Circuit idea: Injecting energy into circuits in the same manner as the respective "positive" impedance elements absorb it
True negative impedance... what is it? ^{[nb 1]} Is it possible? Does it violate natural laws? Does it exist at all? If so, how do we make it? What is the use of the true negative impedance? What is the difference and what is the common between the true and differential negative resistance? You will find answers to all these questions in this exciting circuit story about the mystic phenomenon...
The true negative resistance is closely related to the differential negative resistance. To see the close connection between them, a lot of reciprocal links are placed inside the two stories.
Contents
How to inject energy into circuitsEdit
Essence. There is absolutely no mystery about the true (absolute) negative impedance! It is just a property of some (really, quite odd and exotic but still understandable) oneport electronic circuits to inject a portion of energy into circuits to which they are connected in the same manner as the corresponding "positive" elements (resistors, capacitors or inductors) absorb energy. Thus, from energetic viewpoint, these negative resistors, capacitors and inductors act as a kind of inverted "positive" resistors, capacitors or inductors. They do this "magic" either by producing voltage that depends on the current passing through them or by producing current depending on the voltage applied across them; the first are qualified as current controlled (Sshaped) and the second  as voltage controlled (Nshaped) negative impedance elements. So they actually are dynamic electrical sources...
Special case. The true negative resistor configuration is the most popular negative impedance circuit. It produces voltage/current that is proportional to the current/voltage through/across itself according to Ohm's law (there are also negative nonlinear resistors, e.g. "negative diodes", that do not obey Ohm's law). On the graphical representation (e.g., Fig. 10 below), the IV curve of the absolute negative resistor has a negative slope in the negative resistance region and passes through the origin of the coordinate system as the voltage and the current have opposite signs for every operating point along the curve. As a result, the ratio R = V/I < 0 (the resistance R is negative).
Basic applications. The amazing feature of negative impedance is to neutralize the equivalent "positive" impedance: connecting a currentdriven negative impedance element in series witn the equivalent "positive" impedance element gives zero total impedance (Fig. 1); connecting a voltagedriven negative impedance element in parallel to the equivalent positive impedance element (Fig. 20) gives infinite total impedance. Because of these remarkable properties currentdriven negative impedance elements are used in circuitry (e.g., in telephony line repeaters) to zero the line resistance, internal voltage source resistance and load resistance. Voltagedriven negative impedance elements are used in such excentric circuits as Howland current source, Deboo integrator, load and stray capacitance cancellers to increase up to infinity the internal current source and load impedance. Actually, true negative resistors are oneport amplifiers that can be connected in series or in parallel to the load. They add additional energy to the energy of the input source; so, true negative resistors help the input source while conventional twoport amlifiers replace the input source.
Relations. Although the term negative resistance is frequently used to encompass negative differential resistance as well, the two phenomena are quite different. A true negative resistor is not a resistor; it is actually a dynamic electrical source while a negative differential resistor is a dynamic positive resistor. It cannot be used independently; it may be used in combination with an electrical source to build a true negative resistor.
Negative versus "positive" impedanceEdit
Negative resistance is the most elementary and comprehensible negative impedance manifestation. We can reveal its nature below by comparing an ordinary "positive" resistor having resistance R with a true negative resistor having the same resistance R. For this purpose, two pairs of equivalent electrical circuits are used^{[1]}, in which the resistors are connected in series with the loads so that the same current passes through them or in parallel to the loads so that the same voltage is applied across them.
What the "positive" impedance isEdit
In electrical circuits, passive elements (resistors, capacitors and inductors) impede current by their inherent resistance, capacitance and inductance (more generally, impedance). As a result, voltage drops appear across the passive elements that represent the energy losses in them. Passive elements absorb this energy from the exciting electrical source: resistors dissipate energy from themselves to outside environment^{[2]} while capacitors and inductors accumulate energy into themselves. From the negative impedance viewpoint it is not important if passive elements dissipate or accumulate energy; it is only important that they absorb energy.
Seriesconnected "positive" impedance elementsEdit
Electrical elements may be connected in series, parallel or mixed. For example, if a resistor R is connected in series with a load (Fig. 2a), voltage drop V_{R} = R.I that is proportional to the current appears across the resistor. When capacitors or inductors are placed at this place, voltage drops changing through time in the corresponding manner appear across these elements.
The graphicanalytical interpretation on Fig. 2b visualizes the "positive" resistor operation by using superimposed IV curves. It is supposed the resistor is supplied by a real current source (its IV curve is green colored). When the input current varies, the crossing operating point slides over an IV curve (orange) representing the positive resistance. It is a real, static, immovable IV curve having a positive slope and passing through the origin of the coordinate system.
Parallelconnected "positive" impedance elementsEdit
Conversely, if the resistor R is connected in parallel with the load (Fig. 3a), current I_{R} = V_{L}/R, that is proportional to the voltage, flows through the resistor. When capacitors or inductors are placed there, currents changing through time in the corresponding manner flow through them.
The graphicanalytical interpretation of this kind of "positive" resistance phenomenon is shown on Fig. 3b. Now it is supposed the resistor is supplied by a real voltage source (its IV curve is red colored). When the input voltage varies, the crossing operating point slides over an IV curve (orange) representing the positive resistance. It is again a real, static, immovable IV curve having a positive slope and passing through the origin of the coordinate system.
What the true negative impedance isEdit
Elements with true negative impedance do the opposite  they inject energy into electrical circuits. Whereas passive elements with positive impedance absorb energy from the input source (they are loads), elements with negative impedance add energy to the input source (they are sources^{[2]}). While voltage drops appear across positive elements, negative elements produce voltage; while positive elements sink current, negative elements produce current. However, they are not ordinary varying (dependent) electrical sources. They are special sources  a kind of "selfvarying", dynamic, "selfdependent" sources (voltage sources whose voltage across them depends on the current through them or current sources whose current depends on the voltage across them). Besides, the relation between the voltage and the current is as in the corresponding passive resistors, capacitors or inductors  linear, nonlinear or timedependent.
Currentdriven negative impedance elementsEdit
In Fig. 4a a negative resistor NR with a resistance of R is connected in series with the load so that voltage V_{NR} = R.I that is proportional to the current appears across the negative resistor. However, while above the "positive" resistor detracts the voltage V_{R} from the input voltage (there V_{R} is a voltage drop), here the negative resistor adds voltage V_{NR} to the input voltage (here V_{NR} is a voltage). The element named "resistor" is really a resistor while the "negative resistor" here is actually a voltage source, whose voltage is proportional to the current passing through it; the resistor is a currenttovoltage drop converter while the negative resistor is a oneport currenttovoltage converter. If negative capacitors or inductors are placed at this place, they produce voltages changing through time.
A negative resistor can be implemented as a selfvarying (dynamic) voltage source, whose voltage is proportional to the current passing through it; this twoterminal currentcontrolled voltage source acts as a currentdriven negative resistor. Negative impedance elements can be implemented as dynamic voltage sources, whose voltage depends on the current in the same manner as the voltage drop across the corresponding passive elements (resistors, capacitors or inductors) depends on the current passing through them.
A graphicanalytical interpretation of this kind of negative resistance phenomenon is shown on Fig. 4b. It illustrates the operation of an Sshaped true negative resistor in the middle part of its IV curve (in blue). In this region, the IV curve has a negative slope and passes through the origin of the coordinate system. The green line represents the IV curve of the input real current source and the red line represents the IV curve of the internal dynamic voltage source. The multitude of lines gives an impression of line movement (animation). It is supposed the negative resistor is driven by a real current source. When the input current varies, its green IV curve moves vertically remaining parallel to itself. The internal voltage source representing the negative "resistor" changes proportionally its voltage. Its IV curve (red colored) moves horizontally and the intersection (operating) point slides over a new dynamic IV curve (blue colored) representing the negative resistance. Note it is not a real IV curve; it is an artificial, imaginary IV curve having a negative slope and passing through the origin of the coordinate system. It lies completely in 2nd and 4th quadrants.
Voltagedriven negative impedance elementsEdit
Dually, when the negative resistor is connected in parallel to the load (Fig. 5a), current I_{NR} = V_{L}/R that is proportional to the voltage drop across the load flows through the negative resistor. However, while above the positive resistor sinks a current from the input current (diverts it from the load current), here the negative resistor adds the same current to the input current (injects additional current into the load). If negative capacitors or inductors are placed there, they inject currents changing through time.
A negative resistor can be implemented also as a varying (dynamic) current source, whose current is proportional to the voltage across it; this twoterminal voltagecontrolled current source acts as a voltagedriven negative resistor. Negative impedance elements can be implemented as dynamic current sources, whose current depends on the voltage in the same manner as the current passing through the corresponding passive elements (resistors, capacitors or inductors) depends on the voltage drop across them.
