Very interesting topic... and a great challenge to me:) But first I need to know what this odd circuit actually does... then to see how it does it... and finally, to build the circuit step-by-step... And as the first circuit consists of two NIC (INIC and VNIC), the second is an element of the first, the question is, "What is GIC compared with NIC?"

Cyril - the classical application is

* active simulation of a LOSSLESS inductance (Z3 or Z5 capacitive) , and

* realization of an artificial block called FDNR (frequency-dependent negative resistor), which is used with very good success in filter systems.

So, it is simply a gyrator?

Yes - it can be used as a gyrator. But it is more versatile.

I forgot to mention that for FDNR applications two impedances in the numerator are capacitive.

Lutz

(bye - until sunday evening).

Lutz, your excellent picture (I have cut only the first circuit and attached it below) helped me to see (I hope:) the idea behind this exotic Antoniou's circuit solution (by the way, I have seen it drawn in a more tidy symmetrical way)... I will briefly outlined it here and then we may consider it thoroughly...

As you have noted in the body of the question above, the grounded load resistor of the first NIC (INIC) is replaced by another NIC (VNIC). So, to understand circuit, we need to reveal: first, the role of the second NIC (VNIC); next, the role of the grounded load resistor of the first NIC (INIC)...

Well, let's begin with considering the second NIC (VNIC) assuming (as you have said) that Z5 is a capacitor (the attached picture shows an INIC type; just swap the op-amp inputs to obtain a VNIC type) . This circuit acts as a negative inductor since the op-amp adds a voltage equal to the voltage drop across the grounded resistor (Vc + VR1- VR1 = Vc)... and this voltage represents the voltage drop across an inductor (this is a property of the simple RC circuit where the "complementary" voltage drop across the resistor behaves through time as the voltage drop across an inductor)...

Now about the role of the grounded load resistor of the VNIC... It acts as an "original" for creation of a "voltage-inverted copy". Briefly speaking about the VNIC operation, the op-amp keeps up the current through the "original element" equal to current through the input source, and "inserts" (adds) a reverse voltage into the circuit equal to the voltage across the "original element". To understand how the op-amp does this magic, think of the VNIC circuit (excluding only the op-amp) as of some kind of a balanced Wheatstone bridge with a varying supply voltage...

Now about the role of the grounded load resistor of the first NIC (INIC) - see the attached picture... It acts as an "original" for creation of a "current-inverted copy": if it has a positive impedance (the usual case), the "copy" will have a negative impedance adn v.v., if it has a negative impedance (our case), the "copy" will have a positive impedance... This op-amp circuit does this "magic":) by inverting the current through the input voltage source; thus the name "negative impedance converter with current inversion" (INIC). I suggest to outline briefly this idea here and consider it thoroughly in the question intended especially for it

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

In brief, the op-amp keeps up a voltage drop across the "original element" equal to the voltage of the input source, and "pushes" a reverse current through the input source equal to the current through the "original element". To understand how the op-amp does this magic, think of the INIC circuit (excluding only the op-amp) again as of some kind of a balanced Wheatstone bridge with a varying supply voltage...

As a result, the whole circuit acts as an "inverted original element"... and, for example, if the "original" is a positive resistor, the circuit will be a negative resistor...

Now, if we look again at the exotic Antoniou's circuit solution, we will see that our negative inductor (the second NIC) is connected in the place of the grounded load resistor... so its negative inductance is converted into a positive inductance... thus the whole GIC circuit acts as a "positive inductor"... So, my final conslusion is:

In this application, the generalized impedance converter (GIC) is an "inverted negative inductor"... or a "double inverted inductor"... what gives a "positive virtual inductor"... or simply an inductor:)... So, my answer to your question about how to explain this circuit to students is in the 3-step succession below:

1. Take a "real capacitor"

2. By using a VNIC, swap the current through and voltage across the capacitor thus obtaining a "virtual negative inductor"

3. By using an INIC, invert the "virtual negative inductor" (more precisely, its current or impedance) thus obtaining a "virtual inductor"...

