Dear Cyril Mechkov,

The Barkhausen criterion is based on the assumption of a linear circuit. It is equivalent to placing a complex pole pair on the imaginary axis. You cannot balance on the razors edge. Linear oscillators must be damped systems. The amplitude of an unstable linear circuit goes to infinite. Steady state oscillators are nonlinear systems. The open loop circuit is just an active filter. When you close the loop you may change the load of the power-source. It is the closed loop circuit which might be an oscillator. The Barkhausen criterion is necessary but not sufficient.

Cyril,

this subject (harmonic oscillator) is fascinating me since several years.

To start the discussion I like to say:

Barkhausen`s condition (with all its restrictions) applies to active LC oscillators as well as to RC oscillators. In both cases, the loop gain DESIGN value for steady state conditions must be unity. However, it will NOT be unity during operation because the amplitude stabilization scheme will cause a continuous movement of the poles between the right and the left half of the s-plane.

That means - I cannot see any fundamental difference between LC and RC oscillators.

Additional remark:

I think, we have designed RC oscillators - as an experimental proof on Barkhausen´s rule - since many years.

Erik and Lutz, thank you for the instant reaction! It seems there is some contradiction between the theoretical Barkhausen arrangement and its implementations? It seems to me as a kind of a science fiction that does not endure the proof of the reality:) I would like to direct the discussion to more intuitive expalanations; I need them to be convinced.

Regards, Cyril.

Cyril,

of course, there is a kind of "difference" (I prefer this expression, rather than "contradiction") between theory and practical implementation.

Barkhausens rule`(loop gain unity) applies to linear and steady-state conditions, which never can be realized in practice. Nevertheless, It is (and was since decades) a good starting point for designing working oscillators using the so called "modified Barkhausen condition". Thus, we require a loop gain slightly larger than unity and use in addition a gain controlling element. As a simpler alternative, the limiting/clipping property of the power supply rail can be exploited. Thus, each oscillator becomes a non-linear circuit. To me, that is the whole story. Did I miss something important?

Let me add another aspect:

In the past there were some publications ( in high-reputative journals) claiming a "failure of the Barkhausen condition". I even have read the sentence "down with Barkhausen".

Apparently, those authors didn`t realize that the Barkhausen condition for oscillation is only a necessary one. Thus, any circuit that does NOT oscillate despite of having a loop gain of unity at one single frequency cannot serve as "counter example" to the Barkhausen condition.

Lutz, thank you. You have said it very well... and most of the readers will be satisfied... but I am a little more different person... I would like to penetrate deeply into the essence of the RC oscillation phenomenon. I just want to understand (at the possibly lowest level) how the sinusoid conceives in an RC oscillator. For this purpose, I follow mentally the "trajectory" of the instant output voltage "point" when "moving" between the power supply rails in the manner below:

http://en.wikibooks.org/wiki/Circuit_Idea/How_do_We_Create_Sinusoidal_Oscillations%3F.

Well, it is obvious that to move the "point" between the rails, the gain has to be greater than one (to have a kind of a self-reinforcing positive feedback). The next question is, "What is the gain at the end, when the "point" reaches the rail?" And the final question is, "What makes the 'point' change (reverse) the direction of the movement?" Why does it not stop moving and remaining in this final position?

I have my own explanation of how and why it reverses direction. Near the rail, the nonlinear feedback (nonlinear voltage divider) begins decreasing the gain or the amplifier begins saturating; the voltage begins slowing its rate of change and the curve "looses its nerve". Finally (at the peak), the loop gain becomes one and there is no more regeneration. The output voltage stops increasing; for a while, at the top of the sine wave, it stays constant. Now I suppose that some small disturbances (that ever exist) slightly "pull" the point in the opposite direction; the gain slightly increases above unity and the "point" begins accelerating in the new direction... and this repeats again... and again... and again...

Regards, Cyril.

Cyril, good morning.

Thank you for the link you have provided. After a short look onto the pages you have created I have decided not to read it immediately. It`s too much - however, I will reserve some time for it in the near future.