The graphicanalytical interpretation of this kind of negative resistance phenomenon is shown on Fig. 5b. Now it is supposed the negative resistor is driven by a real voltage source. When the input voltage varies, the current source representing the negative "resistor" changes proportionally its current. As a result, its IV curve (green colored) moves and the crossing operating point slides over a new dynamic IV curve (blue colored) representing another kind of negative resistance. Note it is not a real IV curve; it is an artificial, imaginary IV curve having a negative slope and passing through the origin of the coordinate system.
How negative impedance elements interact with input sourcesEdit
True negative resistors act as supplemental sources that "overhelp" the basic input sources: currentdriven negative resistors are voltage sources "overhelping" the input voltage source to pass bigger current through the load; voltagedriven negative resistors are current sources "overhelping" the input current source to create higher voltage across the load. Figuratively speaking, the input source has the illusion:) that only it determines the current/voltage through/across the load; but actually, both the sources determine these atrtributes (the output quantity depends on two input quantities).
It is interesting fact that as long as the input quantity is zero, the "helping" quantity is zero as well. So, a negative resistor begins operating after the input quantity appears.
How to create negative impedance (compare with NDR)Edit
We have already convinced that true negative impedance elements are amazing and extremely useful. Unfortunately, they do not exist in nature; there are only ordinary passive elements with "positive" impedance (resistors, capacitors, inductors and memristors). Then how do we create them?
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? In the simplest case, how do we invert the positive resistance?
The answer is simple if only we know the Ohm's law:) 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:
Inverting the voltage polarity. In the case of the Sshaped negative resistance R_{S}, we invert the voltage (R_{S} = V/I = R). This means that if we pass current through the Sshaped 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 voltageinversion negative impedance converters (VNIC). Note the power is also inverted (P_{S} = V.I = P).
Reversing the current direction. In the case of the Nshaped negative resistance R_{N}, we invert the current (R_{N} = V/I = R). This means that if we apply positive voltage across the Nshaped 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 currentinversion negative impedance converters (INIC). Note the power is also inverted (P_{N} = V.I = P).
Now, we have only to answer the questions, "How do we invert the voltage?" and "How do we invert the current?" To do that, we need more than Ohm's law...
How to show the secret of the true negative resistanceEdit
We already know that the truth about circuit phenomena is hidden rather in the movement from simple to complex circuits (in the circuit evolution) than in the final perfect circuit solutions. So, the best way to reveal the NR phenomenon is to show the metamorphosis of the true negative resistor from the ordinary ohmic resistor to the sophisticated negative one, to show the evolution of the bare ohmic resistance... According to this idea, we can first reveal the secret of Sshaped true negative resistors by following the succession of typical resistances in the order high ohmic > decreased > zeroed > Snegative. Then we can reveal the secret of the dual Nshaped true negative resistors by a similar succession of other typical resistances  low ohmic > increased > infinite > Nnegative. To visualize the resistor evolution, we will gradually draw the particular segments of the seesaw NR IV curves and will explain how they are obtained. Note they are straight lines although the whole IV curves represent nonlinear resistances. Let's do it!
An Sshaped negative resistor (compare with Sshaped NDR)Edit
(aka currentcontrolled negative resistor)
Looking for the basic Sshaped NR ideaEdit
We can find the solution in many situations (some of them quite funny and even confusing) extracted from our routine.
Money analogy. For example, think of people as of "resistors"  some of them are consumers spending money (acting as "positive resistors":) while other are producers earning money (acting as "negative resistors":) And to make the situation more concrete and jocular, assume that women spend while men earn money (if you prefer, you may swap the roles:) A funny example is a woman spending some money (e.g., $1000 monthly) and the problem is how to make her begin gaining the same money ($1000) instead. The straight solution  to decrease the drain, is hard and unpleasant... but the woman quickly finds out a "lateral" solution  she just puts in touch with a man earning money:) Thus the new community (but we think the woman) begins spending less, zero or even gaining money. The trick is that we do not see the man (he keeps in the background and only works hard:); we see, as before, only the woman and we think that she has changed from a consumer to a producer:) But this is just an illusion since she continues spending the same money as before...^{[nb 2]}^{[nb 3]}
Water analogy. Another example can be the classical water circuit analogy where a constriction and an input flow pump are connected in series in a closed loop of pipe. In this arrangement, I put (unnoticeably for you) on the same direction another "helping" pressure pump and change pressure when you vary the flow of the input pump. Thus I can give an impression of decreased, zeroed and even inverted water impediment.
More analogies. There are many other similar situations in our routine where we mix two opposite things (quantities) in various proportions: acid and alkali, hot and cold water, pressure and vacuum, bitter and sweet, goodness and wickedness, etc., and the result varies depending from the proportion. So, the general idea is:
A setup for emulating an Sshaped negative resistorEdit
We know that the best way to see how such weird electronic devices do this magic (inverting the resistance what particularly means inverting either the voltage or current) is to place ourselves in their place and begin performing their functions, to emulate their behavior by an empathy (we have already used this technique in negative differential resistance). Then how do we emulate a true negative resistor? Following the general idea above, we have just to connect a variable voltage source (the man:) in series to a constant resistor R (the woman:) to obtain the emulating setup in Fig. 6. And to match the resistor with the woman:), choose R = 1000 Ω. Well, let's do it as a funny game: you will control the input current source; I will control the voltage source (you are the source, I am the negative load:) Thus I, the voltage source and the resistor combine to form a "mancontrolled" Sshaped true negative resistor and you drive it with current.^{[nb 4]}
We can present graphically the circuit operation of this simple electrical circuit (the circuit KVL equation V_{A} = V_{H}  I_{A}.R) if we consider that the voltage across the two elements (the input current source and the "negative resistor") is the same and the current flowing through them is the same as well. For this purpose, we have to superimpose their IV curves on the same coordinate system: the IV curve of the "negative resistor" consisting of the "helping" voltage source and the resistor (the community:) is an inclined line with a slope depending on the resistance; the IV curve of the input current source is a horisontal line vertically shifted from the Xaxis. The intersection point A (alias operating point) represents the instant magnitudes of the current I_{A} and the voltage V_{A}. When the input current increases (changes from the most negative to the most positive value), the IV curve of the input current source moves up; the operating point moves up from point 0 to point 7 along the Sshaped IV curve below and gradually draws the curve. To help the understanding of the operation, we will draw stepbystep the particular segments of the IV curve and will explain how they are obtained. A lightgrey colored guide line will direct us during our "excursion" showing the trajectory of the operating point...
Now connect a voltmeter across the input current source and an ammeter in series with it to monitor the voltage and the current. Are you ready? Let's start the "game"!
A setup for emulating an Nshaped true negative resistor (below)
A setup for emulating an Sshaped negative differential resistor
How do we make decreased, zero and negative resistance: Starting point)
How to investigate the linear mode of VNIC
High positive resistanceEdit
In the beginning, the woman from our funny example is alone and continuously spends money (in the simple case, $1000 monthly)... In routine, when applying some kind of "flow" and experiencing a constant high impediment, the obtained "pressure" is proportional to our efforts. For example, in the water analogy I have initially set zero pressure of the "helping" pump (or I have not still connected it). So, when you increase the input flow, the water pressure begins increasing proportionally (a hydraulic manifestation of the Ohm's law).
According to these life situations, I set the maximum positive voltage V_{H} and keep it constant while you begin increasing (in the sense that you make it more positive^{[nb 5]}) continuously the input current I_{IN}. As a result, voltage drop V_{R} appears across the resistor R. According to Ohm's law, it is proportional to the current passing through the resistor  V_{R} = R.I_{IN}. The difference V_{H}  V_{R} (negative) appears across the whole element (the future "negative resistor"). Note this voltage depends only on the input current and the Ohm's law equation is a function of one variable.
On the graphical representation (fig. 7), when you vary the current I_{IN} of the input current source, its (your) IV curve moves vertically remaining parallel to itself (i.e., it translates). As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of the R IV curve represents graphically the value of the ohmic resistance R. This is a real, static, ohmic, "positive" resistance... But why the IV curve does not pass through the coordinate origin?
The reason is that the IV curve is symmetrically located toward the coordinate origin and we begin moving along the curve from point 0. Actually, in the section 01, we investigate not only the bare resistor R but a network consisting of two connected in series components  the resistor R and the voltage source V_{H}^{[nb 6]}. But the voltage source is static (constant) in this region and it does not affect the whole network resistance. This introduces some (inessential for understanding) difference between the operation of the electrical circuit and the funny analogy.