Or, to impress your students so that they never forget their teacher:), you can say it in such a striking way:

"By one swap and next two consecutive negations, the Antoniou's GIC circuit converts a real capacitor into a virtual inductor."

We can ad also that while the single NIC is usually a "positive-to-negative impedance converter", the compound GIC is a "positive-to-positive impedance converter"...

Lutz, these were my insights about the famous Antoniou's circuit. It would be extremely interesting for me to know whether you approve them... and if so, whether you have met them somewhere... because it would be a great achievement to find an intuitive explanation of such an abstract and sophisticated circuit...

I am looking forward your return. Regards, Cyril.

Dear Lutz and Cyril

Please take a look at the circuit when we assume non-ideal op.amps.

Dear Erik - yes, the formulas in the given reference show the influence of finite opamp gain. These formulas can help to compare the various GIC realizations.

As we know: All possible opamp based realizations of one particular transfer function show - of course - the same behaviour for IDEALIZED conditions.

The differences between these alternatives can be observed for REAL conditions only.

Quote: "It would be extremely interesting for me to know whether you approve them.."

Cyril - thank you for your answer.

Of course, I fully agree to your explanations - however, I am afraid, I did not express myself clear enough. I do not want to find explanations for the first circuit I have shown in my attachement.

For my opinion, the function (for ideal opamps) is clear:

* INIC: Zin=-(Z2/Z3)*Zload

* Replacing Zload by a VNIC with -(Z4/Z5)*Z6

we get Zin=+Z2*Z4*Z6/(Z3*Z5).

You have explained this relation verbally for a special case (with one capacitor).

But this was not my problem.

The great work from A. Antoniou is NOT this circuit which consists of two NIC`s in series. He has invented the SECOND circuit I have scetched in my pdf attachement. And this circuit looks different because the output from one opamp is connected NOT only to its own inputs but also to the inputs of the other opamp.

There is something like cross-coupling between both opamps.

And it can be shown that THIS configuration (and some other are also possible!) is the best one regarding the unwanted influence of opamp non-idealities (finite gain and phase shift).

And only this was my question: Is it possible to SHOW HOW Antonious version can be derived from the most logical version which is a simple series connction of the two NIC`s. Of course, both circuits lead to the same expression for Zin - however, this is not really an explanation. Rather, it is a kind of proof.

Do you now understand what I mean?

PS: May be that a way to derive the 2nd circuit from the first one is possible using "pathological" elements like norator and/or nullator. But I am not sure about this.

Well Lutz, I will try...

I only would note that Zin=+Z2*Z4*Z6/(Z3*Z5) cannot replace the "verbal explanation"; these are two different things... The simple expression cannot reveal the structure (the combination of all these elements and building blocks), the causality and the clever idea behind this unknown circuit; it shows only how much the input impedance is... Revealing the fundamental idea behind an unknown circuit is a creative process that is based on human imagination, intuition and emotions... and above I showed how to do this... This is still a human activity while finding that expression can be done by a "stupid" computer:)

Regards, Cyril

"Revealing the fundamental idea behind an unknown circuit is a creative process that is based on human imagination, intuition and emotions... and above I showed how to do this."

Yes Cyril - exactly!!

That´s what I mean: An explanation like yours as given verbally for the NIC series connection. But - as you said - it is hard to follow such a "creative process".

Regards

Lutz

Lutz, I began thinking about this exotic "mesh" of resistors and op-amps:) but I need to know some exemplary (gyrator) applications. Is the text below valid as above? Are Z3 or Z5 capacitors again?

"The classical application is

* active simulation of a LOSSLESS inductance (Z3 or Z5 capacitive)..."

BTW I looked in an old (1974) and very abstract book of J. C. Marchais (Marshe) translated from French into Russian, and I saw an interesting classification of principles how to build gyrators. One of them was this one - cascading two NICs...

