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{{Short description|Property that an increasing voltage results in a decreasing current}}
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], a device with negative differential resistance.<ref name="Sinclair">{{cite book
{{Use American English|date = April 2019}}
], a device with negative differential resistance.<ref name="Sinclair"/> In operation, an increase in current through the fluorescent tube causes a drop in voltage across it. If the tube were connected directly to the power line, the falling tube voltage would cause more and more current to flow, causing it to ] and destroy itself.<ref name="Sinclair" /><ref name="Aluf" /> To prevent this, fluorescent tubes are connected to the power line through a '']''. The ballast adds positive ] (AC resistance) to the circuit to counteract the negative resistance of the tube, limiting the current.<ref name="Sinclair" />]]
In ], '''negative resistance''' ('''NR''') is a property of some ]s and devices in which an increase in ] across the device's terminals results in a decrease in ] through it.<ref name="Amos"/>
This is in contrast to an ordinary ] in which an increase of applied voltage causes a proportional increase in current due to ], resulting in a positive ].<ref name="Shanefield"/> Under certain conditions it can increase the power of an electrical signal, ] it.<ref name="Aluf" /><ref name="Gottlieb"/><ref name="Kaplan"/>
Negative resistance is an uncommon property which occurs in a few ] electronic components. In a nonlinear device, two types of resistance can be defined: 'static' or 'absolute resistance', the ratio of voltage to current <math>v / i</math>, and ''differential resistance'', the ratio of a change in voltage to the resulting change in current <math>\Delta v/\Delta i</math>. The term negative resistance means '''negative differential resistance''' ('''NDR'''), <math>\Delta v / \Delta i < 0</math>. In general, a negative differential resistance is a two-terminal component which can ],<ref name="Aluf" /><ref name="Suzuki"/> converting ] power applied to its terminals to ] output power to amplify an AC signal applied to the same terminals.<ref name="Carr"/><ref name="Iniewski" /> They are used in ]s and ]s,<ref name="Shahinpoor"/> particularly at ] frequencies. Most microwave energy is produced with negative differential resistance devices.<ref name="Golio"/> They can also have ]<ref name="Kumar2"/> and be ], and so are used in ] and ] circuits.<ref name="Beneking"/> Examples of devices with negative differential resistance are ]s, ]s, and ]s such as ]s, and ]. In addition, circuits containing amplifying devices such as ]s and ]s with ] can have negative differential resistance. These are used in ] and ]s.
Because they are nonlinear, negative resistance devices have a more complicated behavior than the positive "ohmic" resistances usually encountered in ]s. 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.<ref name="Kaplan" /><ref name="Gilmore" />
], a ] with negative differential resistance used in ]s to generate ]s.
| isbn = 0750670738}}</ref> In operation, an increase in current through the fluorescent tube causes a drop in voltage across it. If the tube were connected directly to the power line, the falling tube voltage would cause more and more current to flow, until it destroyed itself.<ref name="Sinclair" /><ref name="Aluf" /> To prevent this, fluorescent tubes are connected to the power line through a '']''. The ballast adds positive ] (AC resistance) to the circuit to counteract the negative resistance of the tube, limiting the current.<ref name="Sinclair" />]]
In ], '''negative resistance''' ('''NR''') is a property of some ]s and devices in which an increase in ] across the device's terminals results in a decrease in ] through it.<ref name="Amos">{{cite book
| isbn = 0750698667}}</ref> This is in contrast to an ordinary ] in which an increase of applied voltage causes a proportional increase in current due to ], resulting in a positive ].<ref name="Shanefield">{{cite book
| isbn = 978-0815514671}}</ref> While a positive resistance consumes power from current passing through it, a negative resistance produces power.<ref name="Groszkowski">{{cite book
| doi =
|last1 = Groszkowski
| id =
|first1 = Janusz
| isbn = 0815514670}}</ref> While a positive resistance consumes power from current passing through it, a negative resistance produces power.<ref name="Carr" /><ref name="Groszkowski">{{cite book
|title = Frequency of Self-Oscillations
| last1 = Groszkowski
|publisher = Pergamon Press - PWN (Panstwowe Wydawnictwo Naukowe)
| first1 = Janusz
|date = 1964
| title = Frequency of Self-Oscillations
|location = Warsaw
| publisher = Pergamon Press - PWN (Panstwowe Wydawnictwo Naukowe)
Negative resistance is an uncommon property which occurs in a few ] electronic components. In a nonlinear device, two types of resistance can be defined: ''static'' or ''absolute resistance'', the ratio of voltage to current <math>\scriptstyle v/i\,</math>, and ''differential resistance'', the ratio of a change in voltage to the resulting change in current <math>\scriptstyle \Delta v/\Delta i</math>. The term negative resistance refers to '''''negative differential resistance'' (NDR)''', <math>\scriptstyle \Delta v/\Delta i\;<\;0</math>. In general, a negative differential resistance is a two-terminal component which can ],<ref name="Aluf" /><ref name="Suzuki">"''In semiconductor physics, it is known that if a two-terminal device shows negative differential resistance it can amplify.''" {{cite journal
|archive-date = August 19, 2014
| last1 = Suzuki
}}</ref>
| first1 = Yoshishige
<ref name="Suzuki">"''In semiconductor physics, it is known that if a two-terminal device shows negative differential resistance it can amplify.''" {{cite journal
| last2 = Kuboda
|last1 = Suzuki
| first2 = Hitoshi
|first1 = Yoshishige
| title = Spin-torque diode effect and its application
|last2 = Kuboda
| journal = Journal of the Physical Society of Japan
|first2 = Hitoshi
| volume = 77
|title = Spin-torque diode effect and its application
| issue = 3
|journal = Journal of the Physical Society of Japan
| pages =
|publisher = PSJ
|volume = 77
|issue = 3
| location = Tokyo
|pages = 031002
| date = March 10, 2008
|date = March 10, 2008
| url = http://jpsj.ipap.jp/link?JPSJ/77/031002/
|url = http://jpsj.ipap.jp/link?JPSJ/77/031002/
| issn =
|doi = 10.1143/JPSJ.77.031002
|doi = 10.1143/JPSJ.77.031002
|access-date = June 13, 2013
| id =
|bibcode = 2008JPSJ...77c1002S
| accessdate = June 13, 2013|bibcode = 2008JPSJ...77c1002S }}</ref> converting ] power applied to its terminals to ] output power to amplify an AC signal applied to the same terminals.<ref name="Carr">{{cite book
| isbn = 0854043357}}</ref> particularly at ] frequencies. Most microwave energy is produced with negative differential resistance devices.<ref name="Golio">{{cite book
|last = Golio
|first = Umesh
|title = Design of an indigenized negative resistance characteristics curve tracer
| isbn = 0412562200}}</ref> Examples of devices with negative differential resistance are ]s, ]s, and ]s such as ]s. In addition, circuits containing amplifying devices such as ]s and ]s with ] can have negative differential resistance. These are used in ] and ]s. A device in which the "absolute resistance", the voltage divided by the current, is negative, is simply a power source,<ref name="Wilson" /><ref name="Simin">{{cite web
|archive-date = 2017-12-21
| last = Simin
}}</ref>]]
| first = Grigory
| authorlink =
| title = Lecture 08: Tunnel Diodes (Esaki diode)
| work = ELCT 569: Semiconductor Electronic Devices
| publisher = Prof. Grigory Simin, Univ. of South Carolina
| isbn = 1118038231}} In this source "negative resistance" refers to negative static resistance.</ref> converting power from some other source to electric power out of its terminals.
Because they are nonlinear, negative resistance devices have a more complicated behavior than the positive "ohmic" resistances usually encountered in ]s. Unlike most positive resistances, negative resistance varies depending on the voltage or current applied to the device, and negative resistance devices can have negative resistance over only a limited portion of their voltage or current range.<ref name="Kaplan" /><ref name="Gilmore" /> Therefore, there is no real "negative resistor" analogous to a positive ], which has a constant negative resistance over an arbitrarily wide range of current.
], a ] with negative differential resistance used in ]s to generate ]s]]
==Negative resistance devices==
]s with negative differential resistance include these devices:
In addition, ] circuits with negative differential resistance can also be built with amplifying devices like ]s and ]s, using ].<ref name="Rybin" /><ref name="Ghadiri">{{cite paper
| first = Aliakbar
| last = Ghadiri
| author =
| title = Design of Active-Based Passive Components for Radio Frequency Applications
| version = PhD Thesis
| publisher = Electrical and Computer Engineering Dept., Univ. of Alberta
| isbn = 0521033330}}</ref> A number of new experimental negative differential resistance materials and devices have been discovered in recent years.<ref name="Franz" /> The physical processes which cause negative resistance are diverse,<ref name="Iniewski" /><ref name="Kapoor" /><ref name="Franz" /> and each type of device has its own negative resistance characteristics, specified by its ].<ref name="Kaplan" /><ref name="Rybin" />
==Definitions==
==Definitions==
]
]
The ] between two terminals of an electrical device or circuit is determined by its current–voltage (''I–V'') curve (]), giving the current <math>\scriptstyle i\,</math> through it for any given voltage <math>\scriptstyle v\,</math> across it.<ref name="Herrick">{{cite book
The ] between two terminals of an electrical device or circuit is determined by its current–voltage (''I–V'') curve (]), giving the current <math>i</math> through it for any given voltage <math>v</math> across it.<ref name="Herrick">{{cite book
| last = Herrick
| last = Herrick
| first = Robert J.
| first = Robert J.
| title = DC/AC Circuits and Electronics: Principles & Applications
| title = DC/AC Circuits and Electronics: Principles & Applications
| isbn = 978-0766820838}}</ref> Most materials, including the ordinary (positive) resistances encountered in electrical circuits, obey ]; the current through them is proportional to the voltage over a wide range.<ref name="Shanefield" /> 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 <math>v</math> is the independent variable) and is constant.
| doi =
| id =
| isbn = 0766820831}}</ref> Most materials, including the ordinary (positive) resistances encountered in electrical circuits, obey ]; the current through them is proportional to the voltage over a wide range.<ref name="Shanefield" /> 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 <math>\scriptstyle v\,</math> is the independent variable) and is constant.
Negative resistance occurs in a few ] (nonohmic) devices.<ref name="Haisch">{{cite web
Negative resistance occurs in a few ] (nonohmic) devices.<ref name="Haisch">{{cite web
| accessdate = March 8, 2014}}</ref> In a nonlinear component the ''I–V'' curve is not a straight line,<ref name="Shanefield" /><ref name="Simpson">{{cite book
| isbn = 0205083773}}</ref> so it does not obey Ohm's law.<ref name="Haisch" /> Resistance can still be defined, but the resistance is not constant; it varies with the voltage or current through the device.<ref name="Aluf" >{{cite book
| title = Optoisolation Circuits: Nonlinearity Applications in Engineering
|archive-date = 2014-08-19
| publisher = World Scientific
|date = 2012
|access-date = 2014-08-18
}}</ref> so it does not obey Ohm's law.<ref name="Haisch" /> Resistance can still be defined, but the resistance is not constant; it varies with the voltage or current through the device.<ref name="Aluf">{{cite book
|title = Optoisolation Circuits: Nonlinearity Applications in Engineering
| doi =
|publisher = World Scientific
| id =
|date = 2012
| isbn = 9814317004}} This source uses the term "absolute negative differential resistance" to refer to active resistance</ref><ref name="Haisch" /> The resistance of such a nonlinear device can be defined in two ways,<ref name="Simpson" /><ref name="Lesurf">{{cite web
| publisher = School of Physics and Astronomy, Univ. of St. Andrews
|date = 2006
|archive-date = 2017-12-21
}} This source uses the term "absolute negative differential resistance" to refer to active resistance</ref><ref name="Haisch" /> The resistance of such a nonlinear device can be defined in two ways,<ref name="Simpson" /><ref name="Lesurf">{{cite web
| url = http://books.google.com/books?id=nZzOAsroBIEC&pg=SA13-PA52&lpg=SA13-PA52&dq=%22Static+resistance%22+%22dynamic+resistance }}</ref> which are equal for ohmic resistances:<ref name="Simin" />
| url = https://books.google.com/books?id=nZzOAsroBIEC&q=%22Static+resistance%22+%22dynamic+resistance&pg=SA13-PA52 }}</ref> which are equal for ohmic resistances:<ref name="Simin">{{cite web
</ref> showing regions representing passive devices ''(white)'' and active devices ''(<span style="color:red;">red</span>)'']]
*'''Static resistance''' (also called ''chordal resistance'', ''absolute resistance'' or just ''resistance'') – This is the common definition of resistance; the voltage divided by the current:<ref name="Aluf" /><ref name="Simin" /><ref name="Herrick" /><ref name="Kaiser" />
*'''Static resistance''' (also called ''chordal resistance'', ''absolute resistance'' or just ''resistance'') – This is the common definition of resistance; the voltage divided by the current:<ref name="Aluf" /><ref name="Herrick" /><ref name="Simin" /> <math display="block">R_\mathrm{static} = \frac{v}{i} .</math> It is the inverse slope of the line (]) from the origin through the point on the ''I–V'' curve.<ref name="Shanefield" /> In a power source, like a ] or ], positive current flows ''out'' of the positive voltage terminal,<ref name="Crisson" /> opposite to the direction of current in a resistor, so from the ] <math>i</math> and <math>v</math> 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'' (<math>R_\text{static} < 0).</math><ref name="Simin" /><ref name="Morecroft" /><ref name="Kouřil">{{cite book
:It is the inverse slope of the line (]) from the origin through the point on the ''I–V'' curve.<ref name="Shanefield" /> In a power source, like a ] or ], positive current flows ''out'' of the positive voltage terminal,<ref name="Crisson" /> opposite to the direction of current in a resistor, so from the ] <math>\scriptstyle i\,</math> and <math>\scriptstyle v\,</math> 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'' (<math>\scriptstyle R_\text{static}\;<\;0).</math><ref name="Wilson">{{cite web
}}</ref> However this term is never used in practice, because the term "resistance" is only applied to passive components.<ref name="Karady">"''...since resistance is always positive...the resultant power must also always be positive. ... means that the resistor always absorbs power.''" {{cite book
}}</ref> However this term is never used in practice, because the term "resistance" is only applied to passive components.<ref name="Karady">"''...since resistance is always positive...the resultant power must also always be positive. ... means that the resistor always absorbs power.''" {{cite book
| isbn = 978-1118498033}}</ref><ref name="Bakshi">"''Since the energy absorbed by a (static) resistance is always positive, resistances are passive devices.''" {{cite book
| doi =
|id =
|last = Bakshi
|first = U.A.
| isbn = 1118498038}}</ref><ref name="Bakshi">"''Since the energy absorbed by a (static) resistance is always positive, resistances are passive devices.''" {{cite book
}}, see footnote p. 116</ref> Static resistance determines the ] in a component.<ref name="Traylor" /><ref name="Bakshi" /> ] devices, which consume electric power, have positive static resistance; while ] devices, which produce electric power, do not.<ref name="Simin" /><ref name="Morecroft">{{cite book
| isbn = 9048194423}}, see footnote p. 116</ref> Static resistance determines the ] in a component.<ref name="Traylor" /><ref name="Bakshi" /> ] devices, which consume electric power, have positive static resistance; while ] devices, which produce electric power, do not.<ref name="Wilson" /><ref name="Simin" /><ref name="Baker" /><ref name="Simpson" /><ref name="Morecroft">{{cite book
*'''Differential resistance''' (also called ''dynamic'',<ref name="Aluf" /><ref name="Kaiser" /> or ''incremental''<ref name="Shanefield" /> resistance) – This is the ] of the voltage with respect to the current; the ratio of a small change in voltage to the corresponding change in current,<ref name="Gottlieb" /> the inverse ] of the ''I–V'' curve at a point:
*'''Differential resistance''' (also called ''dynamic'',<ref name="Aluf" /><ref name="Kaiser" /> or ''incremental''<ref name="Shanefield" /> resistance) – This is the ] of the voltage with respect to the current; the ratio of a small change in voltage to the corresponding change in current,<ref name="Gottlieb" /> the inverse ] of the ''I–V'' curve at a point: <math display="block">r_\mathrm{diff} = \frac {dv}{di} .</math> Differential resistance is only relevant to time-varying currents.<ref name="Gottlieb" /> 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''''' {{nowrap|(<math>r_\text{diff} < 0</math>)}}.<ref name="Aluf" /><ref name="Gottlieb" /><ref name="Simpson" /> Devices of this type can amplify signals,<ref name="Aluf" /><ref name="Suzuki" /><ref name="Shahinpoor" /> and are what is usually meant by the term "negative resistance".<ref name="Aluf" /><ref name="Simpson" />
:Differential resistance is only relevant to time-varying currents.<ref name="Gottlieb" /> 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''''' {{nowrap|(<math>\scriptstyle r_\text{diff}\;<\;0</math>)}}.<ref name="Aluf" /><ref name="Gottlieb" /><ref name="Simin" /><ref name="Simpson" /><ref name="Laplante">{{cite book
| last = Laplante
| first = Philip A.
| authorlink =
| title = Comprehensive Dictionary of Electrical Engineering, 2nd Ed.
| isbn = 1420037803}}</ref> Devices of this type can amplify signals,<ref name="Aluf" /><ref name="Suzuki" /><ref name="Shahinpoor" /> and are what is usually meant by the term "negative resistance".<ref name="Aluf" /><ref name="Simpson" />
Negative resistance, like positive resistance, is measured in ]s.
Negative resistance, like positive resistance, is measured in ]s.
] is the ] of ].<ref name="Herrick2">{{cite book
] is the ] of ].<ref name="Herrick2">{{cite book
|last1 = Herrick
|last1 = Herrick
|first1 = Robert J.
|first1 = Robert J.
|title = DC/AC Circuits and Electronics: Principles & Applications
|title = DC/AC Circuits and Electronics: Principles & Applications
}}</ref> It is measured in ] (formerly ''mho'') which is the conductance of a resistor with a resistance of one ].<ref name="Herrick2" /> Each type of resistance defined above has a corresponding conductance<ref name="Ishii" />
}}</ref> It is measured in ] (formerly ''mho'') which is the conductance of a resistor with a resistance of one ].<ref name="Herrick2" /> Each type of resistance defined above has a corresponding conductance<ref name="Ishii" />
It can be seen that the conductance has the same sign as its corresponding resistance: a negative resistance will have a '''negative conductance'''<ref group=note name="NC">Some microwave texts use this term in a more specialized sense: a ''voltage controlled'' negative resistance device (VCNR) such as a ] is called a "negative conductance" while a ''current controlled'' negative resistance device (CCNR) such as an ] is called a "negative resistance". See the ] section</ref> while a positive resistance will have a positive conductance.<ref name="Kouřil" /><ref name="Ishii" />
It can be seen that the conductance has the same sign as its corresponding resistance: a negative resistance will have a '''negative conductance'''<ref group=note name="NC">Some microwave texts use this term in a more specialized sense: a ''voltage controlled'' negative resistance device (VCNR) such as a ] is called a "negative conductance" while a ''current controlled'' negative resistance device (CCNR) such as an ] is called a "negative resistance". See the ] section</ref> while a positive resistance will have a positive conductance.<ref name="Kouřil" /><ref name="Ishii" />
| caption1 = Fig. 1: ''I–V'' curve withnegative differential resistance''(<spanstyle="color:red;">red</span>region)''.<refname="Simin"/>The differential resistance <math>\scriptstyler_\text{diff}\,</math>atapoint'''''P''''' is the inverseslopeof the linetangenttothegraph at thatpoint
| caption1 = Fig. 1: ''I–V'' curve of linear or "ohmic" resistance, the common type of resistance encountered in electrical circuits. The current is proportional to the voltage, so both the static and differential resistance is positive
Since <math>\scriptstyle \Delta v\;>\;0</math> and <math>\scriptstyle \Delta i\;<\;0</math>, at point '''''P''''' <math>\scriptstyle r_\text{diff}\;<\;0</math> .
| caption2 = Fig. 2: ''I–V'' curve with negative differential resistance ''(<span style="color:red;">red</span> region)''.<ref name="Simin" /> The differential resistance <math>r_\text{diff}</math> at a point '''''P''''' is the inverse slope of the line tangent to the graph at that point
| caption2 = Fig. 2: ''I–V'' curve of a power source.<ref name="Simin" /> In the 2nd quadrant ''(<span style="color:red;">red</span> region)'' current flows out of the positive terminal, so electric power flows out of the device into the circuit. For example at point '''''P''''', <math>\scriptstyle v\;<\;0</math> and <math>\scriptstyle i\;>\;0</math>, so<br />
Since <math>\Delta v\;>\;0</math> and <math>\Delta i < 0</math>, at point '''''P''''' <math>r_\text{diff} < 0</math>.
| caption3 = Fig. 3: ''I–V'' curve of a power source.<ref name="Simin" /> In the 2nd quadrant ''(<span style="color:red;">red</span> region)'' current flows out of the positive terminal, so electric power flows out of the device into the circuit. For example at point '''''P''''', <math>v < 0</math> and <math>i > 0</math>, so<br />
| caption3 = Fig. 3: ''I–V'' curve of a negative linear<ref name="Groszkowski" /> or "active" resistance<ref name="Chua2" /><ref name="Dimopoulos" /><ref name="Pippard2">{{cite book
<math>R_\text{static} = \frac{v}{i} < 0 </math>
| last = Pippard
| first = A. B.
| width3 = 152
| image4 = Active negative resistance definition.svg
| title = The Physics of Vibration
| caption4 = Fig. 4: ''I–V'' curve of a negative linear<ref name="Groszkowski" /> or "active" resistance<ref name="Chua2" /><ref name="Pippard2">{{cite book
| publisher = Cambridge University Press
|date = 2007
|last = Pippard
|first = A. B.