Nshaped true negative resistor: Section 01 (low positive resistance) (below)
Sshaped negative differential resistor: Section 01 (high ohmic resistance)
How do we make decreased, zero and negative resistance: Investigating the ordinary resistance
Voltage inversion NIC: Section 02 (bottom positive resistance region)
Virtually decreased resistanceEdit
Then our problem in the funny situation becomes how to make the woman begin spending less money (for concreteness, $500). And she successfully solves the problem putting in touch with a man that earns less money ($500) than she spends ($1000). Thus the woman is moderately helped by the man and we think she begins spending less money ($1000  $500 = $500). But this is an illusion since she continues spending the same $1000 as before and we are misled because we do not see the moderately working man...
There are many other situations in our life where while we implement our purposes someone begins (inconspicuously) helping us. As a result, we have the illusion that the opposition has decreased so the "flow" increases. In our water analogy, when you increase the input flow, the pressure drop across the constriction proportionally increases but, at the same time, I begin increasing the pressure of the helping pump. As a result, the water flow begins more quickly increasing and you have the feeling that the water resistance is decreased. But this is an illusion since you do not see my helping pump...
Let's now apply this clever trick to decrease virtually the resistance R in such an exotic way. Imagine when you reach the point 1 (fig. 8), I decide to help you along the whole section 12. When you increase the current I_{IN} of the input source from point 1 to point 2, its IV curve translates upward. But, at the same time, I begin moderately decreasing the voltage V_{H} (and its magnitude) thus helping you to increase the current (to decrease its magnitude). The composed V_{H}R IV curve moves to the left remaining parallel to itself (translates). As a result, the operating point A slides along a new more vertical IV curve, which represents the new virtual resistance dR_{1} < R.
Actually, the voltage across the "negative resistor" depends both on the current I_{IN} and the voltage V_{H} and the Ohm's law equation becomes a function of two variables  V_{OUT} = f(I_{IN}, V_{H}). You have the illusion that the resistance R has decreased and you see new, lower dynamic resistance dR_{1} < R; thus the initial ohmic resistance R is converted into a smaller virtual resistance dR_{1}. Note the segment 12 is a straight line and it resembles ordinary ohmic resistance. Looking only at this part of the curve, you may think that you investigate an ohmic resistor... but this is just an illusion...
The virtually decreased resistance can be observed in circuits with imperfect parallel negative feedback implemented by an imperfect inverting amplifier with finite gain (usually realized by discrete transistors): transimpedance amplifier, inverting amplifier, etc.
Nshaped true negative resistor: Section 12 (virtually increased resistance) (below)
Sshaped negative differential resistor: Section 12 (virtually decreased resistance)
How do we make decreased, zero and negative resistance: Varying the voltage slightly
Miller theorem: Obtaining decreased impedance
Virtually zeroed resistanceEdit
Inspired by the power of the clever "female technique":), we decide to go more far and to make the woman spend... no money ($0). As a reaction, she puts in touch with a man earning so much money ($1000) as she spends ($1000). Thus the woman is exactly helped by the man and we think she spends nothing ($1000  $1000 = $0). But this is an illusion again since she continues spending the same $1000 as before... and we are misled once more because we do not see the hard working man...
There are many worldly situations where the dynamizing idea above is enforced so that our helper increases (again, inconspicuously for us) his/her help to such extent that we do not experience any impediment when implementing our purposes. The result is amazing: we have the illusion that the opposition has dissapeared and we realize our purposes without any efforts! For example, in the water analogy, you continue increasing the input flow so that the pressure drop across the impediment proportionally increases but I begin increasing the pressure of the "helping" pump with the same rate of change (i.e., the pressure drop and the "helping" pressure change with the same rate). As a result, the water pressure seen by you stays constant and you have the feeling that there is no any impediment...
As this idea is so wonderful, let's apply it to zero virtually the resistance R in such a marvellous manner. Now imagine when you reach the point 2 (fig. 9), I decide to help exactly you along the whole section 23. You continue increasing (make more positive) the current I_{IN} of the input source from point 2 to point 3 so that its IV curve continues translating up. But now I begin vigorously decreasing the voltage V_{H} so the composed V_{H}R IV curve quckly moves (translates) to the left. As a result, the operating point A slides along a new vertical IV curve, which represents the new zero virtual resistance dR_{2} = 0 and you have the illusion the resistance R has become zero...
This great virtual ground idea may be observed in all kinds inverting opamp circuits with perfect parallel negative feedback: transimpedance amplifier, active ammeter, inverting amplifier, etc. In all these circuits, the opamp acts as the compensating voltage source V_{H}  it adds so much voltage to the input voltage as it loses across the resistor connected between the opamp output and the inverting input. The compensating voltage serves as output voltage supplying the load.
Nshaped true negative resistor: Section 23 (virtual infinite resistance) (below)
Sshaped negative differential resistor: Section 23 (virtually zeroed resistance)
How do we make decreased, zero and negative resistance: Varying the voltage exactly
Miller theorem: Obtaining zeroed impedance
How do we create a virtual ground? reveals the secret of the great circuit phenomenon
How I revealed the secret of parallel negative feedback circuits
Opamp circuit builder is an animated Flash tutorial about building virtual ground circuits
Reinventing the transimpedance amplifier
How do we build an opamp ammeter? converts the imperfect ammeter into an almost ideal one
Opamp inverting summer is an animated Flash tutorial about the famous opamp summing circuit
What is the idea behind the opamp inverting current source?
How do we build an opamp RC integrator? shows the evolution of the humble RC integrating circuit
"Inverted positive" (Sshaped negative) resistanceEdit
Finally, fascinated by the unlimited potentialities of the woman's ingenuity:), we go extremely far and dicide to make the woman earn money ($1000) instead spend them?!? Incredible but true  she takes up the challenge and, in responce to our provocation, she puts in touch with a man earning two times more money ($2000) as she spends ($1000). Thus the woman is overhelped by the man and we think she earns money ($2000  $1000 = $1000) instead to spend them. As though the woman is "inverted"; she is transmuted into a man:)^{[nb 3]} But this is our biggest illusion since she, as usual, continues spending the same $1000... and we are cruelly deceived for the last time because we do not see the extremely hard working man...
In the water analogy, you continue increasing the input flow. But I extremely increase the pressure of my "helping" pump so that the water pressure seen by you not only decreases and becomes zero but imagine it even reverses its sign (you see vacuum instead pressure)! In the same way, we can convert a bad man/woman into a good one by making two times bigger goodness than his/her wickedness, 1 liter acid into 1 liter alkali by adding 2 liters alkali, 1 atm. vacuum into 1 atm. pressure by adding 2 atm. pressure, etc. In all these life situations, the dynamizing idea is enormously reinforced so that our helper goes too far increasing many times more his/her help than needed and the result is very, very surprising. So, the general idea is:
Eureka! This is what we needed to transform the positive into negative resistance  to convert the "bad" voltage drop across the positive resistor R into a "good" voltage by adding two times higher voltage!^{[nb 7]} Then let's put this fairy idea in practice to "invert" the resistance R! The recipe is clear  when you reach the point 3 (fig. 10), I begin overhelping you along the whole section 34. As usual, you are continuously increasing (make more positive) the current I_{IN} of the input source from point 3 to point 4 so that its IV curve is translating up remaining parallel to itself. Following the recipe, I am extremely vigorously decreasing^{[nb 8]} the voltage V_{H} so the composed V_{H}R IV curve extremely quckly is translating to the left. As a result, the operating point A slides up along the new IV curve of the inverted positive resistance that is inclined (folded up) to the left and has a negative slope. You have the illusion the resistance R has become true negative resistance dR_{3} < 0.
This idea is directly implemented in voltageinversion negative impedance converters (VNIC).
Nshaped true negative resistor: Section 34 ("inverted positive", i.e. negative, resistance) (below)
Sshaped negative differential resistor: Section 34 ("decreasing" negative differential resistance)
How do we make decreased, zero and negative resistance: Varying the voltage considerably
Miller theorem: Obtaining negative impedance by inversion the voltage
Voltage inversion NIC: Section 24 (middle negative resistance region)
How to compensate resistive losses by series connected negative resistors: Making a negative resistor
Understanding negative impedance converters with voltage inversion
After the Sshaped NR regionEdit
Unfortunately, there are no unlimited things in this world:( The woman from our funny situation ruthlessly continues icreasing her expenses and, at a given moment, the man is so exhausted that he cannot overcompensate the woman's drain anymore. He slows down the rate of working and begins first exactly, then  moderately and finally  stop compensating the woman's expenses (but the woman does not stop increasing the drain:)...
