Talk:Circuit Idea/Negative Impedance Converter

Untitled
I have copied here this old Wikipedia discussion to start a new discussion about the mystic negative impedance converters. Please, join the talk. Circuit-fantasist (discuss • contribs) 07:59, 30 June 2012 (UTC)

What Is the Basic Idea behind a Negative Impedance Converter (NIC)?
(see also w:User:Circuit_dreamer/Negative_impedance_converter_Talk_14-07-06 and Another fresh viewpoint at negative resistance)

Hi! I would like to contribute in writing of this page. IMO, Negative Impedance Converter is one of the most interesting, odd, strange and even "mystic" electronic circuits, which are still unexplained:( I have not yet met some "human-friendly" explanations of this legendary circuit on the web or in the books (if you have found, let me know). Even, the famous Mr. Horovitz has not explained (although mentioned) it in his bestseller The Art of Electronics.

Alejo2083, as I can see, you have created this page. I like it as it looks tidy and your explanations are right. Only, this is insufficiently. Reading this page, visitors will know what NIC is (for example, "a circuit introducing a shift of 180° between the voltage and the current for any signal generator"), but they will not understand it.

In order to understand really this sophisticated circuit, readers need something more. As human beings (not computers), they first need to know what the general idea behind negative resistance is (that is, what negative resistance is), then, how to obtain negative resistance and finally, how the op-amp does that work (what the op-amp does in the circuit of NIC). So, in order to create a nice page about NIC, we have first to answer these questions by using qualitative means and only then to analyze the circuit.

From many years, I have been thinking about the phenomenon of negative resistance and its circuit implementation as a NIC. Finally, I have managed to grasp the general idea behind them; now, I would like to share my penetration with Wikipedia audience. IMO, the best way to show the idea is to build and reinvent step-by-step the circuits as I have begun doing in my circuit stories about dynamic and negative resistance. Of course, this approach is most suitable for educational purposes; in an encyclopedia we have to use pithy exposition.

The idea is extremely simple; we may expound it by reasoning in the succession below.

1. Comparison between "ordinary" and negative resistance. First, we have to compare an ordinary "positive" resistor R with a negative resistor -R; that is, we have to answer the two questions: "What is an ordinary resistance?" and "What is a negative resistance?"

For this purpose, imagine that we have connected in series the two 2-terminal components in some circuit (having at least another resistor) so that the same current I passes through them. As a result, a voltage drop VR = R.I appears across the resistor R and the same voltage VH = VR = R.I appears across the negative resistor -R (Fig. 1). Only, the resistor R sucks a voltage V = R.I from the circuit (it is a voltage drop) while the negative resistor -R adds a voltage V = R.I into the circuit. So, a resistor acts as a current-to-voltage drop converter while a negative resistor acts as a current-to-voltage converter. The element named "resistor" is really a resistor while the "negative resistor" is actually a voltage source, whose voltage is proportional to the current passing through it. Now, we can answer the main question "What is a negative resistance?"; the answer is simple and clear.

A negative resistor is just a voltage source, whose voltage is proportional to the current passing through it. Shortly, a negative resistor is a "current-controlled voltage source".

Note, that we might connect this additional voltage source in the same or in the opposite direction versus the input voltage source; so, it will act as an "over-helping" (Fig. 1) or as an "over-impeding" (Fig. 2) voltage source.

2. How to make a negative resistor. Once we revealed the secret of negative resistance, we can already create a negative resistor. For this purpose, we have first to sense the current and then to regulate the voltage according to the current. So, we need an ordinary resistor acting as a current-to-voltage converter and an amplifier acting as a voltage-controlled voltage source.

2.1. NIC components.

2.1. An internal resistor. What a paradox! The negative resistor will contain internally an ordinary resistor! Well, add it and make the first conclusion: A negative resistor with resistance -R contains internal resistor with resistance R.

2.2. An internal doubling voltage source. Then, we have to insert (in series with the resistor R) a voltage source that doubles the voltage across the resistor R. Well, add it and make the second conclusion: A negative resistor with resistance -R contains also an internal doubling voltage source (voltage amplifier with k = 2).

Finally, generalize the conclusions:

A negative resistor with resistance -R contains internal resistor with resistance R and a doubling voltage source (voltage amplifier with K = 2).

