Negative resistance

In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.
This is in contrast to an ordinary resistor in which an increase of applied voltage causes a proportional increase in current due to Ohm's law, resulting in a positive resistance. While a positive resistance consumes power from current passing through it, a negative resistance produces power. Under certain conditions it can increase the power of an electrical signal, amplifying it.
Negative resistance is an uncommon property which occurs in a few nonlinear electronic components. In a nonlinear device, two types of resistance can be defined: 'static' or 'absolute resistance', the ratio of voltage to current formula_1, and "differential resistance", the ratio of a change in voltage to the resulting change in current formula_2. The term negative resistance means negative differential resistance (NDR), formula_3. In general, a negative differential resistance is a two-terminal component which can amplify, converting DC power applied to its terminals to AC output power to amplify an AC signal applied to the same terminals. They are used in electronic oscillators and amplifiers, particularly at microwave frequencies. Most microwave energy is produced with negative differential resistance devices. They can also have hysteresis and be bistable, and so are used in switching and memory circuits. Examples of devices with negative differential resistance are tunnel diodes, Gunn diodes, and gas discharge tubes such as neon lamps, and fluorescent lights. In addition, circuits containing amplifying devices such as transistors and op amps with positive feedback can have negative differential resistance. These are used in oscillators and active filters.
Because they are nonlinear, negative resistance devices have a more complicated behavior than the positive "ohmic" resistances usually encountered in electric circuits. Unlike most positive resistances, negative resistance varies depending on the voltage or current applied to the device, and negative resistance devices can only have negative resistance over a limited portion of their voltage or current range. Therefore, there is no real "negative resistor" analogous to a positive resistor, which has a constant negative resistance over an arbitrarily wide range of current.
Definitions.
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 formula_4 through it for any given voltage formula_5 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. So the "I–V" curve of an ohmic resistance is a straight line through the origin with positive slope. The resistance is the ratio of voltage to current, the inverse slope of the line (in "I–V" graphs where the voltage formula_5 is the independent variable) and is constant.
Negative resistance occurs in a few nonlinear (nonohmic) devices. In a nonlinear component the "I–V" curve is not a straight line, so it does not obey Ohm's law. Resistance can still be defined, but the resistance is not constant; it varies with the voltage or current through the device. The resistance of such a nonlinear device can be defined in two ways, which are equal for ohmic resistances:
Negative resistance, like positive resistance, is measured in ohms.
Conductance is the reciprocal of resistance. It is measured in siemens (formerly "mho") which is the conductance of a resistor with a resistance of one ohm. Each type of resistance defined above has a corresponding conductance
It can be seen that the conductance has the same sign as its corresponding resistance: a negative resistance will have a negative conductance while a positive resistance will have a positive conductance.
Operation.
One way in which the different types of resistance can be distinguished is in the directions of current and electric power between a circuit and an electronic component. The illustrations below, with a rectangle representing the component attached to a circuit, summarize how the different types work:
Types and terminology.
In an electronic device, the differential resistance formula_14, the static resistance formula_15, or both, can be negative, so there are three categories of devices "(fig. 2–4 above, and table)" which could be called "negative resistances".
The term "negative resistance" almost always means negative "differential" resistance Negative differential resistance devices have unique capabilities: they can act as "one-port amplifiers", increasing the power of a time-varying signal applied to their port (terminals), or excite oscillations in a tuned circuit to make an oscillator. They can also have hysteresis. It is not possible for a device to have negative differential resistance without a power source, and these devices can be divided into two categories depending on whether they get their power from an internal source or from their port:
Occasionally ordinary power sources are referred to as "negative resistances" (fig. 3 above). Although the "static" or "absolute" resistance formula_15 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:
Electric discharges through gases also exhibit negative differential resistance, including these devices
In addition, active circuits with negative differential resistance can also be built with amplifying devices like transistors and op amps, using feedback. A number of new experimental negative differential resistance materials and devices have been discovered in recent years. The physical processes which cause negative resistance are diverse, and each type of device has its own negative resistance characteristics, specified by its current–voltage curve.
Negative static or "absolute" resistance.
A point of some confusion is whether ordinary resistance ("static" or "absolute" resistance, formula_17) can be negative. In electronics, the term "resistance" is customarily applied only to passive materials and components – such as wires, resistors and diodes. These cannot have formula_18 as shown by Joule's law formula_19. A passive device consumes electric power, so from the passive sign convention formula_20. Therefore, from Joule's law formula_21. In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance. For a passive device to have formula_22 would violate either conservation of energy or the second law of thermodynamics, "(diagram)". Therefore, some authors state that static resistance can never be negative.