In any case, now I know what you mean and what your goal is - beyond the classical way to explain the principle of oscillator operation. You want to strictly stay in the time domain, right?

Nevertheless, my "feeling" still is that there is no fundamental difference between active LC and active RC oscillators. Perhaps it is good to start all considerations with 2nd order high-Q filters. In case of passive LC circuits (2nd order) we have two storage elements, which are active in an alternating manner - and, in principle, the same applies for RC circuits of 2nd order. However, because both storage elements are of the same type we need some positive feedback (that means: an amplifier) for high-Q circuits, which allow a sinusoidal decaying step response (like the passive LC circuit). Thus, in both cases we have a "swing" of energy, which in active RC circuits is supported by a second energy source (amplifier).

Please, consider the above as some preliminary attempts to enter the problem you have formulated. Next, I will try to follow your wikibooks contribution.

Regards

L.

Yes, Lutz; for the purpose of my intuitive approach, I always stay in the time domain. I work with the instantaneous values of the electrical quantities and "enter" the transition process to see what happens. My favorite techniques are to slow (mentally or by real experiments) many times the speed of the process (up to step-by-step operation), to identify with the active element (empathy) and to perform "manually" its functions.

For example, in the picture below (a two input "op-amp" inverting summer), I act as an op-amp:) I am surprised by the positive "jump" of the input voltage Vin1. I have not immediately reacted in the first moment and the virtual ground (point A) is not a ground (it is a positive). After a while, I will recover from the shock:) and will "pull down" the virtual ground through the resistor R to make it zero. Here is a link to this animated story (go to Exploring, step P6-2):

http://www.circuit-fantasia.com/collections/circuit-collection/circuits/old-circuits/v-to-v-active-sum-old.html

Regards, Cyril.

Hello Cyril,

thank you again for the link to your wiki-book. It was really interesting to read the chapter about the LC tank circuit. A complete new view and a new way to treat and explain this basic circuit.

However - back to the topic of this discussion (LC and RC oscillators).

As you know - the behaviour of inductors can be emulated by active RC circuits (e.g. GIC based).

Question: Do you think it is possible to explain the behaviour of such an active-LC circuit in a similar way as you did before for the passive tank?

This could be a possible way to find the fundamental differences and/or similarities between passive and active resonators.

Beyond this example, the remaining question to be answered is:

How and why is an active circuit with feedback able to emulate the role of the inductor in a passive RLC filter?

I believe, the time-dependent model of energy sources you have described in your wiki-book could be a good starting point to come closer to an answer. This is because each dependent voltage source (finite gain amplifier in Sallen-Key filters), which is driven by a varying voltage is such a time-varying source.

Hi, Lutz!

So many valuable thoughts there are in your post... Thank you for your opinion; it means a lot to me. Explaining circuits with this intuitive approach is a thankless job and in the most cases I have not got an approval when sharing my insights with people, especially when they are formal (a typical example are the most orthodox Wikipedia editors ("wikipedians") inhabiting the electronics space of the famous encyclopedia. I have spent a few years of my life to "fight" with them but the results (excluding the page about the Miller theorem) were unsatisfactory.

You have posed another extremely interesting and related to our discussions topic - how to emulate an inductor. And the general question is, "How do we create virtual elements?" Maybe, following the wise advices of Prof Pisupati, we have to extract and pose new questions thus enlarging the scope of our talks about the great circuit phenomena. So, the next questions may be about gyrators (simulated inductors).

A three years ago, it was a great challenge for me to reveal, at the lowest intuitive level, the secret of these so abstract circuits. I first exposed my insights in Wikipedia (under the user name Circuit dreamer):

http://en.wikipedia.org/wiki/Talk:Gyrator#Discussion_about_energy_storage_in_Gyrator_circuit

http://en.wikipedia.org/wiki/Talk:Gyrator#Demystifying_gyrator_circuits

http://en.wikipedia.org/w/index.php?title=Gyrator&oldid=357087255#Implementation:_a_simulated_inductor

Then, my students and I (user name Circuit-fantasist) did the same in the laboratory and in Wikibooks:

http://en.wikibooks.org/wiki/Talk:Circuit_Idea/Demystifying_gyrator_circuits

Great idea... and it is closely related to the bootstrapping and true negative impedance...

