Negative resistance
Negative resistance (or negative differential resistance (NDR) or differential negative resistance (DNR)) is a property of electrical circuit elements composed of certain materials in which, over certain voltage ranges, current is a decreasing function of voltage. This range of voltages is known as a negative resistance region.
Some writers prefer to reserve the term negative resistance for situations in which the negatively-sloping portion of the load line passes through the origin, so that a positive absolute value of voltage is associated with a negative absolute value of current. Such a circuit must contain an energy source, and can be used as a form of amplifier. However, the use of the term negative resistance to encompass negative differential resistance is more common.
Absolute negative resistances without an external energy source cannot exist as they would violate the law of conservation of energy.

# Concepts

## Non-electrical domain
An analogy to the negative resistance phenomenon may be observed in many well-known everyday situations where something (a being, a device etc.) containing an additional power source affects (as a proportional reaction of some disturbance) something else containing the main power source . The additional source may help or impede the main one in three degrees (under-, exact- or over-). Negative resistance represents the last degree when the additional source "over-helps" or "over-impedes" the main one. Examples: fluid (a pneumatic amplifier over-helps the driver when he presses the brake pedal), mechanical ("clicking" mechanisms) etc.
In conclusion, the phenomenon observed is a process of injecting an additional excess power to an existing power source proportionally to some disturbance.

## Electrical domain
According to the basic idea above, electrical negative resistance should be a process of injecting an additional electrical power to an existing power source proportionally to some electrical disturbance (current or voltage). Respectively, a negative resistor should act as a proportional additional power source.

### Voltage source acting as a negative resistor
The nature of electrical negative resistance is clarified below (Fig. 1) by comparing an ordinary "positive" resistor R with a negative resistor -R (click on the pictures to view full-size images). For this purpose, two equivalent electrical circuits are used, in which the two components are connected in series with the loads so that the same current passes through them.
As a result, a voltage drop VR = R.I appears across the "positive" resistor R (Fig. 1a) and the same voltage VH = VR = R.I appears across the negative resistor -R (Fig. 1b). However, the resistor R sucks the voltage V from the circuit (it is a voltage drop) while the negative resistor -R adds the voltage V into the circuit ). Therefore, a resistor acts as a current-to-voltage drop converter while a negative resistor acts as a current-to-voltage converter. The element named "resistor" is really a resistor while the "negative resistor" is actually a voltage source, whose voltage is proportional to the current passing through it. If the additional voltage source is connected in the opposite direction to the input voltage source (Fig. 1c), it will act as an "over-impeding" voltage source.
A negative resistor can be a voltage source, whose voltage is proportional to the current passing through it (a current-controlled voltage source).

### Current source acting as a negative resistor
Dual circuits may be assembled (Fig. 2) where the components are connected in parallel to the loads so that the same voltage is applied across them (click on the pictures to view full-size images).
As a result, a current IR = VL/R passes through the resistor R (Fig. 2a) and the same current IH/ = VL/R passes through the negative resistor -R (Fig. 2b). However, the resistor R conducts the current away from the circuit while the negative resistor -R injects the current into the circuit. The element named "resistor" is really a resistor while here the "negative resistor" is actually a current source, whose current is proportional to the voltage across it. If the additional current source is connected in the opposite direction (Fig. 2c) versus the input current source, it will act as an "over-impeding" current source.
A negative resistor can be also a current source, whose current is proportional to the voltage across it (a voltage-controlled current source).

# Realization
The IV curve of an ohmic (static) resistor is sloped from left to right. The only way to slope it from right to left in a limited region is to "dynamize" sufficiently the ohmic resistor in this region. In this way, the problem of obtaining a negative resistance is reduced to the problem of creating a dynamic resistance .
In electrical circuits, static resistance is the ratio of the voltage across a circuit element to the current through it. However, the ratio of the voltage to the current may vary with either voltage or current. The ratio of the change in voltage to the change in current is known as dynamic resistance.
It is more correct to say that a circuit element has a negative differential resistance region than to say that it exhibits negative resistance because even in this region the static resistance of the circuit element is positive, while it is the slope of the resistance curve which is negative.
There are two techniques for obtaining dynamic (negative) resistance - by varying the resistance  and by varying the voltage . The first produces negative differential resistance, while the second gives absolute negative resistance.

## Resistance variation
This is historically the first and more natural way of creating negative resistance. In electronics, there are a few two-terminal electronic components having negative differential resistance. Some of them have an S-shaped IV curve while other components have an N-shaped IV curve. Electronically-active conductive polymers such as Melanin can also show marked negative differential resistance.