Let's implement this situation in our emulating setup. As usual, you continue increasing (make more positive) the current I_{IN} of the input source from point 4 to point 7 (fig. 11) so that its IV curve is translating up. At point 4, I have already exhausted a great part of the initial voltage and, mimicking the man's behavior, I begin slowing the rate of change along the segment 45. The composed V_{H}R IV curve is quckly translating to the left. As a result, the operating point A slides along the vertical segment 45 of the IV curve, which represents virtually zeroed resistance. Then, at point 5, I slow more down the rate of the voltage change. The V_{H}R IV curve translates slower and the operating point A slides along the inclined segment 56 (virtually decreased resistance). Finally, along the segment 67, I stop voltage changing. The V_{H}R IV curve stops moving and the operating point A slides along the final segment 67 (ohmic resistance).
Let's finally see how we have formed the S shape of the whole IV curve. In the beginning, we took the humble linear IV curve of the ohmic resistance R. Then, in a sertain region, we bent counterclockwise the curve by adding "helping" voltage to the input voltage. When the voltage stops changing, the curve folds up clockwise thus resembling the letter "S". The conclusion is:
The Sshaped true negative resistance IV curve is a modified linear IV curve of high ohmic resistance.
Nshaped true negative resistor: Section 47 (after the Nshaped NR region) (below)
Sshaped negative differential resistor: Sections 47 (after the Sshaped NDR region)
Voltage inversion NIC: Section 46 (top positive resistance region)
Compensating resistive losses by Sshaped negative resistorsEdit
The problemEdit
Internal resistance. Real voltage sources have some internal resistance (Ri in Fig. 12). So, when we conect a load R_{L}, the current flows through the internal resistance and creates the voltage drop V_{Ri} = I.Ri across it. This voltage drop is undesired as it subtracts from the input voltage and, as a result, the voltage applied to the load is less than the input voltage. So, the problem is to remove (to zero) the undesired internal resistance.
The classical solution is to buffer the imperfect voltage source by a powerful voltage follower (i.e., to replace the "original" input source by a "copy" source supplied by its own power supply). More original and clever would be to compensate somehow the source's internal resistance. But how do we do this magic? What do we place between the input source and the load?
Probably, we would want to do it by applying the ubiquitous virtual ground configuration (a transimpedance amplifier). But it is impossible to connect the opamp "sense" (the inverting input) to the left side of the internal resistance since it is distributed inside the source; it is not represented by a separate component. Then we arrive at this so strange, exotic and paradoxical idea to compensate the internal resistance by equivalent negative resistance.
Line resistance. Another similar problem appears if we move the load away from the source. Since the line has some resistance, it adds to the internal source resistance and, as a result, the voltage across the load droops even more. In this case the virtual ground configuration is possible but it is inconvinient to stretch an additional third wire.
How to compensate resistive losses by series connected negative resistors: Step 2
The basic idea of the series NR compensationEdit
It's time to put in practice all the wisdom about the true negative resistance extracted from the routine. For this purpose, we have just to incorporate the emulating arrangement above into the circuit. Let's do it!
As the internal "positive" resistance Ri is not accessible (or the line is too long to stretch a third "sensing" line), we apply another clever trick. We place in series to the "original" resistor Ri another "positive" resistor R (the woman:) with equivalent resistance serving as a duplicate of the unaccessible "resistor" Ri or Rl. Since the current passing through the two series connected resistors is the same and they have equal resistances, the voltage drops across them are also equal. Thus we have successfully copied the voltage drop V_{Ri} across the our resistor R and now we have only to overcompensate it (to invert the resistor R). Following the voman's strategy:), we connect in series an "overhelping" voltage source (the man:) adding twice as much voltage V_{H} = 2V_{Ri}.^{[nb 9]} A half of this voltage compensates the voltage drop across the duplicate resistor R; the rest half (V_{Ri}  2V_{Ri}) appears across the whole "negative resistor" as inverse voltage V_{Ri}. It compensates the voltage drop across the source internal resistance Ri (in other words, the negative resistance R = Ri of the whole element neutralizes the positive internal resistance Ri). As a result, only the load resistance R_{L} remains in the circuit and the whole input voltage is applied to the load (V_{L} = V_{IN}).
How to compensate resistive losses by series connected negative resistors: Step 4
Building a true Sshaped NR (voltageinversion NIC)Edit
To transmute the emulated Sshaped true negative resistor into a real electronic circuit, we have only to realize somehow the "helping" voltage source V_{H}. An amplifier with a fixed gain of two will do this donkey work if only we connect its input in parallel and its output in series to the resistor R. It will amplify two times the voltage drop V_{H} across the resistor R and will add the amplified voltage to it. But this solution would match more with Armstrong's times; now we prefer to make an opamp with giant gain amplify two times (Fig. 14). How do we do it?
We may borrow the idea from the wellknown circuit of a transimpedance amplifier where the opamp keeps its output voltage equal to the voltage drop across the resistor. For our purposes here, we may trick the opamp by connecting a voltage divider (composed by two equal resistors R) between the opamp output and the noniverting input. As a result, the opamp will produce two times higher voltage... what was our purpose. A half of this voltage compensates the voltage drop across the duplicate resistor R (below the opamp); the rest half appears across the circuit terminals as inverse voltage V_{Ri}. So, we may think of this circuit as of an overacting transimpedance amplifier.
The "invented" circuit as though inverts the initial voltage drop across the resistor Ri (the mental reservation is that we do not see the opamp; we see only the resistor R); thus the name negative impedance converter with voltage inversion or voltageinversion negative impedance converter (VNIC).
How to compensate resistive losses by series connected negative resistors: Step 8
Stability (operating mode)Edit
A currentdriven true negative resistor with resistance R connected in series with a positive resistor with total resistance R_{TOT} destroys, eats, neutralizes Rpart of the total positive resistance thus converting it to zero resistance. Only, in order to have a stability (see below), some part of the positive resistance has to remain.
Currentdriven true negative resistors are circuits with positive feedback where a part of the output quantity adds to the input quantity. The gain of the feedback loop is proportional to the ratio between the negative resistance R_{N} and positive resistance R_{P}. In order to have a stability (to operate in active mode), we need the positive resistance to dominate over the negative one (R_{N}/R_{P} < 1). For the opamp INIC from Fig. 14 this means: Ri/(Ri + R) > R/(R + R) = 1/2. Otherwise, the circuit will operate in a bistable (memory) mode.
Mimicking the true Sshaped negative resistanceEdit
In the circuit of a transimpedance amplifier, the opamp "copies" the voltage drop across the resistor R (Ri in Fig. 15) and adds this voltage in series to the resistor. In contrast to a true negative resistor, here the opamp uses an additional third wire to "feel" by its inverting input the difference between the voltage drop across the resistor and its output voltage. The opamp compares its output voltage with the voltage drop across the resistor R and changes it so that to keep (almost) zero difference between them. As a result, the opamp produces output voltage compensating the voltage drop. So, the opamp acts as (mimics ) a true negative resistor that neutralizies the resistance R; in the circuit of a transimpedance amplifier two equal but opposite (positive and negative) resistances are neutralized.
This arrangement can be used to neutralize the line resistance since we can stretch a third wire from the opamp inverting input (the "sense") to the left side of the "positive" line "resistor". Note in this arrangement the opamp has "floating" power supply, in order the load to be grounded.
How to compensate resistive losses by series connected negative resistors: Step 9
Transimpedance amplifier (opamp currenttovoltage converter)
Reinventing transimpedance amplifier
Generalizing the Sshaped negative resistance ideaEdit
In the examples above, the Sshaped negative resistor converts the voltage drop across a "positive" resistor into voltage that is proportional to the current. With the same success it can convert the voltage drop across any "positive" element  time dependant (capacitor, inductor, memristor), nonlinear (diode, varistor), etc., into voltage. Actually, the negative resistor does not "understand" what it converts; it just doubles the voltage drop across a kind of element and "inserts" the same voltage in series to the element.
Negative impedance. For example, if we replace the resistor R with a capacitor C, we will obtain a negative capacitor that will add (instead to subtract) its voltage to the input voltage. If we connect this negative capacitor in series to a "positive" one, it will compensate the voltage drop across the "positive" capacitor and the effective voltage across the network will be zero. A similar arrangement is implemented in the circuit of the opamp inverting integrator where the opamp acts as a negative capacitor:
A negative capacitor
How do we build an opamp RC integrator?
Building an opamp integrator (an animated Flash minitutorial)
Keeping a constant current by adding additional voltage
An Nshaped negative resistor (compare with Nshaped NDR)Edit
(aka voltagecontrolled negative resistor)
Looking for the basic Nshaped NR ideaEdit
Let's now looking for situations in our routine, where we experience some opposition when implementing our purposes. A good example can be another water circuit analogy similar to this above (a constriction and an input pressure pump connected in series in a closed loop of pipe). But now, I put (unnoticeably for you) on the contrary the additional pressure pump (it is opposing here) and change its pressure when you vary the pressure of the input pump. Thus I can give an impression of increased, infinite and even inverted water impediment.