Regards, Cyril.

Hi Cyril, thanks for your long reply.

In the context as discussed up to now perhaps the following is also interesting:

The frequency-determining part of an oscillator can be a parallel RLC combination in series with an other resistor. Closing the loop using a non-inv. opamp (finite gain) results in an oscillator. As a next step, the R||L can be replaced by one opamp emulating a lossy inductor. Thus, we have a two-opamp oscillator (two-port topology).

Now - it is interesting that this twoport-oscillator with feedback can be also seen as a negative-resistance one-port oscillator.

That means: Without any circuit change two different explanations (interpretations) of the same circuit are possible - just by allocating one single resistor to another opamp.

Finally, I have the following request:

Looking into the links you have provided I have clicked a bit around - and suddenly I have arrived on a page with a rather long discussion (between you and somebody else) about the principle of RC oscillators. I remember, you did consider this discussion as not very fruitful.

However, I left the page and I did not mark it. Please, can you give me the link again? Thank you.

L.

Hi Lutz, I was just getting ready to place a question about the connection between the Wien bridge oscillator and the true negative impedance (NIC). It will be something as "Is the Wien bridge oscillator a negative impedance circuit?"

About the "two different explanations (interpretations) of the same circuit"... I posed even three interpretations in the introductory "Background" section of the Wikipedia page about the Wien bridge oscillator (the link below points to this old version of the page):

http://en.wikipedia.org/w/index.php?title=Wien_bridge_oscillator&oldid=444847334#Background

After it was removed, I began these "epic" debates about the nature of RC oscillators:

http://en.wikipedia.org/wiki/Talk:Wien_bridge_oscillator#How_do_RC_oscillators_produce_sine_wave.3F

http://en.wikipedia.org/wiki/Talk:Electronic_oscillator#Relaxation_versus_LC_oscillations

http://en.wikipedia.org/wiki/Talk:Electronic_oscillator#Negative_resistance_LC_oscillator

As you can see, I tried to advance the idea that the inert nonlinear resistor (a bulb, thermistor) has to be not much inert:) so that its resistance to change slightly when the output voltage approaches the rails (its time constant has to be small enough)... but my opponents did not accept it. What do you think about this speculation?

Regards, Cyril.

Lutz, I have extracted from my Wikipedia discussions a possible explanation about how the amp output voltage reverses its direction at the top of the peek:

"The output voltage stops increasing; for a while, at the top of the sine wave, it stays constant positive and equal to the input one; no charging current flows and the voltage across the capacitor stops changing. But note this state is not stable. Why? This is the main question. At this moment, the Wien network is supplied by the constant output voltage. If there was a galvanic connection between the output and the grounded capacitor, the voltage across the latter will stay constant as well. But the upper capacitor disconnects the output from the grounded capacitor and the latter begins discharging through the resistor connected in parallel. The voltage across it begins decreasing, the nonlinear fedback begins increasing the gain and the regeneration begins operating."

It seems, the act of reversing in the Wien bridge oscillator is based on the AC (not galvanic) positive feedback since, because of it, the output voltage cannot hold on the top near the rail... Is it true?

Regards, Cyril

Hi Cyril, thank you again for providing me with the links.In particular, I will read the discussion about RC oscillators and will come back if I have some questions or comments.

1.)Regarding the speed of reaction (inertness) of the non-linear resistors my position is as follows: The amplitude controlling device (non-lin. resistor) must NOT react on each peak (like a diode) but should have a thermal time constant that is much larger than an oscillation period. On the other hand, it should not be too large because it must react upon amplitude changes. As a result, the oscillation signal will be modulated with the amplitude controlling signal. The frequency of this modulation can be observed and should be smaller by a factor of (10...100) if compared with the oscillation frequency. This is the result of my former experiences.