However it is easily shown that the ratio of voltage to current v/i at the terminals of any power source (AC or DC) is negative. For electric power (potential energy) to flow out of a device into the circuit, charge must flow through the device in the direction of increasing potential energy, conventional current (positive charge) must move from the negative to the positive terminal. So the direction of the instantaneous current is "out" of the positive terminal. This is opposite to the direction of current in a passive device defined by the passive sign convention so the current and voltage have opposite signs, and their ratio is negative
This can also be proved from Joule's law
This shows that power can flow out of a device into the circuit if and only if formula_18. Whether or not this quantity is referred to as "resistance" when negative is a matter of convention. The absolute resistance of power sources is negative, but this is not to be regarded as "resistance" in the same sense as positive resistances. The negative static resistance of a power source is a rather abstract and not very useful quantity, because it varies with the load. Due to conservation of energy it is always simply equal to the negative of the static resistance of the attached circuit "(right)".
Work must be done on the charges by some source of energy in the device, to make them move toward the positive terminal against the electric field, so conservation of energy requires that negative static resistances have a source of power. The power may come from an internal source which converts some other form of energy to electric power as in a battery or generator, or from a separate connection to an external power supply circuit as in an amplifying device like a transistor, vacuum tube, or op amp.
Eventual passivity.
A circuit cannot have negative static resistance (be active) over an infinite voltage or current range, because it would have to be able to produce infinite power. Any active circuit or device with a finite power source is "eventually passive". 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
Therefore, the ends of the "I–V" curve will eventually turn and enter the 1st and 3rd quadrants. Thus the range of the curve having negative static resistance is limited, confined to a region around the origin. For example, applying a voltage to a generator or battery "(graph, above)" greater than its open-circuit voltage 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 "V"s 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:
The "I–V" curve is nonmonotonic (having peaks and troughs) with regions of negative slope representing negative differential resistance.
Passive negative differential resistances have positive "static" resistance; they consume net power. Therefore, the "I–V" curve is confined to the 1st and 3rd quadrants of the graph, and passes through the origin. This requirement means (excluding some asymptotic cases) that the region(s) of negative resistance must be limited, and surrounded by regions of positive resistance, and cannot include the origin.
Types.
Negative differential resistances can be classified into two types:
Most devices have a single negative resistance region. However devices with multiple separate negative resistance regions can also be fabricated. These can have more than two stable states, and are of interest for use in digital circuits to implement multivalued logic.
An intrinsic parameter used to compare different devices is the "peak-to-valley current ratio" (PVR), the ratio of the current at the top of the negative resistance region to the current at the bottom "(see graphs, above)":
The larger this is, the larger the potential AC output for a given DC bias current, and therefore the greater the efficiency
Amplification.
A negative differential resistance device can amplify an AC signal applied to it if the signal is biased with a DC voltage or current to lie within the negative resistance region of its "I–V" curve.
The tunnel diode circuit "(see diagram)" is an example. The tunnel diode "TD" has voltage controlled negative differential resistance. The battery formula_30 adds a constant voltage (bias) across the diode so it operates in its negative resistance range, and provides power to amplify the signal. Suppose the negative resistance at the bias point is formula_31. For stability formula_32 must be less than formula_33. Using the formula for a voltage divider, the AC output voltage is
In a normal voltage divider, the resistance of each branch is less than the resistance of the whole, so the output voltage is less than the input. Here, due to the negative resistance, the total AC resistance formula_36 is less than the resistance of the diode alone formula_33 so the AC output voltage formula_38 is greater than the input formula_39. The voltage gain formula_40 is greater than one, and increases without limit as formula_32 approaches formula_33.
Explanation of power gain.
The diagrams illustrate how a biased negative differential resistance device can increase the power of a signal applied to it, amplifying it, although it only has two terminals. Due to the superposition principle the voltage and current at the device's terminals can be divided into a DC bias component and an AC component .
Since a positive change in voltage formula_45 causes a "negative" change in current formula_46, the AC current and voltage in the device are 180° out of phase. This means in the AC equivalent circuit "(right)", the instantaneous AC current Δ"i" flows through the device in the direction of "increasing" AC potential Δ"v", as it would in a generator. Therefore, the AC power dissipation is "negative"; AC power is produced by the device and flows into the external circuit.