2.2. NIC versions.

2.2.1. NIC with voltage inversion. Note that in this first arrangement (Fig. 3) the additional voltage source VH adds its voltage to the voltage of the input voltage source VIN (traversing the circuit clockwise the signs are - VIN +, - VH +); so, VH "helps" VIN. All the op-amp inverting circuits, for example op-amp ammeter, op-amp integrator, op-amp inverting current source and so on use this trick. But here VH even "over-helps" VIN since the overall voltage is two times more than the voltage drop VR. As a result, it reverses the voltage over the resistor R while the current continues flowing in the same direction. That is why, these kinds of negative resistors are named NIC with voltage inversion. The resistance of NIC is RNIC = (-V)/I = -R.

Funny analogy. My wife doesn't know what NIC is; but she makes me act as a NIC:) When she spend some MONEY, she wants I to restore (2 x MONEY); so, finally she get new MONEY:)

Wiki analogy. Imagine, a Wikipedian has made a page about something (for example, NIC:) Then, another "twice-powerful" and eager to change Wikipedian comes and begins improving the existing page in the same direction. In this way, he/she acts as NIC adding his editorial power to the existing one.

2.2.2. NIC with current inversion. If we reverse the additional voltage source (Fig. 4), it will subtract its voltage from the voltage of the input voltage source VIN (traversing the circuit again clockwise the signs are - VIN +, + VI -); now, VI will "over-impede" VIN. As a result, it reverses the current passing through the resistor R while the voltage across the new "resistor" -R keeps the old polarity. That is why, these kinds of negative resistors are named NIC with current inversion (INIC). The resistance of INIC is again RINIC = V/(-I) = -R.

Analogies (not so funny as above). Imagine, you move (in a literal or figurative sense) in some direction. However, someone (two times stronger than you) begins moving against you so that he not only stops you (this case might be analogy of bootstrapping) but he moves you in his direction. Another example of an "over-impeding": During World War II, Germans invaded Russians; then Russians not only chased Germans but even invaded them! There are many examples of this phenomenon in our life where a conflict arises between two power sources - weak and powerful.

Wiki analogy. Imagine (as above) that the Wikipedian has already made his page. Only now, the "twice powerful" Wikipedian improves again the existing page but in the opposite direction. As a result, the new Wikipedian changes totally the direction of the page development.

3. How to make a negative impedance converter. Finally, we have to show how to make a practical op-amp circuit behaving as a negative resistor. Following the recipe above, we have just to connect in series a resistor R and the output part of an op-amp amplifier, which twice the voltage drop across the resistor. Moreover, since there are two types of op-amp amplifiers (inverting and non-inverting) we will obtain two types of NICs as well.

3.1. Op-amp NIC with voltage inversion (NIC) - Fig. 5. In ordinary op-amp inverting amplifier (and in all the inverting op-amp circuits), the op-amp acts as an additional "helping" voltage source, which compensates "exactly" the voltage drop across the resistor R2. Maybe, the most suitable example of this trick is Transimpedance amplifier). Actually, the main task of op-amp in all these circuits is to remove (zero) the resistance R2.

In order to convert this circuit into NIC, we have just to "mislead" the op-amp making it "over-act". Well, if we connect a voltage divider (with K = 0.5) to the non-inverting input, we will make the op-amp "over-compensate" twice the voltage drop VR2; thus we will get a negative resistance -R.

3.2. Op-amp NIC with current inversion (INIC) - Fig. 6. In this case, we have made the op-amp act as an "over-impeding" voltage source. Do you recognize the classical circuit of an op-amp non-inverting amplifier with K = 2? Here, it doubles the voltage VR appearing at the left side of the "ordinary" resistor R and applies it to its right side. As a result, the current I reverses its direction - now it flows into the input voltage source.

Alejo2083, your circuit is INIC (NIC with current inversion). --Circuit-fantasist 15:52, 1 July 2006 (UTC)


 * figure 6 isn't really stable for Ri > R. at this point, the stable node becomes an unstable one.  this is definatly seen in pactice as this same circuit is used to provide hysteresis.  also, fig5 doesn't seem to fit.  for a positive input voltage the current flow is from the source.  and further its more then otherwise.  also, as expected, low values of Ri make the system unstable.