2.) Yes - I also believe that each feedback oscillator - in principle - can also be seen as a negative-resistance oscillator. Sometimes it is easy and sometimes more complicated.

3.) Regarding your last point (reverse of direction) I have no comment at the moment. First, I have to understand and to think about it.

Regards

Lutz

Lutz, your answer ("...it should not be too large because it must react upon amplitude changes...") was what I wanted to see. Now it is more than obvious that the time constant has to be large enough to ensure an average gain close to unity and small enough to ensure a "dynamic" gain during the cycle (more than one when the voltage is between the rails and equal to one when it is close to the rail).

Regards, Cyril

Hello Cyril,

As I am very interested in RC oscillators I went in the mean time through the very long Wiki-discussion on the WIEN bridge oscillator. And I agree with you - at the end, rather disappointing:

*Many misunderstandings, repetitions and polemic statements,

*some confusions using the terms „loop gain“ and „open-loop gain“,

*Confusing sights regarding the role of the light bulb.

However, regarding the WIEN oscillator and the „direction reversal“ (as mentioned by you) now I feel able to comment.

As far as I have understood you have two slightly different explanations.

For my opinion, the first one using the loop gain (your second contribution in this discussion) is a bit „problematic“ because the loop gain is a term defined in the frequency domain using the linear small-signal ac impedances of the WIEN path for f=fo. Thus, we cannot use it in the time domain.

Instead, we only can say: The momentary voltage at the non-inv. terminal is amplified by a factor of 3 (as long as the amplifier acts linear).

On the other hand, I think the second explanation (your last post) is a very good starting point for explaining the oscillator behaviour in the time domain.

When the output voltage does not increase anymore (saturation) the grounded capacitor C2 discharges causing a decrease of the input voltage and, thus, also of the ouput voltage. During this process the remaining charge of C1 causes the input voltage even go to negative values. As a result, the output voltages changes direction and re-charges the capacitors with negative voltages. I think, this can explain the oscillation in the time domain.

However, I feel a bit confused about your statement:

„This Barkhausen assertion can be right in the case of an LC oscillator where an LC tank produces the oscillations and the feedback system only sustains them. But it cannot be right in the case of RC oscillator where the humble RC circuit cannot produce any oscillations without an external control. Do you realize the simple truth that Barkhausen is not always true?“

Do you really believe that Barkhausen does NOT apply to RC oscillators?

Lutz, I need some time to process your data above; it is extremely interesting for me and it has provoked my imagination. I have asked a more general question, "How do sinusoidal oscillations arise in RC oscillators?" where to continue the general discussion about the nature of RC oscillators:

https://www.researchgate.net/post/How_do_sinusoidal_oscillations_arise_in_RC_oscillators#share

Regards, Cyril.

I have asked a separate question about the structure and the operation of the Wien bridge oscillator:

https://www.researchgate.net/post/What_is_the_basic_idea_of_Wien_bridge_oscillator_How_does_it_operate#share

without making the answer real long, 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.

Very interesting thoughts, Tolga... Would you take a look at my Wikibooks stories about LC oscillations

http://en.wikibooks.org/wiki/Circuit_Idea/How_do_We_Create_Sinusoidal_Oscillations%3F,

load canceller

http://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Compensating_resistive_losses_by_N-shaped_negative_resistors,

and negative capacitor

http://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#The_basic_electrical_circuit ?

The last two stories show how to increase the ohmic resistance of the load up to infinity and how to neutralize the stray capacitance what is a good illustration of your " the total impedance of the RC circuit is set to INFINITY" (this means to have a loop gain = 1). Only I am not sure if this condition is sufficient to obtain oscillations in an RC circuit; IMO we have little to transcend it, i.e. to make the RC impedance little negative (to have a loop gain > 1). IMO this condition is sufficient only in the case of an LC circuit since the very LC tank can oscillate and it needs only an additional active circuit sustaining its own oscillations while in the case of an RC circuit it cannot produce oascillations alone.