A setup for emulating an Nshaped negative resistorEdit
We can emulate this type of true negative resistor in a similar way as above (Fig. 16)  we connect a variable voltage source in series to a constant resistor R and contrary to the input voltage source. Well, let's do it again as a funny game: you will control the input voltage source; I will control the additional contrary voltage source (you are the source, I am the negative load again:) Thus I, the contrary voltage source and the resistor combine to form a "mancontrolled" Nshaped true negative resistor and you drive it with voltage (to see the negative resistance region in the IV curve, i.e. to operate in a linear mode, now we have to supply the negative resistor by a voltage source; otherwise, it will operate in the exotic bistable mode).
We can present graphically the circuit operation (the circuit KVL equation V_{A} = V_{O}  I_{A}.R) if we consider as above that the voltage across the two elements is the same and the current flowing through them is the same as well. This allows to superimpose their IV curves on the same coordinate system: the IV curve of the combination "voltage source + resistor" is an inclined line (in green color) with a slope depending on the resistance; the IV curve of the input voltage source is a vertical line (in red) horizontally shifted from the Yaxis. The intersection (operating) point A represents the instant magnitudes of the current I_{A} and the voltage V_{A}. When the input voltage increases (changes from the most negative to the most positive value), the IV curve of the input voltage source moves to right (in contrast to the Stype negative resistor, here the two IV curves move in the same direction); the operating point moves to right from point 0 to point 7 along the Nshaped IV curve and gradually draws the curve. To help the understanding of the operation as above, we will draw stepbystep the particular segments of the IV curve and will explain how they are obtained; the lightgrey line will guide us during our "excursion" showing the trajectory of the operating point...
Now connect a voltmeter across the input current source and an ammeter in series with it to monitor the voltage and the current and start the "game"!
A setup for emulating an Sshaped true negative resistor (above)
A setup for emulating an Nshaped negative differential resistor
Low positive resistanceEdit
In the beginning, imagine that in the water analogy I have initially set zero pressure of the opposing pump (or I have not still connected it). So, when you increase the input pressure, the water flow begins increasing proportionally (another hydraulic manifestation of the Ohm's law).
According to this life situation, I set the maximum negative voltage V_{O} and keep it constant while you begin increasing (in the sense that you make it more positive^{[nb 10]}) continuously the input voltage V_{IN}. As V_{IN} is more negative than V_{O}, the current flows through the resistor R from the left to the right. According to Ohm's law, it is proportional to the difference between the two voltages. Note the current depends only on the input voltage and the Ohm's law equation is a function of one variable.
In the graphical representation (fig. 17), when you vary the voltage V_{IN} of the input voltage source, its (your) IV curve moves horizontally remaining parallel to itself (i.e., it translates). As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of the R IV curve represents graphically the value of the ohmic resistance R. This is a real, static, ohmic, "positive" resistance... But again, why the IV curve does not pass through the coordinate origin?
The reason is as above  the IV curve is symmetrically located toward the coordinate origin and we begin moving along the curve from point 0. Actually, in the section 01, we investigate not only the bare resistor R but a network consisting of two connected in series components  the resistor R and the voltage source V_{O}. The voltage source is static (constant) in this region and it does not affect the whole network resistance. As above, this introduces some (inessential for understanding) difference between the operation of the electrical circuit and the funny analogy.
Sshaped true negative resistor: Section 01 (high positive resistance) (above)
Nshaped negative differential resistor: Section 01 (low ohmic resistance)
Current inversion NIC: Section 02 (left positive resistance region)
Virtually increased resistanceEdit
Remember situations in our life when we implement our purposes but an opposer appears and begins increasing (inconspicuously for us) his/her opposition. As a result, we have the illusion that the opposition has increased so the "flow" decreases. In our water analogy, when you increase the input pressure, I begin increasing the pressure of the opposing pump. As a result, the water flow begins increasing lazier (but continues flowing in the same direction) and you have the feeling that the water resistance is increased. But this is an illusion since you do not see my opposing pump...
Let's now apply this clever "plumbing" trick to increase virtually the resistance R in such an exotic way. Imagine when you reach the point 1 (fig. 18), I decide to oppose you along the whole section 12. When you increase the voltage V_{IN} of the input source from point 1 to point 2, its IV curve translates to the right. But, at the same time, I begin moderately increasing the voltage V_{O} (and its magnitude) thus opposing you to increase the current (to decrease its magnitude). The composed V_{O}R IV curve translates to the right. As a result, the operating point A slides along a new more inclined IV curve, which represents the new virtual resistance dR_{1} > R.
Actually, the current through the "negative resistor" depends both on the input voltage V_{IN} and the opposing voltage V_{O}, and the Ohm's law equation becomes a function of two variables  I_{OUT} = f(V_{IN}, V_{O}). You have the illusion that the resistance R has increased and you see new, higher dynamic resistance dR_{1} > R; as though, the initial ohmic resistance R is converted into a higher virtual resistance dR_{1}. Note the segment 12 is a straight line and it resembles ordinary ohmic resistance. Looking only at this part of the curve, you may think that you investigate an ohmic resistor... but this is just an illusion...
The virtually increased resistance can be observed in imperfect voltage followers made by low gain amplifiers (usually realized by discrete transistors)  emitter, source and cathode followers.
Sshaped true negative resistor: Section 12 (virtually decreased resistance) (above)
Nshaped negative differential resistor: Section 12 (virtually increased resistance)
Miller theorem: Obtaining increased impedance
Virtual infinite resistanceEdit
Now imagine that the dynamizing idea above is enforced so that our opposer increases (again, inconspicuously for us) his/her opposition to such extent that we experience infinite impediment when implementing our purposes. The result is amazing: we have the illusion that the opposition has become infinite, we cannot realize our purposes at all and the "flow" stays constant! In the water analogy, you continue increasing the input pressure but I begin increasing the pressure of the opposing pump with the same rate of change (i.e., the input pressure and the opposing pressure change with the same rate and in the same directions). As a result, the water flow stays constant (there is a flow but there is no change of the flow) and you have the feeling you do nothing:)
What a wonderful idea! Let's then apply it to make virtual infinite resistance in such a marvellous manner. So imagine when you reach the point 2 (fig. 19), I decide to oppose exactly you along the whole section 23. You continue increasing (make more positive) the voltage V_{IN} of the input voltage source from point 2 to point 3 so that its IV curve continues translating to the right. But now I begin vigorously increasing the voltage V_{O} so the composed V_{O}R IV curve quckly moves (translates) to the right as well. As a result, the operating point A slides along a new horizontal IV curve, which represents the new virtual infinite resistance dR_{2} = ∞ and you have the illusion the resistance R has become infinite...
We can see this clever trick in the Ohm's equation  I = (V_{O}  V_{IN})/R. In the numerator, the two voltages change with the same rate and direction; so, their difference and respectively the current, stays constant.
This great idea is known as bootstrapping; we will use it below. The bootstrapping may be observed in perfect opamp voltage followers. It is frequently used in amplifiers and constant current sources to increase extremely their input impedance.
Sshaped true negative resistor: Section 23 (virtually zeroed resistance) (above)
Nshaped negative differential resistor: Section 23 (virtual infinite resistance)
Miller theorem: Obtaining infinite impedance
"Inverted positive" (Nshaped negative) resistanceEdit
In the middle of our "excursion", imagine that the dynamizing idea is enormously reinforced so that our opposer goes too far increasing many times more his/her opposition than needed. The result is very, very surprising  we increase our efforts but the "flow" not only does not increase or stay constant but even reverses its direction and goes against us! In the water analogy, you continue increasing the input pressure but I increase the pressure of the opposing pump more considerably (for concreteness, two times higher than the input pressure) so that the water flow reverses its direction (the water begins flowing from me to you).
Eureka! This is the same inverting idea but here we have inverted the flow instead the pressure as above. So we have another idea how to transform the positive into negative resistance  by inverting the current. The recipe is clear  when you reach the point 3 (fig. 20), I begin "overopposing" you along the whole section 34. As usual, you are continuously increasing (make more positive) the voltage V_{IN} of the input voltage source from point 3 to point 4 so that its IV curve continues translating to the right. But now I am extremely vigorously increasing^{[nb 11]} the voltage V_{O} so the composed V_{O}R IV curve quckly moves (translates) to the right as well. As a result, the operating point A slides down along the new IV curve of the inverted positive resistance that is inclined (folded up) to the right and has a negative slope. You have the illusion the resistance R has become true negative resistance dR_{3} < 0.
This idea is directly implemented in currentinversion negative impedance converters (INIC).
