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.^[10] Any active circuit or device with a finite power source is "eventually passive".^[49][74][75] 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^[74]

$${\displaystyle \exists V,I:|v|>V{\text{ or }}|i|>I\Rightarrow R_{\mathrm {static} }=v/i\geq 0}$$

where ${\displaystyle P_{\max }=IV}$ 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.^[75] Thus the range of the curve having negative static resistance is limited,^[10] 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^[76] 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.

Types

Negative differential resistances can be classified into two types:^[16][77]

• Voltage controlled negative resistance (VCNR, short-circuit stable,^{[77][78][note 2]} 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.^[77] 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,^[54] resonant tunneling diode,^[79] lambda diode, Gunn diode,^[80] and dynatron oscillators.^[43][59]
• Current controlled negative resistance (CCNR, open-circuit stable,^{[77][78][note 2]} 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.^[77] 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,^[80] UJT,^[54] SCRs and other thyristors,^[54] electric arc, and gas discharge tubes .^[43]

Most devices have a single negative resistance region. However devices with multiple separate negative resistance regions can also be fabricated.^[67][81] These can have more than two stable states, and are of interest for use in digital circuits to implement multivalued logic.^[67][81]

An intrinsic parameter used to compare different devices is the peak-to-valley current ratio (PVR),^[67] the ratio of the current at the top of the negative resistance region to the current at the bottom (see graphs, above):

$${\displaystyle {\text{PVR}}=i_{1}/i_{2}}$$

The larger this is, the larger the potential AC output for a given DC bias current, and therefore the greater the efficiency

Amplification

Tunnel diode amplifier circuit. Since ${\displaystyle r>R}$ the total resistance, the sum of the two resistances in series (${\displaystyle R-r}$) is negative, so an increase in input voltage will cause a decrease in current. The circuit operating point is the intersection between the diode curve (black) and the resistor load line ${\displaystyle R}$ (blue).^[82] A small increase in input voltage, ${\displaystyle v_{i}}$ (green) moving the load line to the right, causes a large decrease in current through the diode and thus a large increase in the voltage across the diode ${\displaystyle v_{o}}$.

A negative differential resistance device can amplify an AC signal applied to it^[11][13] if the signal is biased with a DC voltage or current to lie within the negative resistance region of its I–V curve.^[7][12]

The tunnel diode circuit (see diagram) is an example.^[82] The tunnel diode TD has voltage controlled negative differential resistance.^[54] The battery ${\displaystyle V_{b}}$ 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 ${\displaystyle \Delta v/\Delta i=-r}$. For stability ${\displaystyle R}$ must be less than ${\displaystyle r}$.^[36] Using the formula for a voltage divider, the AC output voltage is^[82]

$${\displaystyle v_{o}={\frac {-r}{R-r}}v_{i}={\frac {r}{r-R}}v_{i}}$$

so the voltage gain is

$${\displaystyle G_{v}={\frac {r}{r-R}}}$$

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 ${\displaystyle r-R}$ is less than the resistance of the diode alone ${\displaystyle r}$ so the AC output voltage ${\displaystyle v_{o}}$ is greater than the input ${\displaystyle v_{i}}$. The voltage gain ${\displaystyle G_{v}}$ is greater than one, and increases without limit as ${\displaystyle R}$ approaches ${\displaystyle r}$.

Explanation of power gain

An AC voltage applied to a biased NDR. Since the change in current and voltage have opposite signs (shown by colors), the AC power dissipation ΔvΔi is negative, the device produces AC power rather than consuming it.

AC equivalent circuit of NDR attached to external circuit.^[83] 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 Δi_S through the load R, increasing the output power.^[83]

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 (${\displaystyle V_{bias},\;I_{bias}}$) and an AC component (${\displaystyle \Delta v,\;\Delta i}$).

$${\displaystyle v(t)=V_{\text{bias}}+\Delta v(t)}$$

$${\displaystyle i(t)=I_{\text{bias}}+\Delta i(t)}$$

Since a positive change in voltage ${\displaystyle \Delta v}$ causes a negative change in current ${\displaystyle \Delta i}$, the AC current and voltage in the device are 180° out of phase.^{[7][57][36][84]} 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.^[36] Therefore, the AC power dissipation is negative; AC power is produced by the device and flows into the external circuit.^[85]

$${\displaystyle P_{\text{AC}}=\Delta v\Delta i=r_{\text{diff}}|\Delta i|^{2}<0}$$