With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an amplifier, or excite oscillations in a resonant circuit to make an oscillator. Unlike in a two port amplifying device such as a transistor or op amp, the amplified signal leaves the device through the same two terminals (port) as the input signal enters.
In a passive device, the AC power produced comes from the input DC bias current, the device absorbs DC power, some of which is converted to AC power by the nonlinearity of the device, amplifying the applied signal. Therefore, the output power is limited by the bias power
The negative differential resistance region cannot include the origin, because it would then be able to amplify a signal with no applied DC bias current, producing AC power with no power input. The device also dissipates some power as heat, equal to the difference between the DC power in and the AC power out.
The device may also have reactance and therefore the phase difference between current and voltage may differ from 180° and may vary with frequency. As long as the real component of the impedance is negative (phase angle between 90° and 270°), the device will have negative resistance and can amplify.
The maximum AC output power is limited by size of the negative resistance region (formula_49 in graphs above)
Reflection coefficient.
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" formula_51, which travels toward the device, and the "reflected wave" formula_52, which travels away from the device. A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient formula_53, the ratio of the reflected wave to the incident wave, is greater than one.
The "reflected" (output) signal has larger amplitude than the incident; the device has "reflection gain". The reflection coefficient is determined by the AC impedance of the negative resistance device, formula_56, and the impedance of the circuit attached to it, formula_57. If formula_58 and formula_59 then formula_60 and the device will amplify. On the Smith chart, a graphical aide widely used in the design of high frequency circuits, negative differential resistance corresponds to points outside the unit circle formula_61, the boundary of the conventional chart, so special "expanded" charts must be used.
Stability conditions.
Because it is nonlinear, a circuit with negative differential resistance can have multiple equilibrium points (possible DC operating points), which lie on the "I–V" curve. 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 formula_62 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:
For general negative resistance circuits with reactance, the stability must be determined by standard tests like the Nyquist stability criterion. Alternatively, in high frequency circuit design, the values of formula_70 for which the circuit is stable are determined by a graphical technique using "stability circles" on a Smith chart.
Operating regions and applications.
For simple nonreactive negative resistance devices with formula_71 and formula_72 the different operating regions of the device can be illustrated by load lines on the "I–V" curve "(see graphs)".
The DC load line (DCL) is a straight line determined by the DC bias circuit, with equation
where formula_74 is the DC bias supply voltage and R is the resistance of the supply. The possible DC operating point(s) (Q points) occur where the DC load line intersects the "I–V" curve. For stability
The AC load line ("L"1 − "L"3) is a straight line through the Q point whose slope is the differential (AC) resistance formula_75 facing the device. Increasing formula_75 rotates the load line counterclockwise. The circuit operates in one of three possible regions "(see diagrams)", depending on formula_75.
Active resistors – negative resistance from feedback.
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. The input or output impedance of an amplifier with enough positive feedback applied to it can be negative. If formula_84 is the input resistance of the amplifier without feedback, formula_85 is the amplifier gain, and formula_86 is the transfer function of the feedback path, the input resistance with positive shunt feedback is
So if the loop gain formula_88 is greater than one, formula_89 will be negative. The circuit acts like a "negative linear resistor" over a limited range, with "I–V" curve having a straight line segment through the origin with negative slope "(see graphs)". It has both negative differential resistance and is active
and thus obeys Ohm's law as if it had a negative value of resistance "−R", 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". Applying a voltage across the terminals causes a proportional current "out" of the positive terminal, the opposite of an ordinary resistor. For example, connecting a battery to the terminals would cause the battery to charge rather than discharge.
Considered as one-port devices, these circuits function similarly to the passive negative differential resistance components above, and like them can be used to make one-port amplifiers and oscillators with the advantages that:
The "I–V" curve can have voltage-controlled ("N" type) or current-controlled ("S" type) negative resistance, depending on whether the feedback loop is connected in "shunt" or "series".
Negative reactances "(below)" can also be created, so feedback circuits can be used to create "active" linear circuit elements, resistors, capacitors, and inductors, with negative values. They are widely used in active filters because they can create transfer functions that cannot be realized with positive circuit elements. Examples of circuits with this type of negative resistance are the negative impedance converter (NIC), gyrator, Deboo integrator, frequency dependent negative resistance (FDNR), and generalized immittance converter (GIC).