 * You are right. The input voltage source at Fig. 5 will actually serve as a load (the op-amp will "blow" current into the source making it act as a load). I invite you to join Circuit idea wikibook where we might discuss more circuit phenomena. Circuit-fantasist (talk) 13:41, 7 May 2008 (UTC)


 * well, since you have such an in depth vision of this circuit configuration, just improve the article. Feel free to use your hand-made pictures in the article; if you do, I'll provide an SVG version as soon as I have some time :-) Alessio Damato 19:15, 11 July 2006 (UTC)


 * Thank you for suggestion, Alessio! Wikipedians are grand persons! As you can see, I have decided to follow your emotional appeal. So far, I was exposing my ideas only on my web site of circuit-fantasia.com; now, I am happy to share them with Wikipedia audience. For now, I stay in talk pages, in order to coordinate my viewponts with other ones. Only after that, I will join article pages (by the way, talk pages are quite more interesting than the according articles). So, I might remain there:)


 * Alessio, I would like to ask you some questions about your teachers. Have you ever met a teacher who loves circuit phenomena? Have you ever met a teacher who has own philsophy about his/her subject? If yes, please tell me know. I ask you since I have never met such a person among my former teachers and present colleagues:((( --Circuit-fantasist 06:00, 14 July 2006 (UTC)


 * about my teachers... it's a strange question but, to tell the truth, I don't think any of them had his own philosophy about his subject. Some of them were concerned about philosophy in general, so we made discussions about general topics, such as how much we can really understand by physics and so on, but I don't think it's what you mean.


 * about editing articles, just do it, Be bold. If you feel unsecure start with smaller edits, but work on articles. Talk pages might be interesting, but most of the people will just read the article. Moreover the main point of wikipedia is creating articles, not talk pages! Consider joining the WikiProject Electronics, it's a good starting point for coordination; you could help about the structure of the project itself! Alessio Damato 18:53, 14 July 2006 (UTC)


 * Yes this is a good fiarly comprehensive analysis of NICs and negative resistance. However, if you include all the above in one article, I think it may become over complicated and too large. I suggest the basic explanation of negative resistance should be included on the negative resistance page and the circuit explanation using op amps on this page.--Light current 01:11, 12 July 2006 (UTC)


 * Thank you for comments, Light current. Honestly, it is a real pleasure for me to join Wikipedia society and to meet you. I like your thought "Everything should be as simple as possible..." and will try to use it everywhere.


 * You are right about the page. Yes, the most explanations have to be moved from this page to negative resistance page. Obviously, this page has first to show how to make "circuit" negative resistors (a philosophy and general circuit diagrams) and then to show the according practical circuit solutions.

An answer to an interesting email question about NICs
Today, I received this question about the nature of NICs:

"We have a question regarding the Negative Impedance Converter. Let’s make first Ri=0 to simplify the analysis; I think the input must be on the non inverting side (as suggested by my 2 fellow engineers Darwin & Marvin) of the op amp because if the input is in the inverting side transfer function Vo/Vin is 2A/(A-2) where A is the open loop gain; the A-2 on the denominator suggest an unstable circuit because a discontinuity exist at A=2. If the input is at the non inverting side Vo/Vin is 2A/(A+2) which is considered a stable circuit; with A starting from 0 to positive infinity the transfer function is continuous. Let us know your thoughts."

IMO my answer to this question would be helpful for this page; so I have placed it below. Circuit-fantasist (discuss • contribs) 17:38, 21 July 2012 (UTC)

..........