With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an amplifier,^[36] 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.^[86]

In a passive device, the AC power produced comes from the input DC bias current,^[21] 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^[21]

$${\displaystyle |P_{\text{AC}}|\leq I_{\text{bias}}V_{\text{bias}}}$$

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.^[3][10][21] 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.^[8][42][87] As long as the real component of the impedance is negative (phase angle between 90° and 270°),^[84] the device will have negative resistance and can amplify.^[87][88]

The maximum AC output power is limited by size of the negative resistance region (${\displaystyle v_{1},\;v_{2},\;i_{1},\;and\;i_{2}}$ in graphs above)^[21][89]

$${\displaystyle P_{AC(rms)}\leq {\frac {1}{8}}(v_{2}-v_{1})(i_{1}-i_{2})}$$

Reflection coefficient

General (AC) model of a negative resistance circuit: a negative differential resistance device ${\displaystyle Z_{\text{N}}(j\omega )}$, connected to an external circuit represented by ${\displaystyle Z_{\text{L}}(j\omega )}$ which has positive resistance, ${\displaystyle R_{\text{L}}>0}$. Both may have reactance (${\displaystyle X_{\text{L}},\;X_{\text{N}}}$)

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 ${\displaystyle V_{I}}$, which travels toward the device, and the reflected wave ${\displaystyle V_{R}}$, which travels away from the device.^[90] A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient ${\displaystyle \Gamma }$, the ratio of the reflected wave to the incident wave, is greater than one.^[17][85]

$${\displaystyle |\Gamma |\equiv \left|{\frac {V_{R}}{V_{I}}}\right|>1}$$

where

$${\displaystyle \Gamma \equiv {\frac {Z_{N}-Z_{L}}{Z_{N}+Z_{L}}}}$$

The "reflected" (output) signal has larger amplitude than the incident; the device has "reflection gain".^[17] The reflection coefficient is determined by the AC impedance of the negative resistance device, ${\displaystyle Z_{N}(j\omega )=R_{N}+jX_{N}}$, and the impedance of the circuit attached to it, ${\displaystyle Z_{L}(j\omega )\,=\,R_{L}\,+\,jX_{L}}$.^[85] If ${\displaystyle R_{N}<0}$ and ${\displaystyle R_{L}>0}$ then ${\displaystyle |\Gamma |>0}$ 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 ${\displaystyle |\Gamma |=1}$, the boundary of the conventional chart, so special "expanded" charts must be used.^[17][91]

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.^[92] 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.^[93][94] In contrast, a linear circuit has a single equilibrium point that may be stable or unstable.^[95][96] The equilibrium points are determined by the DC bias circuit, and their stability is determined by the AC impedance ${\displaystyle Z_{L}(j\omega )}$ 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:^[86][97]

• In a CCNR (S-type) negative resistance, the resistance function ${\displaystyle R_{N}}$ is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:${\displaystyle Z_{L}(j\omega )+Z_{N}(j\omega )=0}$.^[98][99]

For nonreactive circuits (${\displaystyle X_{L}=X_{N}=0}$) a sufficient condition for stability is that the total resistance is positive^[100]

$${\displaystyle Z_{L}+Z_{N}=R_{L}+R_{N}=R_{L}-r>0}$$

so the CCNR is stable for^[16][77][97]

Since CCNRs are stable with no load at all, they are called "open circuit stable".^{[77][78][86][101][note 2]}

• In a VCNR (N-type) negative resistance, the conductance function ${\displaystyle G_{N}=1/R_{N}}$ is single-valued. Therefore, stability is determined by the poles of the admittance equation ${\displaystyle Y_{L}(j\omega )+Y_{N}(j\omega )=0}$.^[98][99] For this reason the VCNR is sometimes referred to as a negative conductance.^[16][98][99]
  As above, for nonreactive circuits a sufficient condition for stability is that the total conductance in the circuit is positive^[100]
  $${\displaystyle Y_{L}+Y_{N}=G_{L}+G_{N}={\frac {1}{R_{L}}}+{\frac {1}{R_{N}}}={\frac {1}{R_{L}}}+{\frac {1}{-r}}>0}$$

  
  $${\displaystyle {\frac {1}{R_{L}}}>{\frac {1}{r}}}$$

  
  so the VCNR is stable for^[16][97]

Since VCNRs are even stable with a short-circuited output, they are called "short circuit stable".^{[77][78][101][note 2]}

For general negative resistance circuits with reactance, the stability must be determined by standard tests like the Nyquist stability criterion.^[102] Alternatively, in high frequency circuit design, the values of ${\displaystyle Z_{L}(j\omega )}$ for which the circuit is stable are determined by a graphical technique using "stability circles" on a Smith chart.^[17]

Operating regions and applications

For simple nonreactive negative resistance devices with ${\displaystyle R_{N}\;=\;-r}$ and ${\displaystyle X_{N}\;=\;0}$ the different operating regions of the device can be illustrated by load lines on the I–V curve^[77] (see graphs).