Feedback oscillators.
If an LC circuit is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance formula_91 can cancel the positive loss resistance formula_92 inherent in the tuned circuit. If formula_93 this will create in effect a tuned circuit with zero AC resistance (poles on the "jω" axis). Spontaneous oscillation will be excited in the tuned circuit at its resonant frequency, sustained by the power from the amplifier. This is how feedback oscillators such as Hartley or Colpitts oscillators work. This negative resistance model is an alternate way of analyzing feedback oscillator operation. "All" linear oscillator circuits have negative resistance although in most feedback oscillators the tuned circuit is an integral part of the feedback network, so the circuit does not have negative resistance at all frequencies but only near the oscillation frequency.
Q enhancement.
A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so formula_94) 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 and later in "Q multipliers". It is widely used in active filters. For example, RF integrated circuits use "integrated inductors" to save space, consisting of a spiral conductor fabricated on chip. These have high losses and low Q, so to create high Q tuned circuits their Q is increased by applying negative resistance.
Chaotic circuits.
Circuits which exhibit chaotic behavior can be considered quasi-periodic or nonperiodic oscillators, and like all oscillators require a negative resistance in the circuit to provide power. 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. This is usually synthesized using a negative impedance converter circuit.
Negative impedance converter.
A common example of an "active resistance" circuit is the negative impedance converter (NIC) shown in the diagram. The two resistors formula_95 and the op amp constitute a negative feedback non-inverting amplifier with gain of 2. The output voltage of the op-amp is
So if a voltage formula_5 is applied to the input, the same voltage is applied "backwards" across formula_98, causing current to flow through it out of the input. The current is
So the input impedance to the circuit is
The circuit converts the impedance formula_98 to its negative. If formula_98 is a resistor of value formula_103, within the linear range of the op amp formula_104 the input impedance acts like a linear "negative resistor" of value formula_105. The input port of the circuit is connected into another circuit as if it was a component. An NIC can cancel undesired positive resistance in another circuit, for example they were originally developed to cancel resistance in telephone cables, serving as repeaters.
Negative capacitance and inductance.
By replacing formula_98 in the above circuit with a capacitor , negative capacitances and inductances can also be synthesized. A negative capacitance will have an "I–V" relation and an impedance formula_107 of
where formula_109. Applying a positive current to a negative capacitance will cause it to "discharge"; its voltage will "decrease". Similarly, a negative inductance will have an "I–V" characteristic and impedance formula_110 of
A circuit having negative capacitance or inductance can be used to cancel unwanted positive capacitance or inductance in another circuit. NIC circuits were used to cancel reactance on telephone cables.
There is also another way of looking at them. In a negative capacitance the current will be 180° opposite in phase to the current in a positive capacitance. Instead of leading the voltage by 90° it will lag the voltage by 90°, as in an inductor. Therefore, a negative capacitance acts like an inductance in which the impedance has a reverse dependence on frequency ω; decreasing instead of increasing like a real inductance Similarly a negative inductance acts like a capacitance that has an impedance which increases with frequency. Negative capacitances and inductances are "non-Foster" circuits which violate Foster's reactance theorem. 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, exceeding the Chu–Harrington limit.
Oscillators.
Negative differential resistance devices are widely used to make electronic oscillators. 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. The negative resistance cancels the positive resistance of the resonator, creating in effect a lossless resonator, in which spontaneous continuous oscillations occur at the resonator's resonant frequency.
Uses.
Negative resistance oscillators are mainly used at high frequencies in the microwave range or above, since feedback oscillators function poorly at these frequencies. Microwave diodes are used in low- to medium-power oscillators for applications such as radar speed guns, and local oscillators for satellite receivers. They are a widely used source of microwave energy, and virtually the only solid-state source of millimeter wave and terahertz energy Negative resistance microwave vacuum tubes such as magnetrons produce higher power outputs, in such applications as radar transmitters and microwave ovens. Lower frequency relaxation oscillators can be made with UJTs and gas-discharge lamps such as neon lamps.
The negative resistance oscillator model is not limited to one-port devices like diodes but can also be applied to feedback oscillator circuits with two port devices such as transistors and tubes. 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. So high frequency transistor oscillators are designed by applying a reactive load to one port to give the transistor negative resistance, and connecting the other port across a resonator to make a negative resistance oscillator as described below.
