User:Circuit dreamer/Negative impedance converter Talk 14-07-06

What Is the Basic Idea behind a Negative Impedance Converter (NIC)?

(see also 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.

Fig. 1: An "over-helping" voltage source acting as a negative resistor.

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 V_R = R.I appears across the resistor R and the same voltage V_H = V_R = 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

Fig. 2: An "over-impeding" voltage source acting as a negative resistor.

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.

Fig. 3: NIC with voltage inversion (NIC) is just an "over-helping" negative resistor.

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 V_H adds its voltage to the voltage of the input voltage source V_IN (traversing the circuit clockwise the signs are - V_IN +, - V_H +); so, V_H "helps" V_IN. 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 V_H even "over-helps" V_IN since the overall voltage is two times more than the voltage drop V_R. 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 R_NIC = (-V)/I = -R.

Fig. 4: NIC with current inversion (INIC) is just an "over-impeding" negative resistor.

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 V_IN (traversing the circuit again clockwise the signs are - V_IN +, + V_I -); now, V_I will "over-impede" V_IN. 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 R_INIC = 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.

Fig. 5: In a practical op-amp circuit of NIC with voltage inversion (NIC) the op-amp acts as an "over-helping" voltage source.

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 V_R2; thus we will get a negative resistance -R.

Fig. 6: In a practical op-amp circuit of NIC with current inversion (INIC) the op-amp acts as an "over-impeding" voltage source.

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 V_R 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)

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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.

Light current, I have noted that you have corrected some indentations of my replies. Really, I experience difficulty when I have to decide how many ::: to insert. Where can I read about these conventions? Can you give me some rules how to indent the text when reply? Why have you begun this reply with "::" instead only":"? Is it right in this case to begin my reply with ":::" as I have done?

By the way, I have noted that I have been becoming more and more a Wikipediholic:) --Circuit-fantasist 10:43, 14 July 2006 (UTC)

about the "::", as far as I know there isn't any general guide-line, just add one ":" more than the previous comments. If they become too many, it might make sense to start again removing them or to start a new discussion, but it's up to you! Alessio Damato 18:53, 14 July 2006 (UTC)

Transfer function derivation

Negative impedance converter

First, assigning V_a = voltage at + terminal = V_s and assigning V_b = voltage at - terminal and assigning V_o = voltage at output

$I_{s}+{\frac {V_{o}-V_{s}}{R_{3}}}=0,$ Kirchoff's current law

$V_{a}=V_{o}+{I_{s}}{R_{3}},$ rearranging to express $V_{a}$ in terms of $V_{o}$ and $I_{s}$

$V_{o}=V_{s}-{I_{s}}{R_{3}},$ rearranging to express $V_{o}$ in terms of $V_{s}$ and $I_{s}$

${\frac {V_{b}}{R_{1}}}+{\frac {V_{b}-V_{o}}{R_{2}}}=0,$ Kirchoff's current law

$V_{b}={\frac {{V_{o}}{R_{1}}}{R_{1}+R_{2}}},$ rearranging

$V_{o}=(V_{a}-V_{b})\alpha ,$ where $\alpha$= the opamp open loop gain

$(V_{o}+{I_{s}}{R_{3}}-{\frac {{V_{o}}{R_{1}}}{R_{1}+R_{2}}})\alpha =V_{o},$ substituting for $V_{a}$ and $V_{b}$

$\alpha {V_{o}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})+\alpha {I_{s}}{R_{3}}=V_{o},$ rearranging

${V_{o}}(\alpha (1-{\frac {R_{1}}{R_{1}+R_{2}}})-1)=-\alpha {I_{s}}{R_{3}},$ rearranging

${V_{o}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})=-{I_{s}}{R_{3}},$ cancelling out the $\alpha$ term and simplifying

$V_{o}=V_{s}-{I_{s}}{R_{3}},$ shown earlier

$V_{s}(1-{\frac {R_{1}}{R_{1}+R_{2}}})-{I_{s}}{R_{3}}(1-{\frac {R_{1}}{R_{1}+R_{2}}})+{I_{s}}{R_{3}}=0,$ substituting for $V_{o}$

$V_{s}({\frac {R_{2}}{R_{1}+R_{2}}})=-{I_{s}}{R_{3}}({\frac {R_{1}}{R_{1}+R_{2}}})),$

${\frac {V_{s}}{I_{s}}}=-{\frac {{R_{3}}{R_{1}}}{R_{2}}}$

Zen-in (talk) 02:29, 15 December 2013 (UTC)