### S-shaped constant-voltage dynamic resistance
By dynamically decreasing the resistance of an ordinary ohmic resistor , three degrees of dynamic resistance may be obtained (Fig. 3a): decreased (section 1-2), zeroed (section 2-3) and S-negative differential resistance (section 3-4). As the section 2-3 represents a voltage-stable dynamic resistor (for example, a zener diode), a conclusion may be derived:
An S-shaped negative differential resistor is actually an "over-acting" voltage-stable dynamic resistor.
An example of an electronic component exhibiting a negative differential resistance region is the medium within a gas discharge lamp which, as current increases, ionizes to a greater extent, thereby carrying more current.  If such a lamp were allowed to draw power without limit, it would instantly burn itself out.  Limiting the possible current is one of the roles of the ballast in a fluorescent lamp.

### N-shaped constant-current dynamic resistance
Dually, by dynamically increasing the resistance of an ordinary ohmic resistor (fig. 3b), three other degrees of dynamic resistance may be obtained: increased (section 1-2), infinite (section 2-3) and N-negative differential resistance (section 3-4).  As the section 2-3 represents a current-stable dynamic resistor (for example, a barreter or the collector-emitter part of a transistor), another conclusion may be derived:
An N-shaped negative differential resistor is actually an "over-acting" current-stable dynamic resistor.
An example of an electronic component exhibiting an N-shaped negative differential resistance region is the tunnel diode.  Such a device, when biased into its negative differential resistance region, acts as an amplifier.  See also Gunn diode.
Negative differential resistor is an "over-acting" dynamic resistor (a dynamic resistor with extremely varying resistance).
In compliance with the law of conservation of energy, a plot of the negative differential resistance region of a passive component cannot pass through the origin.

## Absolute negative resistance
The negative differential resistor is not a true negative resistor as it does not contain a source; it is just a part of a true negative resistor. In order to get an absolute negative resistor, an additional constant voltage source has to be connected in series:
Actually, the combination of the two components constitutes the varying voltage source needed. By applying this approach, a tunnel diode amplifier is built (see applications).

## Varying voltage source
In op-amp circuitry, there are perfect voltage-controlled voltage sources - operational amplifiers. That is why, it is preferred to make dynamic resistors rather by varying the voltage  than by varying the resistance. Following this approach, excellent "circuit" true negative resistors are built by connecting in series two circuit elements (Fig. 4): a steady "positive" (ohmic) resistor and an "over-acting" varying voltage source (an amplifier):
It is a paradox that a negative resistor -R contains a "positive" resistor R. However, this idea is used in many cases of human routine where, in order to begin creating something positive (-R), they first need something negative (R). Then, they produce two (or many) times more positive quantity (-2R) in order not only to compensate the negative quantity (R) but also to get the desired positive quantity (-2R + R = -R).
According to this idea, the current-sensing resistor R converts the current into proportional voltage drop VR = R.I (Fig. 4a); then, the "over-helping" varying voltage source VH (the amplifier) doubles this voltage drop thus producing an additional voltage VH = -2R.I. Half the voltage (VR) compensates the voltage drop VR; the rest half (VR) adds to the voltage of the excitation input voltage source VIN. As a result, the whole circuit acts as a current-controlled voltage source producing a voltage -VR. Since the "positive" resistance R is converted into negative one -R, these kinds of circuits are named negative impedance converters (NIC).
If the additional voltage source is reversed (Fig. 4b), it subtracts its voltage from the voltage of the input voltage source VIN. In this case, VI "over-impedes" VIN.

# Operation modes
Negative resistors are two-terminal elements, which inputs and outputs are the same - the voltage applied across (or the current passed through) the two terminals of the negative resistor controls the resistance/current/voltage between the same two terminals. Therefore, a feedback exists naturally in the negative resistance circuits. The kind of this feedback (negative or positive) determines the circuit behavior.