Gunn diode oscillator.
The common Gunn diode oscillator "(circuit diagrams)" illustrates how negative resistance oscillators work. The diode "D" has voltage controlled ("N" type) negative resistance and the voltage source formula_112 biases it into its negative resistance region where its differential resistance is formula_113. The choke "RFC" prevents AC current from flowing through the bias source. formula_103 is the equivalent resistance due to damping and losses in the series tuned circuit formula_115, plus any load resistance. Analyzing the AC circuit with Kirchhoff's Voltage Law gives a differential equation for formula_116, the AC current
Solving this equation gives a solution of the form
This shows that the current through the circuit, formula_116, varies with time about the DC Q point, formula_121. When started from a nonzero initial current formula_122 the current oscillates sinusoidally at the resonant frequency ω of the tuned circuit, with amplitude either constant, increasing, or decreasing exponentially, depending on the value of α. Whether the circuit can sustain steady oscillations depends on the balance between formula_103 and formula_124, the positive and negative resistance in the circuit:
Practical oscillators are designed in region (3) above, with net negative resistance, to get oscillations started. A widely used rule of thumb is to make formula_129. When the power is turned on, electrical noise in the circuit provides a signal formula_130 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 formula_124 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 formula_124 becomes smaller, and thus the total resistance formula_133 and the damping formula_134 becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when formula_135.
Gunn diodes have negative resistance in the range −5 to −25 ohms. In oscillators where formula_103 is close to formula_124; 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 formula_103 is far below formula_124, the swing extends further into the nonlinear part of the curve, the clipping distortion of the output sine wave is more severe, and the frequency will be increasingly dependent on the supply voltage.
Types of circuit.
Negative resistance oscillator circuits can be divided into two types, which are used with the two types of negative differential resistance – voltage controlled (VCNR), and current controlled (CCNR)
Conditions for oscillation.
Most oscillators are more complicated than the Gunn diode example, since both the active device and the load may have reactance ("X") as well as resistance ("R"). Modern negative resistance oscillators are designed by a frequency domain technique due to K. Kurokawa. 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 formula_140 depends on frequency "ω" but is also nonlinear, in general declining with the amplitude of the AC oscillation current "I"; while the resonator part formula_141 is linear, depending only on frequency. The circuit equation is formula_142 so it will only oscillate (have nonzero "I") at the frequency "ω" and amplitude "I" for which the total impedance formula_143 is zero. This means the magnitude of the negative and positive resistances must be equal, and the reactances must be conjugate
For steady-state oscillation the equal sign applies. During startup the inequality applies, because the circuit must have excess negative resistance for oscillations to start.
Alternately, the condition for oscillation can be expressed using the reflection coefficient. The voltage waveform at the reference plane can be divided into a component "V"1 travelling toward the negative resistance device and a component "V"2 travelling in the opposite direction, toward the resonator part. The reflection coefficient of the active device formula_146 is greater than one, while that of the resonator part formula_147 is less than one. During operation the waves are reflected back and forth in a round trip so the circuit will oscillate only if
As above, the equality gives the condition for steady oscillation, while the inequality is required during startup to provide excess negative resistance. The above conditions are analogous to the Barkhausen criterion for feedback oscillators; they are necessary but not sufficient, so there are some circuits that satisfy the equations but do not oscillate. Kurokawa also derived more complicated sufficient conditions, which are often used instead.
Amplifiers.
Negative differential resistance devices such as Gunn and IMPATT diodes are also used to make amplifiers, particularly at microwave frequencies, but not as commonly as oscillators. Because negative resistance devices have only one "port" (two terminals), unlike two-port devices such as transistors, the outgoing amplified signal has to leave the device by the same terminals as the incoming signal enters it. Without some way of separating the two signals, a negative resistance amplifier is "bilateral"; it amplifies in both directions, so it suffers from sensitivity to load impedance and feedback problems. To separate the input and output signals, many negative resistance amplifiers use nonreciprocal devices such as isolators and directional couplers.
Reflection amplifier.
One widely used circuit is the "reflection amplifier" in which the separation is accomplished by a "circulator". 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 formula_149 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.