VCNR (N type) load lines and stability regions

CCNR (S type) load lines and stability regions

The DC load line (DCL) is a straight line determined by the DC bias circuit, with equation

$${\displaystyle V=V_{S}-IR}$$

where ${\displaystyle V_{S}}$ 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^[103]

• VCNRs require a low impedance bias (${\displaystyle R\;<\;r}$), such as a voltage source.
• CCNRs require a high impedance bias (${\displaystyle R\;>\;r}$) such as a current source, or voltage source in series with a high resistance.

The AC load line (L₁ − L₃) is a straight line through the Q point whose slope is the differential (AC) resistance ${\displaystyle R_{L}}$ facing the device. Increasing ${\displaystyle R_{L}}$ rotates the load line counterclockwise. The circuit operates in one of three possible regions (see diagrams), depending on ${\displaystyle R_{L}}$.^[77]

• Stable region (green) (illustrated by line L₁): When the load line lies in this region, it intersects the I–V curve at one point Q₁.^[77] For nonreactive circuits it is a stable equilibrium (poles in the LHP) so the circuit is stable. Negative resistance amplifiers operate in this region. However, due to hysteresis, with an energy storage device like a capacitor or inductor the circuit can become unstable to make a nonlinear relaxation oscillator (astable multivibrator) or a monostable multivibrator.^[104]
  • VCNRs are stable when ${\displaystyle R_{L}<r}$.
  • CCNRs are stable when ${\displaystyle R_{L}>r}$.
• Unstable point (Line L₂): When ${\displaystyle R_{L}=r}$ 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^[105] ${\displaystyle r=R_{L}}$.
• Bistable region (red) (illustrated by line L₃): In this region the load line can intersect the I–V curve at three points.^[77] The center point (Q₁) is a point of unstable equilibrium (poles in the RHP), while the two outer points, Q₂ and Q₃ are stable equilibria. So with correct biasing the circuit can be bistable, it will converge to one of the two points Q₂ or Q₃ and can be switched between them with an input pulse. Switching circuits like flip-flops (bistable multivibrators) and Schmitt triggers operate in this region.
  • VCNRs can be bistable when ${\displaystyle R_{L}>r}$
  • CCNRs can be bistable when ${\displaystyle R_{L}<r}$

Feedback oscillators

If an LC circuit is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance ${\displaystyle R_{\text{if}}}$ can cancel the positive loss resistance ${\displaystyle r_{\text{loss}}}$ inherent in the tuned circuit.^[114] If ${\displaystyle R_{\text{if}}\;=\;-r_{\text{loss}}}$ this will create in effect a tuned circuit with zero AC resistance (poles on the jω axis).^[39][107] 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.^[41][115] This negative resistance model is an alternate way of analyzing feedback oscillator operation.^{[14][36][104][108][116][117][118]} All linear oscillator circuits have negative resistance^{[36][84][104][117]} 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.^[119]

Q enhancement

A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so ${\displaystyle |R_{\text{if}}|\;<\;r_{\text{loss}}}$) 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.^{[114][120][121][122]} Q enhancement, also called regeneration, was first used in the regenerative radio receiver invented by Edwin Armstrong in 1912^[107][121] and later in "Q multipliers".^[123] It is widely used in active filters.^[122] 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.^[120][122]

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.^[124] 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.^[124] This is usually synthesized using a negative impedance converter circuit.^[124]

Negative impedance converter

Negative impedance converter (left) and I–V curve (right). It has negative differential resistance in red region and sources power in grey region.

