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Also, this viewpoint was presented by Simone Orcioni in the question below:

As far as I understand, this "negative resistance viewpoint at voltage sources" is the following. A voltage source is connected to a load (a resistor)... so the voltage V (V_G in he  Simone's figure) across them and the current I through them are the same... and therefore the ratio V/I (the resistance) for each element is the same (see the first attached picture below)... Thus the resistor has a resistance R_L = V/I and the voltage source has a "negative resistance" R_S = V/-I = -R_L... so the sum of the two resistances (voltages, according to KVL) is zero... It sounds temptingly simple but...

In this arrangement, there is only one "main" voltage source and one resistor (the load)... and this is the possibly simplest electric circuit still from 19-th century - a source driving a load. But the popular belief is that "negative resistance" is a "supplemental" concept... It implies another (supplemental, "helping") voltage source (B_H in the second attached picture)... and this is not an ordinary constant but "dynamic" voltage source whose voltage is proportional to the current flowing through it (a 2-terminal current-to voltage converter)... and so it will act as a negative resistor with resistance -Ri. This negative resistance compensates another positive resistance Ri (e.g., the source internal resistance or the line resistance) thus giving as a result zero total resistance between the main source V_IN and the load R_L... and this 4-component circuit is reduced to the Simone's initial 2-component circuit (source and load)... The sense of this "trick" is that the unwanted resistance Ri (the voltage drop across it) is neutralized by an equivalent voltage:

If this supplemental voltage source was an ordinary constant voltage source, it would still compensate the voltage drop across Ri... but only for one value of the current; maybe because of that they name this kind of "negative resistance" with the name "static negative resistance". Really, it can compensate also the relatively steady voltage drop across a constant-voltage nonlinear resistor (diode, LED, Zener diode, etc)... but this is just another special case...

Note that, in contrast with an ordinary source, this exotic voltage source will not independently produce voltage if there is no input voltage V_IN; it starts acting after the main (input) voltage source begins increasing its voltage from zero.

IMO the word "resistor"/"negative resistor" has the meaning of something that resists/"helps" the current flowing through it... so it implies some initial current produced by another (main, input) voltage source... Therefore, this main source is simply a source, not a negative resistor... and maybe this viewpoint is just a misconseption as many others in the field of negative impedance phenomena?

I would add here also the questions asked by Lutz Wangenheim: "Does it make sense to interpret this scenario as a connection of a positive and a negative resistance of the same value? More than that, are voltage and current directions of the voltage source in accordance with the DEFINIONS of a negative resistance? If this would be true, we could treat each voltage source in each circuit as a negative resistance, couldn´t we?"

... and second, to answer the question of Erik Lindberg asked at the end of this discussion.

In the discussed arrangement (an RC circuit with various leakages), there are three resistances in parallel - R, Rc and Rv, and the equivalent resistance is Re = R||Rc||Rv. My idea is to connect a variable (N-shaped) negative resistor in parallel to Re and begin adjusting its resistance RN. Depending on its value, it will "eat" some part of Re and it will dissapear (become infinite):

1. RN = Rv (Rv is neutralized) or RN = Rc (Rc is neutralized)

2. RN = Rc||Rv (both Rc and Rv are neutralized)

3. RN = Rc||Rv||R (all the positive resistances are neutralized)

In case 2 (a load canceller), I thought we should obtain a perfect exponential shape... and this should solve the leakage problem. The next my idea was that if we continue decreasing this "destroying" negative resistance beyond this point of exact leakage neutralization, it will begin "eating" a part of the positive resistance of the "useful" resistor R... and finally (case 3), it will destroy all the resistance R. This means that the resistor R as though already has an infinite resistance... and behaves as an ideal current source... Actually, this is the idea of the Howland current source and its special case here - Deboo integrator. But while in the classic Deboo integrator the negative resistor (INIC) neutralizes only the positive resistance R, here it neutralizes all the resistances in parallel (the useful R and harmful leakages).

So, my question now is, "What happens if we try to neutralize all the positive resistances by an equivalent negative resistance (case 3)?"

My doubt is that, as a result of this 100% neutralization, this circuit will become unstable, and if the negative resistance begins dominating over the equivalent positive resistance, the effective resistance (the result of the neutralization) would become fully negative. And here, I suppose, the voltage across the capacitor will begin self-increasing in an avalanche like manner... From other side, the reactance of the capacitor C still remains... and it is a kind of a positive "resistance" (impedance)... and it turns out the circuit should remain stable...

The same problem exists in the Wien bridge oscillator... and it is solved there by applying a non-linear negative feedback in the INIC... Maybe it is possible to keep the circuit stable in a similar way?

