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 in applied voltage causes a proportional increase in current in accordance with Ohm's law, resulting in a positive resistance. Under certain conditions, negative resistance 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 v / i {\displaystyle v/i} , and differential resistance, the ratio of a change in voltage to the resulting change in current Δ v / Δ i {\displaystyle \Delta v/\Delta i} . The term negative resistance means negative differential resistance (NDR), Δ v / Δ i < 0 {\displaystyle \Delta v/\Delta i<0} . 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.
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 i {\displaystyle i} through it for any given voltage v {\displaystyle v} 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 v {\displaystyle v} 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:
Static resistance (also called chordal resistance, absolute resistance or just resistance) – This is the common definition of resistance; the voltage divided by the current: R s t a t i c = v i . {\displaystyle R_{\mathrm {static} }={\frac {v}{i}}.} It is the inverse slope of the line (chord) from the origin through the point on the I–V curve. In a power source, like a battery or electric generator, positive current flows out of the positive voltage terminal, opposite to the direction of current in a resistor, so from the passive sign convention i {\displaystyle i} and v {\displaystyle v} have opposite signs, representing points lying in the 2nd or 4th quadrant of the I–V plane (diagram right). Thus power sources formally have negative static resistance ( R static < 0 ) . {\displaystyle R_{\text{static}}<0).} 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. Passive devices, which consume electric power, have positive static resistance; while active devices, which produce electric power, do not. Differential resistance (also called dynamic, or incremental resistance) – This is the derivative of the voltage with respect to the current; the ratio of a small change in voltage to the corresponding change in current, the inverse slope of the I–V curve at a point: r d i f f = d v d i . {\displaystyle r_{\mathrm {diff} }={\frac {dv}{di}}.} Differential resistance is only relevant to time-varying currents. Points on the curve where the slope is negative (declining to the right), meaning an increase in voltage causes a decrease in current, have negative differential resistance ( r diff < 0 {\displaystyle r_{\text{diff}}<0} ). Devices of this type can amplify signals, and are what is usually meant by the term "negative resistance". 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
Static conductance G s t a t i c = 1 R s t a t i c = i v {\displaystyle G_{\mathrm {static} }={\frac {1}{R_{\mathrm {static} }}}={\frac {i}{v}}}
Differential conductance g d i f f = 1 r d i f f = d i d v {\displaystyle g_{\mathrm {diff} }={\frac {1}{r_{\mathrm {diff} }}}={\frac {di}{dv}}}
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 r diff {\displaystyle r_{\text{diff}}} , the static resistance R static {\displaystyle R_{\text{static}}} , 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 r diff < 0 {\displaystyle r_{\text{diff}}<0} . 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:
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. The I–V curve, including the negative resistance region, lies in the 1st and 3rd quadrant of the plane so the device has positive static resistance. Examples are gas-discharge tubes, tunnel diodes, and Gunn diodes. These devices have no internal power source and in general work by converting external DC power from their port to time varying (AC) power, so they require a DC bias current applied to the port in addition to the signal. To add to the confusion, some authors call these "active" devices, since they can amplify. This category also includes a few three-terminal devices, such as the unijunction transistor. They are covered in the Negative differential resistance section below.
Active negative differential resistance devices (fig. 4): Circuits can be designed in which a positive voltage applied to the terminals will cause a proportional "negative" current; a current out of the positive terminal, the opposite of an ordinary resistor, over a limited range, Unlike in the above devices, the downward-sloping region of the I–V curve passes through the origin, so it lies in the 2nd and 4th quadrants of the plane, meaning the device sources power. Amplifying devices like transistors and op-amps with positive feedback can have this type of negative resistance, and are used in feedback oscillators and active filters. Since these circuits produce net power from their port, they must have an internal DC power source, or else a separate connection to an external power supply. In circuit theory this is called an "active resistor". Although this type is sometimes referred to as "linear", "absolute", "ideal", or "pure" negative resistance to distinguish it from "passive" negative differential resistances, in electronics it is more often simply called positive feedback or regeneration. These are covered in the Active resistors section below.