## Linear (analog) operation
Applying a dominating negative feedback. By applying only a negative feedback, at the best case two kinds of dynamical resistances can be obtained: a dynamical zero resistance (having a vertical IV curve - ) and an infinite resistance (having a horizontal IV curve). Further, in order to get a negative resistance (to fold the IV curves), an additional positive feedback has to be added. However, in order to ensure a linear mode (a stability), the negative feedback has to dominate over the positive one. An example of a linear negative resistance device using the both kinds of feedback is a negative impedance converter.
Applying a depressed positive feedback. The same effect might be achieved by using only a small enough positive feedback (e.g., by using a non-inverting amplifier with small gain A or a feedback attenuator with large ratio B). In these cases, the loop gain is kept A.B < 1.
Many circuit topologies are capable of producing absolute negative resistance (which requires that an energy source be included). The simplest case requires a non-inverting amplifier with voltage gain greater than one. If a resistor R is connected from input to output, the input current, i_i, for a given input voltage v_i is:
i_i = \frac{v_i - v_o}{R}
Where v_o is the output voltage. This assumes an ideal amplifier with infinite input impedance and zero output impedance. If the voltage gain, A_v, of the amplifier is defined as:
A_v = \frac{v_o}{v_i}
The input resistance, R_i is:
R_i = \frac{v_i}{i_i} = \frac{R}{1-A_v}
The input resistance is negative for values of A_v > 1.
In the case of a non-ideal amplifier, negative resistance is still possible as long as the amplifier input impedance is sufficiently high. The net resistance is reduced to:
R_i = \frac{1}{\frac{1}{Z_{i}} + \frac{1-A_v}{R + Z_{o}}}
where Z_i is the amplifier input impedance and Z_o is the amplifier output impedance.

# Applications

## Linear operation

### Compensating resistive losses
If a voltage source drives a distant low-resistive load through a long line (a thin wire) having significant resistance Rl, a problem arises - a voltage drop VRl = I.Rl appears across the line resistance Rl. The local voltages along the line decrease gradually from left to right; the voltage distribution along the line is shown on Fig. 5 where each local voltage drop is represented by a local bar with corresponding height (for simplicity, the envelope of the voltage diagram is drawn). Actually, the line and the load resistance constitute a voltage divider with ratio K = RL/(Rl + RL). As a result, the output voltage VOUT drops (VOUT = VIN - VRl).
Putting the line resistance into a negative feedback loop is a classic perfect solution. However, it needs an additional (third) voltage sense wire. Another remedy may be derived from human routine when they have to compensate some harmful but inaccessible quantity (R). In these cases, they first create a copy of the harmful quantity (R). Then, they produce two times bigger useful quantity (-2R), in order to compensate both the copy and the original quantity (R + R - 2R = 0).
In terms of electronics, that means to connect in series a negative resistor with resistance -Rl consisting of a resistor and a voltage source (Fig. 5). The "copy" resistor R converts the flowing current I into proportional "mirror" voltage drop VR = Rl.I, which drives the compensating voltage source BH (an amplifier with K = 2). As a result, the doubling voltage source BH produces an additional "helping" voltage VH= -2.Rl.I. A portion of this voltage compensates the voltage drop across the "copy" resistor R; the rest part compensates the voltage drop across the line resistance Rl. As a result, the voltage source with negative internal resistance raises its output voltage so that the load voltage VL stays equal to the input voltage - VOUT = VIN.
This idea may be implemented by an op-amp circuit of a negative impedance converter   (NIC). In this arrangement, the op-amp compares its output voltage with the "mirror" voltage drop across the "copy" resistance instead with the "original" line resistance (it "supposes" that the two resistances are equal). However, if the "original" resistance varies, the op-amp will be misled and an error will appear. Another disadvantage: an NIC consumes two time more power PNIC = 2VRl.I than needed since it has to compensate two voltage drops VRl. Only, a negative resistance solution has an advantage - it has only two terminals; therefore, it needs only two wires (it is a 1-port amplifier).
In contrast to this circuit solution, the op-amp of a transimpedance amplifier compares its output voltage directly with the "original" voltage drop across the "harmful" resistance . In this way, it compensates exactly the resistance even when it varies (for example, because of temperature or length variations). As a result, a transimpedance amplifier consumes power only of PTA = VRl.I since it has to compensate only one voltage drop VRl. However, for this purpose the op-amp needs an additional voltage sense wire, in order to "observe" the virtual ground by its inverting input. Unfortunately, in many cases, this point is inaccessible.
Example 1: Compensating the load resistance. In nature, real sources (for example, human beings) have a limited power. Therefore, if they are loaded (for example, by a weight), they droop. A similar problem exists in electronics (electricity) when imperfect voltage sources are loaded. For example, the simplest varying voltage source (Fig. 6) consists of a steady voltage source VIN and a potentiometer P (a voltage divider r1-r2). If there is no load connected, this real voltage source works well - VOUT = r2/(r1 + r2). However, when a load RL is connected, it "sucks" a current IL and the output voltage VL drops. The classic solution is to connect an op-amp voltage follower (a unity-gain amplifier acting as a buffer amplifier) before the load, in order to decrease the current IL (to increase the load resistance RL). Unfortunately, this remedy introduces some errors inherent for this circuit .
In mechanics, there is another powerful idea that is referred to as anti-weight or anti-load (Fig. 7). It is widely used in the lift systems, cranes etc. In electronics, this exotic solution is implemented by a voltage-driven negative resistor (a negative load); it is connected in parallel to the "positive" load (Fig. 6), in order to "help" the imperfect voltage source ,.
The compensating voltage source BH (a non-inverting amplifier with K = 2) produces two times higher "helping" voltage VH= 2VL. It makes a current IH = (VH - VL)/R = (2VL - VL)/RL = VL/RL = IL flow through the load. In this way, the whole load current IL is provided by the "helping" current source IH (the negative resistor -RL) instead by the real voltage source. The load does not consume any energy from the input source since it is supplied completely by the "helping" source. In order to  put this circuit into practice, they use usually a negative impedance converter acting as a voltage-driven negative resistor , .
There is a striking resemblance between the mechanical and electrical versions, which helps understanding the phenomenon. According to this intuitive viewpoint, the load "pulls" the point A down toward the ground while the resistor R "pulls" the point A up toward the voltage VH. 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 between the point B and point A since the whole right part of the circuit (RL, R and VH) behaves as a load with infinite internal resistance. This phenomenon is referred to as 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).
Example 2: Howland current source. In the circuit above (fig. 6), the parallel connected negative resistor with resistance -RL gives the whole load current IL = VL/RL. In another implementation of this powerful idea - the famous circuit of Howland current source  - the negative resistor adds only a "correcting" supplementary current IS = VL/R, in order to keep a constant current through the load: IL = IIN + IS = (V/R - VL/R) + VL/R = V/R.
Example 3: Deboo integrator. Another eccentric circuit called Deboo integrator exploits exactly the same idea , : a negative resistor "helps" an imperfect current source driving a capacitor (a Howland current source drives a capacitor with a constant current).
Example 4: Compensating the losses in the LC tank. Electrical negative resistance is often used to design oscillators. The main part of classic oscillators - tuned circuit - is formed by a parallel connection of an inductor and a capacitor. If energised, it oscillates at its natural resonance frequency but the intensity of oscillation decays as a function of time because of energy losses in the circuit . To maintain continuous oscillation, sufficient energy must be continuously fed into the tuned circuit to balance the energy lost. For this purpose, a negative resistor is usually connected in parallel to the tuned circuit. As a result, the energy lost in the tuned circuit is supplied by the negative resistance circuit. One might imagine the negative resistance as cancelling the positive loss resistance and, in effect, making the shunt loss resistance look like infinity.
Many topologies are possible, such as the Dynatron oscillator, Colpitts oscillator, Hartley oscillator, Wien bridge oscillator, and some types of relaxation oscillators. Negative resistance characteristics of Gunn diodes are often used in microwave frequencies as well.