* Emulating by (varying) voltage. First, we may replace the original elements by varying voltage sources and this is the most natural way of making emulated capacitors and inductors (as they behave as varying through time voltage sources). Op-amp gyrator, multiplying, memcapacitive and meminductive circuits do it in this way. In these circuits, the op-amp output voltage represents the voltage across the according capacitor or inductor.

In the case of the true negative resistor, the ordinary ohmic resistor is replaced again with a voltage source (exactly as in the case of gyrators and multipliers) but it has the same polarity as the input voltage source so that it adds an additional  voltage to the input voltage. For example, the negative impedance converter with voltage inversion (VNIC) is a dynamic voltage source emulating a negative resistor by adding a voltage that is equal to the voltage drop across a real ohmic resistor.

It is interesting that we can change the properties of the ordinary constant voltage source by  properly varying its voltage (as in the attached picture below).

The emulation by including an additional voltage source is the basis of the Miller theorem (see https://en.wikipedia.org/wiki/Miller_theorem#Applications).

* Emulating by (varying) resistance. But a memristor can do the same by replacing the voltage by an equivalent voltage drop across a dynamic time-dependent resistor. Transistor gyrator and multiplying circuits do it in a similar way.

It is interesting that we can change the properties of the ordinary ohmic resistor by properly varying its resistance. A good example of this technique is the creation of the negative differential resistor:

So, to emulate passive elements (to create virtual elements), we may replace the elements behaving as resistors with (properly varying) resistors, and elements behaving as sources - with (properly varying) sources. But it seems we can do it by swapping these correspondences - replacing the elements behaving as sources with resistors, and elements behaving as resistors with sources... Am I right? Please, discuss.

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I was inspired to ask this question mainly by the numerous discussions between me and Prof. Lutz von Wangenheim mostly in the questions below...

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

...and especially, by his idea about the inverse substitution theorem (IST) proposed by him in the question below:

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

It is well known that when we connect (in a circuit) a passive element (resistor, capacitor or inductor) in series to a voltage source, a voltage drop (a part of the input voltage) appears across the element and the resulting voltage is less than the input voltage. The resistor immediately dissipates the energy from itself to outside environment and the voltage drop across it stays constant through time; in contrast, both capacitors and inductors accumulate energy into themselves and the voltage drops across them vary through time. It is obvious how the resistor opposes the input voltage source by continuous energy dissipation... but how do inductor and capacitor oppose the input voltage source without energy dissipation?

IMO both the reactive elements can be considered as (are?) voltage sources that take a part of the input voltage and then oppose their (emf?) voltage to the input voltage... though they are some "ungrateful" sources "fed" from the input source... and after, as a sign of "gratitude", they stand against their "benefactor":) They do this in two opposite (dual) ways:

INDUCTOR. Imagine a circuit of three elements in series - an input voltage source, inductor and resistive load. The input source has been producing some constant (DC) voltage; the inductor is charged so the voltage across it is zero (it is just a piece of wire:) Now increase sharply the input voltage to a higher value (as though you connect a new source in series with voltage equal to the voltage change). The inductor immediately converts the voltage change to an equivalent "antivoltage" and applies it contrary to the input voltage change; figuratively speaking, the voltage "produced" by the inductor "jumps" with a magnitude equal to the input voltage change. Thus we have two voltage sources (an "original" and "cloned":) contrary connected in series and neutralizing each other; as a result, the total (effective) voltage and accordingly the current do not change. Over time this opposition decreases and the inductor acts as a time-variable voltage source; finally, the voltage across the inductor becomes zero again (it is a piece of wire again), and the current increases to the new value. By the way, the op-amp in the circuit of a voltage follower with negative feedback uses the same "trick" to neutralize the input voltage - it produces a voltage that is equivalent and opposing to the input voltage (known as "bootstrapping")... but there this voltage is constant through time...

CAPACITOR. While the inductor creates a decreasing through time voltage opposition, the capacitor does the opposite - it creates an increasing through time voltage opposition. In contrast to the "nimble" inductor, the capacitor is "lazy" - so when the input voltage "jumps", the capacitor does not react in the first moment. Then it begins gradually increasing its voltage acting as a time-variable voltage source and, after a long time, it becomes a voltage source "producing" an equivalent "antivoltage" contrary to the input voltage change... like the op-amp in the circuit of a negative feedback voltage follower...

So, what is your opinion about how inductor and capacitor oppose input voltage sources? Can we consider them as time-variable voltage sources? Are they, in the first moment, ideal current/voltage sources? What is across them - a voltage (emf) or a voltage drop? Can we model them in such a way (by voltage sources)? Don't you think there is something "mystic" in the inductor - it behaves as a current source trying to keep up a steady current... but yet it is a voltage source inside... while the capacitor behaves as a voltage source trying to keep up a steady voltage?