Occasionally ordinary power sources are referred to as "negative resistances" (fig. 3 above). Although the "static" or "absolute" resistance R static {\displaystyle R_{\text{static}}} 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 IMPATT diode, TRAPATT diode and other diodes using the impact ionization mechanism Some NPN transistors with E-C reverse biased, known as negistor unijunction transistor (UJT) thyristors triode and tetrode vacuum tubes operating in the dynatron mode 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 thyratron tubes neon lamp fluorescent lamp other gas discharge tubes 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, R static = v / i {\displaystyle R_{\text{static}}=v/i} ) 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 R static < 0 {\displaystyle R_{\text{static}}<0} as shown by Joule's law P = i 2 R static {\displaystyle P=i^{2}R_{\text{static}}} . A passive device consumes electric power, so from the passive sign convention P ≥ 0 {\displaystyle P\geq 0} . Therefore, from Joule's law R static ≥ 0 {\displaystyle R_{\text{static}}\geq 0} . In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance. For a passive device to have R static = v / i < 0 {\displaystyle R_{\text{static}}=v/i\;<\;0} 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
R s t a t i c = v i < 0 {\displaystyle R_{\mathrm {static} }={\frac {v}{i}}<0}
This can also be proved from Joule's law
P = i v = i 2 R s t a t i c {\displaystyle P=iv=i^{2}R_{\mathrm {static} }}
This shows that power can flow out of a device into the circuit ( P < 0 {\displaystyle P<0} ) if and only if R static < 0 {\displaystyle R_{\text{static}}<0} . 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
∃ V , I : | v | > V or | i | > I ⇒ R s t a t i c = v / i ≥ 0 {\displaystyle \exists V,I:|v|>V{\text{ or }}|i|>I\Rightarrow R_{\mathrm {static} }=v/i\geq 0}
where P max = I V {\displaystyle P_{\max }=IV} is the maximum power the device can produce. 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 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:
r d i f f = d v d i < 0 {\displaystyle r_{\mathrm {diff} }={\frac {dv}{di}}<0}
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:
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. In the most common type there is only one negative resistance region, and the graph is a curve shaped generally like the letter "N". As the voltage is increased, the current increases (positive resistance) until it reaches a maximum (i1), then decreases in the region of negative resistance to a minimum (i2), then increases again. Devices with this type of negative resistance include the tunnel diode, resonant tunneling diode, lambda diode, Gunn diode, and dynatron oscillators. Current controlled negative resistance (CCNR, open-circuit stable, or "S" type): In this type, the dual of the VCNR, the voltage is a single valued function of the current, but the current is a multivalued function of the voltage. In the most common type, with one negative resistance region, the graph is a curve shaped like the letter "S". Devices with this type of negative resistance include the IMPATT diode, UJT, SCRs and other thyristors, electric arc, and gas discharge tubes . 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):
PVR = i 1 / i 2 {\displaystyle {\text{PVR}}=i_{1}/i_{2}}
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 V b {\displaystyle V_{b}} 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 Δ v / Δ i = − r {\displaystyle \Delta v/\Delta i=-r} . For stability R {\displaystyle R} must be less than r {\displaystyle r} . Using the formula for a voltage divider, the AC output voltage is
v o = − r R − r v i = r r − R v i {\displaystyle v_{o}={\frac {-r}{R-r}}v_{i}={\frac {r}{r-R}}v_{i}} so the voltage gain is G v = r r − R {\displaystyle G_{v}={\frac {r}{r-R}}}
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 r − R {\displaystyle r-R} is less than the resistance of the diode alone r {\displaystyle r} so the AC output voltage v o {\displaystyle v_{o}} is greater than the input v i {\displaystyle v_{i}} . The voltage gain G v {\displaystyle G_{v}} is greater than one, and increases without limit as R {\displaystyle R} approaches r {\displaystyle r} .