Now I try to assimilate your speculation, "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..."; it sounds too tempting... Actually, It is the case of a resonance in an LC tank (I have given intuitive explanations like yours of this phenomenon in the Wikipedia discussion below)

http://en.wikipedia.org/wiki/Talk:Electrical_resonance#Revealing_the_truth_about_electrical_resonance_phenomenon

I am deeply impressed by your text about the relation between the LC and RC circuits:

"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."

Honestly, I do not like this approach - to present the active circuit supplying the storage capacitor as an inductor... but it is interesting to me to find an intuitive explanation of this trick...and I will try...

Regards, Cyril

I actually do not like the explanation of (making an L out of C) either. I was using the following steps to explain the RC oscillator: STEP 1) Ok, I will prove to you that, an LC will oscillate, STEP 2) I will prove to you that, just because it is an RC, it doesn't mean that, it is deeply passive. Once you have an OPAMP involved, I can make the "impedance" look like anything ... STEP 3) Ok, let's see, by connecting things here and there, if I can make the RC impedance look like an LC ... STEP 4) Write the formulas, if things line up, so, the impedance looks like the same way it looked in the case of LC, I can expect the same behavior.

My philosophy in explaining the phenomenon was to start with a very simple example, and compare the following more sophisticated things to it ...

Let's give the UNTOLD HERO (the inductor) the credit it deserves. There is no passive circuit like an LC that will create oscillations. OPAMPs are a way to get the amazing behavior that inductors have from a capacitor. On the other hand, this is no small result. If you can use an OPAMP and a CAP to make an inductor, you can synthesize an inductor in a VLSI IC. This technique is widely used in IC's ... However, remember, an OPAMP consumes power, so, you consuming power, just so you can imitate the behavior of this amazing circuit element called the inductor ! So, this "imitation" is not free ! Also, inductor and capacitor may oscillate at 10 mV. , whereas, an OPAMP will require , I don't know, Vcc=3 to 30 Volts to operate. Nothing can match the beauty of an inductor, but it IS possible to imitate him.

Also note, a perfect oscillation will NEVER happen, since an L is never an L, and a C is never a C. They have an ESR and EPR (Equal Series and Parallel resistances). So, the oscillations are doomed to die unless they are "revived" with an active circuitry like an OPAMP. So, we can't really be too harsh on the OPAMP either :) He is doing a gravely important job here too.

I love the way you explained the oscillations with the up and down sequences. For this to make sense to anyone, they should first experiment with an L (some absurdly high value, like 1 Henry) , and a capacitor, like 100 Farads (which is trivial these days with a supercapacitor). Since things will happen very slow with such high values, they will get an intuition into what an inductor does, and what a capacitor does.

Although mathematically the sL and 1/sC sound excellent, FEELING what it means requires you to do the experiments with a GIANT inductor and a GIANT capacitor to understand the behavior. Otherwise, it is a bunch of equations, which we are all good at solving ! You can't imagine how many of my students think that, a current is flowing THROUGH a capacitor, and have no idea that, the net flow through a capacitor is ZERO ! I am actually talking about Ph.D. students :)

Anyways, nice talking to you. Don't get me going, I won't shut up :)

An amazing way of thinking what I have not seen before... congratulations! Great topic... great way of thinking... great manner of expression...! I can only be happy that there are people like you - Tolga, here in ResearchGate! Regards, Cyril

The last contributions from Tolga and Cyril remind me on the following:

* Each sinusoidal oscillator based on the negative-resistance principle can be redrawn (without additional parts, of course) to a topology that shows a classical feedback loop. Thus, we can say: Both concepts are identical - the only difference is: Both topologies look different or can be explained in a different way.

* As a special example:

Consider an oscillator realized with a lossy parallel LC block in the positive feedback loop of a finite gain opamp stage.

Without any redrawing, just by another interpretation of the circuit it is possible to explain the oscillator function based on the negative resistance principle.

Lutz, however what is the active circuit in an RC oscillator - a negative resistor (an "element" with negative impedance) or an 'inductor"?

Good morning, Cyril.

I must confess, I don`t get the meaning of your question.