Regards, Cyril

Here are some initial observations about the second Antoniou's GIC circuit solution...

It is interesting that there is some circuit symmetry - it consists of three pairs of resistive (impedance) elements (ZIN-Z2, Z4-Z3 and Z5-Z6)... thus it resembles a 3-leg bridge. The upper and the bottom pairs are stretched between an op-amp output and ground; the middle pair is stretched between the two op-amp outputs...

Another interesting fact is that the three middle voltages (of the common points between the resistive elements) are equal... as though this "triple bridge" is balanced by the two op-amps...

I think the bridge properties (the voltage drop equalities) can help revealing the secret of this weird circuit... as they helped me to reveal the secret of this gyrator (the attached picture):

It seems VZin = VZ6, VZ4 = VZ5, VZ3 = VZ2, and if Z4 = Z3 then VZ5 = VZ2

Another observation (I have said it before) is that the voltage across Z6 is proportional to the current through Z5... and the current through Zin is proportional to the voltage across Z5 (since VZ2 = VZ5). So, if we use Z5 as an "original" element (a capacitor, the rest are resistors) the whole circuit (seen from the side of the input source) will act as a "reverse Z5" (with swapped current and voltage)...

I remind that I have already asked a question about gyrators (this was in May):

https://www.researchgate.net/post/Can_we_swap_the_voltage_across_and_current_through_electrical_elements_What_is_the_use_of_such_a_swapping_technique_Can_you_show_some_examples?

Lutz, perhaps you understand that my idea here is not only to reveal the essential circuit idea... but also to realize and to show how I think when processing step-by-step an unknown circuit... Of course, this should be also interesting for psychologists and educators... but, as you can see, we "teaching engineers" finally do this work...

Now it is interesting to see how we should think when trying to grasp the new idea (in a Bono manner)... and so I try to see what I do now when trying to understand this unknown (for me) circuit...

I think I do only one simple think - in this "mesh" of components, I try to see something already known (for me)... to break the new solid circuit into smaller well-known building blocks ("bricks")... to see the familiar in the unknown...

Thinking about gyrators and NICs, I have realized the simple truth that "a balanced Wheatstone bridge swaps and inverts the original impedance"...

To understand my idea, imagine a bridge of four elements (3 resistances and one capacitance) and an input voltage source connected in series with the left lower element; the right upper element serves as "original capacitance". As the bridged is balanced, the voltage across the right lower resistor (that represents inductance), is equal to the input voltage. The voltage across the left upper resistor is equal to the voltage across the "original capacitor" so the curent flowing through the resistor and the input source is proportional to the voltage across the "original". The result is the voltage and current are swapped, and the current is inverted... so, by means of this "swapping INIC", we have created a "negative inductor"...

If the right lower (or the left upper element) serves as an "original", then its impedance is only inverted (without swapping)... and this is the classic INIC...

In my opinion, this "bridge viewpoint" is a powerful tool for understanding these exotic circuits... although, 3 years ago, I didn't manage to convince the (clever) Wikipedians (and administrators) inhabitting this page of this simple truth:

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

Quote Cyril: "It seems VZin = VZ6, VZ4 = VZ5, VZ3 = VZ2, and if Z4 = Z3 then VZ5 = VZ2".

Yes Cyril - I think this observation is the key to understand the circuits operation.

Quote Cyril: "So, if we use Z5 as an "original" element (a capacitor, the rest are resistors) the whole circuit (seen from the side of the input source) will act as a "reverse Z5" (with swapped current and voltage).."

Again YES - this explains verbally why this circuit can mimic an active inductor if we chosse Z5 as capacitive.

I have the feeling that we (together) are approaching the secrets of Antoniou`s circuit.

Thanks and regards

Lutz

Lutz, thank you for the warm response! I wonder why only "we (together) are approaching the secrets of Antoniou`s circuit"... but this kind of circuits are so interesting, mysterious and impressive... so, beleive me, I think of them as of living beings!