What is a negative impedance converter? It is just an op-amp implementation of a true negative resistor. But what is a true negative resistor? It is the opposite "element" of the ordinary "positive" resistor; it is a circuit adding (injecting) the same energy that the equivalent "positive" resistor would dissipate. So, it is nothing else than a source... but this is not the ordinary constant source; it is a "self-varying" (dynamic) source. And as there are two kinds of sources (in contrast with the only one kind of "positive" resistor), there are two kinds of true negative resistors (S-shaped and N-shaped), accordingly, there are two kinds of their op-implementations (NICs) as well. First, a NIC can behave as a dynamic voltage source producing voltage that is proportional to the current passing through it (named voltage inversion NIC or VNIC) or as a dynamic current source producing current that is proportional to the voltage across it (named current inversion NIC or INIC). You can consider the VNIC as a 1-port current-to-voltage (to voltage, not to voltage drop!) converter and INIC - as a 1-port voltage-to-current converter. What is the use of negative impedance converters? NICs are not simply sources; they are "faithful" sources that "help" the input source in its striving to pass the desired current through or to apply the desired voltage across the load. When connected in series to a "positive" resistor, a VNIC with the equivalent but negative resistance adds the same voltage as the voltage drop across the resistor. This voltage has an opposite polarity in comparison to the voltage drop across the "positive" resistor; so it adds to the input voltage; it "helps" the input voltage source. When connected in parallel to a "positive" resistor (supplied by an input current source), an INIC adds current that is proportional to the voltage across it. This current has an opposite direction in comparison to the current that would be consumed by the "positive" resistor; so it adds to the current of the input current source; it "helps" the source. Thus a VNIC neutralizes the equivalent series "positive" resistance and the effective resistance is zero while an INIC neutralizes the equivalent parallel "positive" resistance and the effective resistance is infinity. How do we make negative impedance converters? The idea is extremely simple but a little paradoxical - we create negative resistance on the base of equivalent "positive" resistance; as though a NIC converts a positive resistor into a negative one. This is the paradox - to create a negative resistor, we need a "positive" one. So, a NIC (no matter VNIC or INIC) consists of two elements connected in series - an ordinary ohmic resistor and a varying voltage source implemented as an amplifier with a gain of 2. In the case of VNIC, the "doubling" voltage source is connected in the same direction as the input source so that its voltage adds to the input exciting voltage; in the case of INIC, the "doubling" voltage source is connected in an opposite direction to the input source so that its voltage subtracts from the input voltage. The VNIC amplifier doubles the voltage drop VR across the internal resistor while the INIC amplifier doubles the input voltage VR of the whole circuit. The result of these connections is amazing: the voltage across the VNIC becomes an inverted copy of the voltage drop across the resistor as though it changes its sign (thus the name voltage inversion NIC or VNIC); the current through the INIC becomes an inverted copy of the current through the resistor as though it changes its direction (thus the name current inversion NIC or INIC). Generally speaking, as though the humble resistor has become a source. It is interesting to see, from another viewpoint, what the function of the resistor is - in the VNIC, the resistor acts as a current-to-voltage converter while in the INIC it acts as a voltage-to-current converter. From this viewpoint, in an INIC we have actually assembled a current source by a voltage source and a voltage-to-current converter.

