his "movement", others (colleagues, friends, etc.) begin to help him ... so he rapidly rises... they further help him ... and he confidently "moves up"... and it appears that he has surprisingly many friends:) But at some point, for some (e.g., a healthy) reason, it slows his "movement" up... stops... and even starts to turn back... Then those who have helped him so far to grow up, move away from him and even began to "help" him rapidly fall down ... and then he realizes he had no friends... In life, this concept is known as positive feedback...
IMPLEMENTATION. Roughly speaking, this type of oscillations is created only through two operations – a "self-sustaining movement" and a "smooth reversal"; the "movement" is obtained by a "self-drawing" and the "reversal" - by a "self-return". We can implement these operations by a system of two elements – a storage pressure-like element (in electronics - a capacitor) and a controlled source (in electronics – an amplifier), connected in a positive feedback loop. So the general conclusion is:
Sine oscillations are obtained by a smooth loading and unloading of a pressure-like element (a capacitor) implemented by the means of a positive feedback... (to be continued)...
Regards, Cyril
Cyril - I appreciate your aim to describe the behaviour of RC oscillators in the time domain. However, I am afraid that each attempt to describe and explain the rising/falling voltage for one single period depends on the specific circuit. That means: It differs from circuit to circuit and it will be very problematic to find a GENERAL explanation.
As an example, I think that such an approach will not work for an oscillator based on a frequency dependent block that is realized by a delay line.
Dear Prof. Lutz can you please elaborate more on the role of thermal noise in resistance of RC phase shift circuit which is well known source of the oscillations as we already know.
Lutz, it seems here I (a 59 years old "junior student":) and Erik (a 76 years old "senior student":) are the optimists and you - the pessimist:) But that's okay because there is a counterbalance... otherwise we, optimists, will fly too high in the sky and threaten to fall from high:)
In my opinion, this "time domain explanation" covers all types of “self-pulling”oscillators with positive feedback.
Lutz and Erik thank you for such a beautiful and creative discussion on this topic.
I posted this answer somewhere else also. If you read it, apologies for repetition.
==========================
I will start by saying that, there is no way an RC can make oscillations alone. What indeed happens is that, with the help of an OPAMP and a positive feedback, the total impedance of the RC circuit is set to INFINITY, which is the condition to oscillate. To gain insight into this, consider a parallel L and a C:
http://en.wikipedia.org/wiki/LC_circuit
Their impedances are sL, and 1/sC respectively, where s=j*2*PI*f (j being the complex value, sqrt(-1)). So, when you connect an L and a C in parallel, use the same formula you would use to calculate the resistance of two parallel RESISTORs, you arrive at Z=sL/(s^2LC+1). So, this is the impedance (Z) of the LC circuit. Looking at this, it is clear that, if s^2LC+1=0 then the LC circuit will have an infinite impedance. This means that, even if there is a little glitch that gets into the circuit, it will be sustained as an oscillation forever. So, this means that, s^2LC+1 = 0 yields f=1/(2*PI*sqrt(LC)) . f is the frequency at which, the circuit will sustain any sine wave that gets into it. You usually "excite" the circuit, or, you don't really even need to most of the times, since the noise around will do that for you. Of course, this analysis is assuming ideal L and ideal C. There is no way this will happen, since every element is non-ideal in real life. So, the oscillations will slowly die (dampen) unless they are fed back by an active circuitry.
Why sine wave ? Because, the derivative of the sin() is cos() and derivative of cos() yields -sin() (i.e., they are the same FAMILY of waves) ... Since capacitor is a VOLTAGE DERIVATIVE machine, and the inductor is an VOLTAGE INTEGRATOR machine, when you connect them in parallel , you will arrive at a frequency at which the impedance is infinity (in other words, you can transfer the energy from one to the other infinitely without losing it).
So, how does this relate to an RC circuit ? Well, since it IS possible to MAKE AN INDUCTOR USING A CAPACITORs+RESISTORs+OPAMP, this is how you can make an oscillator by using (C as a C) and (R+C+OPAMP as an L). This allows us to turn a DERIVATIVE ENGINE (C) into an INTEGRATOR ENGINE (L). So, C starts acting like an L. Then, we will have the infinite impedance at a certain frequency.
How about Wien bridge oscillator ? http://en.wikipedia.org/wiki/Wien_bridge_oscillator
Same idea. Here, due to the placement of the parts, you get a complex transfer function very similar to the one I showed above, except, it is something like
s/(s^2+3s+1), instead of the s/(s^2+1) I showed above. So, This CAN be set to infinite resistance by setting s^2+3s+1=0. In other words, thinking super simply, look at C1 on the Wiki page, and look at the rest of the STUFF. If the rest of the STUFF can be made to act like an inductor (VOLTAGE INTEGRATOR), you got yourself an oscillator.
@Abhijit Khadke
Quote: "Dear Prof. Lutz can you please elaborate more on the role of thermal noise in resistance of RC phase shift circuit which is well known source of the oscillations as we already know."
I think, it is not easy to answer this question.
For my opinion, it is not thermal noise but the inrush jump (power switch-on) that is responsible for starting oscillations. This is a result of all my simulations with many different oscillator topologies. I think, the amplitudes of this switch-on transient is much larger than any noise.
Another situation arises if - as a first step - the oscillator is designed with a pole pair in the LHP (decreasing oscillations after power switch-on). Now we can tune one of the gain determining resistors and, thus, shift the poles to the RHP. In this case, I see no other stimulus than noise to start oscillations.
I shall try to find a suitable method to simulate such a time-depending resistor.
Quote: "Lutz, it seems here I (a 59 years old "junior student":) and Erik (a 76 years old "senior student":) are the optimists and you - the pessimist:)
In my opinion, this "time domain explanation" covers all types of “self-pulling”oscillators with positive feedback."
No Cyril - in general, I have an optimistic attitude. However, when I have some doubts regarding technical realisations/explanations I am afraid an optimistic attitude does not help too much.
And I repeat my doubts: Imagine a simple feedback loop having a gain of unity and a phase shift of -360 deg for one frequency fo only - realized by an ideal delay line with a delay T=1/fo. How can you explain the oscillation build-up as you did for RC oscillators?
Lutz, I have no experience with delay lines... I have first to create a notion about them in my mind... Maybe it can be thought sooner as a distributed than a lumped element? Or as an integrator with an input quantity - the velocity, and an output quantity - the displacement?
Cyril - same to me, no specific experience with delay lines.
But - for my purpose it is sufficient to know that such a thing does exist and what it does: To delay. And - yes, I suppose it is a distributed technology.
However, I have to correct myself:
The delay must be T=1/2fo causing a phase shift at fo which is only 180deg.
In this case, we need an inverting amplifier (gain: -1). Otherwise, we have unity feedback for DC and - of course - no oscillation.
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