| location =
|title = The Physics of Vibration
| pages = 350, fig. 36; p. 351, fig. 37a; p. 352 fig. 38c; p. 327, fig. 14c
| isbn = 0521033330}} In some of these graphs, the curve is reflected in the vertical axis so the negative resistance region appears to have positive slope.</ref><ref name="Butler" /> ''(AR, <span style="color:red;">red</span>)'' contrasted with an ordinary ohmic resistance ''(OR, black)''. It has negative differential resistance and is an active device:
}} In some of these graphs, the curve is reflected in the vertical axis so the negative resistance region appears to have positive slope.</ref><ref name="Butler" /> ''(AR, <span style="color:red;">red</span>)''. It has negative differential resistance and negative static resistance (is active):<math display="block">R = \frac{\Delta v}{\Delta i} = \frac{v}{i} < 0</math>
| width4 = 187
}}
}}
==How it works==
==Operation==
One way in which the different types of resistance can be distinguished is in the directions that current and electric power flow 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:
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:
{|
{|
|-
|-
| The voltage '''''v''''' and current '''''i''''' variables in an electrical component must be defined according to the ]; positive ] is defined to enter the positive voltage terminal; this means power '''''P''''' flowing from the circuit into the component is defined to be positive, while power flowing from the component into the circuit is negative.<ref name="Traylor" /><ref name="Glisson" /> This applies to both DC and AC current. The diagram shows the directions for positive values of the variables. || ]
| The voltage '''''v''''' and current '''''i''''' variables in an electrical component must be defined according to the ]; positive ] is defined to enter the positive voltage terminal; this means power '''''P''''' flowing from the circuit into the component is defined to be positive, while power flowing from the component into the circuit is negative.<ref name="Traylor" /><ref name="Glisson" /> This applies to both DC and AC current. The diagram shows the directions for positive values of the variables. || ]
|-
|-
| In a '''positive static resistance''', <math>\scriptstyle R_\text{static}\;=\;v/i\;>\;0</math>, so '''''v''''' and '''''i''''' have the same sign.<ref name="Chua2">{{cite book
| In a '''positive static resistance''', <math>R_\text{static}\;=\;v/i\;>\;0</math>, so '''''v''''' and '''''i''''' have the same sign.<ref name="Chua2">{{cite book
| isbn = 0071166505}}</ref> Therefore, from the passive sign convention above, conventional current (positive charge) flows through the device from the positive to the negative terminal, in the direction of the ] '''''E''''' (decreasing ]).<ref name="Traylor" /> <math>\scriptstyle P\;=\;vi\;>\;0</math> so the charges lose ] doing ] on the device, and electric power flows from the circuit into the device,<ref name="Chua2" /><ref name="Karady" /> where it is converted to heat or some other form of energy ''(yellow)''. If AC voltage is applied, <math>\scriptstyle v</math> and <math>\scriptstyle i</math> periodically reverse direction, but the instantaneous <math>\scriptstyle i</math> always flows from the higher potential to the lower potential. || ]
}},</ref> Therefore, from the passive sign convention above, conventional current (flow of positive charge) is through the device from the positive to the negative terminal, in the direction of the ] '''''E''''' (decreasing ]).<ref name="Traylor" /> <math>P = vi\;>\;0</math> so the charges lose ] doing ] on the device, and electric power flows from the circuit into the device,<ref name="Chua2" /><ref name="Karady" /> where it is converted to heat or some other form of energy ''(yellow)''. If AC voltage is applied, <math>v</math> and <math>i</math> periodically reverse direction, but the instantaneous <math>i</math> always flows from the higher potential to the lower potential. || ]
|-
|-
| In a '''power source''', <math>\scriptstyle v/i\;<\;0</math>,<ref name="Simin" /> so <math>\scriptstyle v</math> and <math>\scriptstyle i</math> have opposite signs.<ref name="Chua2" /> This means current is forced to flow from the negative to the positive terminal.<ref name="Simin" /> The charges gain potential energy, so power flows out of the device into the circuit:<ref name="Simin" /><ref name="Chua2" /> <math>\scriptstyleP\;=\;vi\;<\;0</math>. Work ''(yellow)'' must be done on the charges by some power source in the device to make them move in this direction against the force of the electric field. || ]
| In a '''power source''', <math>R_\text{static} = v/i\;<\;0</math>,<ref name="Simin" /> so <math>v</math> and <math>i</math> have opposite signs.<ref name="Chua2" /> This means current is forced to flow from the negative to the positive terminal.<ref name="Simin" /> The charges gain potential energy, so power flows out of the device into the circuit:<ref name="Simin" /><ref name="Chua2" /> <math>P = vi\;<\;0</math>. Work ''(yellow)'' must be done on the charges by some power source in the device to make them move in this direction against the force of the electric field. || ]
|-
|-
| In a passive '''negative differential resistance''', <math>\scriptstyle r_\text{diff}\;=\;\Delta v/\Delta i\;<\;0</math>, only the ''AC component'' of the current flows in the reverse direction. The static resistance is positive<ref name="Shanefield" /><ref name="Gottlieb" /><ref name="Lesurf" /> so the current flows from positive to negative: <math>\scriptstyleP\;=\;vi\;>\;0</math>. But the current (rate of charge flow) decreases as the voltage increases. So when a time-varying (AC) voltage is applied in addition to a DC voltage ''(right)'', the time-varying current <math>\scriptstyle \Delta i\,</math> and voltage <math>\scriptstyle \Delta v\,</math> components have opposite signs, so <math>\scriptstyle P_\text{AC}\;=\;\Delta v\Delta i\;<\;0</math>.<ref name="Ghadiri" /> This means the instantaneous <math>\scriptstyle \Delta i\,</math> flows through the device in the direction of increasing <math>\scriptstyle \Delta v\,</math>, so AC power flows out of the device into the circuit. The device consumes DC power, some of which is converted to AC signal power which can be delivered to a load in the external circuit,<ref name="Carr" /><ref name="Ghadiri" /> enabling the device to amplify the AC signal applied to it.<ref name="Suzuki" /> || ]
| In a passive '''negative differential resistance''', <math>r_\text{diff} = \Delta v / \Delta i\;<\;0</math>, only the ''AC component'' of the current flows in the reverse direction. The static resistance is positive<ref name="Shanefield" /><ref name="Gottlieb" /><ref name="Lesurf" /> so the current flows from positive to negative: <math>P = vi\;>\;0</math>. But the current (rate of charge flow) decreases as the voltage increases. So when a time-varying (AC) voltage is applied in addition to a DC voltage ''(right)'', the time-varying current <math>\Delta i</math> and voltage <math>\Delta v</math> components have opposite signs, so <math>P_\text{AC} = \Delta v\Delta i\;<\;0</math>.<ref name="Ghadiri" /> This means the instantaneous AC current <math>\Delta i</math> flows through the device in the direction of increasing AC voltage <math>\Delta v</math>, so AC power flows out of the device into the circuit. The device consumes DC power, some of which is converted to AC signal power which can be delivered to a load in the external circuit,<ref name="Carr" /><ref name="Ghadiri" /> enabling the device to amplify the AC signal applied to it.<ref name="Suzuki" /> || ]
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In an electronic device, the differential resistance <math>\scriptstyle r_\text{diff}\,</math>, the static resistance <math>\scriptstyle R_\text{static}\,</math>, or both, can be nonpositive,<ref name="Chua2" /> so there are three categories of devices ''(fig. 1–3 above, and table)'' which could be called "negative resistances".
In an electronic device, the differential resistance <math>r_\text{diff}</math>, the static resistance <math>R_\text{static}</math>, or both, can be negative,<ref name="Chua2" /> 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 {{nowrap|<math>\scriptstyle r_\text{diff}\;<\;0</math>.}}<ref name="Aluf" /><ref name="Gilmore" /><ref name="Simpson" /> Negative differential resistance devices have unique capabilities: they can act as ''one-port amplifiers'',<ref name="Aluf" /><ref name="Suzuki" /><ref name="Shahinpoor" /><ref name="Razavi" /> increasing the power of a time-varying signal applied to their port (terminals), or excite oscillations in a ] to make an oscillator.<ref name="Ghadiri" /><ref name="Razavi" /><ref name="Solymar" /> They can also have ].<ref name="Kumar2" /><ref name="Beneking" /> It is not possible for a device to have negative differential resistance without a power source,<ref name="Reich" /> and these devices can be divided into two categories depending on whether they get their power from an internal source or from their port:<ref name="Beneking" /><ref name="Ghadiri" /><ref name="Solymar" /><ref name="Prasad">{{cite book
The term "negative resistance" almost always means negative ''differential'' resistance {{nowrap|<math>r_\text{diff} < 0</math>.}}<ref name="Aluf" /><ref name="Gilmore" /><ref name="Simpson" /> Negative differential resistance devices have unique capabilities: they can act as ''one-port amplifiers'',<ref name="Aluf" /><ref name="Suzuki" /><ref name="Shahinpoor" /><ref name="Razavi" /> increasing the power of a time-varying signal applied to their port (terminals), or excite oscillations in a ] to make an oscillator.<ref name="Ghadiri" /><ref name="Razavi" /><ref name="Solymar" /> They can also have ].<ref name="Kumar2" /><ref name="Beneking" /> It is not possible for a device to have negative differential resistance without a power source,<ref name="Reich">{{cite book
}} on Peter Millet's {{webarchive|url=https://web.archive.org/web/20150324164313/http://www.tubebooks.org/ |date=2015-03-24 }} website</ref> and these devices can be divided into two categories depending on whether they get their power from an internal source or from their port:<ref name="Beneking" /><ref name="Ghadiri" /><ref name="Solymar" /><ref name="Prasad">{{cite book
| last = Prasad
| last = Prasad
| first = Sheila
| first = Sheila
|author2=Hermann Schumacher |author3=Anand Gopinath
|author2=Hermann Schumacher |author3=Anand Gopinath
| title = High-Speed Electronics and Optoelectronics: Devices and Circuits
| title = High-Speed Electronics and Optoelectronics: Devices and Circuits
*<u>Passive negative differential resistance devices</u> (fig. 1 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.<ref name="Prasad" /><ref name="Deliyannis" /> The ''I–V'' curve, including the negative resistance region, lies in the 1st and 3rd quadrant of the plane<ref name="Kumar2" /> so the device has positive static resistance.<ref name="Lesurf" /> Examples are ]s, ]s, and ]s.<ref name="Rybin" /> These devices have no power source and in general work by converting external DC power from their port to time varying (AC) power,<ref name="Carr" /> so they require a DC bias current applied to the port in addition to the signal.<ref name="Ghadiri" /><ref name="Solymar" /> To add to the confusion, some authors<ref name="Gilmore" /><ref name="Rybin" /><ref name="Solymar" /> call these "active" devices, since they can amplify. This category also includes a few three-terminal devices, such as the ].<ref name="Rybin" /> They are covered in the ''']''' section below.
*<u>Passive negative differential resistance devices</u> (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.<ref name="Prasad" /><ref name="Deliyannis" /> The ''I–V'' curve, including the negative resistance region, lies in the 1st and 3rd quadrant of the plane<ref name="Kumar2" /> so the device has positive static resistance.<ref name="Lesurf" /> Examples are ]s, ]s, and ]s.<ref name="Rybin" /> These devices have no internal power source and in general work by converting external DC power from their port to time varying (AC) power,<ref name="Carr" /> so they require a DC bias current applied to the port in addition to the signal.<ref name="Ghadiri" /><ref name="Solymar" /> To add to the confusion, some authors<ref name="Gilmore" /><ref name="Rybin" /><ref name="Solymar" /> call these "active" devices, since they can amplify. This category also includes a few three-terminal devices, such as the unijunction transistor.<ref name="Rybin" /> They are covered in the ''']''' section below.
{{clear}}
]
]
*<u>Active negative differential resistance devices</u> (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,<ref name="Aluf" /><ref name="Crisson" /><ref name="Wilson">{{cite web
|last = Wilson
*<u>Active negative differential resistance devices</u> (fig. 3): 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,<ref name="Aluf" /><ref name="Wilson" /><ref name="Crisson" /><ref name="HorowitzVideo">{{cite web
|last = Horowitz
|first = Marcus
|title = Negative Resistance
| first = Paul
|work = Sciblog 2010 Archive
| title = Negative Resistor – Physics 123 demonstration with Paul Horowitz
|publisher = Science Media Center
| work = Video lecture, Physics 123, Harvard Univ.
| accessdate = November 20, 2012 }} In this video Prof. Horowitz demonstrates that negative static resistance actually exists. He has a black box with two terminals, labelled "−10 kilohms" and shows with ordinary test equipment that it acts like a linear negative resistor (active resistor) with a resistance of −10 KΩ: a positive voltage across it causes a proportional ''negative'' current through it, and when connected in a voltage divider with an ordinary resistor the output of the divider is greater than the input, it can amplify. At the end he opens the box and shows it contains an op-amp negative impedance converter circuit and battery.</ref><ref name="Hickman" /> 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.<ref name="Chua2" /> Amplifying devices like ]s and ]s with positive ] can have this type of negative resistance,<ref name="Ghadiri" /><ref name="Pippard3" /><ref name="Crisson">{{cite journal
|archive-date = October 4, 2012
}}, </ref><ref name="HorowitzVideo">{{cite web
|last = Horowitz
|first = Paul
|title = Negative Resistor – Physics 123 demonstration with Paul Horowitz
}} In this video Prof. Horowitz demonstrates that negative static resistance actually exists. He has a black box with two terminals, labelled "−10 kilohms" and shows with ordinary test equipment that it acts like a linear negative resistor (active resistor) with a resistance of −10 KΩ: a positive voltage across it causes a proportional ''negative'' current through it, and when connected in a voltage divider with an ordinary resistor the output of the divider is greater than the input, it can amplify. At the end he opens the box and shows it contains an op-amp negative impedance converter circuit and battery.</ref><ref name="Hickman" /> 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.<ref name="Chua2" /> Amplifying devices like ]s and ]s with positive ] can have this type of negative resistance,<ref name="Ghadiri" /><ref name="Pippard3" /><ref name="Crisson">{{cite journal
| last = Crisson
| last = Crisson
| first = George
| first = George
Line 734:
Line 547:
| issue = 3
| issue = 3
| pages = 485–487
| pages = 485–487
| publisher = AT&T
| location = USA
| date = July 1931
| date = July 1931
| url = https://archive.org/details/bstj10-3-485
| url = https://archive.org/details/bstj10-3-485
| issn =
| doi = 10.1002/j.1538-7305.1931.tb01288.x
| doi = 10.1002/j.1538-7305.1931.tb01288.x
| access-date = December 4, 2012}}</ref><ref name="Deliyannis" /> and are used in ] and ]s.<ref name="Deliyannis" /><ref name="Hickman" /> 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.<ref name="Chua2" /><ref name="Crisson" /><ref name="Wilson" /> In ] this is called an "active resistor".<ref name="Chua2" /><ref name="Kouřil" /><ref name="Popa">{{cite book
| id =
| accessdate = December 4, 2012}}</ref><ref name="Deliyannis" /> and are used in ] and ]s.<ref name="Deliyannis" /><ref name="Hickman" /> 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.<ref name="Wilson" /><ref name="Chua2" /><ref name="Crisson" /> In ] this is called an "active resistor".<ref name="Chua2" /><ref name="Kouřil" /><ref name="Hoekstra" /><ref name="Popa">{{cite book
| last = Popa
| last = Popa
| first = Cosmin Radu
| first = Cosmin Radu
| authorlink =
| title = Synthesis of Analog Structures for Computational Signal Processing
| title = Synthesis of Analog Structures for Computational Signal Processing
| isbn = 978-1-4614-0403-3| chapter = Active Resistor Circuits
| doi =
}}</ref><ref name="Miano" /> Although this type is sometimes referred to as "linear",<ref name="Chua2" /><ref name="Dimopoulos" /> "absolute",<ref name="Aluf" /> "ideal", or "pure" negative resistance<ref name="Aluf" /><ref name="Hickman" /> to distinguish it from "passive" negative differential resistances, in electronics it is more often simply called ] or '']''. These are covered in the ''']''' section below.
| id =
{{clear}}
| isbn = 978-1-4614-0403-3}}</ref><ref name="Miano" /> It is often simply referred to as "negative resistance"<ref name="Aluf" /><ref name="Hickman" /> in electronics, although sometimes terms like "linear",<ref name="Chua2" /><ref name="Dimopoulos" /> "absolute",<ref name="Aluf" /> "ideal", or "pure" are added to distinguish this type from "passive" negative differential resistances. These are covered in the ''']''' section below.
] has negative static resistance<ref name="Simin" /><ref name="Baker" /> '']'' over its normal operating range, but positive differential resistance.]]
] has negative static resistance<ref name="Simpson" /><ref name="Simin" /><ref name="Baker" /> (red) over its normal operating range, but positive differential resistance.]]
Occasionally ordinary power sources are referred to as "negative resistances"<ref name="Baker" /><ref name="Simpson" /><ref name="Morecroft" /><ref name="Fett">{{cite journal
Occasionally ordinary power sources are referred to as "negative resistances"<ref name="Simpson" /><ref name="Morecroft" /><ref name="Baker" /><ref name="Fett">{{cite journal
| last = Fett
| last = Fett
| first = G. H.
| first = G. H.
| title = Negative Resistance as a Machine Parameter
| title = Negative Resistance as a Machine Parameter
| accessdate = December 2, 2012|bibcode = 1943JAP....14..674F }}, abstract.</ref> (fig. 2). Although the "static" or "absolute" resistance <math>\scriptstyle R_\text{static}\,</math> of active devices (power sources) can be considered negative (see ''']''' section below) most ordinary power sources (AC or DC), such as ], ], and (non positive feedback) amplifiers, have positive ''differential'' resistance (their ]).<ref name="Babin">{{cite web
}}, abstract.</ref> (fig. 3 above). Although the "static" or "absolute" resistance <math>R_\text{static}</math> of active devices (power sources) can be considered negative (see ''']''' section below) most ordinary power sources (AC or DC), such as ], ], and (non positive feedback) amplifiers, have positive ''differential'' resistance (their ]).<ref name="Babin">{{cite web
| last = Babin
|first=Perry
|last = Babin
|first = Perry
| title = Output Impedance
|title = Output Impedance
| work = Basic Car Audio Electronics website
|work = Basic Car Audio Electronics website
| publisher =
|date = 1998
|date = 1998
|url = http://www.bcae1.com/outptimp.htm
|url = http://www.bcae1.com/outptimp.htm
|access-date = December 28, 2014
| doi =
|url-status = live
| accessdate = December 28, 2014}}</ref><ref name="Glisson2" ></ref> Therefore, these devices cannot function as one-port amplifiers or have the other capabilities of negative differential resistances.
}}</ref><ref name="Glisson2"> {{webarchive|url=https://web.archive.org/web/20160413171402/https://books.google.com/books?id=7nNjaH9B0_0C&pg=PA96&dq=%22output+resistance%22+battery+generator+amplifier |date=2016-04-13 }}</ref> Therefore, these devices cannot function as one-port amplifiers or have the other capabilities of negative differential resistances.
==Negativestaticor "absolute" resistance==
==List of negative resistance devices==
]s with negative differential resistance include these devices:
| isbn = 978-1425972431}}</ref> and other diodes using the transferred electron mechanism<ref name="Kapoor" />
*],<ref name="Rybin" /><ref name="Radmanesh" /> TRAPATT diode and other diodes using the impact ionization mechanism<ref name="Kapoor" />
*Some ] with E-C reverse biased, known as ]<ref>url = {{cite web |url=http://www.keelynet.com/zpe/negistor.htm |title=KeelyNet on negative resistance - 04/07/00 |access-date=2006-09-08 |url-status=dead |archive-url=https://web.archive.org/web/20060906055849/http://www.keelynet.com/zpe/negistor.htm |archive-date=2006-09-06 }}</ref>
*] (UJT)<ref name="Fogiel" /><ref name="Rybin" />
*]s<ref name="Fogiel" /><ref name="Rybin" />
*] and ] vacuum tubes operating in the ] mode<ref name="Gottlieb" /><ref name="Whitaker">{{cite book
In addition, ] circuits with negative differential resistance can also be built with amplifying devices like ]s and ]s, using ].<ref name="Rybin" /><ref name="Ghadiri">{{cite journal
|first = Aliakbar
|last = Ghadiri
|title = Design of Active-Based Passive Components for Radio Frequency Applications
|version = PhD Thesis
|publisher = Electrical and Computer Engineering Dept., Univ. of Alberta
}}</ref> A number of new experimental negative differential resistance materials and devices have been discovered in recent years.<ref name="Franz" /> The physical processes which cause negative resistance are diverse,<ref name="Iniewski" /><ref name="Kapoor" /><ref name="Franz" /> and each type of device has its own negative resistance characteristics, specified by its ].<ref name="Kaplan" /><ref name="Rybin" />
== Negative static or "absolute" resistance <span class="anchor" id="NegativeStaticResistance"></span>==
{{multiple image
{{multiple image
| align = right
| align = right
| direction = horizontal
| direction = horizontal
| image1 = Thermodynamics and positive resistance.svg
| image1 = Thermodynamics and positive resistance.svg
| image2 = Thermodynamics and negative resistance.svg
| image2 = Thermodynamics and negative resistance.svg
| width = 180
| width = 180
| footer = A positive static resistor ''(left)'' converts electric power to heat,<ref name="Simin" /> warming its surroundings. But a negative static resistance cannot function like this in reverse ''(right)'', converting ambient heat from the environment to electric power, because it would violate the ].<ref name="Wilson" /><ref name="Solymar" /><ref name="Thompson">{{cite journal
| footer = A positive static resistor ''(left)'' converts electric power to heat,<ref name="Simin" /> warming its surroundings. But a negative static resistance cannot function like this in reverse ''(right)'', converting ambient heat from the environment to electric power, because it would violate the ]<ref name="Solymar" /><ref name="Wilson" /><ref name="Thompson">{{cite journal
|last = Thompson
|last = Thompson
|first = Sylvanus P.
|first = Sylvanus P.
|title = On the properties of a body having a negative electric resistance
|title = On the properties of a body having a negative electric resistance
| accessdate = June 7, 2014}} also see editorial, "Positive evidence and negative resistance", p. 312</ref><ref name="Grant" /><ref name="Cole">{{Cite news
}} also see editorial, "Positive evidence and negative resistance", p. 312</ref><ref name="Grant" /><ref name="Cole">{{cite news
| last = Cole
|last = Cole
| first = K.C.
| title = Experts Scoff at Claim of Electricity Flowing With 'Negative Resistance'
|first = K.C.
|title = Experts Scoff at Claim of Electricity Flowing With 'Negative Resistance'
| accessdate = December 8, 2012}} on . In this article the term "negative resistance" refers to negative static resistance.</ref><ref name="Klein">{{cite book
}} on {{webarchive|url=https://web.archive.org/web/20130802073839/http://www.latimes.com/ |date=2013-08-02 }}. In this article the term "negative resistance" refers to negative static resistance.</ref><ref name="Klein">{{cite book
| isbn = 978-1139498180 }}</ref> which requires a temperature ''difference'' to produce work. Therefore a negative static resistance must have some other source of power.
| doi =
| id =
| isbn = 1139498185}}</ref> which requires a temperature ''difference'' to produce work. Therefore a negative static resistance must have some other source of power.
}}
}}
A point of some confusion is whether ordinary resistance ("static" or "absolute" resistance, <math>\scriptstyle R_\text{static}\;=\;v/i</math>) can be negative.<ref name="Thompson" /><ref name="PhysicsForums">{{cite web
A point of some confusion is whether ordinary resistance ("static" or "absolute" resistance, <math>R_\text{static} = v / i</math>) can be negative.<ref name="Thompson" /><ref name="PhysicsForums">{{cite web
| accessdate = August 17, 2014}}</ref> In electronics, the term "resistance" is customarily applied only to ] materials and components<ref name="Bakshi" /> – such as wires, ]s and ]s. These cannot have <math>\scriptstyle R_\text{static}\;<\;0</math> as shown by ] <math>\scriptstyle P\;=\;i^2 R_\text{static}</math>.<ref name="Karady" /> A passive device consumes electric power, so from the ] <math>\scriptstyle P\;\ge\;0</math>. Therefore, from Joule's law <math>\scriptstyle R_\text{static}\;\ge\;0</math>.<ref n<ref name="Simin" /><ref name="Morecroft" /><ref name="Karady" /> In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance.<ref name="Shanefield" /><ref name="Gibilisco">{{cite book
|archive-date = August 19, 2014
| last = Gibilisco
}}</ref> In electronics, the term "resistance" is customarily applied only to ] materials and components<ref name="Bakshi" /> – such as wires, ]s and ]s. These cannot have <math>R_\text{static} < 0</math> as shown by ] {{nowrap|<math>P = i^2 R_\text{static}</math>.}}<ref name="Karady" /> A passive device consumes electric power, so from the ] <math>P \ge 0</math>. Therefore, from Joule's law {{nowrap|<math>R_\text{static} \geq 0</math>.}}<!--<ref name=""/>--><ref name="Simin" /><ref name="Morecroft" /><ref name="Karady" /> In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance.<ref name="Shanefield" /><ref name="Gibilisco">{{cite book
}}</ref> For a passive device to have <math>R_\text{static} = v/i\;<\;0</math> would violate either ]<ref name="Aluf" /> or the ],<ref name="Solymar" /><ref name="Wilson" /><ref name="Thompson" /><ref name="Klein" /> ''(diagram)''. Therefore, some authors<ref name="Shanefield" /><ref name="Karady" /><ref name="Grant">{{cite web
| id =
|last = Grant
| isbn = 0071412123}}</ref> For a passive device to have <math>\scriptstyle R_\text{static}\;=\;v/i\;<\;0</math> would violate either ]<ref name="Aluf" /> or the ],<ref name="Wilson" /><ref name="Solymar" /><ref name="Thompson" /><ref name="Klein" /> ''(diagram)''. Therefore, some authors<ref name="Shanefield" /><ref name="Karady" /><ref name="Grant">{{cite web
|last = Grant
|first = Paul M.
|title = Journey Down the Path of Least Resistance
| first = Paul M.
|title = JourneyDown the Path ofLeastResistance
|work = OutPost on the Endless Frontier blog
|publisher = EPRI News, Electric Power Research Institute
| work = OutPost on the Endless Frontier blog
|date = July 17, 1998
| publisher = EPRI News, Electric Power Research Institute
| accessdate = December 8, 2012}} on </ref> state that static resistance can never be negative.
}} on {{webarchive|url=https://web.archive.org/web/20130722172704/http://w2agz.com/ |date=2013-07-22 }}</ref> state that static resistance can never be negative.