A common example of an "active resistance" circuit is the negative impedance converter (NIC)^{[45][46][115][125]} shown in the diagram. The two resistors ${\displaystyle R_{\text{1}}}$ and the op amp constitute a negative feedback non-inverting amplifier with gain of 2.^[115] The output voltage of the op-amp is

$${\displaystyle v_{o}=v(R_{1}+R_{1})/R_{1}=2v}$$

So if a voltage ${\displaystyle v}$ is applied to the input, the same voltage is applied "backwards" across ${\displaystyle Z}$, causing current to flow through it out of the input.^[46] The current is

$${\displaystyle i={\frac {v-v_{o}}{Z}}={\frac {v-2v}{Z}}=-{\frac {v}{Z}}}$$

So the input impedance to the circuit is^[76]

$${\displaystyle z_{\text{in}}={\frac {v}{i}}=-Z}$$

The circuit converts the impedance ${\displaystyle Z}$ to its negative. If ${\displaystyle Z}$ is a resistor of value ${\displaystyle R}$, within the linear range of the op amp ${\displaystyle V_{\text{S}}/2<v<-V_{\text{S}}/2}$ the input impedance acts like a linear "negative resistor" of value ${\displaystyle -R}$.^[46] 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,^[126] for example they were originally developed to cancel resistance in telephone cables, serving as repeaters.^[115]

Negative capacitance and inductance

By replacing ${\displaystyle Z}$ in the above circuit with a capacitor (${\displaystyle C}$) or inductor (${\displaystyle L}$), negative capacitances and inductances can also be synthesized.^[37][46] A negative capacitance will have an I–V relation and an impedance ${\displaystyle Z_{\text{C}}(j\omega )}$ of

$${\displaystyle i=-C{dv \over dt}\qquad \qquad Z_{C}=-1/j\omega C}$$

where ${\displaystyle C\;>\;0}$. 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 ${\displaystyle Z_{\text{L}}(j\omega )}$ of

$${\displaystyle v=-L{di \over dt}\qquad \qquad Z_{L}=-j\omega L}$$

A circuit having negative capacitance or inductance can be used to cancel unwanted positive capacitance or inductance in another circuit.^[46] 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.^[46] 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^[46] 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.^[127] 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.^[128] This would allow the creation of small compact antennas that would have broad bandwidth,^[128] exceeding the Chu–Harrington limit.

Uses

Negative resistance oscillators are mainly used at high frequencies in the microwave range or above, since feedback oscillators function poorly at these frequencies.^[14][116] 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^[132] and terahertz energy^[129] Negative resistance microwave vacuum tubes such as magnetrons produce higher power outputs,^[117] 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.^{[116][117][118][133]} 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.^{[37][88][116]} 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.^[116][118]

Gunn diode oscillator

Gunn diode oscillator circuit

AC equivalent circuit

Gunn diode oscillator load lines.
DCL: DC load line, which sets the Q point.
SSL: negative resistance during startup while amplitude is small. Since ${\displaystyle r\;<\;R}$ poles are in RHP and amplitude of oscillations increases.
LSL: large-signal load line. When the current swing approaches the edges of the negative resistance region (green), the sine wave peaks are distorted ("clipped") and ${\displaystyle r}$ decreases until it equals ${\displaystyle R}$.

The common Gunn diode oscillator (circuit diagrams)^[21] illustrates how negative resistance oscillators work. The diode D has voltage controlled ("N" type) negative resistance and the voltage source ${\displaystyle V_{\text{b}}}$ biases it into its negative resistance region where its differential resistance is ${\displaystyle dv/di\;=\;-r}$. The choke RFC prevents AC current from flowing through the bias source.^[21] ${\displaystyle R}$ is the equivalent resistance due to damping and losses in the series tuned circuit ${\displaystyle LC}$, plus any load resistance. Analyzing the AC circuit with Kirchhoff's Voltage Law gives a differential equation for ${\displaystyle i(t)}$, the AC current^[21]

$${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {R-r}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0}$$

Solving this equation gives a solution of the form^[21]

$${\displaystyle i(t)=i_{0}e^{\alpha t}\cos(\omega t+\phi )}$$

where

$${\displaystyle \alpha ={\frac {r-R}{2L}}\quad \omega ={\sqrt {{\frac {1}{LC}}-\left({\frac {r-R}{2L}}\right)^{2}}}}$$

This shows that the current through the circuit, ${\displaystyle i(t)}$, varies with time about the DC Q point, ${\displaystyle I_{\text{bias}}}$. When started from a nonzero initial current ${\displaystyle i(t)=i_{0}}$ 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 ${\displaystyle R}$ and ${\displaystyle r}$, the positive and negative resistance in the circuit:^[21]