Sshaped true negative resistor: Section 34 ("inverted positive" resistance) (above)
Nshaped negative differential resistor: Section 34 ("increasing" NDR)
Miller theorem: Obtaining negative impedance by inversion the current
Current inversion NIC: Section 24 (middle negative resistance region)
After the Nshaped NR regionEdit
But there are no unlimited things in this world; so, finally the opposer is so exhausted that he/she cannot overoppose to our efforts and the "flow" begins increasing again from us to him/her. In the water analogy, you continue increasing the input pressure but I slow down the pressure of the opposing pump  exactly, moderately and finally  stop changing it.
Let's see this situation in our emulating setup. As usual, you continue increasing (make more positive) the voltage V_{IN} of the input source from point 4 to point 7 (Fig. 21) so that its IV curve is translating to the right. At point 4, I have already exhausted a great part of the initial opposing voltage and begin slowing the rate of change along the segment 45. The composed V_{O}R IV curve is quckly translating to the right. As a result, the operating point A slides along the horizontal segment 45 of the IV curve, which represents virtual infinite resistance. Then, at point 5, I slow more down the rate of the voltage change and the operating point A slides along the inclined segment 56 (virtually increased resistance). Finally, along the segment 67, I stop voltage changing and the operating point A slides along the final segment 67 (ohmic resistance).
Let's finally see how we have formed the N shape of the whole IV curve. In the beginning, we took the humble linear IV curve of the ohmic resistance R. Then, in a sertain region, we bent the curve clockwise by adding "opposing" voltage to the input voltage. When the voltage stops changing, the curve folds up counterclockwise thus resembling the letter "N". The conclusion is:
The Nshaped true negative resistance IV curve is a modified linear IV curve of low ohmic resistance.
Sshaped true negative resistor: Section 47 (after the Sshaped NR region) (above)
Nshaped negative differential resistor: Sections 47 (after the Nshaped NDR region)
Current inversion NIC: Section 46 (right positive resistance region)
Compensating resistive losses by Nshaped negative resistorsEdit
(neutralizing the load resistance)
The problemEdit
In nature, real sources (motors, beings, etc.) have a limited power but if they are not loaded, they behave perfectly.
An example of this situation in electricity and electronics is the simplest varying voltage source on Fig. 16. It is composed of a steady voltage source V and a potentiometer P (a voltage divider r1r2). If there is no load connected, this real voltage source works well  it produces exactly V_{OUT} = r2/(r1 + r2).
How to compensate resistive losses by parallel connected negative resistors: Step 1
If the natural sources are loaded (for example, if we try to raise a big weight  Fig. 17), they droop. A similar problem exists in electronics (electricity) when imperfect voltage sources are loaded. For example, when a load R_{L} is connected to the voltage divider in Fig. 18, it "sucks" a current I_{L} and the output voltage V_{L} drops. We may generalize the problem if we complicate a bit the dual currentsupplied electrical circuit shown on Fig. 5a by adding another element with positive resistance PE2 (see Fig. 20 below). Now it contains two elements with positive resistance connected in parallel: the first element PE1 (the load) is useful; the second element PE2 (e.g., leakage resistance, voltmeter internal resistance, etc.) is undesired. Well, what do we do to remove the disturbance?
How to compensate resistive losses by parallel connected negative resistors: Step 2
The basic idea of the parallel NR compensationEdit
The classic remedy is to connect a voltage follower (a unitygain amplifier acting as a buffer amplifier) before the load, in order to decrease the current I_{L} (to increase the load resistance R_{L}). Unfortunately, this solution introduces some errors inherent for this circuit ^{[3]}. Then let's look for a remedy in our routine.
What can we do in real life when some object (being, machine etc.) supplied by a real power source droops? We can just help it. For this purpose, we usually use an additional power source, which "helps" the main source by compensating the losses caused by the load. For example, if someone has to raise a heavy loaded cage, we can help it by an equivalent "antiweght" (a powerful idea of mechanics that is widely used in the lift systems, cranes etc.  Fig. 19).
According to this powerful "neutralization" idea, the positive resistance of the undesired element can be eliminated (can be made infinite) by connecting in parallel an additional voltagedriven element (Nshaped) NE with the same negative resistance. The current flowing through the undesired "positive" resistance element PE2 (Fig. 20) is proportional to the voltage across it. In order to eliminate this disturbing current, the same current (proportional in the same way to the voltage) has to be produced by the compensating negative resistance element NE. As a result, the disturbing element PE2 will not consume any current from the input source; the compensating negative resistance element will provide all the needed current for PE2.
As this idea is so wonderful then let's realize it. How do we create the needed Nshaped negative resistor? We can use various building "scenarios" to do it, let's begin...
Scenario 1. To make a currentdriven (Sshaped) negative resistor, we produced voltage that is proportional to the current flowing through it. Now, to make a voltagedriven (Nshaped) negative resistor, we have to do the opposite  to produce "helping" current that is proportional to the voltage across it. For this purpose, we connect in series a voltage source (the output of an amplifier) and a "positive" resistor R acting as a voltagetocurrent converter (Fig. 21). The voltage source has to keep the same voltage V_{R} across the resistor as the voltage V_{L} across the load; that means it to produce two times higher voltage V_{H}= 2V_{L} producing the "helping" current.
Scenario 2. If a voltage V_{L} is applied across the "positive" load resistor R_{L}, it will consume a load current I_{L} = V_{L}/R_{L}. Conversely, if we apply the same voltage V_{L} across an identical negative resistor R with resistance R_{L}, it has to produce the same current I_{H} = V_{L}/R_{L} = I_{L}. So, we have to "lift" the right end of the resistor (originally connected to ground) with voltage V_{L} toward its left end that is connected to the load. For this purpose, we connect a compensating voltage source B_{H} (a noninverting amplifier with K = 2) in series with the "copy" positive resistor R having the same resistance as the "original" positive resistor R_{L} (Fig. 21).
The voltage source makes a current I_{H} = (V_{H}  V_{L})/R = (2V_{L}  V_{L})/R_{L} = V_{L}/R_{L} = I_{L} that is equal to the load current I_{L} flow through the load. In this way, the whole load current I_{L} is provided only by the "helping" current source I_{H} (the negative resistor R_{L}) instead by the real input voltage source. The load does not consume any energy from the input source since it is supplied completely by the "helping" source. Figuratively speaking, the load "pulls" the point A down toward the ground while the resistor R "pulls" the point A up toward the voltage V_{H}. As a result of this "stretching", the point A experiences "weightlessness" (as it pulls itself up) and it follows easily the point B. There is no current flowing through the "bridge" connecting the point B and point A since the whole right part of the circuit (R_{L}, R and V_{H}) behaves as a load with infinite internal resistance. This is the wellknown phenomenon of bootstrapping and it is put in practice for the first time by Baron Munchhausen (the legend says that he was using his own boot straps to pull himself out of the sea:)
Note the current flowing through the composed voltagedriven negative resistor R has an opposite direction to the current flowing through the initial "positive" resistor R as though the circuit has inverted the initial current.
Building a true Nshaped NR (currentinversion NIC)Edit
Fixed gain amplifier. We need a doubling voltage source; a noninverting amplifier A having a gain of (only) +2 can act as such a "helping" voltage source (Fig. 22). We have just to connect the amp's input to the point A and the amp's output in place of the "helping" voltage source B_{H}. The amplifier is singlesupplied since here the input voltage is only positive.
The amplifier doses the voltage +V of the power supply, in order to produce the voltage needed (V_{A} = 2V_{RL}). Actually, the steady voltage source +V and the amplifier A constitute the varying voltage source needed. The combination of this composed voltage source and the resistor R acts as a "helping" current source. It injects a current I_{H} through the resistor R into the point A and raises its voltage; as a result, the point A "pulls itself up". The output voltage affects the input voltage as a part of the output voltage adds to the input voltage. This great phenomenon is referred to as positive feedback.
Opamp amplifier with negative feedback. In electronics, we realize such amplifiers with fixed gain (in this case we need G = 2) by operational amplifiers. There are perfect opamps having extremely large but unstable voltage gain (typically 200000). By applying a negative feedback we can make an opamp amplify exactly two times needed. How do we do this magic?
Negative feedback systems have a nice feature to reverse the causality in electronic circuits. For example, if we put a passive circuit (an integrator, differentiator, attenuator, etc.) into the feedback loop, we will obtain the opposite active circuit (a differentiator, integrator, amplifier, etc.) According to this idea, let's build a voltage divider having a ratio 0.5 by connecting in series two equal resistors R1 and R2; then, let's connect it between the opamp's output and the inverting input (Fig. 23). As a result, we obtain an opamp noninverting amplifier having the stable gain of 2 needed.