], the static resistance of a power source (''R''<sub>S</sub>), such as a battery, is always equal to the negative of the static resistance of its load (''R''<sub>L</sub>).<ref name="Morecroft" /><ref name="Deliyannis" /> ]]
], the static resistance of a power source (''R''<sub>S</sub>), such as a battery, is always equal to the negative of the static resistance of its load (''R''<sub>L</sub>).<ref name="Morecroft" /><ref name="Deliyannis" /> ]]
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.<ref name="Morecroft" /> For electric power (]) to flow out of a device into the circuit, charge must flow through the device in the direction of increasing potential energy, ] (positive charge) must move from the negative to the positive terminal.<ref name="Wilson" /><ref name="Simin" /><ref name="Butler" /> 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 ] so the current and voltage have opposite signs, and their ratio is 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.<ref name="Morecroft" /> For electric power (]) to flow out of a device into the circuit, charge must flow through the device in the direction of increasing potential energy, ] (positive charge) must move from the negative to the positive terminal.<ref name="Simin" /><ref name="Butler" /><ref name="Wilson" /> 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 ] so the current and voltage have opposite signs, and their ratio is negative
This can also be proved from ]<ref name="Simin" /><ref name="Morecroft" /><ref name="Thompson"/>
This can also be proved from ]<ref name="Simin" /><ref name="Morecroft" /><ref name="Thompson"/>
:<math>P = iv = i^2 R_{static} \,</math>
<math display="block">P = iv = i^2 R_\mathrm{static} </math>
This shows that power can flow out of a device into the circuit {{nowrap|(<math>\scriptstyleP\;<\;0</math>)}} if and only if <math>\scriptstyle R_\text{static}\;<\;0</math>.<ref name="Simin" /><ref name="Baker" /><ref name="Chua2" /><ref name="Reich" /><ref name="Hoekstra" /><ref name="Thompson" /> 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,<ref name="Aluf" /><ref name="Chua2" /> 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 ] it is always simply equal to the negative of the static resistance of the attached circuit ''(right)''.<ref name="Morecroft" /><ref name="Deliyannis" />
This shows that power can flow out of a device into the circuit {{nowrap|(<math>P <0</math>)}} if and only if <math>R_\text{static} < 0</math>.<ref name="Simin" /><ref name="Chua2" /><ref name="Baker" /><ref name="Thompson" /> 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,<ref name="Aluf" /><ref name="Chua2" /> 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 ] it is always simply equal to the negative of the static resistance of the attached circuit ''(right)''.<ref name="Morecroft" /><ref name="Deliyannis" />
] 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 ] requires that negative static resistances have a source of power.<ref name="Aluf" /><ref name="Wilson" /><ref name="Simin" /><ref name="Solymar">{{cite book
] 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 ] requires that negative static resistances have a source of power.<ref name="Aluf" /><ref name="Simin" /><ref name="Solymar">{{cite book
| last = Solymar
| last = Solymar
| first = Laszlo
| first = Laszlo
|author2=Donald Walsh
| author2 = Donald Walsh
| title = Electrical Properties of Materials, 8th Ed.
| title = Electrical Properties of Materials, 8th Ed.
| isbn = 978-0199565917 }}</ref><ref name="Wilson" /> 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<ref name="Wilson" /> as in an amplifying device like a ], ], or ].
| doi =
| id =
| isbn = 0199565910}}</ref> The power may come from an internal source which converts some other form of energy to electric power<ref name="Hoekstra" /> as in a battery or generator, or from a separate connection to an external power supply circuit<ref name="Wilson" /> as in an amplifying device like a ], ], or ].
===Eventual passivity===
===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.<ref name="Kaplan" /> Any active circuit or device with a finite power source is "''eventually passive''".<ref name="Miano">{{cite book
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.<ref name="Kaplan" /> Any active circuit or device with a finite power source is "''eventually passive''".<ref name="Miano">{{cite book
| isbn = 0121897109}} This source calls negative differential resistances "passive resistors" and negative static resistances "active resistors".</ref><ref name="Chen">{{cite book
}} This source calls negative differential resistances "passive resistors" and negative static resistances "active resistors".</ref><ref name="Chen">{{cite book
| accessdate = September 17, 2012 }} Definitions 6 & 7, fig. 27, and Theorem 10 for precise definitions of what this condition means for the circuit solution.</ref> This property means if a large enough external voltage or current of either polarity is applied to it, its static resistance becomes positive<ref name="Chen" />
}} Definitions 6 & 7, fig. 27, and Theorem 10 for precise definitions of what this condition means for the circuit solution.</ref> 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<ref name="Chen" />
:<math>\exists V,I: |v| > V \text{ or } |i| > I \Rightarrow R_\mathrm{static} = v/i \ge 0 \,</math>
<math display="block">\exists V,I: |v| > V \text{ or } |i| > I \Rightarrow R_\mathrm{static} = v/i \ge 0 </math>
where <math>P_{\max} = IV </math> is the maximum power the device can produce.
:where <math>P_{max} = IV \,</math> 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.<ref name="Chua" /> Thus the range of the curve having negative static resistance is limited,<ref name="Kaplan" /> 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<ref name="Muthuswamy">{{cite conference
Therefore, the ends of the ''I–V'' curve will eventually turn and enter the 1st and 3rd quadrants.<ref name="Chua" /> Thus the range of the curve having negative static resistance is limited,<ref name="Kaplan" /> 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<ref name="Muthuswamy">{{cite conference
| first = Bharathwaj
| first = Bharathwaj
| last = Muthuswamy
| last = Muthuswamy
|author2=Joerg Mossbrucker
| author2 = Joerg Mossbrucker
| title = A framework for teaching nonlinear op-amp circuits to junior undergraduate electrical engineering students
| title = A framework for teaching nonlinear op-amp circuits to junior undergraduate electrical engineering students
| booktitle = 2010 Conference Proceedings
| book-title = 2010 Conference Proceedings
| pages =
| publisher = American Society for Engineering Education
| publisher = American Society for Engineering Education
}}{{Dead link|date=April 2020 |bot=InternetArchiveBot |fix-attempted=yes }}, Appendix B. This derives a slightly more complicated circuit where the two voltage divider resistors are different to allow scaling, but it reduces to the text circuit by setting ''R2'' and ''R3'' in the source to ''R1'' in the text, and ''R1'' in source to ''Z'' in the text. The ''I–V'' curve is the same.</ref> 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''<sub>s</sub> will cause the amplifier to saturate, also making its resistance positive.
| id =
| accessdate = October 18, 2012}}, Appendix B. This derives a slightly more complicated circuit where the two voltage divider resistors are different to allow scaling, but it reduces to the text circuit by setting ''R2'' and ''R3'' in the source to ''R1'' in the text, and ''R1'' in source to ''Z'' in the text. The ''I–V'' curve is the same.</ref> 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''<sub>s</sub> will cause the amplifier to saturate, also making its resistance positive.
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:<ref name="Lesurf" />
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:<ref name="Lesurf" />
The ''I–V'' curve is ] (having peaks and troughs) with regions of negative slope representing negative differential resistance.
The ''I–V'' curve is ] (having peaks and troughs) with regions of negative slope representing negative differential resistance.
{{multiple image
{{multiple image
| align = right
| align = right
| direction = vertical
| direction = vertical
| header = Negative differential resistance
| header = Negative differential resistance
| image1 = Voltage controlled negative resistance.svg
| image1 = Voltage controlled negative resistance.svg
| caption1 = Voltage controlled (N type)
| caption1 = Voltage controlled (N type)
| width1 = 174
| width1 = 174
| image2 = Current controlled negative resistance.svg
| image2 = Current controlled negative resistance.svg
| caption2 = Current controlled (S type)
| caption2 = Current controlled (S type)
| width2 = 170
| width2 = 170
| footer =
| footer =
}}
}}
] negative differential resistances have positive ''static'' resistance;<ref name="Aluf" /><ref name="Shanefield" /><ref name="Lesurf" /> they consume net power. Therefore, the ''I–V'' curve is confined to the 1st and 3rd quadrants of the graph,<ref name="Kumar2" /> and passes through the origin. This requirement means (excluding some asymptotic cases) that the region(s) of negative resistance must be limited,<ref name="Gilmore" /><ref name="Kumar" /> and surrounded by regions of positive resistance, and cannot include the origin.<ref name="Aluf" /><ref name="Kaplan" />
] negative differential resistances have positive ''static'' resistance;<ref name="Aluf" /><ref name="Shanefield" /><ref name="Lesurf" /> they consume net power. Therefore, the ''I–V'' curve is confined to the 1st and 3rd quadrants of the graph,<ref name="Kumar2" /> and passes through the origin. This requirement means (excluding some asymptotic cases) that the region(s) of negative resistance must be limited,<ref name="Gilmore" /><ref name="Kumar" /> and surrounded by regions of positive resistance, and cannot include the origin.<ref name="Aluf" /><ref name="Kaplan" />
===Types===
===Types===
Negative differential resistances can be classified into two types:<ref name="Beneking" /><ref name="Kumar">{{cite book
Negative differential resistances can be classified into two types:<ref name="Beneking" /><ref name="Kumar">{{cite book
}}</ref><ref group=note name="Confusion">The terms "''open-circuit stable''" and "''short-circuit stable''" have become somewhat confused over the years, and are used in the opposite sense by some authors. The reason is that in ]s if the load line crosses the I-V curve of the NR device at one point, the circuit is stable, while in nonlinear switching circuits that operate by ] the same condition causes the circuit to become unstable and oscillate as an ], and the ] region is considered the "stable" one. This article uses the former "linear" definition, the earliest one, which is found in the Abraham, Bangert, Dorf, Golio, and Tellegen sources. The latter "switching circuit" definition is found in the Kumar and Taub sources.</ref> or "'''N'''" type): In this type the current is a ], ] of the voltage, but the voltage is a ] of the current.<ref name="Kumar" /> 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 (''i''<sub>1</sub>), then decreases in the region of negative resistance to a minimum (''i''<sub>2</sub>), then increases again. Devices with this type of negative resistance include the ],<ref name="Fogiel" /> ],<ref name="Kidner">{{cite conference
| id =
|first = C.
| accessdate = September 21, 2012}}</ref><ref group=note name="Confusion">The terms "''open-circuit stable''" and "''short-circuit stable''" have become somewhat confused over the years, and are used in the opposite sense by some authors. The reason is that in ]s if the load line crosses the I-V curve of the NR device at one point, the circuit is stable, while in nonlinear switching circuits that operate by ] the same condition causes the circuit to become unstable and oscillate as an ], and the ] region is considered the "stable" one. This article uses the former "linear" definition, the earliest one, which is found in the Abraham, Bangert, Dorf, Golio, and Tellegen sources. The latter "switching circuit" definition is found in the Kumar and Taub sources.</ref> or "'''N'''" type): In this type the current is a ], ] of the voltage, but the voltage is a ] of the current.<ref name="Kumar" /> 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 (''i''<sub>1</sub>), then decreases in the region of negative resistance to a minimum (''i''<sub>2</sub>), then increases again. Devices with this type of negative resistance include the ],<ref name="Fogiel" /> ],<ref name="Kidner">{{cite conference
|first = C.
|last = Kidner
|last = Kidner
|author2 = I. Mehdi
|author3 = J. R. East
| authorlink =
|author2=I.Mehdi|author3=J.R.East|author4=J. I. Haddad
|author4 = J. I. Haddad
|title = Potential and limitations of resonant tunneling diodes
|title = Potential and limitations of resonant tunneling diodes
| booktitle = First International Symposium on Space Terahertz Technology, March 5–6, 1990, Univ. of Michigan
|book-title = First International Symposium on Space Terahertz Technology, March 5–6, 1990, Univ. of Michigan
|pages = 85
|pages = 85
| publisher = US National Radio Astronomy Observatory
|publisher = US National Radio Astronomy Observatory
| isbn = 0521114039}}</ref> and ]s.<ref name="Rybin" /><ref name="Whitaker" />
}}</ref> and ]s.<ref name="Rybin" /><ref name="Whitaker" />
*'''Current controlled negative resistance''' ('''CCNR''', ''open-circuit stable'',<ref name="Kumar" /><ref name="Tellegen" /><ref group=note name="Confusion" /> 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.<ref name="Kumar" /> 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 ],<ref name="Du" /> UJT,<ref name="Fogiel" /> ] and other ],<ref name="Fogiel" /> ], and ]s .<ref name="Rybin" />
Most devices have a single negative resistance region. However devices with multiple separate negative resistance regions can also be fabricated.<ref name="Franz">{{cite journal
*'''Current controlled negative resistance''' ('''CCNR''', ''open-circuit stable'',<ref name="Kumar" /><ref name="Tellegen" /><ref group=note name="Confusion" /> 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.<ref name="Kumar" /> 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 ],<ref name="Du" /> ],<ref name="Fogiel" /> ] and other ],<ref name="Fogiel" /> ], and ]s like ] tubes, ]s and ]s.<ref name="Rybin" />
|last = Franz
|first = Roger L.
Most devices have a single negative resistance region. However devices with multiple separate negative resistance regions can also be fabricated.<ref name="Franz">{{cite journal
|title = Use nonlinear devices as linchpins to next-generation design
| last = Franz
|journal = Electronic Design Magazine
| first = Roger L.
|publisher = Penton Media Inc.
| title = Use nonlinear devices as linchpins to next-generation design
|title = Overview of Nonlinear Devices and Circuit Applications
| id =
|work = Sustainable Technology
| accessdate = September 17, 2012}} An expanded version of this article with graphs and an extensive list of new negative resistance devices appears in {{cite web
| access-date = September 17, 2012}}</ref> These can have more than two stable states, and are of interest for use in ]s to implement ].<ref name="Franz" /><ref name="Abraham" />
| doi =
| id =
| accessdate = September 17, 2012}}</ref> These can have more than two stable states, and are of interest for use in ]s to implement ].<ref name="Franz" /><ref name="Abraham" />
An intrinsic parameter used to compare different devices is the ''peak-to-valley current ratio'' (PVR),<ref name="Franz" /> the ratio of the current at the top of the negative resistance region to the current at the bottom ''(see graphs, above)'':
An intrinsic parameter used to compare different devices is the ''peak-to-valley current ratio'' (PVR),<ref name="Franz" /> 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
The larger this is, the larger the potential AC output for a given DC bias current, and therefore the greater the efficiency
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| align = right
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| image1 = Tunnel diode amplifier.svg
| image1 = Tunnel diode amplifier.svg
| width1 = 172
| width1 = 172
| image2 = Tunnel diode amplifier graph.svg
| image2 = Tunnel diode amplifier graph.svg
| width2 = 150
| width2 = 150
| footer = Tunnel diode amplifier circuit. Since <math>\scriptstyler\,>\,R</math> the total resistance, the sum of the two resistances in series {{nowrap|(<math>\scriptstyleR\;-\;r</math>)}} 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 ] <math>\scriptstyle R</math> ''<span style="color:blue;">(blue)</span>''.<ref name="Weaver" /> A small increase in input voltage, <math>\scriptstyle v_i</math> ''<span style="color:green;">(green)</span>'' 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 <math>\scriptstyle v_o</math>.
| footer = Tunnel diode amplifier circuit. Since <math>r > R</math> the total resistance, the sum of the two resistances in series {{nowrap|(<math>R - r</math>)}} 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 ] <math>R</math> ''<span style="color:blue;">(blue)</span>''.<ref name="Weaver" /> A small increase in input voltage, <math>v_i</math> ''<span style="color:green;">(green)</span>'' 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 <math>v_o</math>.
}}
}}
A negative differential resistance device can ] an AC signal applied to it<ref name="Suzuki" /><ref name="Shahinpoor" /> if the signal is ] with a DC voltage or current to lie within the negative resistance region of its ''I–V'' curve.<ref name="Carr" /><ref name="Iniewski">{{cite book
A negative differential resistance device can ] an AC signal applied to it<ref name="Suzuki" /><ref name="Shahinpoor" /> if the signal is ] with a DC voltage or current to lie within the negative resistance region of its ''I–V'' curve.<ref name="Carr" /><ref name="Iniewski">{{cite book
| last = Iniewski
| last = Iniewski
| first = Krzysztof
| first = Krzysztof
| title = Wireless Technologies: Circuits, Systems, and Devices
| title = Wireless Technologies: Circuits, Systems, and Devices
| accessdate = December 4, 2012}}</ref> The tunnel diode ''TD'' has voltage controlled negative differential resistance.<ref name="Fogiel" /> The battery <math>\scriptstyle V_b</math> 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 <math>\scriptstyle \Delta v /\Delta i\,=\,-r</math>. For stability <math>\scriptstyle R\,</math> must be less than <math>\scriptstyle r\,</math>.<ref name="Butler" /> Using the formula for a ], the AC output voltage is<ref name="Weaver" />
|archive-date = February 4, 2013
:<math>v_o = \frac{-r}{R-r}v_i = \frac{r}{r-R}v_i \,</math> so the ] is <math>G_v = \frac{r}{r-R} \,</math>
Inanormalvoltagedivider,theresistanceofeachbranchislessthantheresistanceofthewhole,so the output voltage isless than the input.Here,duetothe negative resistance, thetotalACresistance<math>\scriptstyler-R\,</math>isless than the resistance of the diodealone <math>\scriptstyler\,</math> sothe AC output voltage <math>\scriptstyle v_o</math> isgreater than the input <math>\scriptstyle v_i</math>. Thevoltagegain<math>\scriptstyleG_v\,</math>isgreaterthan one, andincreaseswithoutlimit as <math>\scriptstyleR\,</math>approaches <math>\scriptstyle r\,</math>.
}}</ref> The tunnel diode ''TD'' has voltage controlled negative differential resistance.<ref name="Fogiel" /> The battery <math>V_b</math> 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 <math>\Delta v /\Delta i = -r</math>. For stability <math>R</math> must be less than <math>r</math>.<ref name="Butler" /> Using the formula for a ], the AC output voltage is<ref name="Weaver" />
<math display="block">v_o = \frac{-r}{R-r}v_i = \frac{r}{r-R}v_i </math> so the ] is <math display="block">G_v = \frac{r}{r-R} </math>
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 <math>r - R</math> is less than the resistance of the diode alone <math>r</math> so the AC output voltage <math>v_o</math> is greater than the input <math>v_i</math>. The voltage gain <math>G_v</math> is greater than one, and increases without limit as <math>R</math> approaches <math>r</math>.
===Explanation of power gain===
===Explanation of power gain===
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| align = right
| direction = horizontal
| direction = horizontal
| header =
| header =
| image1 = Negative resistance amplification.svg
| image1 = Negative resistance amplification.svg
| caption1 = 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 Δ''v''Δ''i'' is ''negative'', the device produces AC power rather than consuming it.
| caption1 = 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 Δ''v''Δ''i'' is ''negative'', the device produces AC power rather than consuming it.
| width1 = 180
| width1 = 180
| image2 = Negative resistance AC equivalent circuit.svg
| image2 = Negative resistance AC equivalent circuit.svg
| caption2 = AC equivalent circuit of NDR attached to external circuit.<ref name="Lowry">{{cite book
| caption2 = AC equivalent circuit of NDR attached to external circuit.<ref name="Lowry">{{cite book
|last = Lowry
|last = Lowry
|first = H. R.
|first = H. R.
|author2 = J. Georgis
| authorlink =
|author2=J.Georgis|author3=E. Gottlieb
|author3 = E. Gottlieb
|title = General Electric Tunnel Diode Manual, 1st Ed.
|title = General Electric Tunnel Diode Manual, 1st Ed.
| isbn = }}</ref> The NDR acts as a dependent AC ] of value Δ''i'' = Δ''v''/''r''. Because the current and voltage are 180° out of phase, the instantaneous AC current Δ''i'' flows ''out'' of the terminal with positive AC voltage Δ''v''. Therefore it adds to the AC source current Δ''i''<sub>S</sub> through the load ''R'', increasing the output power.<ref name="Lowry" />
}}</ref> The NDR acts as a dependent AC ] of value Δ''i'' = Δ''v''/''r''. Because the current and voltage are 180° out of phase, the instantaneous AC current Δ''i'' flows ''out'' of the terminal with positive AC voltage Δ''v''. Therefore it adds to the AC source current Δ''i''<sub>S</sub> through the load ''R'', increasing the output power.<ref name="Lowry" />
| width2 = 200
| footer =
| width2 = 200
| footer =
}}
}}
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 ] the voltage and current at the device's terminals can be divided into a DC bias component {{nowrap|(<math>\scriptstyle V_{bias},\;I_{bias}</math>)}} and an AC component {{nowrap|(<math>\scriptstyle \Delta v,\;\Delta i</math>)}}.
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 ] the voltage and current at the device's terminals can be divided into a DC bias component {{nowrap|(<math>V_{bias},\;I_{bias}</math>)}} and an AC component {{nowrap|(<math>\Delta v,\;\Delta i</math>)}}.
Since a positive change in voltage <math>\scriptstyle \Delta v\,</math> causes a ''negative'' change in current <math>\scriptstyle \Delta i\,</math>, the AC current and voltage in the device are 180° ].<ref name="Carr">{{cite book
Since a positive change in voltage <math>\Delta v</math> causes a ''negative'' change in current <math>\Delta i</math>, the AC current and voltage in the device are 180° ].<ref name="Carr" /><ref name="Radmanesh" /><ref name="Butler">{{cite web
|last = Carr
|last = Butler
|first = Lloyd
| first = Joseph J.
|title = Negative Resistance Revisited
| title = Microwave & Wireless Communications Technology
|work = Amateur Radio magazine
| publisher = Newnes
|publisher = Wireless Institute of Australia, Bayswater, Victoria
| isbn = 0750697075}}</ref><ref name="Radmanesh" /><ref name="Butler">{{cite web
}} on {{webarchive|url=https://web.archive.org/web/20140819085332/http://users.tpg.com.au/users/ldbutler/index.htm |date=2014-08-19 }}</ref><ref name="Duncan">The requirements for negative resistance in oscillators were first set forth by ] in 1907 in according to {{cite journal
| accessdate = September 22, 2012}} on </ref><ref name="Duncan">The requirements for negative resistance in oscillators, "''For alternating current power to be available in a circuit which has externally applied only continuous voltages, the average power consumption during a cycle must be negative...which demands the introduction of negative resistance '''' requires that the phase difference between voltage and current lie between 90° and 270°...'''' the value 180° must hold... The volt-ampere characteristic of such a resistance will therefore be linear, with a negative slope...''" were first set forth by ] in 1907 in according to {{cite journal
| access-date = July 17, 2013 | bibcode = 1921PhRv...17..302D }}: "''For alternating current power to be available in a circuit which has externally applied only continuous voltages, the average power consumption during a cycle must be negative...which demands the introduction of negative resistance '''' requires that the phase difference between voltage and current lie between 90° and 270°...'''' the value 180° must hold... The volt-ampere characteristic of such a resistance will therefore be linear, with a negative slope...''"</ref> This means in the AC ] ''(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 ].<ref name="Butler" /> Therefore, the AC power dissipation is ''negative''; AC power is produced by the device and flows into the external circuit.<ref name="Frank">{{cite web
| id =
| accessdate = July 17, 2013|bibcode = 1921PhRv...17..302D }}</ref> This means in the AC ] ''(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 ].<ref name="Butler" /> Therefore, the AC power dissipation is ''negative''; AC power is produced by the device and flows into the external circuit.<ref name="Frank">{{cite web
| last = Frank
| last = Frank
| first = Brian
| first = Brian
| title = Microwave Oscillators
| title = Microwave Oscillators
| work =
| work = Class Notes: ELEC 483 – Microwave and RF Circuits and Systems
| publisher = Dept. of Elec. and Computer Eng., Queen's Univ., Ontario
| publisher = Dept. of Elec. and Computer Eng., Queen's Univ., Ontario
:<math>P_\text{AC} = \Delta v \Delta i = r_\text{diff}|\Delta i|^2 < 0 \,</math>
<math display="block">P_\text{AC} = \Delta v \Delta i = r_\text{diff}|\Delta i|^2 < 0 </math>
With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an ],<ref name="Butler" /> or excite oscillations in a resonant circuit to make an ]. Unlike in a ] amplifying device such as a transistor or op amp, the amplified signal leaves the device through the same two terminals (]) as the input signal enters.<ref name="Golio2" />
With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an ],<ref name="Butler" /> or excite oscillations in a resonant circuit to make an ]. Unlike in a ] amplifying device such as a transistor or op amp, the amplified signal leaves the device through the same two terminals (]) as the input signal enters.<ref name="Golio2" />
In a passive device, the AC power produced comes from the input DC bias current,<ref name="Lesurf" /> 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<ref name="Lesurf" />
In a passive device, the AC power produced comes from the input DC bias current,<ref name="Lesurf" /> 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<ref name="Lesurf" />
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.<ref name="Aluf" /><ref name="Kaplan" /><ref name="Lesurf" /> The device also dissipates some power as heat, equal to the difference between the DC power in and the AC power out.