1. 
   ${\displaystyle r<R\Rightarrow \alpha <0}$: (poles in left half plane) If the diode's negative resistance is less than the positive resistance of the tuned circuit, the damping is positive. Any oscillations in the circuit will lose energy as heat in the resistance ${\displaystyle R}$ and die away exponentially to zero, as in an ordinary tuned circuit.^[39] So the circuit does not oscillate.
2. 
   ${\displaystyle r=R\Rightarrow \alpha =0}$: (poles on jω axis) If the positive and negative resistances are equal, the net resistance is zero, so the damping is zero. The diode adds just enough energy to compensate for energy lost in the tuned circuit and load, so oscillations in the circuit, once started, will continue at a constant amplitude.^[39] This is the condition during steady-state operation of the oscillator.
3. 
   ${\displaystyle r>R\Rightarrow \alpha >0}$: (poles in right half plane) If the negative resistance is greater than the positive resistance, damping is negative, so oscillations will grow exponentially in energy and amplitude.^[39] This is the condition during startup.

Practical oscillators are designed in region (3) above, with net negative resistance, to get oscillations started.^[118] A widely used rule of thumb is to make ${\displaystyle R\;=\;r/3}$.^[17][134] When the power is turned on, electrical noise in the circuit provides a signal ${\displaystyle i_{0}}$ 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 ${\displaystyle r}$ 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 ${\displaystyle r}$ becomes smaller, and thus the total resistance ${\displaystyle R\;-\;r}$ and the damping ${\displaystyle \alpha }$ becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when ${\displaystyle r\;=\;R}$.^[21]

Gunn diodes have negative resistance in the range −5 to −25 ohms.^[135] In oscillators where ${\displaystyle R}$ is close to ${\displaystyle r}$; 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 ${\displaystyle R}$ is far below ${\displaystyle r}$, the swing extends further into the nonlinear part of the curve, the clipping distortion of the output sine wave is more severe,^[134] 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)^[91][103]

• Negative resistance (voltage controlled) oscillator: Since VCNR ("N" type) devices require a low impedance bias and are stable for load impedances less than r,^[103] the ideal oscillator circuit for this device has the form shown at top right, with a voltage source V_bias to bias the device into its negative resistance region, and parallel resonant circuit load LC. The resonant circuit has high impedance only at its resonant frequency, so the circuit will be unstable and oscillate only at that frequency.

• Negative conductance (current controlled) oscillator: CCNR ("S" type) devices, in contrast, require a high impedance bias and are stable for load impedances greater than r.^[103] The ideal oscillator circuit is like that at bottom right, with a current source bias I_bias (which may consist of a voltage source in series with a large resistor) and series resonant circuit LC. The series LC circuit has low impedance only at its resonant frequency and so will only oscillate there.

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 Kaneyuki Kurokawa.^{[88][118][136]} 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).^[137] The complex impedance of the negative resistance part ${\displaystyle Z_{N}=R_{N}(I,\omega )+jX_{N}(I,\omega )}$ depends on frequency ω but is also nonlinear, in general declining with the amplitude of the AC oscillation current I; while the resonator part ${\displaystyle Z_{L}=R_{L}(\omega )+jX_{L}(\omega )}$ is linear, depending only on frequency.^{[88][117][137]} The circuit equation is ${\displaystyle (Z_{N}+Z_{L})I=0}$ so it will only oscillate (have nonzero I) at the frequency ω and amplitude I for which the total impedance ${\displaystyle Z_{N}+Z_{L}}$ is zero.^[88] This means the magnitude of the negative and positive resistances must be equal, and the reactances must be conjugate^{[85][117][118][137]}

$${\displaystyle R_{N}\leq -R_{L}}$$

and

$${\displaystyle X_{N}=-X_{L}}$$

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.^{[85][88][118]}

Alternately, the condition for oscillation can be expressed using the reflection coefficient.^[85] The voltage waveform at the reference plane can be divided into a component V₁ travelling toward the negative resistance device and a component V₂ travelling in the opposite direction, toward the resonator part. The reflection coefficient of the active device ${\displaystyle \Gamma _{N}=V_{2}/V_{1}}$ is greater than one, while that of the resonator part ${\displaystyle \Gamma _{L}=V_{1}/V_{2}}$ is less than one. During operation the waves are reflected back and forth in a round trip so the circuit will oscillate only if^{[85][117][137]}

$${\displaystyle |\Gamma _{N}\Gamma _{L}|\geq 1}$$

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,^[118] so there are some circuits that satisfy the equations but do not oscillate. Kurokawa also derived more complicated sufficient conditions,^[136] which are often used instead.^[88][118]

