Negative resistance

thumb|upright=1.3|[[Fluorescent lamp, a device with negative differential resistance.

While a positive resistance consumes power from current passing through it, a negative resistance produces power.

]]

Definitions

thumb|upright=0.8|An I–V curve, showing the difference between static resistance (inverse slope of line B) and differential resistance (inverse slope of line C) at a point (A).

The resistance between two terminals of an electrical device or circuit is determined by its current–voltage (I–V) curve (characteristic curve), giving the current <math>i</math> through it for any given voltage <math>v</math> across it. Most materials, including the ordinary (positive) resistances encountered in electrical circuits, obey Ohm's law; the current through them is proportional to the voltage over a wide range. In a nonlinear component the I–V curve is not a straight line, so it does not obey Ohm's law. which are equal for ohmic resistances:

thumb|The quadrants of the I–V plane, showing regions representing passive devices (white) and active devices (red)

Static resistance (also called chordal resistance, absolute resistance or just resistance) – This is the common definition of resistance; the voltage divided by the current: However this term is never used in practice, because the term "resistance" is only applied to passive components. Static resistance determines the power dissipation in a component.

Differential resistance (also called dynamic, It is measured in siemens (formerly mho) which is the conductance of a resistor with a resistance of one ohm. while a positive resistance will have a positive conductance. Therefore, from the passive sign convention above, conventional current (flow of positive charge) is through the device from the positive to the negative terminal, in the direction of the electric field E (decreasing potential). and these devices can be divided into two categories depending on whether they get their power from an internal source or from their port:

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Passive negative differential resistance devices (fig. 2 above): These are the most well-known type of "negative resistances"; passive two-terminal components whose intrinsic I–V curve has a downward "kink", causing the current to decrease with increasing voltage over a limited range. (fig. 3 above). Although the "static" or "absolute" resistance <math>R_\text{static}</math> of active devices (power sources) can be considered negative (see Negative static resistance section below) most ordinary power sources (AC or DC), such as batteries, generators, and (non positive feedback) amplifiers, have positive differential resistance (their source resistance). Therefore, these devices cannot function as one-port amplifiers or have the other capabilities of negative differential resistances.

List of negative resistance devices

Electronic components with negative differential resistance include these devices:

tunnel diode, resonant tunneling diode and other semiconductor diodes using the tunneling mechanism
Gunn diode and other diodes using the transferred electron mechanism
unijunction transistor (UJT)
Some magnetron tubes and other microwave vacuum tubes
maser
parametric amplifier

Electric discharges through gases also exhibit negative differential resistance, including these devices

electric arc
neon lamp A number of new experimental negative differential resistance materials and devices have been discovered in recent years. which requires a temperature difference to produce work. Therefore a negative static resistance must have some other source of power.

A point of some confusion is whether ordinary resistance ("static" or "absolute" resistance, <math>R_\text{static} = v / i</math>) can be negative. In electronics, the term "resistance" is customarily applied only to passive materials and components For a passive device to have <math>R_\text{static} = v/i\;<\;0</math> would violate either conservation of energy state that static resistance can never be negative.

thumb|upright=0.7|From [[Kirchhoff's voltage law|KVL, the static resistance of a power source (RS), such as a battery, is always equal to the negative of the static resistance of its load (RL). This property means if a large enough external voltage or current of either polarity is applied to it, its static resistance becomes positive and it consumes power will reverse the direction of current flow, making its static resistance positive so it consumes power. Similarly, applying a voltage to the negative impedance converter below greater than its power supply voltage Vs will cause the amplifier to saturate, also making its resistance positive.

Negative differential resistance

In a device or circuit with negative differential resistance (NDR), in some part of the I–V curve the current decreases as the voltage increases:

Voltage controlled negative resistance (VCNR, short-circuit stable, or "N" type): In this type the current is a single valued, continuous function of the voltage, but the voltage is a multivalued function of the current. lambda diode, Gunn diode, and dynatron oscillators. These can have more than two stable states, and are of interest for use in digital circuits to implement multivalued logic.