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.<ref name="Aluf" /><ref name="Kaplan" /><ref name="Lesurf" /> 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 ] and therefore the phase difference between current and voltage may differ from 180° and may vary with frequency.<ref name="Groszkowski" /><ref name="Deliyannis" /><ref name="Chang">{{cite book
The device may also have ] and therefore the phase difference between current and voltage may differ from 180° and may vary with frequency.<ref name="Groszkowski" /><ref name="Deliyannis" /><ref name="Chang">{{cite book
| isbn = 978-0471351993}}</ref> As long as the real component of the impedance is negative (phase angle between 90° and 270°),<ref name="Duncan" /> the device will have negative resistance and can amplify.<ref name="Chang" /><ref name="Maas">{{cite book
| doi =
|id =
|last = Maas
|first = Stephen A.
| isbn = 0471351997}}</ref> As long as the real component of the impedance is negative (phase angle between 90° and 270°),<ref name="Duncan" /> the device will have negative resistance and can amplify.<ref name="Chang" /><ref name="Maas">{{cite book
|title = Nonlinear Microwave and RF Circuits, 2nd Ed.
| last = Maas
|publisher = Artech House
| first = Stephen A.
|date = 2003
| title = Nonlinear Microwave and RF Circuits, 2nd Ed.
The maximum AC output power is limited by size of the negative resistance region (<math>\scriptstyle v_1,\; v_2,\; i_1,\; and\; i_2</math> in graphs above)<ref name="Lesurf" /><ref name="Mazda">{{cite book
The maximum AC output power is limited by size of the negative resistance region (<math>v_1,\; v_2,\; i_1,\; and\; i_2</math> in graphs above)<ref name="Lesurf" /><ref name="Mazda">{{cite book
The reason that the output signal can leave a negative resistance through the same port that the input signal enters is that from ] theory, the AC voltage or current at the terminals of a component can be divided into two oppositely moving waves, the ''incident wave'' <math>\scriptstyle V_I\,</math>, which travels toward the device, and the ''reflected wave'' <math>\scriptstyle V_R\,</math>, which travels away from the device.<ref name="Bowick">{{cite book
The reason that the output signal can leave a negative resistance through the same port that the input signal enters is that from ] theory, the AC voltage or current at the terminals of a component can be divided into two oppositely moving waves, the ''incident wave'' <math>V_I</math>, which travels toward the device, and the ''reflected wave'' <math>V_R</math>, which travels away from the device.<ref name="Bowick">{{cite book
| isbn = 978-0750685184}}</ref> A negative differential resistance in a circuit can amplify if the magnitude of its ] <math>\Gamma </math>, the ratio of the reflected wave to the incident wave, is greater than one.<ref name="Gilmore">{{cite book
| doi =
| id =
| last1 = Gilmore
| first1 = Rowan
| isbn = 0750685182}}</ref> A negative differential resistance in a circuit can amplify if the magnitude of its ] <math>\scriptstyle \Gamma \,</math>, the ratio of the reflected wave to the incident wave, is greater than one.<ref name="Gilmore">{{cite book
The "reflected" (output) signal has larger amplitude than the incident; the device has "reflection gain".<ref name="Gilmore" /> The reflection coefficient is determined by the AC impedance of the negative resistance device, <math>Z_N(j\omega) = R_N + jX_N</math>, and the impedance of the circuit attached to it, <math>Z_L(j\omega)\,=\,R_L\,+\,jX_L</math>.<ref name="Frank" /> If <math>R_N < 0</math> and <math>R_L > 0</math> then <math>|\Gamma| > 0</math> and the device will amplify. On the ], a graphical aide widely used in the design of high frequency circuits, negative differential resistance corresponds to points outside the unit circle <math>|\Gamma| = 1</math>, the boundary of the conventional chart, so special "expanded" charts must be used.<ref name="Gilmore" /><ref name="Rhea" />
| isbn = 9781580535229}}</ref><ref name="Frank" />
The "reflected" (output) signal has larger amplitude than the incident; the device has "reflection gain".<ref name="Gilmore" /> The reflection coefficient is determined by the AC impedance of the negative resistance device, <math>\scriptstyle Z_N(j\omega)\,=\,R_N\,+\,jX_N</math>, and the impedance of the circuit attached to it, <math>\scriptstyle Z_L(j\omega)\,=\,R_L\,+\,jX_L</math>.<ref name="Frank" /> If <math>\scriptstyle R_N\,<\,0</math> and <math>\scriptstyle R_L\,>\,0\,</math> then <math>\scriptstyle |\Gamma|\,>\,0\,</math> and the device will amplify.
On the ], a graphical aide widely used in the design of high frequency circuits, negative differential resistance corresponds to points outside the unit circle <math>\scriptstyle |\Gamma|\,=\,1\,</math>, the boundary of the conventional chart, so special "expanded" charts must be used.<ref name="Gilmore" /><ref name="Rhea" />
===Stability conditions===
===Stability conditions===
Because it is nonlinear, a circuit with negative differential resistance can have multiple ]s (possible DC operating points), which lie on the ''I–V'' curve.<ref name="Chen2">{{cite book
Because it is nonlinear, a circuit with negative differential resistance can have multiple ]s (possible DC operating points), which lie on the ''I–V'' curve.<ref name="Chen2">{{cite book
| isbn = 0080477488}}</ref> An equilibrium point will be ], so the circuit converges to it within some neighborhood of the point, if its ]s are in the left half of the ] (LHP), while a point is unstable, causing the circuit to ] or "latch up" (converge to another point), if its poles are on the ''jω'' axis or right half plane (RHP), respectively.<ref name="Dorf">{{cite book
}}</ref> An equilibrium point will be ], so the circuit converges to it within some neighborhood of the point, if its ]s are in the left half of the ] (LHP), while a point is unstable, causing the circuit to ] or "latch up" (converge to another point), if its poles are on the ''jω'' axis or right half plane (RHP), respectively.<ref name="Dorf">{{cite book
}}</ref> In contrast, a linear circuit has a single equilibrium point that may be stable or unstable.<ref name="Ballard">{{cite book
| isbn = 0203912659 }}</ref> The equilibrium points are determined by the DC bias circuit, and their stability is determined by the AC impedance <math>\scriptstyle Z_L(j\omega)\,</math> of the external circuit.
| last1 = Ballard
However, because of the different shapes of the curves, the condition for stability is different for VCNR and CCNR types of negative resistance:<ref name="Golio2">Golio (2000) '''', pp. 7.25–7.26, 7.29</ref><ref name="Crisson2">Crisson (1931) '''', pp. 488–492</ref>
| first1 = Dana H.
*In a CCNR (S-type) negative resistance, the resistance function <math>\scriptstyle R_N\,</math> is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:<math>\scriptstyle Z_L(j\omega)\;+\;Z_N(j\omega)\;=\;0\,</math>.<ref name="Karp">{{cite paper
| title = An Introduction to Natural Computation
| first = M. A.
| publisher = MIT Press
| last = Karp
| date = 1999
| title = A transistor D-C negative immittance converter
}}</ref><ref name=Vukic1></ref> The equilibrium points are determined by the DC bias circuit, and their stability is determined by the AC impedance <math>Z_L(j\omega)</math> of the external circuit.
| pages = 3, 25–27
However, because of the different shapes of the curves, the condition for stability is different for VCNR and CCNR types of negative resistance:<ref name="Golio2">Golio (2000) '''', pp. 7.25–7.26, 7.29</ref><ref name="Crisson2">Crisson (1931) '' {{webarchive|url=https://web.archive.org/web/20131216161632/http://www3.alcatel-lucent.com/bstj/vol10-1931/articles/bstj10-3-485.pdf |date=2013-12-16 }}'', pp. 488–492</ref>
*In a CCNR (S-type) negative resistance, the resistance function <math>R_N</math> is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:<math>Z_L(j\omega) + Z_N(j\omega) = 0</math>.<ref name="Karp">{{cite journal
| accessdate = December 3, 2012}} on US website</ref><ref name="Giannini">{{cite book
| last1 = Giannini
|first = M. A.
|last = Karp
|title = A transistor D-C negative immittance converter
}} on US {{webarchive|url=https://web.archive.org/web/20090316085603/http://www.dtic.mil/dpmo/pmkor/korwald.htm |date=2009-03-16 }} website</ref><ref name="Giannini">{{cite book
:For nonreactive circuits {{nowrap|(<math>\scriptstyle X_L\;=\;X_N\;=\;0\,</math>)}} a sufficient condition for stability is that the total resistance is positive<ref name="Yngvesson">{{cite book
:For nonreactive circuits {{nowrap|(<math>X_L = X_N = 0</math>)}} a sufficient condition for stability is that the total resistance is positive<ref name="Yngvesson">{{cite book
:Since CCNRs are stable with no load at all, they are called '''''"open circuit stable"'''''.<ref name="Kumar" /><ref name="Tellegen" /><ref name="Golio2" /><ref name="Bangert" /><ref group=note name="Confusion" />
:Since CCNRs are stable with no load at all, they are called '''''"open circuit stable"'''''.<ref name="Kumar" /><ref name="Tellegen" /><ref name="Golio2" /><ref name="Bangert" /><ref group=note name="Confusion" />
*In a VCNR (N-type) negative resistance, the ] function <math>\scriptstyleG_N\;=\;1/R_N</math> is single-valued. Therefore, stability is determined by the poles of the admittance equation <math>\scriptstyle Y_L(j\omega)\;+\;Y_N(j\omega)\;=\;0</math>.<ref name="Karp" /><ref name="Giannini" /> For this reason the VCNR is sometimes referred to as a '''negative conductance'''.<ref name="Beneking" /><ref name="Karp" /><ref name="Giannini" />
*In a VCNR (N-type) negative resistance, the ] function <math>G_N = 1/R_N</math> is single-valued. Therefore, stability is determined by the poles of the admittance equation <math>Y_L(j\omega) + Y_N(j\omega) = 0</math>.<ref name="Karp" /><ref name="Giannini" /> For this reason the VCNR is sometimes referred to as a '''negative conductance'''.<ref name="Beneking" /><ref name="Karp" /><ref name="Giannini" />{{pb}}As above, for nonreactive circuits a sufficient condition for stability is that the total ] in the circuit is positive<ref name="Yngvesson" /> <math display="block">Y_L + Y_N = G_L + G_N = \frac{1}{R_L} + \frac{1}{R_N} = \frac{1}{R_L} + \frac{1}{-r} > 0 </math> <math display="block">\frac{1}{R_L} > \frac{1}{r}</math> so the VCNR is stable for<ref name="Beneking" /><ref name="Crisson2" />
:As above, for nonreactive circuits a sufficient condition for stability is that the total ] in the circuit is positive<ref name="Yngvesson" />
:Since VCNRs are even stable with a short-circuited output, they are called '''''"short circuit stable"'''''.<ref name="Kumar" /><ref name="Tellegen" /><ref name="Bangert">{{cite journal
:Since VCNRs are even stable with a short-circuited output, they are called '''''"short circuit stable"'''''.<ref name="Kumar" /><ref name="Tellegen" /><ref name="Bangert">{{cite journal
| last = Bangert
| last = Bangert
| first = J. T.
| first = J. T.
| title = The Transistor as a Network Element
| title = The Transistor as a Network Element
| journal = Bell System Tech. J.
| journal = Bell System Tech. J.
| volume = 33
| volume = 33
| issue = 2
| issue = 2
| page = 330
| page = 330
|date=March 1954
| publisher = American Tel. and Tel.
| url = https://archive.org/details/bstj33-2-329
| location = USA
| doi = 10.1002/j.1538-7305.1954.tb03734.x
|date=March 1954
| access-date = June 20, 2014| bibcode = 1954ITED....1....7B
| url = https://archive.org/details/bstj33-2-329
| issn =
| s2cid = 51671649
}}</ref><ref group=note name="Confusion" />
| doi = 10.1002/j.1538-7305.1954.tb03734.x
| id =
| accessdate = June 20, 2014}}</ref><ref group=note name="Confusion" />
For general negative resistance circuits with ], the stability must be determined by standard tests like the ].<ref name="Gilmore2">{{cite book
For general negative resistance circuits with ], the stability must be determined by standard tests like the ].<ref name="Gilmore2">{{cite book
| last1 = Gilmore
| last1 = Gilmore
| first1 = Rowan
| first1 = Rowan
| last2 = Besser
| last2 = Besser
| first2 = Les
| first2 = Les
| author2-link = Les Besser
| title = Practical RF Circuit Design for Modern Wireless Systems
| title = Practical RF Circuit Design for Modern Wireless Systems
}}</ref> Alternatively, in high frequency circuit design, the values of <math>Z_L(j\omega)</math> for which the circuit is stable are determined by a graphical technique using "stability circles" on a ].<ref name="Gilmore" />
}}</ref> Alternatively, in high frequency circuit design, the values of <math>\scriptstyle Z_L(j\omega)</math> for which the circuit is stable are determined by a graphical technique using "stability circles" on a ].<ref name="Gilmore" />
===Operating regions and applications===
===Operating regions and applications===
For simple nonreactive negative resistance devices with <math>\scriptstyle R_N\;=\;-r</math> and <math>\scriptstyle X_N\;=\;0</math> the different operating regions of the device can be illustrated by ]s on the ''I–V'' curve<ref name="Kumar" /> ''(see graphs)''.
For simple nonreactive negative resistance devices with <math>R_N\;=\;-r</math> and <math>X_N\;=\;0</math> the different operating regions of the device can be illustrated by ]s on the ''I–V'' curve<ref name="Kumar" /> ''(see graphs)''.
{{multiple image
{{multiple image
| align = right
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| direction = horizontal
| direction = horizontal
| image1 = Negative resistance stability regions VCNR.svg
| image1 = Negative resistance stability regions VCNR.svg
| caption1 = VCNR (N type) load lines and stability regions
| caption1 = VCNR (N type) load lines and stability regions
| image2 = Negative resistance stability regions CCNR.svg
| image2 = Negative resistance stability regions CCNR.svg
| caption2 = CCNR (S type) load lines and stability regions
| caption2 = CCNR (S type) load lines and stability regions
| width = 180
| width = 180
}}
}}
The DC load line (DCL) is a straight line determined by the DC bias circuit, with equation
The DC load line (DCL) is a straight line determined by the DC bias circuit, with equation <math display="block">V = V_S - IR </math> where <math>V_S</math> is the DC bias supply voltage and R is the resistance of the supply. The possible DC operating point(s) (]s) occur where the DC load line intersects the ''I–V'' curve. For stability<ref name="Krugman" />
*VCNRs require a low impedance bias {{nowrap|(<math>R\;<\;r</math>)}}, such as a ].
:<math>V = V_S - IR\,</math>
*CCNRs require a high impedance bias {{nowrap|(<math>R\;>\;r</math>)}} such as a ], or voltage source in series with a high resistance.
where <math>\scriptstyle V_S</math> is the DC bias supply voltage and R is the resistance of the supply. The possible DC operating point(s) (]s) occur where the DC load line intersects the ''I–V'' curve. For stability<ref name="Krugman" />
The AC load line (''L''<sub>1</sub> − ''L''<sub>3</sub>) is a straight line through the Q point whose slope is the differential (AC) resistance <math>R_L</math> facing the device. Increasing <math>R_L</math> rotates the load line counterclockwise. The circuit operates in one of three possible regions ''(see diagrams)'', depending on <math>R_L</math>.<ref name="Kumar" />
*VCNRs require a low impedance bias {{nowrap|(<math>\scriptstyle R\;<\;r</math>)}}, such as a ].
*'''<span style="color:green;">Stable region (green)</span>''' (illustrated by line ''L''<sub>1</sub>): When the load line lies in this region, it intersects the ''I–V'' curve at one point ''Q''<sub>1</sub>.<ref name="Kumar" /> For nonreactive circuits it is a ] (]s in the LHP) so the circuit is stable. Negative resistance ]s operate in this region. However, due to ], with an energy storage device like a capacitor or inductor the circuit can become unstable to make a nonlinear ] (]) or a ].<ref name="Gottlieb2"> {{webarchive|url=https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |date=2016-05-15 }}</ref>
*CCNRs require a high impedance bias {{nowrap|(<math>\scriptstyle R\;>\;r</math>)}} such as a ], or voltage source in series with a high resistance.
**VCNRs are stable when <math>R_L < r</math>.
The AC load line (''L''<sub>1</sub> − ''L''<sub>3</sub>) is a straight line through the Q point whose slope is the differential (AC) resistance <math>\scriptstyle R_L\,</math> facing the device. Increasing <math>\scriptstyle R_L\,</math> rotates the load line counterclockwise. The circuit operates in one of three possible regions ''(see diagrams)'', depending on <math>\scriptstyle R_L\,</math>.<ref name="Kumar" />
**CCNRs are stable when <math>R_L > r</math>.
*'''<span style="color:green;">Stable region (green)</span>''' (illustrated by line ''L''<sub>1</sub>): When the load line lies in this region, it intersects the ''I–V'' curve at one point ''Q''<sub>1</sub>.<ref name="Kumar" /> For nonreactive circuits it is a ] (]s in the LHP) so the circuit is stable. Negative resistance ]s operate in this region. However, due to ], with an energy storage device like a capacitor or inductor the circuit can become unstable to make a nonlinear ] (]) or a ].<ref name="Gottlieb2"></ref>
*'''Unstable point''' (Line ''L''<sub>2</sub>): When <math>R_L = r</math> the load line is tangent to the ''I–V'' curve. The total differential (AC) resistance of the circuit is zero (poles on the ''jω'' axis), so it is unstable and with a ] can oscillate. Linear ] 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<ref name="Nahin" /> <math>r = R_L</math>.
**VCNRs are stable when <math>\scriptstyle R_L\;<\;r</math>.
*'''<span style="color:red;">Bistable region (red)</span>''' (illustrated by line ''L''<sub>3</sub>): In this region the load line can intersect the ''I–V'' curve at three points.<ref name="Kumar" /> The center point (''Q''<sub>1</sub>) is a point of ] (poles in the RHP), while the two outer points, ''Q''<sub>2</sub> and ''Q''<sub>3</sub> are ]. So with correct biasing the circuit can be ], it will converge to one of the two points ''Q''<sub>2</sub> or ''Q''<sub>3</sub> and can be switched between them with an input pulse. Switching circuits like ]s (]s) and ]s operate in this region.
**CCNRs are stable when <math>\scriptstyle R_L\;>\;r</math>.
**VCNRs can be bistable when <math>R_L > r</math>
*'''Unstable point''' (Line ''L''<sub>2</sub>): When <math>\scriptstyle R_L\;=\;r</math> the load line is tangent to the ''I–V'' curve. The total differential (AC) resistance of the circuit is zero (poles on the ''jω'' axis), so it is unstable and with a ] can oscillate. Linear ] 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<ref name="Nahin" /> <math>\scriptstyle r\;=\;R_L</math>.
**CCNRs can be bistable when <math>R_L < r</math>
*'''<span style="color:red;">Bistable region (red)</span>''' (illustrated by line ''L''<sub>3</sub>): In this region the load line can intersect the ''I–V'' curve at three points.<ref name="Kumar" /> The center point (''Q''<sub>1</sub>) is a point of ] (poles in the RHP), while the two outer points, ''Q''<sub>2</sub> and ''Q''<sub>3</sub> are ]. So with correct biasing the circuit can be ], it will converge to one of the two points ''Q''<sub>2</sub> or ''Q''<sub>3</sub> and can be switched between them with an input pulse. Switching circuits like ]s (]s) and ]s operate in this region.
**VCNRs can be bistable when <math>\scriptstyle R_L\;>\;r</math>
**CCNRs can be bistable when <math>\scriptstyle R_L\;<\;r</math>
== Active resistors – negative resistance from feedback <span class="anchor" id="ActiveResistors"></span>==
{{Anchor|ActiveResistors}} <!-- please keep with section below -->
==Active resistors – negative resistance from feedback==
{{multiple image
{{multiple image
| align = right
| align = right
| direction = horizontal
| direction = horizontal
| header =
| header =
| image1 = Active negative resistance - voltage controlled.svg
| image1 = Active negative resistance - voltage controlled.svg
| width1 = 150
| width1 = 150
| image2 = Active negative resistance - current controlled.svg
| image2 = Active negative resistance - current controlled.svg
| width2 = 150
| width2 = 150
| image3 = Negative resistance vs loop gain.svg
| image3 = Negative resistance vs loop gain.svg
| width3 = 150
| width3 = 150
| footer = Typical ''I–V'' curves of "active" negative resistances:><ref name="Pippard2" /><ref name="Spangenberg">{{cite book
| footer = Typical ''I–V'' curves of "active" negative resistances:<ref name="Pippard2" /><ref name="Spangenberg">{{cite book
}}, fig. 20.20</ref> N-type ''(left)'', and S-type ''(center)'', generated by feedback amplifiers. These have negative differential resistance ''(<span style="color:red;">red</span> region)'' and produce power ''(grey region)''. Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive ''('''black''' portion of curve)'', and above the supply voltage rails <math>\pm V_S</math> the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain <math>A\beta </math> ''(right)''.
| id =
| isbn =
}}, fig. 20.20</ref> N-type ''(left)'', and S-type ''(center)'', generated by feedback amplifiers. These have negative differential resistance ''(<span style="color:red;">red</span> region)'' and produce power ''(grey region)''. Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive ''('''black''' portion of curve)'', and above the supply voltage rails <math>\scriptstyle \pm V_S\,</math> the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain <math>\scriptstyle A\beta \,</math> ''(right)''.
}}
}}
[[File:Negative resistance by positive feedback.svg|thumb|upright=1.2|An example of an amplifier with positive feedback that has negative resistance at its input. The input current '''''i''''' is<br />
[[File:Negative resistance by positive feedback.svg|thumb|upright=1.2|An example of an amplifier with positive feedback that has negative resistance at its input. The input current '''''i''''' is
If <math>A > 1 + R_1/R_\text{in} </math> it will have negative input resistance.]]
In addition to the passive devices with intrinsic negative differential resistance above, circuits with ] devices like transistors or op amps can have negative resistance at their ports.<ref name="Aluf" /><ref name="Ghadiri" /> The ] or ] of an amplifier with enough ] applied to it can be negative.<ref name="Pippard3" /><ref name="Razavi">{{cite book
If <math>A > 1 + R_1/R_\text{in} \,</math> it will have negative input resistance.]]
| last = Razavi
| first = Behzad
In addition to the passive devices with intrinsic negative differential resistance above, circuits with ] devices like transistors or op amps can have negative resistance at their ports.<ref name="Aluf" /><ref name="Ghadiri" /> The ] or ] of an amplifier with enough ] applied to it can be negative.<ref name="Pippard3" /><ref name="Razavi">{{cite book
| title = Design of Analog CMOS Integrated Circuits
| last = Razavi
| publisher = The McGraw-Hill Companies
| first = Behzad
| date = 2001
| title = Design of Analog CMOS Integrated Circuits
}}</ref> If <math>R_i</math> is the input resistance of the amplifier without feedback, <math>A</math> is the ], and <math>\beta(j\omega)</math> is the ] of the feedback path, the input resistance with positive shunt feedback is<ref name="Aluf" /><ref name="Singh">{{cite book
| doi = 10.1109/jrproc.1922.219822
| id =
| last1 = Singh
| first1 = Balwinder
| accessdate = September 9, 2013}}. "Regeneration" means "positive feedback"</ref><ref name="SSBmanual">{{cite book
| last2 = Dixit
| title = Technical Manual no. 11-685: Fundamentals of Single-Sideband Communication
| first2 = Ashish
| publisher = US Dept. of the Army and Dept. of the Navy
| isbn = }}</ref> If <math>\scriptstyle R_i\,</math> is the input resistance of the amplifier without feedback, <math>\scriptstyle A\,</math> is the amplifier ], and <math>\scriptstyle \beta(j\omega)\,</math> is the ] of the feedback path, the input resistance with positive shunt feedback is<ref name="Aluf" /><ref name="Singh">{{cite book
So if the ] <math>A\beta </math> is greater than one, <math>R_{if}</math> will be negative. The circuit acts like a "negative linear resistor"<ref name="Aluf" /><ref name="HorowitzVideo" /><ref name="Dimopoulos">{{cite book
| last = Singh
|last = Dimopoulos
| first = Balwinder
|first = Hercules G.
| last2 = Dixit
|title = Analog Electronic Filters: Theory, Design and Synthesis
So if the ] <math>\scriptstyle A\beta \,</math> is greater than one, <math>\scriptstyle R_{if}\,</math> will be negative. The circuit acts like a "negative linear resistor"<ref name="Aluf" /><ref name="HorowitzVideo" /><ref name="Dimopoulos">{{cite book
| last = Dimopoulos
| first = Hercules G.
| title = Analog Electronic Filters: Theory, Design and Synthesis
| isbn = 978-0521266734}} This source uses "negative resistance" to mean active resistance</ref> over a limited range,<ref name="Deliyannis" /> with ''I–V'' curve having a straight line segment through the origin with negative slope ''(see graphs)''.<ref name="Franz" /><ref name="Chua2" /><ref name="Crisson" /><ref name="Pippard2" /><ref name="Spangenberg" /> It has both negative differential resistance and is active
and thus obeys ] as if it had a negative value of resistance −''R'',<ref name="Franz" /><ref name="Hickman">{{cite book
| isbn = 0521266734}} This source uses "negative resistance" to mean active resistance</ref> over a limited range,<ref name="Deliyannis" /> with ''I–V'' curve having a straight line segment through the origin with negative slope ''(see graphs)''.<ref name="Chua2" /><ref name="Franz" /><ref name="Crisson" /><ref name="Pippard2" /><ref name="Spangenberg" /> It has both negative differential resistance and is active
| isbn = 1483105350}}</ref> over its linear range (such amplifiers can also have more complicated negative resistance ''I–V'' curves that do not pass through the origin).