The active circuit always is the opamp, is it not?

Hi, Lutz! It is so wonderful when our discussions reanimate... My question is connected with the Tolga's approach above - to explain the RC oscillator with the means of an LC oscillator (introducing an inductor "imitation"). As I can see, his approach is to use the LC network as a base of understanding the RC oscillating network... as a kind of an electrical analogy of an electrical circuit...

Regarding the two alternatives (I have mentioned before) for explaining the oscillator function I have decided to present a corresponding circuit diagram.

1.) The opamp including Rs, RR, Ro can be seen as a negative resistance RN for compensating the damping resistor Rp.

2.) The opamp - together with the feedback resistors RR and Ro - form a non-inv. amplifier that has in addition a positive feedback path consisting of a series connection of Rs and a lossy RLC block.

Cyril - how do you manage it that a pdf picture appears directly with the text (not as an attachement)?

Exactly Lutz, I completely second you; really, we have considered this topic. I would only add that "the opamp including Rs, RR, Ro can be seen as a negative resistance RN" is a current-inversion negative impedance converter (INIC). Nice picture, nice idea...

About the pictures... The RG software allows to attach only one file (by clicking the "add attachment" below the text); if it is a picture, it is visible. In this case, a link is created and you can use this link as you want. For example, in the "whiteboard story", I have first attached the 10 pictures and then I have copied their links from the browser URL address window and inserted them in the old text above. See for example how I have copied the link to your pdf file and inserted it here

https://www.researchgate.net/file.PostFileLoader.html?id=521b3626d3df3e006f9ae493&key=50463521b362654ba0

(I have right-clicked on the link below the text and chosen "Copy shortcut")

You can insert only links (not pictures) inside the text; so you have first to upload somehow the picture on the web and then to copy&paste the link into the text. I am not sure if I have explained it well; so do not hesitate to ask me more about the topic.

Best regards, Cyril

Cyril - thank you. I`ll try it next time.

In a previous comment Prof. Tolga Soyata mentioned the role of the inductor and some related realizations resp. interpretations based on a capacitor.

In this context, I have another illustrative example as described in my contribution

https://www.researchgate.net/publication/233860070_A_novel_harmonic_Oscillator_GIC_Resonator?ev=prf_pub

Here, it is shown that an active circuit designed as an two-pole oscillator can be interpreted twofold:

1.) As a classical RLC oscillator (however, with active L-realization) - using a negative resistor for compensating losses, or

2.) as a ZD-R oscillator, where a frequency-dependent negative resistor (active FDNR realization) produces a resonant effect if combined with an ohmic resistor.

Data A novel harmonic Oscillator: GIC Resonator

Hi Tolga,

In your statement above (I mean your last comment; I will also reply to your previous comment but now it is too late) there are several interesting approaches to the understanding and studying circuits which I really like because I myself use similar techniques:

* having your own philosophy in explaining circuit phenomena

* step-by-step building and investigating circuits - I always do it

* using a similar (analogical) arrangement - just mandatory

* personalizing and spiritualizing the active elements (empathy) - wonderful!

* slowing the speed of the circuit operation by using elements with extremely high values ("a GIANT inductor and a GIANT capacitor ") - this is my favorite technique in the laboratory

Remarkable thoughts:

* "Once you have an OPAMP involved, I can make the "impedance" look like anything"

* "By connecting things here and there, if I can make the RC impedance look like an LC"

* "There is no passive circuit like an LC that will create oscillations"

* "OPAMPs are a way to get the amazing behavior that inductors have from a capacitor. "

* "... so, you consuming power, just so you can imitate the behavior of this amazing circuit element called the inductor ! So, this "imitation" is not free !"

* "... we can't really be too harsh on the OPAMP either :)"

* "Otherwise, it is a bunch of equations, which we are all good at solving !"

Here you have touched the extremely interesting issues concerning the creation of virtual elements, e.g., a gyrator (simulated inductor). IMO we have to dedicate a special question to them...