The tunnel diode circuit (see diagram) is an example. The tunnel diode TD has voltage controlled negative differential resistance. The NDR acts as a dependent AC current source of value Δi = Δv/r. Because the current and voltage are 180° out of phase, the instantaneous AC current Δi flows out of the terminal with positive AC voltage Δv. Therefore it adds to the AC source current ΔiS through the load R, increasing the output power. This means in the AC equivalent circuit (right), the instantaneous AC current Δi flows through the device in the direction of increasing AC potential Δv, as it would in a generator.

<math display="block">P_\text{AC} = \Delta v \Delta i = r_\text{diff}|\Delta i|^2 < 0 </math>

With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an amplifier, As long as the real component of the impedance is negative (phase angle between 90° and 270°),

The maximum AC output power is limited by size of the negative resistance region (<math>v_1,\; v_2,\; i_1,\; and\; i_2</math> in graphs above)

Reflection coefficient

thumb|General (AC) model of a negative resistance circuit: a negative differential resistance device <math>Z_\text{N}(j\omega)</math>, connected to an external circuit represented by <math>Z_\text{L}(j\omega)</math> which has positive resistance, <math>R_\text{L} > 0</math>. Both may have [[reactance (physics)|reactance ]]

The reason that the output signal can leave a negative resistance through the same port that the input signal enters is that from transmission line theory, the AC voltage or current at the terminals of a component can be divided into two oppositely moving waves, the incident wave <math>V_I</math>, which travels toward the device, and the reflected wave <math>V_R</math>, which travels away from the device. A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient <math>\Gamma </math>, the ratio of the reflected wave to the incident wave, is greater than one. An equilibrium point will be stable, so the circuit converges to it within some neighborhood of the point, if its poles are in the left half of the s plane (LHP), while a point is unstable, causing the circuit to oscillate or "latch up" (converge to another point), if its poles are on the jω axis or right half plane (RHP), respectively. In contrast, a linear circuit has a single equilibrium point that may be stable or unstable. The equilibrium points are determined by the DC bias circuit, and their stability is determined by the AC impedance <math>Z_L(j\omega)</math> of the external circuit.

However, because of the different shapes of the curves, the condition for stability is different for VCNR and CCNR types of negative resistance:

In a CCNR (S-type) negative resistance, the resistance function <math>R_N</math> is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:<math>Z_L(j\omega) + Z_N(j\omega) = 0</math>.

:For nonreactive circuits a sufficient condition for stability is that the total resistance is positive <math display="block">Z_L + Z_N = R_L + R_N = R_L - r > 0 </math> so the CCNR is stable for Alternatively, in high frequency circuit design, the values of <math>Z_L(j\omega)</math> for which the circuit is stable are determined by a graphical technique using "stability circles" on a Smith chart.

VCNRs are stable when <math>R_L < r</math>.
CCNRs are stable when <math>R_L > r</math>.
Unstable point (Line L2): When <math>R_L = r</math> the load line is tangent to the I–V curve. The total differential (AC) resistance of the circuit is zero (poles on the jω axis), so it is unstable and with a tuned circuit can oscillate. Linear oscillators operate at this point. Practical oscillators actually start in the unstable region below, with poles in the RHP, but as the amplitude increases the oscillations become nonlinear, and due to eventual passivity the negative resistance r decreases with increasing amplitude, so the oscillations stabilize at an amplitude where N-type (left), and S-type (center), generated by feedback amplifiers. These have negative differential resistance (red region) and produce power (grey region). Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive (black portion of curve), and above the supply voltage rails <math>\pm V_S</math> the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain <math>A\beta </math> (right).

[[File:Negative resistance by positive feedback.svg|thumb|upright=1.2|An example of an amplifier with positive feedback that has negative resistance at its input. The input current i is

so the input resistance is

If <math>A > 1 + R_1/R_\text{in} </math> it will have negative input resistance.]]