### Amplification
Basic idea. An amplification is nothing else than controlled attenuation. According to this paradoxical idea, an amplifier consists of two components: a controlled regulating element and a power source. In electronics, the classic 3-terminal regulating element (tube, transistor etc.) acts as an electrically controlled resistor with separate input and output ports. The voltage (current) applied across (through) the input port controls the resistance between the two terminals of the output port.
The odd 2-terminal regulating element (for example, a tunnel diode) acts as an electrically controlled resistor, which input and output are the same. The voltage (current) applied across (through) the two terminals of the element controls the resistance between the same two terminals. In order to do that, the 2-terminal regulating element is actually an "over-acting" dynamic resistor (that is, a negative resistor).
Tunnel diode amplifier. In order to build such a 1-port amplifier, four components have to be connected in series (Fig. 8): a constant-voltage power supply V, an input voltage source VIN, a "positive" resistor R and a negative differential resistor NDR (for example, a tunnel diode). Actually, the two resistors constitute a "dynamic" voltage divider supplied by a varying composed voltage source (V + VIN). When the input voltage varies slightly, the negative differential resistor reacts vigorously to this "intervention"; it changes considerably its resistance according to the input voltage, which makes the voltage divider change noticeably its ratio. As a result, the voltage drops across the "positive" and negative resistors vary considerably; therefore, some of them may be used as an output voltage. In this arrangement, the differential negative resistor is not an amplifier; it is just a part of an amplifier (the differential negative resistor is just a 2-terminal regulating element). The combination of the differential negative resistor acting as a regulating element and the power supply constitutes true amplifier:

### RF antenna design
Another concept of negative resistance exists in the domain of radio frequency antenna design.  This is also known as negative impedance.  It is not uncommon for an antenna containing multiple driven elements to exhibit apparent negative impedance in one or more of the driven elements.