As a result, the same current flows through the two elements - the resistor and the op-amp output, and the same voltage appears across them; so, they process the same energy and they have the same impedance. But while the first of them is a passive element that consumes energy (voltage) from the input voltage source, the second is an active element that adds the same energy (voltage) to the input voltage source. Then, if the first element has a "positive" resistance, the second element will show a negative resistance!

So, we can conclude that in all the op-amp inverting circuits, the combination of the op-amp and the power supply actually acts as an element with negative impedance that neutralizes the positive impedance of the element connected between the output and the inverting input. As a result, the whole combination of the "positive" element (the resistor R in the picture) and negative element (the supplied op-amp) behaves just as... a piece of wire with zero impedance!

From this negative impedance viewpoint, in the circuit of a capacitive integrator the op-amp is a "negative capacitor" producing the voltage Vc, in a diode log converter - a "negative diode" producing the voltage Vf, etc.

It is interesting to compare this negative "resistor" with the true negative resistance circuit (voltage-inversion negative impedance converter - VNIC)...

...I have finally arrived at the conclusion that we have to consider the bare Wien network as a key point of understanding the Wien oscillator. For this purpose, we have to reveal the role of each element constituting the whole network. Here are my speculations.

The Wien network consists of four elements - C1, R1, C2 and R2, but they can be grouped into two parts (impedance elements) - the lower part consists of the two connected in parallel C1 and R1; the upper part consists of the two connected in series C2 and R2. Thus, the Wien network can be thought as of a frequency- or time-dependent voltage divider. Let's think freely over and even dream about this ubiquitous arrangement to stir our imagination...

The voltage divider configuration consists of two impedances connected in series. The input voltage is applied across the whole network. The output voltage is taken between the common point and some reference point: the ground, the voltage supply rail or a middle point (virtual ground). In the last case, we obtain a bridge configuration (the Wien bridge in our case).

FREQUENCY DOMAIN. The two elements (resistive, nonlinear, reactive, etc.) of the voltage divider configuration have an opposite influence over the transfer ratio and the output voltage. If only one of them is frequently dependent (a capacitor), it will act as "loosing" or "pulling" element when the frequency increases from zero up to infinity. Thus we obtain the classic integrating (low-pass) and differentiating (high-pass) circuits.

Now assume both the elements are frequency-dependent and we have taken the voltage drop across the lower element as an output (the case of the Wien bridge oscillator). When the frequency increases, the impedance of the lower element decreases up to zero and the ratio (the output voltage) decreases up to zero as well. Contrary, if the frequency decreases, the impedance of the upper element increases up to infinity and the ratio (the output voltage) decreases up to zero again. Only at some ("resonant") frequency the ratio of this frequency-dependent voltage divider reaches its maximum of 1/3 and the circuit behaves as some "mixture" of integrating and differentiating circuits (band-pass). It is interesting to see how such a behavior is achieved... to imagine how Wien was thinking when inventing this clever passive circuits... to put ourselves in his place...

If the two elements had opposite frequency-dependent behavior (a capacitor and an inductor), there was no a problem to create a band-pass voltage divider. The problem here is that we have only one kind of a frequency-dependent element - a capacitor, having a different behavior at different frequencies: it has a low impedance at high frequency and high impedance at low frequency. Then, how do we make it to have the same behavior (not to pass the signal) at both the frequencies - high and low?

If we look closely at the Wien network, we can discern an integrating (low-pass) circuit or a differentiating (high-pass) circuit or both the elementary circuits inside it. First, we can think of it as of an integrating circuit R2-C1 that is "neutralized" at extremely low frequencies by connecting the capacitor C2 in series with the resistor R2 and the resistor R1 in parallel to the capacitor C1. Then, with the same success, we can think of it as of a differentiating circuit C2-R1 that is "neutralized" at extremely high frequencies by connecting the resistor R2 in series with the capacitor C2 and the capacitor C1 in parallel to the resistor R1.

Thus, at the very high frequencies, the behavior of the upper capacitor C2 is reversed and made similar to the behavior of an inductor by connecting in series the resistor R2 (the capacitive reactance has gradually disapeared with the frequency increase and the resistance R2 has gradually dominated). At the very low frequencies, the behavior of the lower capacitor C1 is reversed and made similar to the behavior of an inductor by connecting in parallel the resistor R1 (the capacitive reactance has gradually increased up to infinity with the frequency decrease and the resistance R1 has gradually dominated).

So, this was the great Wien's idea - to transmute a capacitor into an "inductor": at the very high frequencies - by connecting in series a resistor; at the very low frequencies - by connecting in parallel a resistor. Am I right?