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 ( V b i a s , I b i a s {\displaystyle V_{bias},\;I_{bias}} ) and an AC component ( Δ v , Δ i {\displaystyle \Delta v,\;\Delta i} ).
v ( t ) = V bias + Δ v ( t ) {\displaystyle v(t)=V_{\text{bias}}+\Delta v(t)}
i ( t ) = I bias + Δ i ( t ) {\displaystyle i(t)=I_{\text{bias}}+\Delta i(t)}
Since a positive change in voltage Δ v {\displaystyle \Delta v} causes a negative change in current Δ i {\displaystyle \Delta i} , 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 Δi flows through the device in the direction of increasing AC potential Δv, 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.
P AC = Δ v Δ i = r diff | Δ i | 2 < 0 {\displaystyle P_{\text{AC}}=\Delta v\Delta i=r_{\text{diff}}|\Delta i|^{2}<0}
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
| P AC | ≤ I bias V bias {\displaystyle |P_{\text{AC}}|\leq I_{\text{bias}}V_{\text{bias}}}
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 ( v 1 , v 2 , i 1 , a n d i 2 {\displaystyle v_{1},\;v_{2},\;i_{1},\;and\;i_{2}} in graphs above)
P A C ( r m s ) ≤ 1 8 ( v 2 − v 1 ) ( i 1 − i 2 ) {\displaystyle P_{AC(rms)}\leq {\frac {1}{8}}(v_{2}-v_{1})(i_{1}-i_{2})}
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 V I {\displaystyle V_{I}} , which travels toward the device, and the reflected wave V R {\displaystyle V_{R}} , which travels away from the device. A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient Γ {\displaystyle \Gamma } , the ratio of the reflected wave to the incident wave, is greater than one.
| Γ | ≡ | V R V I | > 1 {\displaystyle |\Gamma |\equiv \left|{\frac {V_{R}}{V_{I}}}\right|>1} where Γ ≡ Z N − Z L Z N + Z L {\displaystyle \Gamma \equiv {\frac {Z_{N}-Z_{L}}{Z_{N}+Z_{L}}}}
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, Z N ( j ω ) = R N + j X N {\displaystyle Z_{N}(j\omega )=R_{N}+jX_{N}} , and the impedance of the circuit attached to it, Z L ( j ω ) = R L + j X L {\displaystyle Z_{L}(j\omega )\,=\,R_{L}\,+\,jX_{L}} . If R N < 0 {\displaystyle R_{N}<0} and R L > 0 {\displaystyle R_{L}>0} then | Γ | > 0 {\displaystyle |\Gamma |>0} 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 | Γ | = 1 {\displaystyle |\Gamma |=1} , 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 Z L ( j ω ) {\displaystyle Z_{L}(j\omega )} 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 R N {\displaystyle R_{N}} is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation: Z L ( j ω ) + Z N ( j ω ) = 0 {\displaystyle Z_{L}(j\omega )+Z_{N}(j\omega )=0} . For nonreactive circuits ( X L = X N = 0 {\displaystyle X_{L}=X_{N}=0} ) a sufficient condition for stability is that the total resistance is positive Z L + Z N = R L + R N = R L − r > 0 {\displaystyle Z_{L}+Z_{N}=R_{L}+R_{N}=R_{L}-r>0} so the CCNR is stable for
Since CCNRs are stable with no load at all, they are called "open circuit stable". In a VCNR (N-type) negative resistance, the conductance function G N = 1 / R N {\displaystyle G_{N}=1/R_{N}} is single-valued. Therefore, stability is determined by the poles of the admittance equation Y L ( j ω ) + Y N ( j ω ) = 0 {\displaystyle Y_{L}(j\omega )+Y_{N}(j\omega )=0} . For this reason the VCNR is sometimes referred to as a negative conductance.As above, for nonreactive circuits a sufficient condition for stability is that the total conductance in the circuit is positive Y L