Actually, this opamp circuit converts the positive resistance R of the duplicate resistor into a negative resistance R, i.e. it acts as a negative impedance converter (NIC). As the current flowing through this kind of negative resistance circuit has an opposite direction to the current flowing through the initial "positive" resistor, this circuit is named negative impedance converter with current inversion (INIC).
When the opamp output voltage approaches supply rails, it stops changing as the opamp saturates and begins acting as an ordinary constant voltage source. The "magic" of negative resistance ceases.
Stability (operating mode)Edit
A voltagedriven true negative resistor with resistance R connected in parallel to a positive resistor with a total resistance R_{TOT} destroys, eats, neutralizes Rpart of the total positive resistance thus converting it to infinite resistance. Only, in order to have a stability (see below), some portion of negative resistance has to remain.
Voltagedriven true negative resistors are also circuits with positive feedback where a part of the output quantity adds to the input quantity. Here, the gain of the feedback loop is proportional to the ratio between the positive resistance R_{P} and negative resistance R_{N}. So, in order to have a stability (to operate in active mode), now we need the negative resistance to dominate over the positive one ( R_{P}/R_{N} < 1). For the opamp INIC from Fig. 21 this means: R_{L}/(R + R_{L}) < R_{2}/(R_{1} + R_{2}). Otherwise, the circuit will operate in bistable (memory) mode.
Generalizing the Nshaped negative resistance ideaEdit
(neutralizing the stray capacitance or creating a negative capacitor)
So far we have been using linear ohmic resistors as initial, passive elements with "positive" resistance to make dual active elements with voltagecontrolled negative resistance. But with the same success we can transmute every nonlinear "positive" resistor into a negative one (e.g., a diode into a negative diode). Finally, by using the same technique, we can create various timedependent elements with negative impedance, e.g. a voltagecontrolled negative capacitor.
The concept of negative capacitance is abstract enough; so, let's consider a typical application  neutralizing a stray capacitance by a negative capacitance. Although this brilliant idea is proposed as far back as in early 60s maybe we might find its origin in Armstrong's radio times. It seems paradoxical but there are not still clear, simple and intuitive explanations of the capacitive neutralization idea. So, it is worth unveiling the mystery of this clever trick.
Negative capacitors are AC circuits driven by sine wave input voltage. In order to really understand how they exactly operate, we will show what the voltages are and where the currents flow in the circuits at one given (arbitrary chosen) moment of the sine wave. So, think of the pictures of voltage bars and current loops superimposed on the figures below as kinds of snapshots.
The problem caused by the stray capacitanceEdit
Imagine a sine wave generator with output resistance R_{IN} drives a load with infinite input resistance (Fig. 24a). As no current flows through the resistance there is no voltage drop across it and, as a result, the output voltage is equal to the input one (V_{OUT} = V_{IN}).
If the load has a significant stray capacitance C_{STR}, it constitutes (in conjunction with the resistance R_{IN}) an integrating circuit (Fig. 24b). As a result, the output voltage begins lagging and thus differing from the input one.
The problem is that the capacitor draws a current from the input source; it is a passive element that absorbs energy from the exciting electrical source and accumulates the "stolen" energy into itself.
The basic electrical circuitEdit
Exactly in the same way as above, we may neutralize the positive impedance of the stray capacitance C_{STR} by connecting in parallel an additional voltagedriven negative capacitor with the same but negative impedance. While the ordinary "positive" capacitor consumes energy from the input source (it is a load); the negative capacitor does the opposite  it injects energy into the circuit (it is a source). Speaking more concrete, while a series connected "positive" capacitor detracts a voltage drop from the input voltage, a currentdriven negative capacitor adds voltage to the input voltage (it is a voltage source); while a parallel connected "positive" capacitor "sucks" current, a voltagedriven negative capacitor produces current (it is a current source). As above, a current flows through the undesired stray capacitance (Fig. 25) that is a differential of the voltage across it. In order to eliminate this disturbing current, the negative capacitor has to produce the same current (depending in the same way on the voltage through time). As a result, the stray capacitance will not consume any current from the input source; the negative capacitor will provide all the current needed to charge the stray capacitance. It is wonderful but yet... how do we make a negative capacitor?
We may use the same trick as above  to convert a "positive" capacitor into a negative one. For this purpose, we connect a "helping" voltage source V_{H} = 2.V_{STR} (a noninverting amplifier with K = 2) in series with a "positive" capacitor C having the same as the stray capacitance C_{STR}. The voltage source makes a current I_{H} flow that is equal to the current I_{C} flowing through the stray capacitance. In this way, the whole current I_{C} is provided only by the "helping" voltage source V_{H} instead by the real input voltage source. The load does not consume any energy from the input source since it is supplied completely by the "helping" source. The input voltage source works at ideal load conditions; it has the "feeling" that there is not a capacitive load connected and the output voltage V_{STR} is equal to the input one V_{IN}. The situation is exactly as it is shown in Fig. 24a.
But... where do we take the output voltage from?
We may use, as usual, V_{STR} as an output voltage (OUT1 serves as an output). Only, if the load has some resistance R_{L}, it will constitute a voltage divider with the internal resistance R_{IN} and the output voltage will droop  V_{OUT} = V_{IN}.R_{L}/(R_{L} + R_{IN} (note that the negative capacitor compensates only the stray capacitance; it does not compensate the load resistance). But we have the unique possibility to use the compensating voltage V_{H} = 2V_{STR} as an output voltage (OUT2 serves as an output)! As a result, the stray capacitance will consume energy from the helping voltage source instead from the input voltage source and the output voltage will not droop. In addition, it will be amplified two times (whether we wish it or not). this clever trick is widely used in opamp inverting circuits with parallel negative feedback (with a virtual ground configuraion).
Opamp implementationEdit
The opamp implementation of a negative capacitor (Fig. 27) is similar to the opamp circuit of a negative resistor (Fig. 21) with only one difference  a capacitor C is connected instead the resistor R. As above, the opamp and the voltage divider (the resistors R_{1} and R_{2}) constitute a noninverting amplifier with gain of 2 that serves as a compensating voltage source.
Finally, let's look at the scanned image on Fig. 26 giving thanks to pioneers. It is an extract from page 8 of the remarkable genuine paper written by Dan Sheingold in the enthusiastic issue The Lightning Empiricist of Philbrick Researches in the distant 60's. As you can see, the basic idea behind this exotic circuit solution is thoroughly hidden there...and it was staying hidden as many as 45 years...and we have finally managed to reveal it relying only on our human intuition and common sense!
General considerations about creating the NREdit
How to bend the positive resistor IV curveEdit
Let's finally generalize how we have obtained such exotic IV curves having regions with negative slope in their middle parts. In both the cases, we use as an initial material the linear IV curve of an ordinary ohmic ("positive") resistor. Then, in a certain region, we bent the curve by adding/subtracting additional proportional voltage to/from the input voltage: if we add "helping" voltage, the IV curve folds up counterclockwise thus resembling the letter "S"; if we subtract "opposing" voltage, the IV curve folds up clockwise thus resembling the letter "N". Figuratively speaking, at given moment we change the movement (trajectory) of the operating point (by adding "helping" voltage we make it turn to the left and by subtracting "opposing" voltage we make it turn to the right). The general conclusion is:
The nonlinear true negative resistance IV curve is a modified linear ohmic resistance IV curve.
"Helping" or "opposing" negative resistors?Edit
Probably, you already ask yourself why in some cases we consider negative resistors (e.g., Nshaped) as opposing while in other cases we consider them as helping. The answer is simple  negative resistors can be both "opposing" and "helping"; this depends on the way of connection.
If we connect an Sshaped negative resistor in series to the load, it adds its voltage to the input voltage; thus it "helps" the input voltage source in its desire to pass the current through the load. Conversely, if we connect the negative resistor in parallel to the load, it "opposes" the input source.
If we connect an Nshaped negative resistor in parallel to the load, it adds its current to the input current; thus it "helps" the exciting input voltage source (that creates the current) in its desire to create the voltage across the load. Conversely, if we connect the negative resistor in series to the load, it "opposes" the input source.









Quantitative considerationsEdit
In the arrangements above, by using a "positive" resistor with resistance R and an amplifier with a gain of 2, we have created a negative resistor wtih equivalent negative resistance R. So, we can change the value of the negative resistance in two possible ways:
 by changing the value of the initial "positive" resistance R
 by changing the value of the amplifier gain К
The value of the negative resistance R_{NEG} can be defined by applying the Miller theorem:
For example, in the case of the Nshaped NR above where K = 2, R_{NEG} = R.
General properties of negative impedance elementsEdit
True negative versus "positive" impedance elementsEdit
 True negative impedance elements are sources injecting energy into circuits while the according "positive" impedance elements (resistors, capacitors and inductors) absorb energy from circuits.