In addition to the passive devices with intrinsic negative differential resistance above, circuits with amplifying devices like transistors or op amps can have negative resistance at their ports. If <math>R_i</math> is the input resistance of the amplifier without feedback, <math>A</math> is the amplifier gain, and <math>\beta(j\omega)</math> is the transfer function of the feedback path, the input resistance with positive shunt feedback is

So if the loop gain <math>A\beta </math> is greater than one, <math>R_{if}</math> will be negative. The circuit acts like a "negative linear resistor" over a limited range, over its linear range (such amplifiers can also have more complicated negative resistance I–V curves that do not pass through the origin).

In circuit theory these are called "active resistors". Examples of circuits with this type of negative resistance are the negative impedance converter (NIC), gyrator, Deboo integrator, frequency dependent negative resistance (FDNR),

Feedback oscillators

If an LC circuit is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance <math>R_\text{if}</math> can cancel the positive loss resistance <math>r_\text{loss}</math> inherent in the tuned circuit. If <math>R_\text{if}\;=\;-r_\text{loss}</math> this will create in effect a tuned circuit with zero AC resistance (poles on the jω axis). This negative resistance model is an alternate way of analyzing feedback oscillator operation.

Q enhancement

A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so <math>|R_\text{if}|\;<\;r_\text{loss}</math>) will not oscillate, but the negative resistance will decrease the damping in the circuit (moving its poles toward the jω axis), increasing its Q factor so it has a narrower bandwidth and more selectivity. Q enhancement, also called regeneration, was first used in the regenerative radio receiver invented by Edwin Armstrong in 1912 It is widely used in active filters. Chua's circuit, a simple nonlinear circuit widely used as the standard example of a chaotic system, requires a nonlinear active resistor component, sometimes called Chua's diode. shown in the diagram. The two resistors <math>R_\text{1}</math> and the op amp constitute a negative feedback non-inverting amplifier with gain of 2. for example they were originally developed to cancel resistance in telephone cables, serving as repeaters. One application being researched is to create an active matching network which could match an antenna to a transmission line over a broad range of frequencies, rather than just a single frequency as with current networks. This would allow the creation of small compact antennas that would have broad bandwidth, In a negative resistance oscillator, a negative differential resistance device such as an IMPATT diode, Gunn diode, or microwave vacuum tube is connected across an electrical resonator such as an LC circuit, a quartz crystal, dielectric resonator or cavity resonator with a DC source to bias the device into its negative resistance region and provide power. A resonator such as an LC circuit is "almost" an oscillator; it can store oscillating electrical energy, but because all resonators have internal resistance or other losses, the oscillations are damped and decay to zero. and terahertz energy In addition, in modern high frequency oscillators, transistors are increasingly used as one-port negative resistance devices like diodes. At microwave frequencies, transistors with certain loads applied to one port can become unstable due to internal feedback and show negative resistance at the other port. When the power is turned on, electrical noise in the circuit provides a signal <math>i_0</math> to start spontaneous oscillations, which grow exponentially. However, the oscillations cannot grow forever; the nonlinearity of the diode eventually limits the amplitude.

At large amplitudes the circuit is nonlinear, so the linear analysis above does not strictly apply and differential resistance is undefined; but the circuit can be understood by considering <math>r</math> to be the "average" resistance over the cycle. As the amplitude of the sine wave exceeds the width of the negative resistance region and the voltage swing extends into regions of the curve with positive differential resistance, the average negative differential resistance <math>r</math> becomes smaller, and thus the total resistance <math>R\;-\;r</math> and the damping <math>\alpha</math> becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when <math>r\;=\;R</math>. In oscillators where <math>R</math> is close to <math>r</math>; just small enough to allow the oscillator to start, the voltage swing will be mostly limited to the linear portion of the I–V curve, the output waveform will be nearly sinusoidal and the frequency will be most stable. In circuits in which <math>R</math> is far below <math>r</math>, the swing extends further into the nonlinear part of the curve, the clipping distortion of the output sine wave is more severe,