}}</ref> over its linear range (such amplifiers can also have more complicated negative resistance ''I–V'' curves that do not pass through the origin).
Theseareoften called "active resistors".<ref name="Chua2" /><ref name="Kouřil" /><ref name="Hoekstra" /><ref name="Popa" /><ref name="Miano" /> Applying a voltage across the terminals causes a proportional current ''out'' of the positive terminal, the opposite of an ordinary resistor.<ref name="Crisson" /><ref name="HorowitzVideo" /><ref name="Hickman" /> For example, connecting a battery to the terminals would cause the battery to ] rather than discharge.<ref name="Wilson" />
In circuit theory these are called "active resistors".<ref name="Chua2" /><ref name="Kouřil" /><ref name="Popa" /><ref name="Miano" /> Applying a voltage across the terminals causes a proportional current ''out'' of the positive terminal, the opposite of an ordinary resistor.<ref name="Crisson" /><ref name="HorowitzVideo" /><ref name="Hickman" /> For example, connecting a battery to the terminals would cause the battery to ] rather than discharge.<ref name="Wilson" />
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<ref name="Aluf" /><ref name="Suzuki" /> with the advantages that:
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<ref name="Aluf" /><ref name="Suzuki" /> with the advantages that:
*because they are active devices they do not require an external DC bias to provide power, and can be ],
*because they are active devices they do not require an external DC bias to provide power, and can be ],
*the amount of negative resistance can be varied by adjusting the ],
*the amount of negative resistance can be varied by adjusting the ],
*they can be linear circuit elements;<ref name="Groszkowski" /><ref name="Dimopoulos" /><ref name="Deliyannis" /> the voltage is proportional to the current, so they do not cause ].
*they can be linear circuit elements;<ref name="Groszkowski" /><ref name="Deliyannis" /><ref name="Dimopoulos" /> if operation is confined to the straight segment of the curve near the origin the voltage is proportional to the current, so they do not cause ].
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".<ref name="Crisson" />
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".<ref name="Crisson" />
Negative ]s ''(below)'' can also be created, so feedback circuits can be used to create "active" linear circuit elements, resistors, capacitors, and inductors, with negative values.<ref name="Ghadiri" /><ref name="Hickman" /> They are widely used in ]s<ref name="Dimopoulos" /><ref name="Deliyannis" /> because they can create ]s that cannot be realized with positive circuit elements.<ref name="Podell">{{cite conference
Negative ]s ''(below)'' can also be created, so feedback circuits can be used to create "active" linear circuit elements, resistors, capacitors, and inductors, with negative values.<ref name="Ghadiri" /><ref name="Hickman" /> They are widely used in ]s<ref name="Deliyannis" /><ref name="Dimopoulos" /> because they can create ]s that cannot be realized with positive circuit elements.<ref name="Podell">{{cite conference
| last = Podell
| last = Podell
| first = A.F.
| first = A.F.
|author2=Cristal, E.G.
|author2=Cristal, E.G.
| title = Negative-Impedance Converters (NIC) for VHF Through Microwave Circuit Applications
| title = Negative-Impedance Converters (NIC) for VHF Through Microwave Circuit Applications
| booktitle = Microwave Symposium Digest, 1971 IEEE GMTT International 16–19 May 1971
| book-title = Microwave Symposium Digest, 1971 IEEE GMTT International 16–19 May 1971
| pages = Abstract.
| pages = 182–183
| publisher = Institute of Electrical and Electronic Engineers
| publisher = Institute of Electrical and Electronics Engineers
}} on IEEE website</ref> Examples of circuits with this type of negative resistance are the ] (NIC), ], Deboo integrator,<ref name="Dimopoulos" /><ref name="Simons">{{cite web
| doi =
|id =
|last = Simons
|first = Elliot
| accessdate = December 19, 2012}} on IEEE website</ref> Examples of circuits with this type of negative resistance are the ] (NIC), ], Deboo integrator,<ref name="Dimopoulos" /><ref name="Simons">{{cite web
|title = Consider the "Deboo" integrator for unipolar noninverting designs
| last = Simons
|work = Electronic Design magazine website
| first = Elliot
|publisher = Penton Media, Inc.
| title = Consider the "Deboo" integrator for unipolar noninverting designs
If an ] is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance <math>\scriptstyle R_\text{if}</math> can cancel the positive loss resistance <math>\scriptstyle r_\text{loss}</math> inherent in the tuned circuit.<ref name="Peters">this property was often called "resistance neutralization" in the days of vacuum tubes, see {{cite journal
If an ] is connected across the input of a positive feedback amplifier like that above, the negative differential input resistance <math>R_\text{if}</math> can cancel the positive loss resistance <math>r_\text{loss}</math> inherent in the tuned circuit.<ref name="Peters">this property was often called "resistance neutralization" in the days of vacuum tubes, see {{cite journal
|last = Bennett
|last = Bennett
|first = Edward
|first = Edward
|author2=Leo James Peters
|author2 = Leo James Peters
|title = Resistance Neutralization: An application of thermionic amplifier circuits
|title = Resistance Neutralization: An application of thermionic amplifier circuits
|journal = Jour. of the AIEE
|journal = Journal of the AIEE
|volume = 41
|volume = 41
|issue = 1
|issue = 1
|pages = 234–248
|pages = 234–248
| publisher = American Institute of Electrical Engineers
|publisher = American Institute of Electrical Engineers
}}</ref> If <math>R_\text{if}\;=\;-r_\text{loss}</math> this will create in effect a tuned circuit with zero AC resistance (] on the ''jω'' axis).<ref name="Solymar" /><ref name="Armstrong" /> Spontaneous oscillation will be excited in the tuned circuit at its ], sustained by the power from the amplifier. This is how ]s such as ] or ]s work.<ref name="Prasad" /><ref name="Lee">{{cite book
| doi =
| id =
| isbn = }}</ref> If <math>\scriptstyle R_\text{if}\;=\;-r_\text{loss}</math> this will create in effect a tuned circuit with zero AC resistance (] on the ''jω'' axis).<ref name="Solymar" /><ref name="Armstrong" /> Spontaneous oscillation will be excited in the tuned circuit at its ], sustained by the power from the amplifier. This is how ]s such as ] or ]s work.<ref name="Prasad" /><ref name="Lee">{{cite book
| isbn = 978-0521835398}}</ref> This negative resistance model is an alternate way of analyzing feedback oscillator operation.<ref name="Golio" /><ref name="Butler" /><ref name="Gottlieb2" /><ref name="SSBmanual" /><ref name="Kung">{{cite web
| doi =
|id =
|last = Kung
|first = Fabian Wai Lee
| isbn = 0521835399}}</ref> This negative resistance model is an alternate way of analyzing feedback oscillator operation.<ref name="Golio" /><ref name="Butler" /><ref name="Gottlieb2" /><ref name="SSBmanual" /><ref name="Kung">{{cite web
|title = Lesson 9: Oscillator Design
| last = Kung
|work = RF/Microwave Circuit Design
| first = Fabian Wai Lee
|publisher = Prof. Kung's website, Multimedia University
| accessdate = October 17, 2012}}, Sec. 3 Negative Resistance Ocillators, pp. 9–10, 14</ref><ref name="Räisänen" /><ref name="Ellinger" /> ''All'' linear oscillator circuits have negative resistance<ref name="Butler" /><ref name="Duncan" /><ref name="Gottlieb2" /><ref name="Räisänen" /> 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.<ref name="Gottlieb1"></ref>
}}, Sec. 3 Negative Resistance Oscillators, pp. 9–10, 14,</ref><ref name="Räisänen" /><ref name="Ellinger" /> ''All'' linear oscillator circuits have negative resistance<ref name="Butler" /><ref name="Duncan" /><ref name="Gottlieb2" /><ref name="Räisänen" /> 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.<ref name="Gottlieb1"> {{webarchive|url=https://web.archive.org/web/20160515053022/https://books.google.com/books?id=e_oZ69GAuxAC |date=2016-05-15 }}</ref>
===Q enhancement===
===Q enhancement===
A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so <math>\scriptstyle |R_\text{if}|\;<\;r_\text{loss}</math>) will not oscillate, but the negative resistance will decrease the damping in the circuit (moving its ]s toward the ''jω'' axis), increasing its ] so it has a narrower ] and more ].<ref name="Peters" /><ref name="Li">{{cite conference
A tuned circuit connected to a negative resistance which cancels some but not all of its parasitic loss resistance (so <math>|R_\text{if}|\;<\;r_\text{loss}</math>) will not oscillate, but the negative resistance will decrease the ] in the circuit (moving its ]s toward the ''jω'' axis), increasing its ] so it has a narrower ] and more ].<ref name="Peters" /><ref name="Li">{{cite conference
| first = Dandan
| first = Dandan
| last = Li
| last = Li
| authorlink =
|author2=Yannis Tsividis
|author2=Yannis Tsividis
| title = Active filters using integrated inductors
| title = Active filters using integrated inductors
| booktitle = Design of High Frequency Integrated Analogue Filters
| book-title = Design of High Frequency Integrated Analogue Filters
| pages = 58
| pages = 58
| publisher = Institution of Engineering and Technology (IET)
| publisher = Institution of Engineering and Technology (IET)
| isbn = 978-0852969762}}</ref> Q enhancement, also called ''regeneration'', was first used in the ] invented by ] in 1912<ref name="Armstrong" /><ref name="Rembovsky" /> and later in "Q multipliers".<ref name="Carr2">{{cite book
| doi =
| id =
| isbn = 0852969767}}</ref> Q enhancement, also called ''regeneration'', was first used in the ] invented by ] in 1912<ref name="Armstrong" /><ref name="Rembovsky" /> and later in "Q multipliers".<ref name="Carr2">{{cite book
| isbn = 978-0080493886}}</ref> It is widely used in active filters.<ref name="Sun" /> 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.<ref name="Li" /><ref name="Sun" />
| doi =
| id =
| isbn = 0080493882}}</ref> Today it is widely used in active filters.<ref name="Sun" /> 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.<ref name="Li" /><ref name="Sun" />
===Chaotic circuits===
===Chaotic circuits===
Circuits which exhibit ] behavior can be considered quasi-periodic or nonperiodic oscillators, and like all oscillators require a negative resistance in the circuit to provide power.<ref name="Kennedy">{{cite journal
Circuits which exhibit ] behavior can be considered quasi-periodic or nonperiodic oscillators, and like all oscillators require a negative resistance in the circuit to provide power.<ref name="Kennedy">{{cite journal
|last = Kennedy
|last = Kennedy
|first = Michael Peter
|first = Michael Peter
|title = Three Steps to Chaos: Part 1 – Evolution
|title = Three Steps to Chaos: Part 1 – Evolution
|journal = IEEE Trans. on Circuits and Systems
|journal = IEEE Transactions on Circuits and Systems
|volume = 40
|volume = 40
|issue = 10
|issue = 10
|page = 640
|page = 640
|date = October 1993
| publisher = Inst. of Electrical and Electronic Engineers
| accessdate = February 26, 2014}}</ref> ], a simple nonlinear circuit widely used as the standard example of a chaotic system, requires a nonlinear active resistor component, sometimes called ].<ref name="Kennedy" /> This is usually synthesized using a negative impedance converter circuit.<ref name="Kennedy" />
}}</ref> ], a simple nonlinear circuit widely used as the standard example of a chaotic system, requires a nonlinear active resistor component, sometimes called ].<ref name="Kennedy" /> This is usually synthesized using a negative impedance converter circuit.<ref name="Kennedy" />
===Negative impedance converter===
===Negative impedance converter===
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| align = right
| align = right
| direction = horizontal
| direction = horizontal
| header =
| header =
| image1 = General negative impedance circuit.svg
| image1 = General negative impedance circuit.svg
| width1 = 150
| width1 = 150
| image2 = Negative impedance converter IV curve.svg
| image2 = Negative impedance converter IV curve.svg
| width2 = 200
| width2 = 200
| caption2 = Negative impedance converter ''(left)'' and ''I–V'' curve ''(right)''. It has negative differential resistance in ''<span style="color:red;">red</span>'' region and sources power in grey region.
| caption2 = Negative impedance converter ''(left)'' and ''I–V'' curve ''(right)''. It has negative differential resistance in ''<span style="color:red;">red</span>'' region and sources power in grey region.
}}
}}
A common example of an "active resistance" circuit is the ] (NIC)<ref name="HorowitzVideo" /><ref name="Hickman" /><ref name="Lee" /><ref name="Linvill">{{cite journal
A common example of an "active resistance" circuit is the ] (NIC)<ref name="HorowitzVideo" /><ref name="Hickman" /><ref name="Lee" /><ref name="Linvill">{{cite journal
|last = Linvill
|last = Linvill
|first = J.G.
|first = J.G.
|title = Transistor Negative-Impedance Converters
|title = Transistor Negative-Impedance Converters
|journal= Proceedings of the IRE
|journal= Proceedings of the IRE
|pages = 725–729
|pages = 725–729
|year = 1953
|year = 1953
|doi = 10.1109/JRPROC.1953.274251
|doi = 10.1109/JRPROC.1953.274251
|volume = 41
|volume = 41
|issue = 6|s2cid = 51654698
|issue = 6}}</ref> shown in the diagram. The two resistors <math>\scriptstyle R_\text{1}</math> and the op amp constitute a negative feedback non-inverting amplifier with gain of 2.<ref name="Lee" /> The output voltage of the op-amp is
}}</ref> shown in the diagram. The two resistors <math>R_\text{1}</math> and the op amp constitute a negative feedback non-inverting amplifier with gain of 2.<ref name="Lee" /> The output voltage of the op-amp is
So if a voltage <math>\scriptstyle v\,</math> is applied to the input, the same voltage is applied "backwards" across <math>\scriptstyle Z</math>, causing current to flow through it out of the input.<ref name="Hickman" /> The current is
So if a voltage <math>v</math> is applied to the input, the same voltage is applied "backwards" across <math>Z</math>, causing current to flow through it out of the input.<ref name="Hickman" /> The current is
The circuit converts the impedance <math>\scriptstyle Z</math> to its negative. If <math>\scriptstyle Z</math> is a resistor of value <math>\scriptstyle R</math>, within the linear range of the op amp <math>\scriptstyle V_\text{S}/2\;<\;v\;<\;-V_\text{S}/2</math> the input impedance acts like a linear "negative resistor" of value <math>\scriptstyle -R</math>.<ref name="Hickman" /> 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,<ref name="Maxim">{{cite web
The circuit converts the impedance <math>Z</math> to its negative. If <math>Z</math> is a resistor of value <math>R</math>, within the linear range of the op amp <math>V_\text{S}/2 < v < -V_\text{S}/2</math> the input impedance acts like a linear "negative resistor" of value <math>-R</math>.<ref name="Hickman" /> 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,<ref name="Maxim">{{cite web
| access-date = October 8, 2014}}</ref> for example they were originally developed to cancel resistance in telephone cables, serving as ]s.<ref name="Lee" />
| format =
| doi =
| accessdate = October 8, 2014}}</ref> for example they were originally developed to cancel resistance in telephone cables, serving as ]s.<ref name="Lee" />
===Negative capacitance and inductance===
===Negative capacitance and inductance===
By replacing <math>\scriptstyle Z</math> in the above circuit with a capacitor {{nowrap|(<math>\scriptstyle C</math>) or inductor (<math>\scriptstyle L</math>)}}, negative capacitances and inductances can also be synthesized.<ref name="Ghadiri" /><ref name="Hickman" /> A negative capacitance will have an ''I–V'' relation and an ] <math>\scriptstyle Z_\text{C}(j\omega)</math> of
By replacing <math>Z</math> in the above circuit with a capacitor {{nowrap|(<math>C</math>) or inductor (<math>L</math>)}}, negative capacitances and inductances can also be synthesized.<ref name="Ghadiri" /><ref name="Hickman" /> A negative capacitance will have an ''I–V'' relation and an ] <math>Z_\text{C}(j\omega)</math> of
where <math>\scriptstyle C\;>\;0</math>. 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 <math>\scriptstyle Z_\text{L}(j\omega)</math> of
where <math>C\;>\;0</math>. 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 <math>Z_\text{L}(j\omega)</math> of
A circuit having negative capacitance or inductance can be used to cancel unwanted positive capacitance or inductance in another circuit.<ref name="Hickman" /> NIC circuits were used to cancel reactance on telephone cables.
A circuit having negative capacitance or inductance can be used to cancel unwanted positive capacitance or inductance in another circuit.<ref name="Hickman" /> 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.<ref name="Hickman" /> 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<ref name="Hickman" /> 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 ].<ref name="Hansen">{{cite book
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.<ref name="Hickman" /> 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<ref name="Hickman" /> 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 ].<ref name="Hansen">{{cite book
| isbn = 978-0470890837}}</ref> One application being researched is to create an active ] which could match an ] to a ] over a broad range of frequencies, rather than just a single frequency as with current networks.<ref name="Aberle">{{cite book
| doi =
|id =
|last = Aberle
|first = James T.
| isbn = 0470890835}}</ref> One application being researched is to create an active ] which could match an ] to a ] over a broad range of frequencies, rather than just a single frequency as with current networks.<ref name="Aberle">{{cite book
|author2 = Robert Loepsinger-Romak
| last = Aberle
|title = Antennas With Non-Foster Matching Networks
| first = James T.
|publisher = Morgan & Claypool
|author2=Robert Loepsinger-Romak
|date = 2007
| title = Antennas With Non-Foster Matching Networks
| access-date = October 17, 2012}}</ref> In a negative resistance oscillator, a negative differential resistance device such as an ], ], or microwave vacuum tube is connected across an electrical ] such as an ], a ], ] or ]<ref name="Räisänen">{{cite book
| doi =
|id =
|last = Räisänen
|first = Antti V.
| accessdate = October 17, 2012}}</ref> In a negative resistance oscillator, a negative differential resistance device such as an ], ], or microwave vacuum tube is connected across an electrical ] such as an ], a ], ] or ]<ref name="Räisänen">{{cite book
|author2 = Arto Lehto
| last = Räisänen
|title = Radio Engineering for Wireless Communication and Sensor Applications
| first = Antti V.
|publisher = Artech House
|author2=Arto Lehto
|date = 2003
| title = Radio Engineering for Wireless Communication and Sensor Applications
| isbn = 0121709604}}</ref> 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 ] and decay to zero.<ref name="Lesurf" /><ref name="Solymar" /><ref name="Lee" /> 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 ].<ref name="Lesurf" /><ref name="Solymar" />
}}</ref> 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.<ref name="Lesurf" /><ref name="Solymar" /><ref name="Lee" /> 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 ].<ref name="Lesurf" /><ref name="Solymar" />
===Uses===
===Uses===
Negative resistance oscillators are mainly used at high ] in the ] range or above, since ] function poorly at these frequencies.<ref name="Golio" /><ref name="Kung" /> Microwave diodes are used in low- to medium-power oscillators for applications such as ]s, and ]s for ]s. They are a widely used source of microwave energy, and virtually the only solid-state source of ]<ref name=" Du">{{cite book
Negative resistance oscillators are mainly used at high ] in the ] range or above, since ] function poorly at these frequencies.<ref name="Golio" /><ref name="Kung" /> Microwave diodes are used in low- to medium-power oscillators for applications such as ]s, and ]s for ]s. They are a widely used source of microwave energy, and virtually the only solid-state source of ]<ref name="Du2">{{cite book
| last = Du
| last = Du
| first = Ke-Lin,
| first = Ke-Lin
|author2=M. N. S. Swamy
| authorlink =
|author2=M. N. S. Swamy
| title = Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies
| title = Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies
| isbn = 978-0521114035}}</ref> and ] energy<ref name="Haddad" /> Negative resistance microwave ]s such as ]s produce higher power outputs,<ref name="Räisänen" /> in such applications as ] transmitters and ]s. Lower frequency ]s can be made with UJTs and gas-discharge lamps such as ]s.
| doi =
| id =
| isbn = 0521114039}}</ref> and ] energy<ref name="Haddad" /> Negative resistance microwave ]s such as ]s produce higher power outputs,<ref name="Räisänen" /> in such applications as ] transmitters and ]s. Lower frequency ]s can be made with ]s and gas-discharge lamps such as ]s.
The negative resistance oscillator model is not limited to one-port devices like diodes but can also be applied to feedback oscillator circuits with ] devices such as transistors and ].<ref name="Gottlieb2">{{cite book
The negative resistance oscillator model is not limited to one-port devices like diodes but can also be applied to feedback oscillator circuits with ] devices such as transistors and ].<ref name="Kung" /><ref name="Räisänen" /><ref name="Ellinger">{{cite book
|last = Gottlieb
|last = Ellinger
|first = Frank
| first = Irving M.
|title = Radio Frequency Integrated Circuits and Technologies, 2nd Ed.
| isbn = 3540693246}}</ref> 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.<ref name="Ghadiri" /><ref name="Maas" /><ref name="Kung" /> 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.<ref name="Kung" /><ref name="Ellinger" />
}}</ref> 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.<ref name="Ghadiri" /><ref name="Maas" /><ref name="Kung" /> 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.<ref name="Kung" /><ref name="Ellinger" />
===Gunn diode oscillator===
===Gunn diode oscillator===
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| align = right
| align = right
| direction = horizontal
| direction = horizontal
| image1 = Gunn diode oscillator circuit.svg
| image1 = Gunn diode oscillator circuit.svg
| caption1 = Gunn diode oscillator circuit
| caption1 = Gunn diode oscillator circuit
| width1 = 150
| width1 = 150
| image2 = Gunn diode oscillator AC circuit.svg
| image2 = Gunn diode oscillator AC circuit.svg
| caption2 = AC equivalent circuit
| caption2 = AC equivalent circuit
| width2 = 102
| width2 = 102
| footer =
| footer =
}}
}}
{{main | Gunn oscillator }}
]s.<br />
]s.<br />
''DCL'': DC load line, which sets the Q point.<br />
''DCL'': DC load line, which sets the Q point.<br />
''SSL'': negative resistance during startup while amplitude is small. Since <math>\scriptstyle r\;<\;R</math> poles are in RHP and amplitude of oscillations increases.<br />
''SSL'': negative resistance during startup while amplitude is small. Since <math>r\;<\;R</math> poles are in RHP and amplitude of oscillations increases.<br />
''LSL'': large-signal load line. When the current swing approaches the edges of the negative resistance region ''<span style="color:green;">(green)</span>'', the sine wave peaks are distorted ("clipped") and <math>\scriptstyle r</math> decreases until it equals <math>\scriptstyle R</math>. ]]
''LSL'': large-signal load line. When the current swing approaches the edges of the negative resistance region ''<span style="color:green;">(green)</span>'', the sine wave peaks are distorted ("clipped") and <math>r</math> decreases until it equals <math>R</math>. ]]
The common ] oscillator ''(circuit diagrams)''<ref name="Lesurf" /> illustrates how negative resistance oscillators work. The diode ''D'' has voltage controlled ("N" type) negative resistance and the voltage source <math>\scriptstyle V_\text{b}</math> biases it into its negative resistance region where its differential resistance is <math>\scriptstyle dv/di\;=\;-r</math>. The ] ''RFC'' prevents AC current from flowing through the bias source.<ref name="Lesurf" /> <math>\scriptstyle R</math> is the equivalent resistance due to damping and losses in the series tuned circuit <math>\scriptstyle LC</math>, plus any load resistance. Analyzing the AC circuit with ] gives a differential equation for <math>\scriptstyle i(t)</math>, the AC current<ref name="Lesurf" />
The common ] oscillator ''(circuit diagrams)''<ref name="Lesurf" /> illustrates how negative resistance oscillators work. The diode ''D'' has voltage controlled ("N" type) negative resistance and the voltage source <math>V_\text{b}</math> biases it into its negative resistance region where its differential resistance is <math>dv/di\;=\;-r</math>. The ] ''RFC'' prevents AC current from flowing through the bias source.<ref name="Lesurf" /> <math>R</math> is the equivalent resistance due to damping and losses in the series tuned circuit <math>LC</math>, plus any load resistance. Analyzing the AC circuit with ] gives a differential equation for <math>i(t)</math>, the AC current<ref name="Lesurf" />
This shows that the current through the circuit, <math>\scriptstyle i(t)</math>, varies with time about the DC ], <math>\scriptstyle I_\text{bias}</math>. When started from a nonzero initial current <math>\scriptstyle i(t)\;=\;i_0</math> the current oscillates ]ly at the ] '''''ω''''' of the tuned circuit, with amplitude either constant, increasing, or decreasing ], depending on the value of '''''α'''''. Whether the circuit can sustain steady oscillations depends on the balance between <math>\scriptstyle R</math> and <math>\scriptstyle r</math>, the positive and negative resistance in the circuit:<ref name="Lesurf" />
This shows that the current through the circuit, <math>i(t)</math>, varies with time about the DC ], <math>I_\text{bias}</math>. When started from a nonzero initial current <math>i(t) = i_0</math> the current oscillates ]ly at the ] '''''ω''''' of the tuned circuit, with amplitude either constant, increasing, or decreasing ], depending on the value of '''''α'''''. Whether the circuit can sustain steady oscillations depends on the balance between <math>R</math> and <math>r</math>, the positive and negative resistance in the circuit:<ref name="Lesurf" />
#]<math>r < R \Rightarrow \alpha < 0\,</math>: (]s 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 <math>\scriptstyle R</math> and die away exponentially to zero, as in an ordinary tuned circuit.<ref name="Solymar" /> So the circuit does not oscillate.