I have already some notion about the circuit operation but I still "can't see the wood for the trees"... I cannot see sub-circuits as INIC, VNIC inside this circuit... so I continue thinking...

Hi Cyril - what do you think about the following explanation?

1.) Applying classical methods for circuit analysis, the input impedance of the first circuit (two NIC`s in series) can be derived as Zi=+Z2*Z4*Z6/Z3*Z5.

2.) Based on your finding that VZin = VZ6, VZ4 = VZ5, VZ3 = VZ2, we can say that the voltages at both non-inverting opamp inputs are equal (idealized opamp properties).

3.) Thus, we are allowed to swap both opamp input voltages without expecting a dramatic change in circuit properties:

* Connect the node between Zi and Z2 to the lower opamp input, and

* connect the node between Z5 and Z6 to the upper opamp input.

4.) After rearranging the parts (redrawing without any further change) we arrive at Antoniou`s famous circuit. As expected, the input impedance has not changed.

______________

The remaining task would be to proof that this new circuit - in case of REAL amplifiers - has an improved performance if compared with the original circuit.

(A circuit simulation can do this job; I don`t know if A. Antoniou or anybody else has done this proof theoretically).

* Remark: This is another example for the following general rule:

The various circuit alternatives to implement a certain function (transfer function) using operational amplifiers have identical performances as long as IDEALIZED opamp models are used. The differences between the circuit realizations can be seen only in case of REAL opamps (frequency-dependent and finite gain).

Hi Lutz! It is so marvellous to think about such a sophisicated circuit solution... and to hope that we together will demistify it!

This morning a I did a "mental gymnastics" exploring in my mind the circuit opeation. Trying to break the circuit into smaller functional blocks, I began seeing in the combination of Z3, Z4, Z5, Z6 and the lower op-amp, an op-amp differential amplifier operating at a common-mode input signal. But, I thought, this should be an unbalanced differential amplifier (Z3 < Z4 if Z5 = Z6) since, if it was balanced (Z3 = Z4 and Z5 = Z6) its output voltage would not change at all... it would stay constant...

About your second remark... I think "the voltages at both non-inverting opamp inputs are equal" since I suppose the whole feedback circuit is stable (the overall feedback is negative... or the negative feedback dominates over the positive one). So according to op-amp "golden rules", both the op-amps manage to keep (almost) zero voltage difference between their inputs... thus all the four op-amp inputs are at equal voltages with respect to ground...

But Lutz, I can't understand what circuit you have transformed - the first or the second? While waiting for an answer, I will begin to explore mentallly the operation of the second circuit...

We always begin investigating the circuit operation at zero input voltage (VIN = 0 V)... so all the circuit voltages (both input and output) are zero as well...

Then we begin increasing the (positive) input voltage VIN applied to the upper op-amp non-inverting input... its (positive) output voltage VOUT1 begins increasing... the output voltage VOUT2 of the lower op-amp (the "differential amplifier") increases but less (because of the mixed feedback)... and the circuit continuosly reaches the equilibrium point... As a result, the input voltage and the four op-amp input voltages "move" simultaneously (stay equal)...

The voltage drop VZ4 always stays equal to the voltage drop VZ5 across the original element Z5 (a capacitor in our example)... Thus the voltage drop VZ4 "copies" the "original" capacitor's voltage VZ5; so the current through the resistor Z4 (acting as a voltage-to-current converter) is proportional to the voltage across the capacitor. The same current flows through the resistor Z3 (acting as the dual current-to-voltage converter) and creates another voltage drop VZ3... so it is proportional to the "original" voltage drop VZ5. Note that the current flows through Z3 from the left to the right (from the upper op-amp output to the lower op-amp output); this is important to obtain finally a "positive" inductance). The voltage drop across Z2 "copies" the voltage VZ3; so the curent through Z2 is proportional to the voltage across the "original" capacitor Z5... and this current is drawn from the input source (i.e., it flows through Z2 from the left to the right and enters the lower op-amp output). The result is that a current proportional to the voltage across the capacitor Z5 is drawn from the input source...