## Bi-stable operation

### Non-electrical examples
There are many mechanical systems that exhibit ranges of negative differential resistance. In fact, this is a common design element in systems that are designed to have "detents" or a "positive action" or a "click." A popular example is the well-known pen clicker. Good examples are also the keys on a computer keyboard and on a computer mouse, taking the key position and upward force to be analogous to voltage and current, respectively. As a key is pressed downward, it initially presents a firm and increasing upward force. Beyond a critical point, a zone is entered in which the upward force decreases, which feels like a "sudden" yielding. This is often referred to as a "collapse action" mechanism. There are several keyboard technology that give such collapse action, such as buckling spring switches. A general characteristic of negative resistance systems is that by driving them "firmly" it is possible to traverse the negative resistance region continuously (linear applications), but bistable switching action occurs if the system is driven "loosely" (bi-stable applications).

### Electrical examples
Switching circuits. Negative resistance is also useful in certain switching and comparator circuits, such as the op-amp non-inverting Schmitt trigger. Specialized diodes, such as the step recovery diode also exhibit negative resistance. In this case, a very sharp pulse can be generated that produces a broad spectrum of harmonics. This can be used as a frequency multiplier at gigahertz frequencies. This is sometimes used in certain frequency synthesiser designs.
Memory circuits.

# Components with negative differential resistance
- plasma (plasma channel)
- electric arc*
- fluorescent lamp
- tunnel diode
- IMPATT diode
- unijunction transistor
- resonant tunneling diode
- resonant tunneling transistor

# Circuits with negative differential resistance
- Lambda diode
- negative impedance converter
- generalized impedance converter
- frequency-dependent negative resistor.

# History
Examples of the use of negative resistances in analogue computing can be found in the works of Gabriel Kron.  While a scientist for General Electric, Kron 
used negative resistors (circuits like those described above) for the US Navy's "Network Analyser" in the 1930s. 
For example, 
refers to the use of active negative resistances with network analysers, and also shows how these can be replaced by inductors and capacitors in AC simulations.

# Scientific sensations: Deborah Chung's 'apparent negative resistance'
In July 1998, Deborah Chung and Shoukai Wang of the University of Buffalo presented the results of an experiment that showed an apparent absolute negative resistance in bare carbon fibers held together by pressure.
In the experiment, two bundles of carbon fibers are arranged in a cross shape, with the ends of each bundle shorted with copper foil and silver paint (at A, B, C, and D in the image).  A current is driven through one branch, and a voltage is measured across the other branch.  In the paper, the voltage divided by current is referred to as an "apparent resistance".  (A real electrical resistance requires both the current and voltage to be measured at the same points.)
The paper describes how the apparent contact resistance of the interface changes from positive to negative when the fibers are compressed.  The current-voltage characteristic of the measured "negative resistance" is then a straight line of negative slope through the origin.  The apparent negative resistance was also observed in metal wires (silver-coated copper), but was not observed for a single fiber crossing another single fiber.  The paper claims that this phenomenon is useful because the forward flow and backflow of electrons in the same piece of material can be reproducibly controlled by external forces.
It was initially reported on July 9, 1998 by the University as a breakthrough in room temperature superconductor research, in the press release Superconduction At Room Temperature: Negative Electrical Resistance Seen In Carbon Composites, claiming that the discoveries "have enabled carbon-fiber materials to superconduct at room temperature", because of measurements of "zero apparent resistance" at certain pressures.  This was quickly seized upon by the free energy community as a working example of a device that supplies energy with no apparent source, claiming it to be a true, absolute negative resistance, and was reported in the popular press as a breakthrough.  The original press release was later pulled from UB's website, on July 16, 1998, and replaced with one which stated "her findings do not indicate that the combination is itself a superconductor."
Chung's paper itself says:
True negative resistance in the former sense is not possible due to energy consideration. However, apparent negative resistance in the former sense is reported here. ... Although the negative resistance reported here is apparent rather than true, its mechanism resembles that of true negative resistance (which actually does not occur due to energetics) in that the electrons flow in the unexpected direction relative to the applied current/voltage.
It never claims that the device is a source of energy.