#]<math>r < R \Rightarrow \alpha < 0</math>: (]s 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 <math>R</math> and die away exponentially to zero, as in an ordinary tuned circuit.<ref name="Solymar" /> So the circuit does not oscillate.
#]<math>r = R \Rightarrow \alpha = 0\,</math>: (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.<ref name="Solymar" /> This is the condition during steady-state operation of the oscillator.
#]<math>r = R \Rightarrow \alpha = 0</math>: (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.<ref name="Solymar" /> This is the condition during steady-state operation of the oscillator.
#]<math>r > R \Rightarrow \alpha > 0\,</math>: (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.<ref name="Solymar" /> This is the condition during startup.
#]<math>r > R \Rightarrow \alpha > 0</math>: (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.<ref name="Solymar" /> This is the condition during startup.
{{clear}}
Practical oscillators are designed in region (3) above, with net negative resistance, to get oscillations started.<ref name="Ellinger" /> A widely used rule of thumb is to make <math>R\;=\;r/3</math>.<ref name="Gilmore" /><ref name="Kung2">{{cite web
|last = Kung
|first = Fabian Wai Lee
|title = Lesson 9: Oscillator Design
|work = RF/Microwave Circuit Design
|publisher = Prof. Kung's website, Multimedia University
}}, Sec. 3 Negative Resistance Oscillators, p. 21</ref> When the power is turned on, ] in the circuit provides a signal <math>i_0</math> to start spontaneous oscillations, which grow exponentially. However, the oscillations cannot grow forever; the nonlinearity of the diode eventually limits the amplitude.
At large amplitudes the circuit is nonlinear, so the linear analysis above does not strictly apply and differential resistance is undefined; but the circuit can be understood by considering <math>r</math> to be the "average" resistance over the cycle. As the amplitude of the sine wave exceeds the width of the negative resistance region and the voltage swing extends into regions of the curve with positive differential resistance, the average negative differential resistance <math>r</math> becomes smaller, and thus the total resistance <math>R\;-\;r</math> and the damping <math>\alpha</math> becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when <math>r\;=\;R</math>.<ref name="Lesurf" />
Practical oscillators are designed in region (3) above, with net negative resistance, to get oscillations started.<ref name="Ellinger" /> A widely used rule of thumb is to make <math>\scriptstyle R\;=\;r/3</math>.<ref name="Gilmore" /><ref name="Kung2">{{cite web
| last = Kung
| first = Fabian Wai Lee
| title = Lesson 9: Oscillator Design
| work =
| publisher = Prof. Kung's website, Multimedia University
| accessdate = October 17, 2012}}, Sec. 3 Negative Resistance Ocillators, p. 21</ref> When the power is turned on, ] in the circuit provides a signal <math>\scriptstyle i_0</math> to start spontaneous oscillations, which grow exponentially. However, the oscillations cannot grow forever; the nonlinearity of the diode eventually limits the amplitude.
At large amplitudes the circuit is nonlinear, so the linear analysis above does not strictly apply and differential resistance is undefined; but the circuit can be understood by considering <math>\scriptstyle r</math> to be the "average" resistance over the cycle. As the amplitude of the sine wave exceeds the width of the negative resistance region and the voltage swing extends into regions of the curve with positive differential resistance, the average negative differential resistance <math>\scriptstyle r</math> becomes smaller, and thus the total resistance <math>\scriptstyle R\;-\;r</math> and the damping <math>\scriptstyle \alpha</math> becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when <math>\scriptstyle r\;=\;R</math>.<ref name="Lesurf" />
Gunn diodes have negative resistance in the range −5 to −25 ohms.<ref name="Kshetrimayum">{{cite web
Gunn diodes have negative resistance in the range −5 to −25 ohms.<ref name="Kshetrimayum">{{cite web
|last = Kshetrimayum
|last = Kshetrimayum
|first = Rakhesh Singh
|first = Rakhesh Singh
|title = Experiment 5: Study of ''I–V'' Characteristics of Gunn Diodes
|title = Experiment 5: Study of ''I–V'' Characteristics of Gunn Diodes
|work = EC 341 Microwave Laboratory
|work = EC 341 Microwave Laboratory
|publisher =, Guwahati, India
|publisher = Electrical Engineering Dept., Indian Institute of Technology, Guwahati, India
| accessdate = January 8, 2013}}</ref> In oscillators where <math>\scriptstyle R</math> is close to <math>\scriptstyle r</math>; just small enough to allow the oscillator to start, the voltage swing will be mostly limited to the linear portion of the ''I–V'' curve, the output waveform will be nearly sinusoidal and the frequency will be most stable. In circuits in which <math>\scriptstyle R</math> is far below <math>\scriptstyle r</math>, the swing extends further into the nonlinear part of the curve, the clipping distortion of the output sine wave is more severe,<ref name="Kung2" /> and the frequency will be increasingly dependent on the supply voltage.
|archive-date = January 24, 2014
}}</ref> In oscillators where <math>R</math> is close to <math>r</math>; just small enough to allow the oscillator to start, the voltage swing will be mostly limited to the linear portion of the ''I–V'' curve, the output waveform will be nearly sinusoidal and the frequency will be most stable. In circuits in which <math>R</math> is far below <math>r</math>, the swing extends further into the nonlinear part of the curve, the clipping distortion of the output sine wave is more severe,<ref name="Kung2" /> and the frequency will be increasingly dependent on the supply voltage.
===Types of circuit===
===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)<ref name="Rhea">{{cite book
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)<ref name="Rhea">{{cite book
}} reprinted on {{webarchive|url=https://web.archive.org/web/20141223065727/http://www.vias.org/about.html |date=2014-12-23 }} website</ref>
]
]
*'''Negative resistance (voltage controlled) oscillator:''' Since VCNR ("N" type) devices require a low impedance bias and are stable for load impedances less than ''r'',<ref name="Krugman" /> the ideal oscillator circuit for this device has the form shown at top right, with a voltage source ''V''<sub>bias</sub> to bias the device into its negative resistance region, and ] 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 resistance (voltage controlled) oscillator:''' Since VCNR ("N" type) devices require a low impedance bias and are stable for load impedances less than ''r'',<ref name="Krugman" /> the ideal oscillator circuit for this device has the form shown at top right, with a voltage source ''V''<sub>bias</sub> to bias the device into its negative resistance region, and ] 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.
{{clear}}
]
]
*'''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''.<ref name="Krugman" /> The ideal oscillator circuit is like that at bottom right, with a current source bias ''I''<sub>bias</sub> (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.
*'''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''.<ref name="Krugman" /> The ideal oscillator circuit is like that at bottom right, with a current source bias ''I''<sub>bias</sub> (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.
{{clear}}
===Conditions for oscillation===
===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 ] technique due to K. Kurokawa.<ref name="Maas" /><ref name="Ellinger" /><ref name="Kurokawa">{{cite journal
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 ] technique due to Kaneyuki Kurokawa.<ref name="Maas" /><ref name="Ellinger" /><ref name="Kurokawa">{{cite journal
| last = Kurokawa
| last = Kurokawa
| first = K.
| first = Kaneyuki
| title = Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits
| title = Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits
| journal = Bell System Tech. J.
| journal = Bell System Tech. J.
| volume = 48
| volume = 48
| issue = 6
| issue = 6
| pages = 1937–1955
| pages = 1937–1955
| publisher = American Tel. & Tel.
| location = USA
| date = July 1969
| date = July 1969
| url = https://archive.org/details/bstj48-6-1937
| url = https://archive.org/details/bstj48-6-1937
| issn =
| doi = 10.1002/j.1538-7305.1969.tb01158.x
| doi = 10.1002/j.1538-7305.1969.tb01158.x
| access-date = December 8, 2012}} Eq. 10 is the necessary condition for oscillation, eq. 12 is sufficient condition.</ref> The circuit diagram is imagined to be divided by a "''reference plane''" ''<span style="color:red;">(red)</span>'' which separates the negative resistance part, the active device, from the positive resistance part, the resonant circuit and output load ''(right)''.<ref name="Rohde">{{cite book
| id =
|last = Rohde
| accessdate = December 8, 2012}} Eq. 10 is the necessary condition for oscillation, eq. 12 is sufficient condition.</ref> The circuit diagram is imagined to be divided by a "''reference plane''" ''<span style="color:red;">(red)</span>'' which separates the negative resistance part, the active device, from the positive resistance part, the resonant circuit and output load ''(right)''.<ref name="Rohde">{{cite book
|first = Ulrich L.
| last = Rohde
|first = UlrichL.
|author2 = Ajay K. Poddar
|author2=AjayK.Poddar|author3=Georg Böck
|author3 = Georg Böck
|title = The Design of Modern Microwave Oscillators for Wireless Applications:Theory and Optimization
|title = The Design of Modern Microwave Oscillators for Wireless Applications:Theory and Optimization
| isbn = 0471727164}}</ref> The ] of the negative resistance part <math>\scriptstyle Z_N = R_N(I, \omega) + jX_N(I, \omega) \,</math> depends on frequency ''ω'' but is also nonlinear, in general declining with the amplitude of the AC oscillation current ''I''; while the resonator part <math>\scriptstyle Z_L = R_L(\omega) + jX_L(\omega) \,</math> is linear, depending only on frequency.<ref name="Maas" /><ref name="Räisänen" /><ref name="Rohde" /> The circuit equation is <math>\scriptstyle (Z_N + Z_L)I = 0\,</math> so it will only oscillate (have nonzero ''I'') at the frequency ''ω'' and amplitude ''I'' for which the total impedance <math>\scriptstyle Z_N + Z_L \,</math> is zero.<ref name="Maas" /> This means the magnitude of the negative and positive resistances must be equal, and the reactances must be ]<ref name="Frank" /><ref name="Räisänen" /><ref name="Ellinger" /><ref name="Rohde" />
|archive-date = 2017-09-21
}}</ref> The ] of the negative resistance part <math>Z_N = R_N(I, \omega) + jX_N(I, \omega) </math> depends on frequency ''ω'' but is also nonlinear, in general declining with the amplitude of the AC oscillation current ''I''; while the resonator part <math>Z_L = R_L(\omega) + jX_L(\omega) </math> is linear, depending only on frequency.<ref name="Maas" /><ref name="Räisänen" /><ref name="Rohde" /> The circuit equation is <math>(Z_N + Z_L)I = 0</math> so it will only oscillate (have nonzero ''I'') at the frequency ''ω'' and amplitude ''I'' for which the total impedance <math>Z_N + Z_L </math> is zero.<ref name="Maas" /> This means the magnitude of the negative and positive resistances must be equal, and the reactances must be ]<ref name="Frank" /><ref name="Räisänen" /><ref name="Ellinger" /><ref name="Rohde" />
<math display="block">R_N \le -R_L</math> and <math display="block">X_N = -X_L </math>
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.<ref name="Frank" /><ref name="Maas" /><ref name="Ellinger" />
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.<ref name="Frank" /><ref name="Maas" /><ref name="Ellinger" />
Alternately, the condition for oscillation can be expressed using the ].<ref name="Frank" /> The voltage waveform at the reference plane can be divided into a component ''V''<sub>1</sub> travelling toward the negative resistance device and a component ''V''<sub>2</sub> travelling in the opposite direction, toward the resonator part. The reflection coefficient of the active device <math>\scriptstyle \Gamma_N = V_2/V_1 \,</math> is greater than one, while that of the resonator part <math>\scriptstyle \Gamma_L = V_1/V_2 \,</math> is less than one. During operation the waves are reflected back and forth in a round trip so the circuit will oscillate only if<ref name="Frank" /><ref name="Räisänen" /><ref name="Rohde" />
Alternately, the condition for oscillation can be expressed using the ].<ref name="Frank" /> The voltage waveform at the reference plane can be divided into a component ''V''<sub>1</sub> travelling toward the negative resistance device and a component ''V''<sub>2</sub> travelling in the opposite direction, toward the resonator part. The reflection coefficient of the active device <math>\Gamma_N = V_2/V_1 </math> is greater than one, while that of the resonator part <math>\Gamma_L = V_1/V_2 </math> is less than one. During operation the waves are reflected back and forth in a round trip so the circuit will oscillate only if<ref name="Frank" /><ref name="Räisänen" /><ref name="Rohde" />
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 ] for feedback oscillators; they are necessary but not sufficient,<ref name="Ellinger" /> so there are some circuits that satisfy the equations but do not oscillate. Kurokawa also derived more complicated sufficient conditions,<ref name="Kurokawa" /> which are often used instead.<ref name="Maas" /><ref name="Ellinger" />
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 ] for feedback oscillators; they are necessary but not sufficient,<ref name="Ellinger" /> so there are some circuits that satisfy the equations but do not oscillate. Kurokawa also derived more complicated sufficient conditions,<ref name="Kurokawa" /> which are often used instead.<ref name="Maas" /><ref name="Ellinger" />
==Amplifiers==
==Amplifiers==
Negative differential resistance devices such as Gunn and IMPATT diodes are also used to make ]s, particularly at microwave frequencies, but not as commonly as oscillators.<ref name="Golio2" /> Because negative resistance devices have only one ''port'' (two terminals), unlike ] devices such as ]s, the outgoing amplified signal has to leave the device by the same terminals as the incoming signal enters it.<ref name="Iniewski" /><ref name="Golio2" /> 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.<ref name="Golio2" /> To separate the input and output signals, many negative resistance amplifiers use ] devices such as ]s and ]s.<ref name="Golio2" />
Negative differential resistance devices such as Gunn and IMPATT diodes are also used to make ]s, particularly at microwave frequencies, but not as commonly as oscillators.<ref name="Golio2" /> Because negative resistance devices have only one ''port'' (two terminals), unlike ] devices such as ]s, the outgoing amplified signal has to leave the device by the same terminals as the incoming signal enters it.<ref name="Iniewski" /><ref name="Golio2" /> 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.<ref name="Golio2" /> To separate the input and output signals, many negative resistance amplifiers use ] devices such as ]s and ]s.<ref name="Golio2" />
===Reflection amplifier===
===Reflection amplifier===
Line 1,983:
Line 1,900:
| align = right
| align = right
| direction = vertical
| direction = vertical
| header =
| header =
| image1 = Reflection amplifier.svg
| image1 = Reflection amplifier.svg
| caption1 = AC equivalent circuit of reflection amplifier
| caption1 = AC equivalent circuit of reflection amplifier
| width1 = 250
| width1 = 250
| image2 = 10Gig Tunnel Amp M.jpg
| image2 = 10Gig Tunnel Amp M.jpg
| caption2 = 8–12 GHz microwave amplifier consisting of two cascaded tunnel diode reflection amplifiers
| caption2 = 8–12 GHz microwave amplifier consisting of two cascaded tunnel diode reflection amplifiers
| width2 = 250
| width2 = 250
| footer =
| footer =
}}
}}
One widely used circuit is the ''reflection amplifier'' in which the separation is accomplished by a '']''.<ref name="Golio2" /><ref name="Das">{{cite book
One widely used circuit is the ''reflection amplifier'' in which the separation is accomplished by a '']''.<ref name="Golio2" /><ref name="Das">{{cite book
}}</ref> A circulator is a ] ] component with three ] (connectors) which transfers a signal applied to one port to the next in only one direction, port 1 to port 2, 2 to 3, and 3 to 1. In the reflection amplifier diagram the input signal is applied to port 1, a biased VCNR negative resistance diode ''N'' is attached through a filter ''F'' to port 2, and the output circuit is attached to port 3. The input signal is passed from port 1 to the diode at port 2, but the outgoing "reflected" amplified signal from the diode is routed to port 3, so there is little coupling from output to input. The characteristic impedance <math>Z_0</math> of the input and output ]s, 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.<ref name="Button" /><ref name="Linkhart" />
}}</ref> A circulator is a ] ] component with three ] (connectors) which transfers a signal applied to one port to the next in only one direction, port 1 to port 2, 2 to 3, and 3 to 1. In the reflection amplifier diagram the input signal is applied to port 1, a biased VCNR negative resistance diode ''N'' is attached through a filter ''F'' to port 2, and the output circuit is attached to port 3. The input signal is passed from port 1 to the diode at port 2, but the outgoing "reflected" amplified signal from the diode is routed to port 3, so there is little coupling from output to input. The characteristic impedance <math>\scriptstyle Z_0</math> of the input and output ]s, 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.<ref name="Button" /><ref name="Linkhart" />
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 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
So the ] <math>\scriptstyle G_P</math> of the amplifier is the square of the reflection coefficient<ref name="Das" /><ref name="Button" /><ref name="Linkhart">{{cite book
So the ] <math>G_P</math> of the amplifier is the square of the reflection coefficient<ref name="Das" /><ref name="Button" /><ref name="Linkhart">{{cite book
<math>\scriptstyle R_\text{N}</math> is the negative resistance of the diode '''''−r'''''. Assuming the filter is matched to the diode so <math>\scriptstyleX_1\;=\;-X_N</math><ref name="Button" /> then the gain is
<math>R_\text{N}</math> is the negative resistance of the diode '''−''r'''''. Assuming the filter is matched to the diode so <math>X_1 = -X_N</math><ref name="Button" /> then the gain is
The VCNR reflection amplifier above is stable for <math>\scriptstyle R_1\;<\;r</math>.<ref name="Button" /> while a CCNR amplifier is stable for <math>\scriptstyle R_1\;>\;r</math>. It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as <math>\scriptstyle R_\text{1}</math> approaches the point of oscillation at <math>\scriptstyle r</math>.<ref name="Button" /> This is a characteristic of all NR amplifiers,<ref name="Willardson" /> 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.
The VCNR reflection amplifier above is stable for <math>R_1 < r</math>.<ref name="Button" /> while a CCNR amplifier is stable for <math>R_1 > r</math>. It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as <math>R_1</math> approaches the point of oscillation at <math>r</math>.<ref name="Button" /> This is a characteristic of all NR amplifiers,<ref name="Willardson" /> 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.
]s and ]s are extremely low noise NR amplifiers that are also implemented as reflection amplifiers; they are used in applications like ]s.<ref name="Linkhart" />
]s and ]s are extremely low noise NR amplifiers that are also implemented as reflection amplifiers; they are used in applications like ]s.<ref name="Linkhart" />
==Switching circuits==
==Switching circuits==
Negative differential resistance devices are also used in ]s in which the device operates nonlinearly, changing abruptly from one state to another, with ].<ref name="Kumar2" /> The advantage of using a negative resistance device is that a ], ] or memory cell can be built with a single active device,<ref name="Abraham" /> whereas the standard logic circuit for these functions, the ], requires two active devices (transistors). Three switching circuits built with negative resistances are
Negative differential resistance devices are also used in ]s in which the device operates nonlinearly, changing abruptly from one state to another, with ].<ref name="Kumar2" /> The advantage of using a negative resistance device is that a ], ] or memory cell can be built with a single active device,<ref name="Abraham" /> whereas the standard logic circuit for these functions, the ], requires two active devices (transistors). Three switching circuits built with negative resistances are
*'']'' – a circuit with two unstable states, in which the output periodically switches back and forth between the states. The time it remains in each state is determined by the time constant of an RC circuit. Therefore, it is a ], and can produce ]s or ]s.
*'']'' – a circuit with two unstable states, in which the output periodically switches back and forth between the states. The time it remains in each state is determined by the time constant of an RC circuit. Therefore, it is a ], and can produce ]s or ]s.
*'']'' – is a circuit with one unstable state and one stable state. When in its stable state a pulse is applied to the input, the output switches to its other state and remains in it for a period of time dependent on the time constant of the RC circuit, then switches back to the stable state. Thus the monostable can be used as a timer or delay element.
*'']'' – is a circuit with one unstable state and one stable state. When in its stable state a pulse is applied to the input, the output switches to its other state and remains in it for a period of time dependent on the time constant of the RC circuit, then switches back to the stable state. Thus the monostable can be used as a timer or delay element.
*'']'' or '']'' – is a circuit with two stable states. A pulse at the input switches the circuit to its other state. Therefore, bistables can be used as memory circuits, and ]s.
*'']'' or '']'' – is a circuit with two stable states. A pulse at the input switches the circuit to its other state. Therefore, bistables can be used as memory circuits, and ]s.
==Other applications==
==Other applications==
===Neuronal models===
===Neuronal models===
Some instances of neurons display regions of negative slope conductances (RNSC) in voltage-clamp experiments.<ref name="MacLean">{{cite journal
Some instances of neurons display regions of negative slope conductances (RNSC) in voltage-clamp experiments.<ref name="MacLean">{{cite journal
| last1 = MacLean
| last1 = MacLean
| first1 = Jason N.
| first1 = Jason N.
| last2 = Schmidt
| last2 = Schmidt
| first2 = Brian J.
| first2 = Brian J.
| title = Voltage-Sensitivity of Motoneuron NMDA Receptor Channels Is Modulated by Serotonin in the Neonatal Rat Spinal Cord
| title = Voltage-Sensitivity of Motoneuron NMDA Receptor Channels Is Modulated by Serotonin in the Neonatal Rat Spinal Cord
| journal = Jour. of Neurophysiology
| journal = Journal of Neurophysiology
| volume = 86
| volume = 86
| issue = 3
| issue = 3
| pages = 1131–1138
| pages = 1131–1138
| date = September 2001
| publisher = American Physiological Society
| doi =10.1152/jn.2001.86.3.1131
| location =
| pmid = 11535663
|date=September 2001
| s2cid = 8074067
}}</ref> The negative resistance here is implied were one to consider the neuron a typical ] style circuit model.
| accessdate = August 18, 2014}}</ref> The negative resistance here is implied were one to consider the neuron a typical ] style circuit model.