Reflection amplifier

AC equivalent circuit of reflection amplifier

8–12 GHz microwave amplifier consisting of two cascaded tunnel diode reflection amplifiers

One widely used circuit is the reflection amplifier in which the separation is accomplished by a circulator.^{[86][138][139][140]} 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 ${\displaystyle Z_{0}}$ 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.^[140][141]

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

$${\displaystyle P_{\text{in}}=V_{I}^{2}/R_{1}}$$

The output power from the diode is

$${\displaystyle P_{\text{out}}=V_{R}^{2}/R_{1}}$$

So the power gain ${\displaystyle G_{P}}$ of the amplifier is the square of the reflection coefficient^{[138][140][141]}

$${\displaystyle G_{\text{P}}={P_{\text{out}} \over P_{\text{in}}}={V_{R}^{2} \over V_{I}^{2}}=|\Gamma |^{2}}$$

$${\displaystyle |\Gamma |^{2}=\left|{Z_{N}-Z_{1} \over Z_{N}+Z_{1}}\right|^{2}}$$

$${\displaystyle |\Gamma |^{2}=\left|{R_{N}+jX_{N}-(R_{1}+jX_{1}) \over R_{N}+jX_{N}+R_{1}+jX_{1}}\right|^{2}}$$

${\displaystyle R_{\text{N}}}$ is the negative resistance of the diode −r. Assuming the filter is matched to the diode so ${\displaystyle X_{1}=-X_{N}}$^[140] then the gain is

$${\displaystyle G_{\text{P}}=|\Gamma |^{2}={(r+R_{1})^{2}+4X_{N}^{2} \over (r-R_{1})^{2}}}$$

The VCNR reflection amplifier above is stable for ${\displaystyle R_{1}<r}$.^[140] while a CCNR amplifier is stable for ${\displaystyle R_{1}>r}$. It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as ${\displaystyle R_{1}}$ approaches the point of oscillation at ${\displaystyle r}$.^[140] This is a characteristic of all NR amplifiers,^[139] 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.^[141]

Neuronal models

Some instances of neurons display regions of negative slope conductances (RNSC) in voltage-clamp experiments.^[142] The negative resistance here is implied were one to consider the neuron a typical Hodgkin–Huxley style circuit model.

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.^[143][149] In the same year Elihu Thomson built a negative resistance oscillator by connecting an LC circuit to the electrodes of an arc,^[105][150] 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.^{[105][143][147]} 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.^{[105][143][147]} To demonstrate his invention Duddell wired several tuned circuits to an arc and played a tune on it.^[143][147] Duddell's "singing arc" oscillator was limited to audio frequencies.^[105] 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,^[151] inventing the Poulsen arc radio transmitter, which was widely used until the 1920s.^[105][143]

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.^[143] Heinrich Barkhausen in 1907 showed that oscillators must have negative resistance.^[84] 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.^[143]

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.^[9][36][152] 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.^[9] 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 magnetron invented by Hull in 1920.^[60]

The negative impedance converter originated from work by Marius Latour around 1920.^[153][154] He was also one of the first to report negative capacitance and inductance.^[153] A decade later, vacuum tube NICs were developed as telephone line repeaters at Bell Labs by George Crisson and others,^[26][127] which made transcontinental telephone service possible.^[127] Transistor NICs, pioneered by Linvill in 1953, initiated a great increase in interest in NICs and many new circuits and applications developed.^[125][127]

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^[155][156] and G. W. Pickard.^[156][157] 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.^[157] 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.^{[157][158][159][160][161]} He used these to build solid-state amplifiers, oscillators, and amplifying and regenerative radio receivers, 25 years before the invention of the transistor.^{[155][159][161][162]} Later he even built a superheterodyne receiver.^[161] 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),^[162] and it was forgotten.^[161]

The first widely used solid-state negative resistance device was the tunnel diode, invented in 1957 by Japanese physicist Leo Esaki.^[67][163] 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,^[164] 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.^[136] Currently negative differential resistance diode oscillators are the most widely used sources of microwave energy,^[80] and many new negative resistance devices have been discovered in recent decades.^[67]