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Negative resistance (voltage controlled) oscillator: Since VCNR ("N" type) devices require a low impedance bias and are stable for load impedances less than r, The circuit diagram is imagined to be divided by a "reference plane" (red) which separates the negative resistance part, the active device, from the positive resistance part, the resonant circuit and output load (right). The complex impedance of the negative resistance part <math>Z_N = R_N(I, \omega) + jX_N(I, \omega) </math> depends on frequency ω but is also nonlinear, in general declining with the amplitude of the AC oscillation current I; while the resonator part <math>Z_L = R_L(\omega) + jX_L(\omega) </math> is linear, depending only on frequency. A circulator is a nonreciprocal solid-state component with three ports (connectors) which transfers a signal applied to one port to the next in only one direction, port 1 to port 2, 2 to 3, and 3 to 1. In the reflection amplifier diagram the input signal is applied to port 1, a biased VCNR negative resistance diode N is attached through a filter F to port 2, and the output circuit is attached to port 3. The input signal is passed from port 1 to the diode at port 2, but the outgoing "reflected" amplified signal from the diode is routed to port 3, so there is little coupling from output to input. The characteristic impedance <math>Z_0</math> of the input and output transmission lines, usually 50Ω, is matched to the port impedance of the circulator. The purpose of the filter F is to present the correct impedance to the diode to set the gain. At radio frequencies NR diodes are not pure resistive loads and have reactance, so a second purpose of the filter is to cancel the diode reactance with a conjugate reactance to prevent standing waves.

<math display="block">G_\text{P} = {P_\text{out} \over P_\text{in = {V_R^2 \over V_I^2} = |\Gamma|^2</math>

<math display="block">|\Gamma|^2 = \left|{Z_N - Z_1 \over Z_N + Z_1}\right|^2</math>

<math display="block">|\Gamma|^2 = \left|{R_N + jX_N - (R_1 + jX_1)\over R_N + jX_N + R_1 + jX_1}\right|^2</math>

<math>R_\text{N}</math> is the negative resistance of the diode −r. Assuming the filter is matched to the diode so <math>X_1 = -X_N</math> The negative resistance here is implied were one to consider the neuron a typical Hodgkin–Huxley style circuit model.

History

Negative resistance was first recognized during investigations of electric arcs, which were used for lighting during the 19th century. In 1881 Alfred Niaudet had observed that the voltage across arc electrodes decreased temporarily as the arc current increased, but many researchers thought this was a secondary effect due to temperature. Beginning in 1895 Hertha Ayrton, extending her husband William's research with a series of meticulous experiments measuring the I–V curve of arcs, established that the curve had regions of negative slope, igniting controversy. Frith and Rodgers in 1896 with the support of the Ayrtons In the same year Elihu Thomson built a negative resistance oscillator by connecting an LC circuit to the electrodes of an arc, perhaps the first example of an electronic oscillator. William Duddell, a student of Ayrton at London Central Technical College, brought Thomson's arc oscillator to public attention. inventing the Poulsen arc radio transmitter, which was widely used until the 1920s. In a vacuum tube when electrons strike the plate electrode they can knock additional electrons out of the surface into the tube. This represents a current away from the plate, reducing the plate current. He was also one of the first to report negative capacitance and inductance. and G. W. Pickard. They noticed that when junctions were biased with a DC voltage to improve their sensitivity as radio detectors, they would sometimes break into spontaneous oscillations. He used these to build solid-state amplifiers, oscillators, and amplifying and regenerative radio receivers, 25 years before the invention of the transistor. Later he even built a superheterodyne receiver. Because they have lower parasitic capacitance than vacuum tubes due to their small junction size, diodes can function at higher frequencies, and tunnel diode oscillators proved able to produce power at microwave frequencies, above the range of ordinary vacuum tube oscillators. Its invention set off a search for other negative resistance semiconductor devices for use as microwave oscillators, resulting in the discovery of the IMPATT diode, Gunn diode, TRAPATT diode, and others. In 1969 Kurokawa derived conditions for stability in negative resistance circuits.