But Tolga, I would like to repeat the question taht I have posed above, "Yet what is the active circuit in an RC oscillator - a negative resistor (an 'element' with negative impedance) or a 'simulated inductor' (gyrator)?"

My answer (like Lutz's) is "a negative impedance circuit". In addition to the Lutz's pdf drawing, an hour ago, I drew (with a pleasure:) a "whiteboard picture" illustrating this concept. Let's look at the picture.

The Wien network consists of two parts (lower and upper) having respectively impedances Z (C1 in parallel to R1) and 2Z (C2 in series to R2). The voltage drop across the lower part (Z) is Vp and the voltage drop across the upper part (2Z) is 2Vp while the current Ip is the same. See also that the op-amp produces an output voltage Vout = Vp + 2Vp = 3Vp. So the result is amazing - the circuit consisting of the op-amp, R3, R4 and the upper part of the Wien network acts as a negative impedance element with impedance -Z; it neutralizes the "positive impedance" Z of the lower part and the result is an infinite impedance as you have said above (this corresponds to a loop gain = 1).

Now let's return to the main question about the validity of the Barkhausen criterion or simply, "Is the loop gain equal to 1?" which, if it is true, means (in the context of the negative impedance) that the negative impedance is equal to the positive impedance.

IMO this is only at the peaks of the sine curve where the output voltage stops "moving".

Where the output voltage increases...

... or decreases,...

... the gain is > 1 and the negative impedance (conductance) dominates - it is less (more) than the positive impedance.

Quote Cyril: "So the result is amazing - the circuit consisting of the op-amp, R3, R4 and the upper part of the Wien network acts as a negative impedance element with impedance -Z".

Yes - Cyril, this is a very illustrative example for the claim that each feedback oscillator can also be seen as a negative-resistance oscillator.

Ok, let's not make "negative resistance" and "negative impedance" cooler than it sounds. If you look at what RESISTANCE means: R=V/I.

To make it slightly more intuitive, let's say R=DeltaV/DeltaI to denote the instantaneous resistance on the V-I curve.

So, the resistance is the tendency to increase the voltage (DeltaV goes up) when the current is increased through the resistor (when DeltaI goes up).

This means that, if I have a two-terminal circuit whose voltage goes DOWN when I INCREASE the current , it is ACTING LIKE a negative resistance.

If you draw the V-I curve of any two-terminal device, and if you find a negative slope pieces in the curve, you got yourself a negative resistance REGION OF OPERATION. However, don't forget, this might not hold for every single "V" value. This means that, you have to operate your device within the "V" region where you do have these -R SLOPES ... It gets worse, if you come out of those regions during the oscillations, you will hit the positive slope regions ... You are back to +R ...

Clearly, the INFINITE RESISTANCE regions like between +R and -R when the curve changes from up-slope to down-slope ...

Using a negative resistor, I can make an oscillator ... clearly ... The question is : can I make a negative resistance out of PASSIVE elements ?

The answer is : Only with an LC ...

To make a negative resistor from an RC, you need the help of the OPAMP.

Yes - I think everybody will agree.

Left to say that an active negative resistance circuit can be regarded as a voltage controlled current source, whereby the current goes through the controlling source.

Nice and funny explanations... Tolga, what would be interesting discussions going on here if only there were a few more witty and creative thinking people like you! Thus we could seamlessly explain the odd, absurd and paradoxical circuit phenomena, and they will not look like "black magic" in the eyes of the uninitiated:)

Yes, Tolga, we should always strive to do this "boring" (for today's young people) matter "cool." At least I try to do this; as an example I give links to both vast Wikibooks stories devoted to both types of negative resistance - absolute and differential (and also to the virtual zero and infinite resistance)

http://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance and

http://en.wikibooks.org/wiki/Circuit_Idea/Negative_Differential_Resistance,

where you can find amusing analogies illustrating the great phenomenon (negative resistance is my favorite phenomenon and I intend to dedicate it separate questions). Now I will add some comments to your speculations.