TIME DOMAIN. It is even more interesting to investigate the Wien network operation through the time. I would even build a "real-time simulation" arrangement to visualize the operation extremely slowly, in a human friendly manner, like this one:

For this purpose, use high resistances (e.g., 100 kom) and large capacitances (e.g., 100 μF) to obtain an extremely low "resonant" frequency (about 0.01 Hz). Then, drive the network with a varying sine wave voltage source with an extremely low frequency (it is interesting to vary the voltage manually trying to keep a sine wave). Next, connect three Microlab analog inputs (ADCs): the first - to the input voltage; the second - to the voltage of the common point between the upper elements R2 and C2; the third - to the common point between the two network parts (the common point of R1, C1 and R2). Finally, write a program that continuously measures the voltages and draws voltage bars over the respective elements representing the voltages across them like this diagram (in addition, it draws current loops):

My next explanations are closely related to the Lutz's considerations about the network operation, e.g. this one: "...However, always: V(P)<V(N) because charging of Cp with decreasing charge current (Voltage at Cp goes high up to a maximum and then decreases again due to Cs charging and discharging through Rp at the same time)..." Now begin changing up and down (wiggle) the input voltage: first - very slowly, then moderately and finally - very quickly.

1. When you wiggle the input voltage slowly, the role of the capacitor C1 and the resistor R2 is negligible and we may ignore them (to imagine there is no C1 and R2 is a short connection). So, the Wien network acts as a humble differentiating C2-R1 circuit. Note that the discharging current through the resistor R1 dominates over the charging current coming from the upper part of the network and the output voltage "leads" the input one (there is a phase shift); the respective voltage bars on the screen do not move in the same direction. See also:

2. Contrary, when you wiggle the input voltage quickly, the role of the resistor R1 and the capacitor C2 is negligible and we may ignore them (to imagine there is no R1 and C2 is a short connection). Now, the Wien network acts as a humble integrating R2-C1 circuit. Now note that the charging current coming from the upper part of the network dominates over the discharging current through the resistor R1 and the output voltage lags the input one (there is a phase shift again); the respective voltage bars on the screen do not move in the same direction as above. See again:

3. And finally, when you wiggle the input voltage moderately (with a rate of change corresponding to the "resonant" frequency f0), all the elements play a role in the circuit operation, and the Wien network acts as a humble "resistive" voltage divider. There is such a proportion between the charging and discharging currents so that the output voltage follows the input one (there is no phase shift at all). But how do we define this "moderate rate of change"? Just look at the two voltage bars representing respectively the input and the output voltage on the screen and adjust the rate of change of the input voltage so that the output voltage bar to follow the input one.

I have stated several times that we can find the answer of this question in the time domain by following the sine "movement" of the output voltage between the supply rails. I suggest to do it here by investigating the structure and the operation of the ubiquitous Wien bridge oscillator. Let's begin with the structure; here are my speculations:

To realize this exotic circuit solution, we have "to see the forest for the trees":), i.e. to group the particular elements in well-known functional blocks. Thus, we may first group the two resistors Rf, Rb and the op-amp U1 (see the attached picture below) into a low-gain (≈ 3) single-ended nonlinear amplifier (the classic non-inverting op-amp amplifier) with a Wien network (R1 = R2, C1 = C2) connected in the positive feedback loop. Rb (a bulb) self heats and reduces the amplifier gain until the point is reached that there is just enough (maybe, 3?) gain to sustain the sine oscillations without reaching the saturation point of the amplifier. So, from this viewpoint, the Wien bridge oscillator is considered as two connected in a loop devices - a non-inverting amplifier and a Wien network (a non-inverting amplifier with a Wien network positive feedback).

Then, we may group (in a little more exotic way) the non-inverting amplifier above with the upper part (R2, C2) of the Wien network into a current-driven negative impedance circuit (INIC). Its impedance is roughly equal to the "positive" impedance of the lower part (R1, C1) so that the two opposite impedances roughly neutralize each other at the equilibrium point.

The next powerful idea is to see the whole Wien bridge circuit (Rf, Rb, R1, C1, R2, C2) and to consider the Wien bridge oscillator as a combination of an op-amp and a Wien bridge connected in the positive feedback loop between the op-amp output and its differential input. The loop gain is a product of the very high op-amp gain and the very low bridge ratio. At the oscillating frequency, the bridge is slightly unbalanced and has a very small transfer ratio; so, the loop gain is about unity.

The final, and maybe the most popular viewpoint, is to break down the Wien bridge into two half bridges, and to consider the overall feedback as composed of two partial feedbacks - a nonlinear negative feedback (the voltage divider Rb-Rf connected to the inverting op-amp input) and a frequency-dependent positive feedback (the Wien network connected to the non-inverting input). Thus the feedback voltage applied to the op-amp differential input is the difference between the two partial voltages.

Now about the operation...