How do we make op-amp negative impedance converters? It's time to reveal the mistery of this humble but never understood odd circuit of NIC consisting only of three (usually equal) resistors and an op-amp (the four "resistor" is the internal resistance of the input source). I will start with this assertion: Either the inverting or non-inverting input of the op-amp can be used as an input of this NIC. Let's see why and how. VNIC. If we use the inverting op-amp input as a NIC's input, the combination of the op-amp and the other two resistors (the voltage divider connected between the op-amp output and its non-inverting input) constitutes a little odd amplifier having the needed gain of 2. It is odd since its input is connected in parallel to the "positive" resistor (it sounds strange but this is the amplifier input port - between the op-amp inverting input and the output) and its output is connected in series to the resistor. As I have mentioned above, the resistor converts the input current IIN into a voltage drop VR = R.IIN; the amplifier amplifies this voltage drop two times and subtracts its output voltage 2VR from the voltage drop VR. As a result, negative voltage -VR appears across the NIC's input (between the op-amp inverting input and the ground). The sigh of the voltage drop VR as though is inverted; so this NIC is a voltage inversion one (VNIC). Note there is a little trick here: actually, the voltage drop is not inverted; the whole voltage across the combination of the two elejments (the resistor and the "helping" voltage source) is inverted. But it is supposed we do not see this trick and continue thinking of the same voltage drop:) INIC. If we use the non-inverting op-amp input as a NIC's input, the combination of the op-amp and the other two resistors (the voltage divider connected between the op-amp output and its inverting input) constitutes an ordinary non-inverting amplifier having the needed gain of 2. Here, the input of the amplifier serves as a NIC's input. Now, the "positive" resistor to be converted is connected between the op-amp non-inverting input (the amplifier input) and the op-amp output. Thus the amplifier output is connected again in series to the resistor. As I have mentioned above, the resistor converts the difference VR = 2VR - VR = VR between the op-amp output voltage and the input voltage into current IIN = VR/R and this current passes through the input source. As a result, negative current -IIN flows through the NIC's input. The direction of the current is inverted; so this NIC is a current inversion one (INIC). Which resistor is converted into a negative one? They usually assume that in the op-amp NIC the "positive" resistor to be converted is the resistor between the inverting (INIC) or non-inverting (VNIC) op-amp input and the ground. But what do we say if we have realized a NIC by a humble amplifeir with gain of two? There is no these two resitors (forming a voltage divider) in this humble implementation. There is only one resistor and this is the resistor whose "positive" resistance is converted into negative one. So I consider this resistor as the resistor to be converted. When a NIC is stable? Let's finally answer to your question about the NIC's stability. Note that both the modes can be useful - stable (linear) and unstable (bi-stable); in the last case, a NIC acts as a flip-flop. To consider the stability of the op-amp NIC, we have to take into account the internal resistance RG of the input source. It can be zero (voltage sourdce), medium (real source) or infinite (current source). Voltage divider viewpoint. This "resistor" and one of the other resistors form a voltage divider; the other two resistors form another voltage divider. The two voltage dividers introduce two feedbacks - a negative feedback (between the op-amp output and its inverting input) and a positive feedback (between the op-amp output and its non-inverting input). Thus the feedback voltage applied to the op-amp differential input is a difference between the two partial voltages. To work in a linear mode (a stable circuit), the negative feedback has to dominate over the positive one. This means the first ratio has to be higher than the second one: RG/(R + RG) > R/(R + R) for VNIC and RG/(R + RG) < R/(R + R) for INIC. The circuit will be always stable if we drive the VNIC with a current source and if we drive the INIC with a voltage source. Wheatstone bridge viewpoint. You can think also of the four resistors as of a Wheatstone bridge connected in the feedback loop between the op-amp output and its differential input. Depending on the proportion between the resistors, this bridge converter can have negative (negative feedback), zero (no feedback) or positive (positive feedback) transfer ratio. To have a stable circuit, the ratio has to be negative... To become more familiar with my insights about all these amazing negative resistance circuits, I recommend you first to read my Wikipedia story about negative impedance and its implementations. I (Circuit dreamer) created it in last summer but it was removed by the orthodox wikipedians inhabiting this space. See also the Wikipedia negative resistance talk page and the archives; my first Wikipedia talk page about NICs can be also interesting for you. Then you may visit my Circuit idea story about negative impedance (I am Circuit-fantasist there) and the two stories about VNIC and INIC operating in a linear mode where I have scrutinized their circuit operation. I have created also a story about the mystic and never explained bistable mode of the INIC. You probably will be interested in my two circuit stories on the whiteboard about VNIC (How to compensate resistive losses by series connected negative resistors) and INIC (How to compensate resistive losses by parallel connected negative resistors), and the first my story about understanding VNIC. If you want to know the truth about the two legendary NIC's applications - the Howland current source and the Deboo integrator, visit this Circuit idea talk page. I have generalized the idea of negative impedance in the Wikipedia Miller theorem article. I created it in the last summer (see the talk page) but I realized this great idea in the early 90's; it is my great circuit achievement...

..........

Circuit-fantasist (discuss • contribs) 17:38, 21 July 2012 (UTC)

Reconstructing the module
In the beginning of this year, I start reconstructing the page. My intention is to reveal step-by-step the basic ideas behind the two NIC versions. Circuit-fantasist (discuss • contribs) 21:48, 1 January 2014 (UTC)

A question about NIC asked in ResearchGate
Last year, I asked the question below in the scientific network ResearchGate...

"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?"

... and then I used the answers in this Wikibooks page. Here is a more detailed text below the question:

"We have already seen that negative impedance elements are amazing and extremely useful electronic devices (circuits):

[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 What is negative impedance? Does it exist? If so, how can elements with negative impedance be implemented? Are they passive or active?]

But there is only one small problem:) - there are not such elements in nature; there are only humble passive elements with "positive" impedance (resistors, capacitors, inductors and memristors:( So, we have to make them... and maybe this is the most interesting part of our discussion about the negative impedance phenomenon. Then, how do we create negative impedance elements (named "negative impedance converters", shortly NIC)?