At the same time, the voltage drop across the resistor Z6 is proportional to the curent flowing through the capacitor Z5... thus this voltage is complementary to the voltage across the capacitor (Z5 and Z6 constitutes a differentiating circuit)... and it represents the voltage across an inductor. The same voltage is applied at the upper op-amp non-inverting input...

So, if think of the whole ciruit as a to-terminal "element", the current through (drawn from) it is proportional to the voltage across the capacitor, and the voltage across it is proportional to the current flowing through the capacitor... we can think of it as of a capacitor with swapped voltage and current ... or as of an "inductor"...

I cannot only understand how the two op-amps behave (interact) so that to adjust all these input voltages equal... this is some sort of magic:)

And, of course, I still do not understand the meaning of this entwined circuit topology...

But still there is some progress - I begin seeing a current-inversion NIC (INIC) in the combination of Z2, Z3, Z4 and the op-amps... The only difference with the conventional INIC is that here two op-amps are cascaded somehow instead the only one op-amp... No doubt, a devil circuit...

Lutz, I have realized your transformation... Very interesting and original idea... but I still cannot make sense of this swap in the second version (e.g.,how it has overcome the op-amp imperfections)...

Quote 1: "But Lutz, I can't understand what circuit you have transformed - the first or the second? While waiting for an answer, I will begin to explore mentallly the operation of the second circuit..".

Quote 2: "Lutz, I have realized your transformation... Very interesting and original idea... but I still cannot make sense of this swap in the second version (e.g.,how it has overcome the op-amp imperfections)..."

Good morning, Cyril. let me try to answer your questions.

I think, I have described the task in my first post - perhaps not clear enough.

1.) It would be nice to have a versatile impedance converter for realizing many artificial parts or transfer functions.

2.) It has been shown that such a circuit can be built using two NIC in combination (replacing the grounded element in the 1st NIC by the input of the 2nd NIC. This is, of course, the first circuit I have shown in my original attachement.

3.) The input impedance is Zin=Z2*Z4*Z6/Z3*Z5 (opamps ideal).

Thus, by choosing on or more impedances as capacitive we can realize some interesting functions (grounded L, grounded FDNR, C-enlargement,...)

4.) However, this works only sufficiently as long as the opamp non-idealities come into the play.

5.) Now comes A. Antoniou: He has found another (but similar) structure which is less sensitive to opamp non-idealities (frequency-dependent gain).

This is the second circuit I have shown in my attachement.

This Antoniou-topology has (for ideal opamps) the same input impedance, see 3.)

6.) My only question was regarding the connection between both alternative solutions.

When the 2nd circuit has the same Zin as the 1st one it must be possible to DERIVE the 2nd one from the 1st one (thus, perhaps following Antoniou´s thoughts).

7.) And I think that I have found such a connection: By simply swapping both non-inverting opamp inputs within the 1st circuit (which is allowed because they carry the same voltage under ideal conditions) we arrive at the 2nd circuit as proposed by A. Antoniou .

8.) However, as far as I can see today, only circuit simulations can reveal that the 2nd circuit is less sensitive to opamp non-idealities.

______

That`s the whole story. OK?

Regards

Lutz

Hi Lutz! Congratulations again for your clever idea! It is wonderful that still there is people trying to invent or explain sucn sophisticated circuits... and I am glad that I can share my insights with them...

I would like to ask some questions. Whose is the first circuit? Who has invented it? Is not again Antonio? Have you seen also the Deboo patent about a gyrator circuit?

http://www.freepatentsonline.com/3493901.pdf

About your eight note (and at all) I do not think circuit simulations can reveal the fact that "the 2nd circuit is less sensitive to opamp non-idealities"... it can show if it is true... but then we are the ones who have to explain why it is true...