"So, the resistance is the tendency to increase the voltage (DeltaV goes up) when the current is increased through the resistor (when DeltaI goes up)" sounds a bit strange to me (as though the resistor produces voltage, acts as a voltage source:)

"This means that, if I have a two-terminal circuit whose voltage goes DOWN when I INCREASE the current , it is ACTING LIKE a negative resistance. " - it is interesting to see how this "magic" happens... what the negative resistor does...

"Clearly, the INFINITE RESISTANCE regions like between +R and -R when the curve changes from up-slope to down-slope ." Right... but there is another (more exotic) way to obtain an infinite resistance - by "mixing" negative and positive resistances. It is implemented by connecting in parallel two opposite ("positive" and N-shaped negative) resistors... and it is this idea is used in the Wien bridge oscillator (it is illustrated in the left figure of the Lutz's pdf file).

I like your final bold speculations - "The question is : can I make a negative resistance out of PASSIVE elements ? The answer is : Only with an LC ... To make a negative resistor from an RC, you need the help of the OPAMP"... but I cannot quite figure it out... Particularly, how will you "make a negative resistance with an LC"?

Regards, Cyril

oh, here: If you solve the parallel C and L, Z=sL/(s^2LC+1). So, your "critical point" is s^2LC+1=0. This is where the impedance is infinite. So, that's at j^2*4*PI^2*f^2+1=0. In other words, Where can Z be negative ? Solution:

j^2*4*PI^2*f^2+1 < 0

-4*PI^2*f^2+1 < 0

4*PI^2*f^2-1 > 0

4*PI^2*f^2 > 1

f>1/(2*PI)

Now, this is in radians, of course,

Also, remember, since there is sL on top, this is where the IMPEDANCE is negative, not RESISTANCE. So, this LC is moving the current negative when the voltage goes positive, however, with a phase difference.

to simply visualize why this is happening, go to Google, and type plot(1/(x+1)). You will see the asymptote at -1. This is really 1/(2*PI). Any impedance below this is negative, anything above is positive.

So, to clarify the terms:

NEGATIVE PURE RESISTANCE is the tendency for a 2-terminal circuit element to respond with a current DECREASE when its terminal voltage is INCREASED

*** immediately ***

NEGATIVE IMPEDANCE is .... when voltage DECREASES, the current INCREASES a *** little bit later *** ... i.e., phase difference

You can't make a pure negative resistance with strictly passive elements ... IF this was possible, you would have to put together R, L, C's and add up the complex math, and have the complex terms cancel out (the s's) and end up with strictly a real number that is negative. With opamps, no problem. With strictly passive elements, not possible.

The same argument I made about the INFINITE resistance can be achieved in the SERIAL L+C, however, this time, we get ZERO IMPEDANCE, not infinite ... same idea. But, we can never get negative pure resistance with passive elements.

I also want you to remember something: NEGATIVE RESISTANCE is an illusion !!! When you increase the voltage and the current decreases, it looks like you are creating electrons out of nowhere ! NO WAY ! You are taking them from the Vcc of the OPAMP and burning it through the GND of the OPAMP. So, the OVERALL CIRCUIT is still resistive. It is only the isolated two terminal element that you are making ACT LIKE a negative resistor, by ABUSING the OPAMP :) not the entire circuit !!!

If you could get negative resistance from the ENTIRE CIRCUIT, you would be the next guy to get the Nobel prize :) Because, your circuit would be creating electrons out of nowhere :) Conversation of the energy is still a valid phenomenon !

I suggest to continue the discussion about negative impedance (resistance) and negative differential resistance in the new questions below:

https://www.researchgate.net/post/What_is_negative_impedance_Does_can_it_exist_If_so_how_can_elements_with_negative_impedance_be_implemented_Are_they_passive_or_active

https://www.researchgate.net/post/What_is_the_basic_idea_behind_the_negative_impedance_converter_How_is_it_implemented_How_does_it_operate_What_does_the_op-amp_do_in_this_circuit

https://www.researchgate.net/post/What_is_negative_differential_resistance_How_is_it_implemented_How_does_it_operate_What_is_its_relationship_with_the_true_negative_resistance