IMO they are ones of the most interesting, odd, "mystic" and still unexplained electronic circuits... a real nightmare for students... and their teachers:) I have not still met some "human-friendly" explanations of this legendary circuit (if you find, let me know). Even the famous Mr. Horovitz has not explained (although mentioned) the NIC in his bestseller The Art of Electronics (see page 251). Instead, he has afforded this opportunity to his students; maybe, he had hoped they would help him:)?

For me, as a "circuit thinker", the understanding of this clever circuit (in its two versions) was crucial for understanding the phenomenon of negative impedance. It was interesting that I first figured out what the op-amp was doing in this circuit to create a negative impedance and this gave me a chance to figure out what the phenomenon of negative impedance was. Here are my insights.

"REVERSING" THE RESISTANCE. The idea is simple but powerful - we can make negative impedance by "inverting" some initial positive impedance. Thus the original positive elements will serve as shaping elements for creating "mirror" negative elements. But how do we invert the positive impedance (e.g., the positive resistance)?

The answer is simple (but a bit formal) and requires only to know the Ohm's law (how wonderful it sounds!) As we know, it presents the resistance as a ratio between the voltage and the current (R = V/I); so when the two variables are positive, the resistance is positive as well. To make negative resistance, we have to invert one of them - the voltage or the current: How to create negative impedance


 * INVERTING THE VOLTAGE POLARITY. In the case of the S-shaped (current-controlled) negative resistance RS, we invert the voltage (RS = -V/I = -R). This means that if we pass a current through the S-shaped negative resistor, the input terminal becomes negative (instead positive as in the case of the ordinary "positive" resistor). That is why, circuits implementing this technique are named "voltage-inversion negative impedance converters" (VNIC). Note the power is also inverted (PS = -V.I = -P).


 * INVERTING THE CURRENT DIRECTION. In the case of the N-shaped (voltage-controlled) negative resistance RN, we invert the current (RN = V/-I = -R). This means that if we apply positive voltage across the N-shaped negative resistor, the current goes out of the negative resistor and enters the positive terminal of the voltage source (instead to leave the positive terminal of the voltage source and to enter the negative resistor as in the case of the ordinary "positive" resistor). That is why, circuits implementing this technique are named "current-inversion negative impedance converters" (INIC). Note the power is also inverted (PN = V.-I = -P).

But how do we invert an electrical quantity (e.g., the voltage? We can see the solution around us when we invert some (usually "bad") quantity by adding a two times bigger opposite ("good") quantity. So, we may convert the "bad" voltage drop across an initial "reference" positive resistor into a "good" voltage across a new negative resistor by adding a two times higher voltage (by connecting in series a doubling variable voltage source): The basic idea of the series NR compensation

An amplifier with a gain of 2 can serve as such a variable voltage source: Building a true S-shaped NR (voltage-inversion NIC)

and this is the famous circuit of the voltage-inversion negative impedance converter (VNIC). It is shown in the attached picture (Vin and Ri do not belong to the circuit; they represent the input source).

A few years ago I started a discussion in the Wikipedia page about NICs (under the names Circuit-fantasist and Circuit dreamer), where I enthusiastically presented my insights (then I had no notion about wikipedian's manners and customs): [https://en.wikipedia.org/w/index.php?title=Talk:Negative_impedance_converter&oldid=210801768#What_Is_the_Basic_Idea_behind_a_Negative_Impedance_Converter_(NIC)? Talk:Negative impedance converter] (July 1, 2006) is my first material about the fundamental ideas behind NICs

But the discussion did not happen and after several years my suggestions were removed (I hope this will not be repeated in RG:) After that, I created a few stories about NICs in Wikibooks:

https://en.wikibooks.org/wiki/Circuit_Idea/Linear_Mode_of_Voltage_Inversion_NIC https://en.wikibooks.org/wiki/Circuit_Idea/Linear_Mode_of_Current_Inversion_NIC https://en.wikibooks.org/wiki/Circuit_Idea/Bistable_Mode_of_Current_Inversion_NIC

You can see also my stories about NICs uploaded on circuit-fantasia.com:

http://www.circuit-fantasia.com/circuit_stories/understanding_circuits/nic/vnic/vnic.htm http://www.circuit-fantasia.com/circuit_stories/inventing_circuits/ser_nr_comp/ser_neg_res_comp.htm http://www.circuit-fantasia.com/my_work/conferences/cs_2006/paper.htm

So, what does the op-amp do in the circuit of the negative impedance converter?"