I had a problem with understanding your idea above since you had written "Connect the node between Zi and Z2 to the lower opamp input, and connect the node between Z5 and Z6 to the upper opamp input " ("lower" and "upper" belonged to the second circuit but you talked about the first one). Maybe you had to write "right" and "left"... but it is already clear... Now I should think about this swap...

"About your eight note (and at all) I do not think circuit simulations can reveal the fact that "the 2nd circuit is less sensitive to opamp non-idealities"... it can show if it is true... but then we are the ones who have to explain why it is true..."

Yes - that´s exactly what I think.

Regarding the year of invention I will gather some information.

Cyril - don`t be frightened, but the combination of two NIC`s (my first circuit) was described already in 1967 by R.M. Riordan. And A. Antoniou followed already 3 years later (1970). Surprising.

Only described... or explained in such an intuitive manner ("inverted negative swapped capacitor" = "inductor":)?

And what does "Yes - that´s exactly what I think" mean - that "circuit simulations can reveal" or "circuit simulations cannot reveal... and we are the ones who have to explain why the 2nd circuit is less sensitive to opamp non-idealities"?

Surprisingly (???) I have shared your opinion.

I am happy, Lutz! Besides this "devilish" circuit:) I plan also to consider thoroughly (I hope, with your help), in the respective questions, the two NIC versions, the swapping idea (gyrators), the "bridge viewpoint" (since all these circuits are actually balanced bridges as I have noted above) and the general idea of how to create virtual elements... I will start it next week when I will draw a series of pictures on the whiteboard... I have the feeling that we will manage to disclose the secret of the Antoniou's "puzzle" since we are (at least) two who puzzle over this problem... I needed years to unriddle the secrets of the negative impedance, NICs and simple gyrator (simulated inductor)... but I was alone; now perhaps we are two or more... so we need months... or weeks:)?

Again about your clever swapping idea... It is interesting to know if Antoniou has reached in this way to his famous circuit...

By the way - there are other alternatives for "swapping":

* Plus with plus (that`s what we have discussed)

* Minus with minus

* Minus with plus and vice versa.

All these alternatives (described already around 1970) give stable circuits with identical input impedances in case of IDEALIZED opamps (zero input differential voltage).

Lutz, this morning, walking (as usual) through the University park, I finally realized what NICs do and how they do what they do:) But I would like to ask you, "What do you think, can voltage and current sources be inserted in the legs of a bridge circuit? Will it be still a bridge? Can be such a bridge balanced by varying the supplying voltage? Will we manage to balance it (stability)? What are the conditions to reach the equilibrium?" The answers of these questions would explain the operation of all these "satanic" circuits:)

Very interesting... but only if we realize what we do when swapping the parts in all these ways...

Quote: "But I would like to ask you, "What do you think, can voltage and current sources be inserted in the legs of a bridge circuit? Will it be still a bridge? Can be such a bridge balanced by varying the supplying voltage?"

Cyril - there is a little known theorem (the "substitution" theorem) which allows such a replacement (volatage or current source instead of impedance). The only condition is that the voltage resp. current must remain unchanged.

Thus, this should work also within a tuned bridge.

It is interestimg to note that this theorem is applied very often - however, in most cases without knowing.

Lutz, I am extremely interested in this theorem since this is the clever trick to create virtual elements...

I have managed to find a simple explanation of this intricate circuit that is based only on the humble H&H op-amp golden rules.

The general idea is to simulate (emulate, make virtual) an inductor by the dual capacitor... or more precisely, only to imitate the inductor's time behavior with an active RC circuit.

To implement this idea, we have to swap the current and voltage in a capacitor. We can do it if, in an RC network (R4,C here) driven by the input source, we do the following:

• copy the voltage drop (Vc) across the capacitor and convert it into an input current (Iin) of the simulating circuit
• copy the voltage drop (VR4) across the resistor and impose it on the input of the simulating circuit.