==History==
==History==
Negative resistance was first recognized during investigations of ]s, which were used for lighting during the 19th century.<ref name="Hong">{{cite book
Negative resistance was first recognized during investigations of ]s, which were used for lighting during the 19th century.<ref name="Hong">{{cite book
|last = Hong
|last = Hong
|first = Sungook
|first = Sungook
|title = Wireless: From Marconi's Black-Box to the Audion
|title = Wireless: From Marconi's Black-Box to the Audion
| isbn = 0262082985}}</ref> In 1881 Alfred Niaudet<ref>A. Niaudet, ''La Lumiere Electrique'', No. 3, 1881, p. 287, cited in Encyclopaedia Britannica, 11th Ed., Vol. 16, p. 660</ref> 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.<ref name="Britannica" /> 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.<ref name="Thompson" /><ref name="Britannica">], {{cite encyclopedia
|archive-date = 2014-08-19
| title = Lighting
}}</ref> In 1881 Alfred Niaudet<ref>A. Niaudet, ''La Lumiere Electrique'', No. 3, 1881, p. 287, cited in Encyclopædia Britannica, 11th Ed., Vol. 16, p. 660</ref> 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.<ref name="Britannica" /> 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.<ref name="Thompson" /><ref name="Britannica">{{Cite EB1911|wstitle= Lighting |volume= 16 |last= Garcke |first= Emile |author-link= Emile Garcke | pages = 651–673;see pages 660-661 }}</ref><ref name="Heaviside">{{cite journal
| encyclopedia = Encyclopaedia Britannica, 11th Ed
| access-date = December 24, 2012}}, also see letter by Andrew Gray on same page</ref> Beginning in 1895 ], 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.<ref name="Ayrton">{{cite journal
| issn =
| doi =
| id =
| accessdate = December 24, 2012}}, also see letter by Andrew Gray on same page</ref> Beginning in 1895 ], extending her husband William's research with a series of meticulous experiments measuring the ''I–V'' curve of arcs, established that arcs had negative resistance, igniting controversy.<ref name="Ayrton">{{cite journal
| access-date = May 3, 2013}}</ref> with the support of the Ayrtons<ref name="Ayrton" /> 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 ].<ref name="Gethemann" />
| id =
| accessdate = May 3, 2013}}</ref> with the support of the Ayrtons<ref name="Ayrton" /> 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 ].<ref name="Gethemann" />
===Arc transmitters===
===Arc transmitters===
] first realized in 1892 that if the damping resistance in a resonant circuit could be made zero or negative, it would produce continuous oscillations.<ref name="Hong" /><ref name="Fitzgerald">G. Fitzgerald, ''On the Driving of Electromagnetic Vibrations by Electromagnetic and Electrostatic Engines'', read at the January 22, 1892 meeting of the Physical Society of London, in {{cite book
|last = Larmor
] first realized in 1892 that if the damping resistance in a resonant circuit could be made zero or negative, it would produce continuous oscillations.<ref name="Hong" /><ref name="Fitzgerald">G. Fitzgerald, ''On the Driving of Electromagnetic Vibrations by Electromagnetic and Electrostatic Engines'', read at the January 22, 1892 meeting of the Physical Society of London, in {{cite book
|first = Joseph, Ed.
| last = Larmor
|title = The Scientific Writings of the late George Francis Fitzgerald
| first = Joseph, Ed.
|publisher = Longmans, Green and Co.
| title = The Scientific Writings of the late George Francis Fitzgerald
| isbn = }}</ref> In the same year ] built a negative resistance oscillator by connecting an ] to the electrodes of an arc,<ref name="Nahin">{{cite book
}}</ref> In the same year ] built a negative resistance oscillator by connecting an ] to the electrodes of an arc,<ref name="Nahin">{{cite book
|last = Nahin
|last = Nahin
|first = Paul J.
|first = Paul J.
|title = The Science of Radio: With Matlab and Electronics Workbench Demonstration, 2nd Ed.
|title = The Science of Radio: With Matlab and Electronics Workbench Demonstration, 2nd Ed.
| isbn = }}</ref> perhaps the first example of an electronic oscillator. ], a student of Ayrton at London Central Technical College, brought Thomson's arc oscillator to public attention.<ref name="Nahin" /><ref name="Hong" /><ref name="Gethemann" /> Due to its negative resistance, the current through an arc was unstable, and ] would often produce hissing, humming, or even howling noises. In 1899, investigating this effect, Duddell connected an ] across an arc and the negative resistance excited oscillations in the tuned circuit, producing a musical tone from the arc.<ref name="Nahin" /><ref name="Hong" /><ref name="Gethemann" /> To demonstrate his invention Duddell wired several tuned circuits to an arc and played a tune on it.<ref name="Hong" /><ref name="Gethemann" /> Duddell's "]" oscillator was limited to audio frequencies.<ref name="Nahin" /> However, in 1903 Danish engineers ] and P. O. Pederson increased the frequency into the radio range by operating the arc in a hydrogen atmosphere in a magnetic field,<ref name="Poulsen">{{cite conference |first=Valdemar |last=Poulsen |title=System for producing continuous electric oscillations |booktitle=Transactions of the International Electrical Congress, St. Louis, 1904, Vol. 2 |pages=963–971 |publisher=J. R. Lyon Co. |date=12 September 1904 |location= |url=http://books.google.com/books?id=JHgSAAAAYAAJ&pg=PA963 |doi= |id= |accessdate=22 September 2013}}</ref> inventing the ] radio transmitter, which was widely used until the 1920s.<ref name="Nahin" /><ref name="Hong" />
|archive-date = 2016-03-15
}}</ref> perhaps the first example of an electronic oscillator. ], a student of Ayrton at London Central Technical College, brought Thomson's arc oscillator to public attention.<ref name="Nahin" /><ref name="Hong" /><ref name="Gethemann" /> Due to its negative resistance, the current through an arc was unstable, and ] would often produce hissing, humming, or even howling noises. In 1899, investigating this effect, Duddell connected an ] across an arc and the negative resistance excited oscillations in the tuned circuit, producing a musical tone from the arc.<ref name="Nahin" /><ref name="Hong" /><ref name="Gethemann" /> To demonstrate his invention Duddell wired several tuned circuits to an arc and played a tune on it.<ref name="Hong" /><ref name="Gethemann" /> Duddell's "]" oscillator was limited to audio frequencies.<ref name="Nahin" /> However, in 1903 Danish engineers ] and P. O. Pederson increased the frequency into the radio range by operating the arc in a hydrogen atmosphere in a magnetic field,<ref name="Poulsen">{{cite conference |first=Valdemar |last=Poulsen |title=System for producing continuous electric oscillations |book-title=Transactions of the International Electrical Congress, St. Louis, 1904, Vol. 2 |pages=963–971 |publisher=J. R. Lyon Co. |date=12 September 1904 |url=https://books.google.com/books?id=JHgSAAAAYAAJ&pg=PA963 |access-date=22 September 2013 |url-status=live |archive-url=https://web.archive.org/web/20131009040125/http://books.google.com/books?id=JHgSAAAAYAAJ&pg=PA963 |archive-date=9 October 2013 }}</ref> inventing the ] radio transmitter, which was widely used until the 1920s.<ref name="Nahin" /><ref name="Hong" />
===Vacuum tubes===
===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.<ref name="Hong" /> ] in 1907 showed that oscillators must have negative resistance.<ref name="Duncan" /> Ernst Ruhmer and Adolf Pieper discovered that ]s could produce oscillations, and by 1912 AT&T had used them to build amplifying ]s for ]s.<ref name="Hong" />
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.<ref name="Hong" /> ] in 1907 showed that oscillators must have negative resistance.<ref name="Duncan" /> ] and Adolf Pieper discovered that ]s could produce oscillations, and by 1912 AT&T had used them to build amplifying ]s for ]s.<ref name="Hong" />
In 1918 Albert Hull at ] discovered that ]s could have negative resistance in parts of their operating ranges, due to a phenomenon called ].<ref name="Gottlieb" /><ref name="Butler" /><ref name="Hull">{{cite journal
In 1918 Albert Hull at ] discovered that ]s could have negative resistance in parts of their operating ranges, due to a phenomenon called ].<ref name="Gottlieb" /><ref name="Butler" /><ref name="Hull">{{cite journal
| last = Hull
| last = Hull
| first = Albert W.
| first = Albert W.
| authorlink =
| title = The Dynatron – A vacuum tube possessing negative electric resistance
| title = The Dynatron – A vacuum tube possessing negative electric resistance
| accessdate = 2012-05-06}}</ref> In a vacuum tube when electrons strike the ] they can knock additional electrons out of the surface into the tube. This represents a current ''away'' from the plate, reducing the plate current.<ref name="Gottlieb" /> 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 ]. Other negative resistance tube oscillators followed, such as the ] invented by Hull in 1920.<ref name="Gilmour" />
| access-date = 2012-05-06}}</ref> In a vacuum tube when electrons strike the ] they can knock additional electrons out of the surface into the tube. This represents a current ''away'' from the plate, reducing the plate current.<ref name="Gottlieb" /> 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 ]. Other negative resistance tube oscillators followed, such as the ] invented by Hull in 1920.<ref name="Gilmour" />
The negative impedance converter originated from work by Marius Latour around 1920.<ref name="Latour">{{cite journal
The negative impedance converter originated from work by Marius Latour around 1920.<ref name="Latour">{{cite journal
| access-date = December 27, 2012}}</ref><ref name="Merrill">{{cite journal
| issn =
| doi =
| last = Merrill
| id =
| first = J.L. Jr.
| accessdate = December 27, 2012}}</ref><ref name=" Merrill">{{cite journal
| last = Merrill
| first = J.L., Jr.
| title = Theory of the Negative Impedance Converter
| title = Theory of the Negative Impedance Converter
| journal = Bell System Tech. J.
| journal = Bell System Tech. J.
Line 2,260:
Line 2,136:
| issue = 1
| issue = 1
| pages = 88–109
| pages = 88–109
| publisher = American Tel. & Tel.
| location = USA
| date = January 1951
| date = January 1951
| url = https://archive.org/details/bstj30-1-88
| url = https://archive.org/details/bstj30-1-88
| issn =
| doi = 10.1002/j.1538-7305.1951.tb01368.x
| doi = 10.1002/j.1538-7305.1951.tb01368.x
| access-date = December 9, 2012}}</ref> He was also one of the first to report negative capacitance and inductance.<ref name="Latour" /> A decade later, vacuum tube NICs were developed as telephone line ]s at ] by George Crisson and others,<ref name="Crisson" /><ref name="Hansen" /> which made transcontinental telephone service possible.<ref name="Hansen" /> Transistor NICs, pioneered by Linvill in 1953, initiated a great increase in interest in NICs and many new circuits and applications developed.<ref name="Linvill" /><ref name="Hansen" />
| id =
| accessdate = December 9, 2012}}</ref> He was also one of the first to report negative capacitance and inductance.<ref name="Latour" /> A decade later, vacuum tube NICs were developed as telephone line ]s at ] by George Crisson and others,<ref name="Crisson" /><ref name="Hansen" /> which made transcontinental telephone service possible.<ref name="Hansen" /> Transistor NICs, pioneered by Linvill in 1953, initiated a great increase in interest in NICs and many new circuits and applications developed.<ref name="Linvill" /><ref name="Hansen" />
===Solid state devices===
===Solid state devices===
Negative differential resistance in ]s was observed around 1909 in the first point-contact junction ]s, called ]s, by researchers such as ]<ref name="Grebennikov">{{cite book
Negative differential resistance in ]s was observed around 1909 in the first point-contact junction ]s, called ]s, by researchers such as ]<ref name="Grebennikov">{{cite book
| access-date = July 15, 2014}}</ref> and ].<ref name="Pickard" /><ref name="White">{{cite web
| issn =
|doi =
|last = White
|id =
|first = Thomas H.
|title = Section 14 – Expanded Audio and Vacuum Tube Development (1917–1930)
| accessdate = July 15, 2014}}</ref> and ].<ref name="Pickard" /><ref name="QST">{{cite journal
|work = United States Early Radio History
| title = Strays
|publisher = earlyradiohistory.us
| journal = QST magazine
|volume = 6
|date = 2021
|url = https://earlyradiohistory.us/sec014.htm
| issue =
|access-date = May 5, 2021
| page = 44
}}</ref> 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.<ref name="White" /> However the effect was not pursued.
| publisher = American Radio Relay League
| location = USA
| date = March 1920
| url = http://earlyradiohistory.us/1920cry.htm
| issn =
| doi =
| id =
| accessdate = April 5, 2013}}</ref><ref name="White">{{cite web
| last = White
| first = Thomas H.
| authorlink =
| title = Section 14 – Expanded Audio and Vacuum Tube Development (1917–1924)
| work = United States Early Radio History
| publisher = earlyradiohistory.us
| date = 2003
| url = http://earlyradiohistory.us/sec014.htm
| doi =
| accessdate = September 23, 2012}}</ref> 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.<ref name="White" /> However the effect was not pursued.
The first person to exploit negative resistance diodes practically was Russian radio researcher ], who in 1922 discovered negative differential resistance in biased ] (]) point contact junctions.<ref name="White" /><ref name="Losev">{{cite journal
The first person to exploit negative resistance diodes practically was Russian radio researcher ], who in 1922 discovered negative differential resistance in biased ] (]) point contact junctions.<ref name="White" /><ref name="Losev">{{cite journal
}}</ref><ref name="Lee2"></ref> He used these to build solid-state ]s, ]s, and amplifying and regenerative ]s, 25 years before the invention of the transistor.<ref name="Grebennikov" /><ref name="Gabel" /><ref name="Lee2" /><ref name="Gernsback">{{cite journal
| isbn = 3540688315}}</ref><ref name="Lee2"></ref> He used these to build solid-state ]s, ]s, and amplifying and regenerative ]s, 25 years before the invention of the transistor.<ref name="Grebennikov" /><ref name="Gabel" /><ref name="Lee2" /><ref name="Gernsback">{{cite journal
}} and "", pp. 294–295</ref> Later he even built a ].<ref name="Lee2" /> However his achievements were overlooked because of the success of ] technology. After ten years he abandoned research into this technology (dubbed "Crystodyne" by ]),<ref name="Gernsback" /> and it was forgotten.<ref name="Lee2" />
| accessdate = May 23, 2012}} and "", pp. 294–295</ref> Later he even built a ].<ref name="Lee2" /> However his achievements were overlooked because of the success of ] technology. After ten years he abandoned research into this technology (dubbed "Crystodyne" by ]),<ref name="Gernsback" /> and it was forgotten.<ref name="Lee2" />
The first widely used solid-state negative resistance device was the ], invented in 1957 by Japanese physicist ].<ref name="Franz" /><ref name="Esaki">{{cite journal
The first widely used solid-state negative resistance device was the ], invented in 1957 by Japanese physicist ].<ref name="Franz" /><ref name="Esaki">{{cite journal
| last1 = Esaki
| last1 = Esaki
| first1 = Leo
| first1 = Leo
| title = New Phenomenon in Narrow Germanium p−n Junctions
| title = New Phenomenon in Narrow Germanium p−n Junctions
|bibcode = 1958PhRv..109..603E }}</ref> Because they have lower ] than ]s due to their small junction size, diodes can function at higher frequencies, and tunnel diode oscillators proved able to produce power at ] frequencies, above the range of ordinary ] oscillators. Its invention set off a search for other negative resistance semiconductor devices for use as microwave oscillators,<ref name="Ridley">{{cite journal
| id =
| accessdate = December 28, 2014|bibcode = 1958PhRv..109..603E }}</ref> Because they have lower ] than ]s due to their small junction size, diodes can function at higher frequencies, and tunnel diode oscillators proved able to produce power at ] frequencies, above the range of ordinary ] oscillators. Its invention set off a search for other negative resistance semiconductor devices for use as microwave oscillators,<ref name="Ridley">{{cite journal
| last = Ridley
| last = Ridley
| first = B. K.
| first = B. K.
| authorlink =
| title = "Electric bubbles" and the quest for negative resistance
| title = "Electric bubbles" and the quest for negative resistance
}}{{Dead link|date=August 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> resulting in the discovery of the ], ], TRAPATT diode, and others. In 1969 Kurokawa derived conditions for stability in negative resistance circuits.<ref name="Kurokawa" /> Currently negative differential resistance diode oscillators are the most widely used sources of microwave energy,<ref name="Du" /> and many new negative resistance devices have been discovered in recent decades.<ref name="Franz" />
| doi =
| id =
| accessdate = November 15, 2012}}</ref> resulting in the discovery of the ], ], TRAPATT diode, and others. In 1969 Kurokawa derived conditions for stability in negative resistance circuits.<ref name="Kurokawa" /> Currently negative differential resistance diode oscillators are the most widely used sources of microwave energy,<ref name="Du" /> and many new negative resistance devices have been discovered in recent decades.<ref name="Franz" />
==Notes==
==Notes==
{{reflist|group=note}}
{{Reflist|group=note}}
==References==
==References==
{{reflist|2}}
{{Reflist|2}}
==Further reading==
==Further reading==
*{{cite book | last = Gottlieb | first = Irving M. | title = Practical Oscillator Handbook | publisher = Elsevier | date = 1997 | url = https://books.google.com/books?id=e_oZ69GAuxAC&q=%22negative+resistance&pg=PA75 | isbn = 978-0080539386}} How negative differential resistance devices work in oscillators.
*{{cite book | last = Hong | first = Sungook | title = Wireless: From Marconi's Black-Box to the Audion | publisher = MIT Press | date = 2001 | location = USA | url = http://monoskop.org/images/f/f4/Hong_Sungook_Wireless_From_Marconis_Black-Box_to_the_Audion.pdf | isbn = 978-0262082983}}, ch. 6 Account of discovery of negative resistance and its role in early radio.
*{{cite encyclopedia | last = Snelgrove | first = Martin | title = Negative resistance circuits | encyclopedia = AccessScience Online Encyclopedia | publisher = McGraw-Hill | date = 2008 | doi = 10.1036/1097-8542.446710 | url = http://www.accessscience.com/content/negative-resistance-circuits/446710 | access-date = May 17, 2012}} Elementary one-page introduction to negative resistance.
{{Authority control}}
*{{cite book | last = Gottlieb | first = Irving M. | title = Practical Oscillator Handbook | publisher = Elsevier | date = 1997 | location = | pages = | url = http://books.google.com/books?id=e_oZ69GAuxAC&pg=PA75&dq=%22negative+resistance | doi = | id = | isbn = 0080539386}} How negative differential resistance devices work in oscillators.
*{{cite book | last = Hong | first = Sungook | title = Wireless: From Marconi's Black-Box to the Audion | publisher = MIT Press | date = 2001 | location = USA | pages = | url = http://monoskop.org/images/f/f4/Hong_Sungook_Wireless_From_Marconis_Black-Box_to_the_Audion.pdf | doi = | id = | isbn = 0262082985}}, ch. 6 Account of discovery of negative resistance and its role in early radio.
*{{cite web | last = Snelgrove | first = Martin | title = Negative resistance circuits | work = AccessScience Online Encyclopedia | publisher = McGraw-Hill | date = 2008 | url = http://www.accessscience.com/content/negative-resistance-circuits/446710 | doi = | accessdate = May 17, 2012}} Elementary one-page introduction to negative resistance.
]
]
]
]
]
]
]
Latest revision as of 18:33, 24 November 2024
Property that an increasing voltage results in a decreasing current
Fluorescent lamp, a device with negative differential resistance. In operation, an increase in current through the fluorescent tube causes a drop in voltage across it. If the tube were connected directly to the power line, the falling tube voltage would cause more and more current to flow, causing it to arc flash and destroy itself. To prevent this, fluorescent tubes are connected to the power line through a ballast. The ballast adds positive impedance (AC resistance) to the circuit to counteract the negative resistance of the tube, limiting the current.
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. 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 , and differential resistance, the ratio of a change in voltage to the resulting change in current . The term negative resistance means negative differential resistance (NDR), . 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.
An I–V curve, showing the difference between static resistance (inverse slope of line B) and differential resistance (inverse slope of line C) at a point (A).
The resistance between two terminals of an electrical device or circuit is determined by its current–voltage (I–V) curve (characteristic curve), giving the current through it for any given voltage 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 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:
The quadrants of the I–V plane, showing regions representing passive devices (white) and active devices (red)
Static resistance (also called chordal resistance, absolute resistance or just resistance) – This is the common definition of resistance; the voltage divided by the current: 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 and 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 ( 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: 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 (). 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
Differential 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.
Fig. 1: I–V curve of linear or "ohmic" resistance, the common type of resistance encountered in electrical circuits. The current is proportional to the voltage, so both the static and differential resistance is positive
Fig. 2: I–V curve with negative differential resistance (red region). The differential resistance at a point P is the inverse slope of the line tangent to the graph at that point
Since and , at point P.Fig. 3: I–V curve of a power source. In the 2nd quadrant (red region) current flows out of the positive terminal, so electric power flows out of the device into the circuit. For example at point P, and , so Fig. 4: I–V curve of a negative linear or "active" resistance (AR, red). It has negative differential resistance and negative static resistance (is active):
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:
The voltage v and current i variables in an electrical component must be defined according to the passive sign convention; positive conventional current is defined to enter the positive voltage terminal; this means power P flowing from the circuit into the component is defined to be positive, while power flowing from the component into the circuit is negative. This applies to both DC and AC current. The diagram shows the directions for positive values of the variables.
In a positive static resistance, , so v and i have the same sign. Therefore, from the passive sign convention above, conventional current (flow of positive charge) is through the device from the positive to the negative terminal, in the direction of the electric fieldE (decreasing potential). so the charges lose potential energy doing work on the device, and electric power flows from the circuit into the device, where it is converted to heat or some other form of energy (yellow). If AC voltage is applied, and periodically reverse direction, but the instantaneous always flows from the higher potential to the lower potential.
In a power source, , so and have opposite signs. This means current is forced to flow from the negative to the positive terminal. The charges gain potential energy, so power flows out of the device into the circuit: . Work (yellow) must be done on the charges by some power source in the device to make them move in this direction against the force of the electric field.
In a passive negative differential resistance, , only the AC component of the current flows in the reverse direction. The static resistance is positive so the current flows from positive to negative: . But the current (rate of charge flow) decreases as the voltage increases. So when a time-varying (AC) voltage is applied in addition to a DC voltage (right), the time-varying current and voltage components have opposite signs, so . This means the instantaneous AC current flows through the device in the direction of increasing AC voltage , so AC power flows out of the device into the circuit. The device consumes DC power, some of which is converted to AC signal power which can be delivered to a load in the external circuit, enabling the device to amplify the AC signal applied to it.
Types and terminology
rdiff > 0 Positive differential resistance
rdiff < 0 Negative differential resistance
Rstatic > 0 Passive: Consumes net power
Positive resistances:
Resistors
Ordinary diodes
Most passive components
Passive negative differential resistances:
Tunnel diodes
Gunn diodes
Gas-discharge tubes
Rstatic < 0 Active: Produces net power
Power sources:
Batteries
Generators
Transistors
Most active components
"Active resistors" Positive feedback amplifiers used in:
Feedback oscillators
Negative impedance converters
Active filters
In an electronic device, the differential resistance , the static resistance , 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:
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.
A battery has negative static resistance (red) over its normal operating range, but positive differential resistance.
Occasionally ordinary power sources are referred to as "negative resistances" (fig. 3 above). Although the "static" or "absolute" resistance 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.
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 positive static resistor (left) converts electric power to heat, warming its surroundings. But a negative static resistance cannot function like this in reverse (right), converting ambient heat from the environment to electric power, because it would violate the second law of thermodynamics which requires a temperature difference to produce work. Therefore a negative static resistance must have some other source of power.
A point of some confusion is whether ordinary resistance ("static" or "absolute" resistance, ) 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 as shown by Joule's law. A passive device consumes electric power, so from the passive sign convention. Therefore, from Joule's law . In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance. For a passive device to have would violate either conservation of energy or the second law of thermodynamics, (diagram). Therefore, some authors state that static resistance can never be negative.
From KVL, the static resistance of a power source (RS), such as a battery, is always equal to the negative of the static resistance of its load (RL).
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 . 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
where 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:
The I–V curve is nonmonotonic (having peaks and troughs) with regions of negative slope representing negative differential resistance.
Negative differential resistanceVoltage controlled (N type)Current controlled (S type)
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):
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 the total resistance, the sum of the two resistances in series () 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(blue). A small increase in input voltage, (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 .
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 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 . For stability must be less than . Using the formula for a voltage divider, the AC output voltage is
so the voltage gain 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 is less than the resistance of the diode alone so the AC output voltage is greater than the input . The voltage gain is greater than one, and increases without limit as approaches .
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. The NDR acts as a dependent AC current source of value Δi = Δv/r. Because the current and voltage are 180° out of phase, the instantaneous AC current Δi flows out of the terminal with positive AC voltage Δv. Therefore it adds to the AC source current ΔiS through the load R, increasing the output power.
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 causes a negative change in current , 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.
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 ( in graphs above)
Reflection coefficient
General (AC) model of a negative resistance circuit: a negative differential resistance device , connected to an external circuit represented by which has positive resistance, . Both may have reactance ()
The reason that the output signal can leave a negative resistance through the same port that the input signal enters is that from transmission line theory, the AC voltage or current at the terminals of a component can be divided into two oppositely moving waves, the incident wave, which travels toward the device, and the reflected wave, which travels away from the device. A negative differential resistance in a circuit can amplify if the magnitude of its reflection coefficient, the ratio of the reflected wave to the incident wave, is greater than one.
where
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, , and the impedance of the circuit attached to it, . If and then 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 , 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 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 is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:.
For nonreactive circuits () a sufficient condition for stability is that the total resistance is positive 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 is single-valued. Therefore, stability is determined by the poles of the admittance equation . 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 so the VCNR is stable for
Since VCNRs are even stable with a short-circuited output, they are called "short circuit stable".