The filter has only reactive components and so does not absorb any power itself, so power is passed between the diode and the ports without loss. The input signal power to the diode is
The output power from the diode is
So the power gain formula_152 of the amplifier is the square of the reflection coefficient
formula_156 is the negative resistance of the diode −r. Assuming the filter is matched to the diode so formula_157 then the gain is
The VCNR reflection amplifier above is stable for formula_159. while a CCNR amplifier is stable for formula_160. It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as formula_95 approaches the point of oscillation at formula_124. This is a characteristic of all NR amplifiers, contrasting with the behavior of two-port amplifiers, which generally have limited gain but are often unconditionally stable. In practice the gain is limited by the backward "leakage" coupling between circulator ports.
Masers and parametric amplifiers are extremely low noise NR amplifiers that are also implemented as reflection amplifiers; they are used in applications like radio telescopes.
Switching circuits.
Negative differential resistance devices are also used in switching circuits in which the device operates nonlinearly, changing abruptly from one state to another, with hysteresis. The advantage of using a negative resistance device is that a relaxation oscillator, flip-flop or memory cell can be built with a single active device, whereas the standard logic circuit for these functions, the Eccles-Jordan multivibrator, requires two active devices (transistors). Three switching circuits built with negative resistances are
Other applications.
Neuronal models.
Some instances of neurons display regions of negative slope conductances (RNSC) in voltage-clamp experiments. 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. The term "negative resistance" was applied by some to this effect, but the term was controversial because it was known that the resistance of a passive device could not be negative. 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 introduced the concept of "differential" resistance, "dv/di", and it was slowly accepted that arcs had negative differential resistance. In recognition of her research, Hertha Ayrton became the first woman voted for induction into the Institute of Electrical Engineers.
Arc transmitters.
George Francis FitzGerald first realized in 1892 that if the damping resistance in a resonant circuit could be made zero or negative, it would produce continuous oscillations. 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. Due to its negative resistance, the current through an arc was unstable, and arc lights would often produce hissing, humming, or even howling noises. In 1899, investigating this effect, Duddell connected an LC circuit across an arc and the negative resistance excited oscillations in the tuned circuit, producing a musical tone from the arc. To demonstrate his invention Duddell wired several tuned circuits to an arc and played a tune on it. Duddell's "singing arc" oscillator was limited to audio frequencies. However, in 1903 Danish engineers Valdemar Poulsen and P. O. Pederson increased the frequency into the radio range by operating the arc in a hydrogen atmosphere in a magnetic field, inventing the Poulsen arc radio transmitter, which was widely used until the 1920s.
Vacuum tubes.
By the early 20th century, although the physical causes of negative resistance were not understood, engineers knew it could generate oscillations and had begun to apply it. Heinrich Barkhausen in 1907 showed that oscillators must have negative resistance. Ernst Ruhmer and Adolf Pieper discovered that mercury vapor lamps could produce oscillations, and by 1912 AT&T had used them to build amplifying repeaters for telephone lines.
In 1918 Albert Hull at GE discovered that vacuum tubes could have negative resistance in parts of their operating ranges, due to a phenomenon called secondary emission. 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. Under certain conditions increasing the plate voltage causes a "decrease" in plate current. By connecting an LC circuit to the tube Hull created an oscillator, the dynatron oscillator. Other negative resistance tube oscillators followed, such as the biotron invented by John Scott-Taggart in 1919, and the magnetron invented by Hull in 1920.
The negative impedance converter originated from work by Marius Latour around 1920. He was also one of the first to report negative capacitance and inductance. A decade later, vacuum tube NICs were developed as telephone line repeaters at Bell Labs by George Crisson and others, which made transcontinental telephone service possible. Transistor NICs, pioneered by Linvill in 1953, initiated a great increase in interest in NICs and many new circuits and applications developed.
Solid state devices.
Negative differential resistance in semiconductors was observed around 1909 in the first point-contact junction diodes, called cat's whisker detectors, by researchers such as William Henry Eccles 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. However the effect was not pursued.
The first person to exploit negative resistance diodes practically was Russian radio researcher Oleg Losev, who in 1922 discovered negative differential resistance in biased zincite (zinc oxide) point contact junctions. 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. However his achievements were overlooked because of the success of vacuum tube technology. After ten years he abandoned research into this technology (dubbed "Crystodyne" by Hugo Gernsback), and it was forgotten.
The first widely used solid-state negative resistance device was the tunnel diode, invented in 1957 by Japanese physicist Leo Esaki. 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. Currently negative differential resistance diode oscillators are the most widely used sources of microwave energy, and many new negative resistance devices have been discovered in recent decades.