In Antoniou's circuit, we can see two interacting op-amp NFB systems that keep up equal voltages at the three points A, B and C (VA = VB = VC)... and so zero differences (voltages) between them (VA - VB = VB - VC = VA - VC = 0). As a result:

•  VA (the input voltage of the simulated inductor) is equal to VR4 (that is proportional to the current through the initial capacitor); so, the voltage across the virtual inductor is proportional to the current through the initial capacitor
• VR1 (that sets the input current of the simulated inductor) is equal to VC (the voltage across the initial capacitor) since the same current flows through R2 = R3 -> VR2 = VR3 -> VR1 = VC; so, the current through the virtual inductor is proportional to the voltage across the initial capacitor.

In addition to this explanation, it would be very interesting to draw the currents (in a form of closed loops) and voltages (as bars) with their directions and signs.

(the hand-drawn circuit diagram is taken from another forum)

Cyril - I have the following problem:

Let´s assume I present both of your conclusions to another person (replacing the term "inductor" with the neutral word "element"):

1.) The voltage across the virtual element is proportional to the current through the initial capacitor

2.) the current through the virtual element is proportional to the voltage across the initial capacitor.

What will he answer? Most probably he will conclude that I have described nothing else than the ohmic law for a capacitor in two different forms (v=i*Z and i=v/Z).

Where is an inductive behaviour?

Lutz, I do not see what the problem is. When presenting these conclusions to another person, you have only to specify that you talk about the electrical quantities voltage and current in two different elements (like when you talk about "transresistance", in contrast to "resistance").

The inductor and capacitor are dual elements - the voltage across the inductor behaves through time as the current through the capacitor, and the current through the inductor - as the voltage across the capacitor. So, if we create a 2-terminal circuit so that the voltage across it behaves through time as the current through a capacitor, and the current through it - as the voltage across the capacitor, we can name it "inductor".

To create such a circuit, we can use the two complementary voltage drops across the capacitor and resistor in an RC circuit: the voltage drop across the capacitor - to create the current, and the voltage drop across the resistor - to create the voltage of the "virtual inductor".

(I have answered app. 1 hour ago - the text disappeared somehow??).

OK - again:

Cyril - I see. You were speaking about differential quantities (time domain). It was my error to assume voltages and currents in the frequency domain. Thanks for clarification. 

Dear Lutz,

This evening, I finally managed to see the two NICs in the Antoniou's GIC. As you can see, the only difference with the other circuit is that the positive feedback of OA2 (Z2 or R1) is closed through OA1 (not directly to the OA2 non-inverting input as in the case of a conventional INIC). The result is the same - the right end of R1 is held equal to the voltage of the non-inverting input of OA2 as though it is connected to this input. Actually Z6 (R4) does not belong to the VNIC; it serves as an "input" element determining the voltage across the virtual inductor.

It seems the idea is the same - the VNIC (OA1 + C + R3 + R4) acts as a negative inductor. The INIC (OA2 + R1 + R2) inverts this negative inductor into a positive inductor.

Best regards and good night,

Cyril.

Dear Cyril,

great idea.

I think, this is the best and most logical description how Antoniou`s circuit can be derived from the "simple" cascade of two NIC´s.  Thank you.

Thanks, Lutz!

This recognition has a great value for me because I do not know anyone who could appreciate this intuitive explanation better than you.

Of course, great is the Antoniou's idea... we, "mere mortals":), can only give some interpretations to help understanding.. to make it obvious and clear...

I think that such a severe mental activity dedicated to revealing the basic ideas behind circuits and generalizing them in a kind of a "circuit philosophy", is no less significant than the work of a circuit designer daily developing specific circuit solutions...

I think, in principle, Riordan`s GIC realization (as shown in the attached article) can be explained in a similar manner.