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 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 and the different operating regions of the device can be illustrated by load lines on the I–V curve (see graphs).
VCNR (N type) load lines and stability regionsCCNR (S type) load lines and stability regions
The DC load line (DCL) is a straight line determined by the DC bias circuit, with equation where 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
VCNRs require a low impedance bias (), such as a voltage source.
CCNRs require a high impedance bias () 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 facing the device. Increasing rotates the load line counterclockwise. The circuit operates in one of three possible regions (see diagrams), depending on .
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. 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.
VCNRs are stable when .
CCNRs are stable when .
Unstable point (Line L2): When 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 .
Bistable region (red) (illustrated by line L3): In this region the load line can intersect the I–V curve at three points. 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
CCNRs can be bistable when
Active resistors – negative resistance from feedback
Typical I–V curves of "active" negative resistances: N-type (left), and S-type (center), generated by feedback amplifiers. These have negative differential resistance (red region) and produce power (grey region). Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive (black portion of curve), and above the supply voltage rails the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain (right).
An example of an amplifier with positive feedback that has negative resistance at its input. The input current i is so the input resistance is If it will have negative input resistance.
In addition to the passive devices with intrinsic negative differential resistance above, circuits with amplifying devices like transistors or op amps can have negative resistance at their ports. The input or output impedance of an amplifier with enough positive feedback applied to it can be negative. If is the input resistance of the amplifier without feedback, is the amplifier gain, and is the transfer function of the feedback path, the input resistance with positive shunt feedback is
So if the loop gain is greater than one, 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:
because they are active devices they do not require an external DC bias to provide power, and can be DC coupled,
the amount of negative resistance can be varied by adjusting the loop gain,
they can be linear circuit elements; if operation is confined to the straight segment of the curve near the origin the voltage is proportional to the current, so they do not cause harmonic distortion.
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 can cancel the positive loss resistance inherent in the tuned circuit. If 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 ) 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
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) shown in the diagram. The two resistors 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 is applied to the input, the same voltage is applied "backwards" across , 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 to its negative. If is a resistor of value , within the linear range of the op amp the input impedance acts like a linear "negative resistor" of value . 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 in the above circuit with a capacitor () or inductor (), negative capacitances and inductances can also be synthesized. A negative capacitance will have an I–V relation and an impedance of
where . 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 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.
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.
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
Gunn diode oscillator circuitAC equivalent circuit
Main article: Gunn oscillatorGunn diode oscillator load lines. DCL: DC load line, which sets the Q point. SSL: negative resistance during startup while amplitude is small. Since 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 decreases until it equals .
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 biases it into its negative resistance region where its differential resistance is . The chokeRFC prevents AC current from flowing through the bias source. is the equivalent resistance due to damping and losses in the series tuned circuit , plus any load resistance. Analyzing the AC circuit with Kirchhoff's Voltage Law gives a differential equation for , the AC current
Solving this equation gives a solution of the form
where
This shows that the current through the circuit, , varies with time about the DC Q point, . When started from a nonzero initial current the current oscillates sinusoidally at the resonant frequencyω 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 and , the positive and negative resistance in the circuit:
: (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 and die away exponentially to zero, as in an ordinary tuned circuit. So the circuit does not oscillate.
: (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. This is the condition during steady-state operation of the oscillator.
: (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. This is the condition during startup.
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 . When the power is turned on, electrical noise in the circuit provides a signal 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 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 becomes smaller, and thus the total resistance and the damping becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when .
Gunn diodes have negative resistance in the range −5 to −25 ohms. In oscillators where is close to ; 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 is far below , 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)
Negative resistance (voltage controlled) oscillator: Since VCNR ("N" type) devices require a low impedance bias and are stable for load impedances less than r, the ideal oscillator circuit for this device has the form shown at top right, with a voltage source Vbias 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. The ideal oscillator circuit is like that at bottom right, with a current source bias Ibias (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. 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 depends on frequency ω but is also nonlinear, in general declining with the amplitude of the AC oscillation current I; while the resonator part is linear, depending only on frequency. The circuit equation is so it will only oscillate (have nonzero I) at the frequency ω and amplitude I for which the total impedance is zero. This means the magnitude of the negative and positive resistances must be equal, and the reactances must be conjugate
and
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 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 is greater than one, while that of the resonator part 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
AC equivalent circuit of reflection amplifier8–12 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. A circulator is a nonreciprocalsolid-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 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 of the amplifier is the square of the reflection coefficient
is the negative resistance of the diode −r. Assuming the filter is matched to the diode so then the gain is
The VCNR reflection amplifier above is stable for . while a CCNR amplifier is stable for . It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as approaches the point of oscillation at . 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
Astable multivibrator – a circuit with two unstable states, in which the output periodically switches back and forth between the states. The time it remains in each state is determined by the time constant of an RC circuit. Therefore, it is a relaxation oscillator, and can produce square waves or triangle waves.
Monostable multivibrator – is a circuit with one unstable state and one stable state. When in its stable state a pulse is applied to the input, the output switches to its other state and remains in it for a period of time dependent on the time constant of the RC circuit, then switches back to the stable state. Thus the monostable can be used as a timer or delay element.
Bistable multivibrator or flip flop – is a circuit with two stable states. A pulse at the input switches the circuit to its other state. Therefore, bistables can be used as memory circuits, and digital counters.
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 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.
Notes
Some microwave texts use this term in a more specialized sense: a voltage controlled negative resistance device (VCNR) such as a tunnel diode is called a "negative conductance" while a current controlled negative resistance device (CCNR) such as an IMPATT diode is called a "negative resistance". See the Stability conditions section
^ The terms "open-circuit stable" and "short-circuit stable" have become somewhat confused over the years, and are used in the opposite sense by some authors. The reason is that in linear circuits if the load line crosses the I-V curve of the NR device at one point, the circuit is stable, while in nonlinear switching circuits that operate by hysteresis the same condition causes the circuit to become unstable and oscillate as an astable multivibrator, and the bistable region is considered the "stable" one. This article uses the former "linear" definition, the earliest one, which is found in the Abraham, Bangert, Dorf, Golio, and Tellegen sources. The latter "switching circuit" definition is found in the Kumar and Taub sources.
^ Kaplan, Ross M. (December 1968). "Equivalent circuits for negative resistance devices" (PDF). Technical Report No. RADC-TR-68-356. Rome Air Development Center, US Air Force Systems Command: 5–8. Archived from the original (PDF) on August 19, 2014. Retrieved September 21, 2012. {{cite journal}}: Cite journal requires |journal= (help)
^ Haisch, Bernhard (2013). "Nonlinear conduction". Online textbook Vol. 1: DC Circuits. All About Circuits website. Archived from the original on March 20, 2014. Retrieved March 8, 2014.
^ Lesurf, Jim (2006). "Negative Resistance Oscillators". The Scots Guide to Electronics. School of Physics and Astronomy, Univ. of St. Andrews. Archived from the original on July 16, 2012. Retrieved August 20, 2012.
^ Simin, Grigory (2011). "Lecture 08: Tunnel Diodes (Esaki diode)" (PDF). ELCT 569: Semiconductor Electronic Devices. Prof. Grigory Simin, Univ. of South Carolina. Archived from the original (PDF) on September 23, 2015. Retrieved September 25, 2012., pp. 18–19,
^ Traylor, Roger L. (2008). "Calculating Power Dissipation" (PDF). Lecture Notes – ECE112:Circuit Theory. Dept. of Elect. and Computer Eng., Oregon State Univ. Archived (PDF) from the original on 6 September 2006. Retrieved 23 October 2012., archived
^ "Since the energy absorbed by a (static) resistance is always positive, resistances are passive devices." Bakshi, U.A.; V.U.Bakshi (2009). Electrical And Electronics Engineering. Technical Publications. p. 1.12. ISBN978-8184316971. Archived from the original on 2017-12-21.
^ Pippard, A. B. (2007). The Physics of Vibration. Cambridge University Press. pp. 350, fig. 36, p. 351, fig. 37a, p. 352 fig. 38c, p. 327, fig. 14c. ISBN978-0521033336. Archived from the original on 2017-12-21. In some of these graphs, the curve is reflected in the vertical axis so the negative resistance region appears to have positive slope.
^ Wilson, Marcus (November 16, 2010). "Negative Resistance". Sciblog 2010 Archive. Science Media Center. Archived from the original on October 4, 2012. Retrieved September 26, 2012., archived
^ Horowitz, Paul (2004). "Negative Resistor – Physics 123 demonstration with Paul Horowitz". Video lecture, Physics 123, Harvard Univ. YouTube. Archived from the original on December 17, 2015. Retrieved November 20, 2012. In this video Prof. Horowitz demonstrates that negative static resistance actually exists. He has a black box with two terminals, labelled "−10 kilohms" and shows with ordinary test equipment that it acts like a linear negative resistor (active resistor) with a resistance of −10 KΩ: a positive voltage across it causes a proportional negative current through it, and when connected in a voltage divider with an ordinary resistor the output of the divider is greater than the input, it can amplify. At the end he opens the box and shows it contains an op-amp negative impedance converter circuit and battery.
^ see "Negative resistance by means of feedback" section, Pippard, A. B. (2007). The Physics of Vibration. Cambridge University Press. pp. 314–326. ISBN978-0521033336. Archived from the original on 2017-12-21.
^ Miano, Giovanni; Antonio Maffucci (2001). Transmission Lines and Lumped Circuits. Academic Press. pp. 396, 397. ISBN978-0121897109. Archived from the original on 2017-10-09. This source calls negative differential resistances "passive resistors" and negative static resistances "active resistors".
Babin, Perry (1998). "Output Impedance". Basic Car Audio Electronics website. Archived from the original on April 17, 2015. Retrieved December 28, 2014.
Siegman, A. E. (1986). Lasers. University Science Books. pp. 63. ISBN978-0935702118. neon negative resistance glow discharge., fig. 1.54
^ Ayrton, Hertha (August 16, 1901). "The Mechanism of the Electric Arc". The Electrician. 47 (17). London: The Electrician Printing & Publishing Co.: 635–636. Retrieved January 2, 2013.
^ Franz, Roger L. (June 24, 2010). "Use nonlinear devices as linchpins to next-generation design". Electronic Design Magazine. Penton Media Inc. Archived from the original on June 18, 2015. Retrieved September 17, 2012., . An expanded version of this article with graphs and an extensive list of new negative resistance devices appears in Franz, Roger L. (2012). "Overview of Nonlinear Devices and Circuit Applications". Sustainable Technology. Roger L. Franz personal website. Retrieved September 17, 2012.
^ Thompson, Sylvanus P. (July 3, 1896). "On the properties of a body having a negative electric resistance". The Electrician. 37 (10). London: Benn Bros.: 316–318. Archived from the original on November 6, 2017. Retrieved June 7, 2014. also see editorial, "Positive evidence and negative resistance", p. 312
^ see Chua, Leon O. (November 1980). "Dynamic Nonlinear Networks: State of the Art" (PDF). IEEE Transactions on Circuits and Systems. CAS-27 (11). US: Inst. of Electrical and Electronic Engineers: 1076–1077. Archived (PDF) from the original on August 19, 2014. Retrieved September 17, 2012. Definitions 6 & 7, fig. 27, and Theorem 10 for precise definitions of what this condition means for the circuit solution.
^ Muthuswamy, Bharathwaj; Joerg Mossbrucker (2010). "A framework for teaching nonlinear op-amp circuits to junior undergraduate electrical engineering students". 2010 Conference Proceedings. American Society for Engineering Education. Retrieved October 18, 2012., Appendix B. This derives a slightly more complicated circuit where the two voltage divider resistors are different to allow scaling, but it reduces to the text circuit by setting R2 and R3 in the source to R1 in the text, and R1 in source to Z in the text. The I–V curve is the same.
^ Tellegen, B. d. h. (April 1972). "Stability of negative resistances". International Journal of Electronics. 32 (6): 681–686. doi:10.1080/00207217208938331.
Kidner, C.; I. Mehdi; J. R. East; J. I. Haddad (March 1990). "Potential and limitations of resonant tunneling diodes" (PDF). First International Symposium on Space Terahertz Technology, March 5–6, 1990, Univ. of Michigan. Ann Arbor, M: US National Radio Astronomy Observatory. p. 85. Archived (PDF) from the original on August 19, 2014. Retrieved October 17, 2012.
^ The requirements for negative resistance in oscillators were first set forth by Heinrich Barkhausen in 1907 in Das Problem Der Schwingungserzeugung according to Duncan, R. D. (March 1921). "Stability conditions in vacuum tube circuits". Physical Review. 17 (3): 304. Bibcode:1921PhRv...17..302D. doi:10.1103/physrev.17.302. Retrieved July 17, 2013.: "For alternating current power to be available in a circuit which has externally applied only continuous voltages, the average power consumption during a cycle must be negative...which demands the introduction of negative resistance requires that the phase difference between voltage and current lie between 90° and 270°... the value 180° must hold... The volt-ampere characteristic of such a resistance will therefore be linear, with a negative slope..."
^ Frank, Brian (2006). "Microwave Oscillators" (PDF). Class Notes: ELEC 483 – Microwave and RF Circuits and Systems. Dept. of Elec. and Computer Eng., Queen's Univ., Ontario. pp. 4–9. Retrieved September 22, 2012.
Podell, A.F.; Cristal, E.G. (May 1971). "Negative-Impedance Converters (NIC) for VHF Through Microwave Circuit Applications". Microwave Symposium Digest, 1971 IEEE GMTT International 16–19 May 1971. USA: Institute of Electrical and Electronics Engineers. pp. 182–183. doi:10.1109/GMTT.1971.1122957. on IEEE website
^ this property was often called "resistance neutralization" in the days of vacuum tubes, see Bennett, Edward; Leo James Peters (January 1921). "Resistance Neutralization: An application of thermionic amplifier circuits". Journal of the AIEE. 41 (1). New York: American Institute of Electrical Engineers: 234–248. Retrieved August 14, 2013. and Ch. 3: "Resistance Neutralization" in Peters, Leo James (1927). Theory of Thermionic Vacuum Tube Circuits (PDF). McGraw-Hill. pp. 62–87. Archived (PDF) from the original on 2016-03-04.
^ Kung, Fabian Wai Lee (2009). "Lesson 9: Oscillator Design" (PDF). RF/Microwave Circuit Design. Prof. Kung's website, Multimedia University. Archived from the original (PDF) on July 22, 2015. Retrieved October 17, 2012., Sec. 3 Negative Resistance Oscillators, pp. 9–10, 14,
^ Li, Dandan; Yannis Tsividis (2002). "Active filters using integrated inductors". Design of High Frequency Integrated Analogue Filters. Institution of Engineering and Technology (IET). p. 58. ISBN0852969767. Retrieved July 23, 2013.
^ Kung, Fabian Wai Lee (2009). "Lesson 9: Oscillator Design" (PDF). RF/Microwave Circuit Design. Prof. Kung's website, Multimedia University. Archived from the original (PDF) on May 26, 2012. Retrieved October 17, 2012., Sec. 3 Negative Resistance Oscillators, p. 21
Kshetrimayum, Rakhesh Singh. "Experiment 5: Study of I–V Characteristics of Gunn Diodes" (PDF). EC 341 Microwave Laboratory. Electrical Engineering Dept., Indian Institute of Technology, Guwahati, India. Archived (PDF) from the original on January 24, 2014. Retrieved January 8, 2013.
MacLean, Jason N.; Schmidt, Brian J. (September 2001). "Voltage-Sensitivity of Motoneuron NMDA Receptor Channels Is Modulated by Serotonin in the Neonatal Rat Spinal Cord". Journal of Neurophysiology. 86 (3): 1131–1138. doi:10.1152/jn.2001.86.3.1131. PMID11535663. S2CID8074067.
Heaviside, Oliver (July 31, 1892). "Correspondence: Negative Resistance". The Electrician. 37 (14). London: "The Electrician" Printing and Publishing Co.: 452. Retrieved December 24, 2012., also see letter by Andrew Gray on same page
G. Fitzgerald, On the Driving of Electromagnetic Vibrations by Electromagnetic and Electrostatic Engines, read at the January 22, 1892 meeting of the Physical Society of London, in Larmor, Joseph, Ed. (1902). The Scientific Writings of the late George Francis Fitzgerald. London: Longmans, Green and Co. pp. 277–281. Archived from the original on 2014-07-07.{{cite book}}: CS1 maint: multiple names: authors list (link)
Poulsen, Valdemar (12 September 1904). "System for producing continuous electric oscillations". Transactions of the International Electrical Congress, St. Louis, 1904, Vol. 2. J. R. Lyon Co. pp. 963–971. Archived from the original on 9 October 2013. Retrieved 22 September 2013.
^ Pickard, Greenleaf W. (January 1925). "The Discovery of the Oscillating Crystal" (PDF). Radio News. 6 (7). New York: Experimenter Publishing Co.: 1166. Retrieved July 15, 2014.
Losev, O. V. (January 1925). "Oscillating Crystals" (PDF). Radio News. 6 (7). New York: Experimenter Publishing Co.: 1167, 1287. Retrieved July 15, 2014.
^ Gabel, Victor (October 1, 1924). "The Crystal as a Generator and Amplifier" (PDF). The Wireless World and Radio Review. 15. London: Iliffe & Sons Ltd.: 2–5. Archived (PDF) from the original on October 23, 2014. Retrieved March 20, 2014.
, and differential resistance, the ratio of a change in voltage to the resulting change in current 
. The term negative resistance means negative differential resistance (NDR), 
. In general, a negative differential resistance is a two-terminal component which can 
through it for any given voltage 
across it. Most materials, including the ordinary (positive) resistances encountered in electrical circuits, obey 
It is the inverse slope of the line (
However this term is never used in practice, because the term "resistance" is only applied to passive components. Static resistance determines the 
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 (
). Devices of this type can amplify signals, and are what is usually meant by the term "negative resistance".
Fig. 1: I–V curve of linear or "ohmic" resistance, the common type of resistance encountered in electrical circuits. The current is proportional to the voltage, so both the static and differential resistance is positive
Fig. 2: I–V curve with negative differential resistance (red region). The differential resistance at a point P is the inverse slope of the line tangent to the graph at that point
at a point P is the inverse slope of the line tangent to the graph at that point
and 
, at point P 
Fig. 3: I–V curve of a power source. In the 2nd quadrant (red region) current flows out of the positive terminal, so electric power flows out of the device into the circuit. For example at point P, and , so
and 
, so
Fig. 4: I–V curve of a negative linear or "active" resistance (AR, red). It has negative differential resistance and negative static resistance (is active):
, so v and i have the same sign. Therefore, from the passive sign convention above, conventional current (flow of positive charge) is through the device from the positive to the negative terminal, in the direction of the 
so the charges lose 
, so 
. Work (yellow) must be done on the charges by some power source in the device to make them move in this direction against the force of the electric field.
, only the AC component of the current flows in the reverse direction. The static resistance is positive so the current flows from positive to negative: 
and voltage 
components have opposite signs, so 
. This means the instantaneous AC current 
, or both, can be negative, so there are three categories of devices (fig. 2–4 above, and table) which could be called "negative resistances".
A positive static resistor (left) converts electric power to heat, warming its surroundings. But a negative static resistance cannot function like this in reverse (right), converting ambient heat from the environment to electric power, because it would violate the second law of thermodynamics which requires a temperature difference to produce work. Therefore a negative static resistance must have some other source of power.
) can be negative. In electronics, the term "resistance" is customarily applied only to 
as shown by 
. A passive device consumes electric power, so from the 
. Therefore, from Joule's law 
. In other words, no material can conduct electric current better than a "perfect" conductor with zero resistance. For a passive device to have 
This can also be proved from 
This shows that power can flow out of a device into the circuit (
) if and only if 
where 
is the maximum power the device can produce.
The I–V curve is 
Voltage controlled (N type)
Current controlled (S type)
The larger this is, the larger the potential AC output for a given DC bias current, and therefore the greater the efficiency
Tunnel diode amplifier circuit. Since the total resistance, the sum of the two resistances in series () 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 (blue). A small increase in input voltage, (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 .
the total resistance, the sum of the two resistances in series (
) 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 
(blue). A small increase in input voltage, 
(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 
.
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 
. For stability 
. Using the formula for a 
so the 
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 
is less than the resistance of the diode alone 
is greater than one, and increases without limit as 
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. The NDR acts as a dependent AC current source of value Δi = Δv/r. Because the current and voltage are 180° out of phase, the instantaneous AC current Δi flows out of the terminal with positive AC voltage Δv. Therefore it adds to the AC source current ΔiS through the load R, increasing the output power.
) and an AC component (
).
Since a positive change in voltage 
With the proper external circuit, the device can increase the AC signal power delivered to a load, serving as an 
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.
in graphs above)
, connected to an external circuit represented by 
which has positive resistance, 
. Both may have 
)
, which travels toward the device, and the reflected wave 
, which travels away from the device. A negative differential resistance in a circuit can amplify if the magnitude of its 
, the ratio of the reflected wave to the incident wave, is greater than one.
where 
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, 
, and the impedance of the circuit attached to it, 
. If 
and 
then 
and the device will amplify. On the 
, the boundary of the conventional chart, so special "expanded" charts must be used.
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:
is single-valued. Therefore, stability is determined by the poles of the circuit's impedance equation:
.
) a sufficient condition for stability is that the total resistance is positive 
so the CCNR is stable for
is single-valued. Therefore, stability is determined by the poles of the admittance equation 
. 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 
so the VCNR is stable for
and 
the different operating regions of the device can be illustrated by 
VCNR (N type) load lines and stability regions
CCNR (S type) load lines and stability regions
where 
is the DC bias supply voltage and R is the resistance of the supply. The possible DC operating point(s) (
), such as a 
) such as a 
facing the device. Increasing 
.
.
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 
.
Typical I–V curves of "active" negative resistances: N-type (left), and S-type (center), generated by feedback amplifiers. These have negative differential resistance (red region) and produce power (grey region). Applying a large enough voltage or current of either polarity to the port moves the device into its nonlinear region where saturation of the amplifier causes the differential resistance to become positive (black portion of curve), and above the supply voltage rails the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain (right).
the static resistance becomes positive and the device consumes power. The negative resistance depends on the loop gain 
(right).
it will have negative input resistance.
is the input resistance of the amplifier without feedback, 
is the 
is the 
So if the 
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 
can cancel the positive loss resistance 
inherent in the tuned circuit. If 
this will create in effect a tuned circuit with zero AC resistance (
) will not oscillate, but the negative resistance will decrease the 
Negative impedance converter (left) and I–V curve (right). It has negative differential resistance in red region and sources power in grey region.
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 
, 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 
the input impedance acts like a linear "negative resistor" of value 
. 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 
) or inductor (
), negative capacitances and inductances can also be synthesized. A negative capacitance will have an I–V relation and an 
of
where 
. 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 
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.
Gunn diode oscillator circuit
AC equivalent circuit
Main article: Gunn oscillator
poles are in RHP and amplitude of oscillations increases.
biases it into its negative resistance region where its differential resistance is 
. The 
, plus any load resistance. Analyzing the AC circuit with 
, the AC current
Solving this equation gives a solution of the form
where 
This shows that the current through the circuit, 
. When started from a nonzero initial current 
the current oscillates 
: (
: (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. This is the condition during steady-state operation of the oscillator.
: (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. This is the condition during startup.
. When the power is turned on, 
to start spontaneous oscillations, which grow exponentially. However, the oscillations cannot grow forever; the nonlinearity of the diode eventually limits the amplitude.
and the damping 
becomes less negative and eventually turns positive. Therefore, the oscillations will stabilize at the amplitude at which the damping becomes zero, which is when 
.
depends on frequency ω but is also nonlinear, in general declining with the amplitude of the AC oscillation current I; while the resonator part 
is linear, depending only on frequency. The circuit equation is 
so it will only oscillate (have nonzero I) at the frequency ω and amplitude I for which the total impedance 
is zero. This means the magnitude of the negative and positive resistances must be equal, and the reactances must be 
and 
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.
is greater than one, while that of the resonator part 
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 
AC equivalent circuit of reflection amplifier
8–12 GHz microwave amplifier consisting of two cascaded tunnel diode reflection amplifiers
of the input and output 
The output power from the diode is
So the 
of the amplifier is the square of the reflection coefficient
is the negative resistance of the diode −r. Assuming the filter is matched to the diode so 
then the gain is
The VCNR reflection amplifier above is stable for 
. while a CCNR amplifier is stable for 
. It can be seen that the reflection amplifier can have unlimited gain, approaching infinity as 
approaches the point of oscillation at