 True negative impedance elements add as much energy to the input sources as it loses into "positive" impedance elements having the same impedance.
 True negative impedance elements are electronic circuits while "positive" impedance elements are real elements (components).
True negative elements versus sourcesEdit
 True negative impedance elements are dynamic sources while ordinary (constantvoltage and constantcurrent) sources are static.
 Elements with true negative impedance compensate the equivalnt impedance for each value of the input voltage/current; constant (static) sources compensate the equivalnt impedance for only one value of the input quantity (i.e., they have static negative impedance).
Current controlled versus voltage controlled negative impedance elementsEdit
 Both the true negative impedance elements are composed circuits consisting of two connected in series components: an internal "positive" impedance element and (the output of) an amplifier with gain of 2. The amplifier of a currentdriven negative impedance element amplifies the voltage drop across the internal "positive" impedance element; the amplifier of a voltagedriven negative impedance element amplifies the voltage drop across the terminals of the very negative impedance element.
 A currentdriven negative impedance element is a currentdriven voltage source consisting of a currenttovoltage converter that drives a voltage amplifier; a voltagedriven negative impedance element is a voltagedriven current source consisting of a voltage amplifier and a voltagetocurrent converter.
 Currentdriven negative impedance elements are connected in series while voltagedriven negative impedance elements are connected in parallel to "positive" impedance elements.
 Currentdriven negative impedance elements add so much voltage to the input voltage source as it would appear across an equivalent "positive" impedance element; voltagedriven negative impedance elements add so much current to the input current source as it would flow through an equivalent "positive" impedance element.
 A currentdriven true negative resistor with resistance R connected in series with a positive resistor with a total resistance R_{TOT} destroys, "eats", neutralizes Rpart of the total positive resistance; the result of this neutralization is zero resistance. A voltagedriven true negative resistor with resistance R connected in parallel to a positive resistor with a total resistance R_{TOT} neutralizes Rpart of the total positive resistance; the result of this neutralization is infinite resistance.
 The voltage across a currentdriven negative impedance element has an opposite polarity to the voltage drop across the initial "positive" resistor; so, it behaves as a negative impedance converter with voltage inversion (VNIC). The current flowing through a voltagedriven negative impedance element has an opposite direction to the current flowing through the initial "positive" resistor; so, it behaves as a negative impedance converter with current inversion (INIC).
 To operate in active mode, the positive resistance has to dominate over the negative resistance in circuits with currentcontrolled negative resistors while the negative resistance has to dominate over the "positive" resistance in circuits with voltagecontrolled negative resistors. Otherwise, these circuits will operate in bistable mode (acting as Schmitt trigger).
True negative versus differential negative resistorsEdit
 True negative resistors are electronic circuits while negative differential resistors can be elements (components) as well as circuits.
 Both the negative resistors are dynamic electronic elements (circuits).
 True negative resistors are dynamic electrical sources while negative differential resistors are just dynamic resistors that cannot be used independently; they may be used in combination with electrical sources to build true negative resistors.
General rules for using negative impedance elementsEdit
 Currentdriven negative impedance elements:
 Transform a "positive" impedance element into a currentdriven negative one by connecting in series an amplifier with gain of 2 that amplifies the voltage drop across the "positive" impedance element.
 Connect currentdriven negative impedance elements in series with "positive" impedance elements to decrease their impedance.
 Make the negative impedance equal to some part of the positive one to destroy it and to give zero impedance.
 Reserve some positive resistance to operate in an active mode.
 Voltagedriven negative impedance elements:
 Transform a "positive" impedance element into a voltagedriven negative one by connecting in series an amplifier with gain of 2 that amplifies the voltage drop across the terminals of the very negative impedance element.
 Connect voltagedriven negative impedance elements in parallel to positive impedance elements to decrease their impedance.
 Make the negative impedance equal to some part of the positive one to annihilate it and to give infinite impedance.
 Reserve some negative resistance to operate in an active mode.
NotesEdit
 ↑ In this area of electronics, negative impedance has the meaning of antiimpedance as it symbolizes the ability of special active electronic circuits (NICs) to produce energy in the same manner as the according passive electrical elements with "positive" impedance consume energy. In contrast to this "with regard to energy" viewpoint, the classical AC electricity uses a "with regard to time" viewpoint at the same terms. There, negative impedance and positive impedance terms symbolize the two opposite kinds of behavior (reactance) that reactive elements show through time  increasing voltage opposition (capacitors) or decreasing voltage opposition (inductors). As a result, in negative impedance circuitry both capacitors and inductors have positive impedance while in AC electricity capacitors have negative impedance but inductors have positive impedance.
 ↑ Our situation (according to the emulating setup below) is even more complicated since the woman will continuously increase expenses and the man will accordingly increase his gains.
 ↑ ^{a} ^{b} With the same success we can make the opposite transformation  to make the man spend money ($1000) instead earn them, to "invert" the man and to transmute him to a woman:). For this purpose, he has only to put in touch with a woman spending two times more money ($2000) as he earns ($1000). Thus the man is oversqueezed by the woman and we think he spends money ($1000  $2000 = $1000) instead to earn them. While the direct example above helped us to convert positive into negative resistance, this reverse example can help us to convert negative into positive resistance (of course, the former is more useful and frequently use). Actually, they represent the same resistance neutralization idea.
 ↑ To see the negative resistance regions in the IV curves of negative resistors, they have to operate in a linear mode (to be stable). For this purpose, we have to supply Sshaped negative resistors by a current source; otherwise, they will operate in the more exotic bistable mode.
 ↑ Actually, the magnitude of the current decreases.
 ↑ You can think of this combination as of a real source with voltage V_{H} and internal resistance R.
 ↑ The proportion in Fig. 10 is different for reasons of clarity.
 ↑ Note that if you consider the current and the voltage as signed quantities, when the current increases, the voltage decreases in this region (the classic negative resistance definition). But if you mean their magnitudes (the more natural viewpoint), when the current increases the voltage increases as well.
 ↑ If you have more sense of humor, you can see in this arrangement a more extended version of the funny analogy. You may think of the internal "resistor" Ri as of another woman (e.g., a wife); so, the three persons form a kind of a love triangle:) The second (the "other") woman (R) "copies" (makes the same expenses as) the first one (Ri). The man observes the second woman and earns two times more money than she spends; thus he actually compensates (works for) the both of them.
 ↑ Actually, the magnitude of the voltage decreases.
 ↑ Note again that if you consider the voltage and current as signed quantities, when the voltage increases the current decreases in this region (the classic negative resistance definition). But if you mean their magnitudes (the more natural viewpoint), when the voltage increases the current increases as well.
ReferencesEdit
 ↑ Mechkov C. (2006) A heuristic approach to teaching negative resistance phenomenon. COMPUTER SCIENCE' 2006, Istanbul, Turkey.
 ↑ ^{a} ^{b} Negative resistance process versus positive resistance process (an expressive picture) Invalid
<ref>
tag; name "Bearden" defined multiple times with different content  ↑ Negativeresistance load canceller helps drive heavy loads shows a typical voltagedriven negative resistance application.
See alsoEdit
Negative Impedance Converter
Investigating the linear mode of negative impedance converters with voltage inversion
Investigating the linear mode of negative impedance converters with current inversion
Negative resistance is based on an old Wikipedia article dedicated to the phenomenon.
Negative impedance converter considers NIC with current inversion (INIC).
External linksEdit
Negative Impedance: What It Is and How It Works by Tom Bearden.
Understanding negative impedance converters (VNIC)  reveals in three consecutive steps the basic idea behind negative impedance converters with voltage inversion (VNIC).
Negative Resistance Revived  condensed version of article originally published in Amateur Radio, November 1995.
Negativeresistance circuits  nice material from Answers.com.
Negative Resistance
Negative Resistor Tutorial by Adam Siembida
Handbook of operational amplifier active RC networks  a formal but wellwritten electronic book.
How I revealed the secret of parallel negative feedback circuits reveals the philosophy of circuits with parallel negative feedback.
How do we create a virtual ground? reveals the secret of the great circuit phenomenon.
Reinventing transimpedance amplifier is a story about the opamp currenttovoltage converter.
Opamp inverting summer is an animated Flash tutorial about the famous opamp summing circuit.
How do we build an opamp RC integrator?
Analog electronics 2004, Class 2: Elementary passive converters with voltage output
Forum discussionsEdit
Yahoo answers: Is it possible to have a resistor with negative resistance? Can an electronic circuit have negative impedance?
Edaboard: What is "negative impedance"?
All about circuits: Negative resistance
Physics forums: Negative resistance  the negative resistance material acts as a source
Physics stop: Negative resistance
ResearchGate: Why are there no practical applications of negative impedance?