Dear Lutz,

I have never supposed how great is the impact of one good word on the creative activity of a human being... In particular, with respect to me, you have an incredible ability to awaken the creativity of my mind... So elated by your praise, I instantly saw the idea of this bizarre Riordan's circuit... Let me first explain it by means of the first way - "seeing only the trees, and without seeing the forest":)

As in the other GIC circuits, here the humble R3C network (particularly, the capacitor C) serves as an initial element for creating an artificial virtual inductor: As you can see:

• The voltage drop across the resistor R4 is kept equal to the voltage drop across the capacitor C.. but the current through R4 is the input current I1. So, the input current I1 flowing through (consumed by) the virtual inductor, is proportional to the voltage across the capacitor C 
• The voltage drop across the resistor R3 (proportional to the current through the capacitor) is kept equal to the input voltage V1(VR3 = VOA1 - V1 = 2V1 - V1 = V1). So, the input voltage drop V1 across the virtual inductor is proportional to the current through the capacitor C

As a result, the voltage across and the current through the initial capacitor are swapped in the virtual inductor.

Now let's try "to see the forest for the trees"... to discern some familiar functional blocks (NICs) in this unknown circuit (Riordan's GIC)...

In contrast to the Antoniou's GICs, here I cannot see a VNIC... I see only INICs... Let me explain what I exactly see...

If the left end of R3 was grounded, then the combination OA2 + R3 + R4 + C would be a INIC acting as a negative virtual inductor. But here the left end of R3 is driven by the non-inverting amplifier OA1 + R1 + R2 (gain of 2). So, the whole combination OA1 + R1 + R2 + R3 is a INIC acting as a negative resistor -R. As a result, the negative virtual inductor on the right becomes a positive inductor.

Here is a more conventional explanation of the inductor sign. As a non-inverting and inverting configuration are cascaded, the whole combination is inverting. Thus, when the input voltage is positive, the output voltage of OA2 is negative and the OA2's output sinks the input current I1 via the resistor R4... so the circuit acts as a positive element (inductor).

Here is another simpler intuitive explanation of the Riordan's GIC. I will present it in a form of a building "scenario" (for clarity, I will briefly repeat some of the basic explanations above).

1. Negative virtual inductor. Imagine that OA1, R1 and R2 are not connected... and a classic negative inductance circuit is built by OA2, C, R3 and R4. The op-amp OA2 changes its output voltage Vo so that to keep up (almost) zero voltage between its inputs. It does it by means of two feedbacks - negative (implemented by C and R2) and positive (implemented by R4 and Rin). As a result:

• The voltage drop VR3 across the resistor R3 is proportional to the current through the capacitor (VR3 = Ic*R3), and is equal to the input voltage V1. So, the input voltage V1 is proportional to the current through the capacitor C (V1 = Ic*R3) 
• The voltage drop VR4 across the resistor R4 is equal to the voltage drop across the capacitor C (VR4 = I1*R4 = Vc). So, the input current I1 flowing through R4 is proportional to the voltage across the capacitor C (I1 = Vc/R4). Only note this is a reverse current that is produced instead to be consumed by the circuit... it is "pushed" by the circuit back into the input source...
• As a result, the voltage across and the current through the initial capacitor are swapped in the circuit... and it behaves through time as an artificial virtual inductor. But the problem is that it is a negative virtual inductor producing instead to consume the current... it is a functional (inductor-like) INIC...

2. Positive virtual inductor. We can make the negative inductor become "positive", if we reverse (invert) the current through it so that to be consumed (instead produced) by the circuit. For this purpose, we should connect a current inverter (INIC) before R3 to reverse the current through it, and accordingly, through R4 (see the second picture below).

The humble non-inverting amplifer (OA1 and the voltage divider R1-R2) can do this donkey work (the first picture below). It "lifts" the left end of R3 with two times bigger than the input voltage... and the current through it (IR3 = V1/R3) reverses its direction (IR3 = (2V1 - V1)/R3 = -V1/R3). This causes the input current (flowing through R4) to reverse its direction as well.

And what about nullors theory?

Josef - yes, most probably this will be another method for find possible structure alternatives. However, up to now I didn`t try it.

Perhaps also another technique called  "relocation" would work? This is similar to the  "complemetary transformation" introduced by N. Fliege.